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+{"text":"\\section{Introduction}\nSubmodular functions arise in a wide range of applications: graph theory, optimization, economics, game theory, to name a few. \nA function $f: 2^V\\to \\mathbb{R}$ on a ground set $V$ is {\\em submodular} if\n$f(X)+f(Y)\\geq f(X\\cap Y)+f(X\\cup Y)$ for all sets $X,Y\\subseteq\nV$. Submodularity can also be interpreted as a decreasing marginals property.\n\nThere has been significant interest in submodular optimization in the machine learning and computer vision communities. The {\\em submodular function minimization} (SFM) problem arises\nin problems in image segmentation or MAP inference tasks in Markov Random Fields. Landmark results in combinatorial optimization give polynomial-time exact algorithms for SFM. However, the high-degree polynomial dependence in the running time is prohibitive for large-scale problem instances. The main objective in this context is to develop fast and scalable SFM algorithms.\n\nInstead of minimizing arbitrary submodular functions, several recent papers aim to exploit special structural properties of submodular functions arising in practical applications. A popular model is {\\em decomposable submodular functions}: these can be written as sums of several ``simple'' submodular functions defined on small supports.\n\nSome definitions are in order. Let $f:2^V\\to\\mathbb{R}$ be a submodular function, and let $n:=|V|$. We can assume w.l.o.g. that $f(\\emptyset)=0$. We are interested in solving the {\\em submodular function minimization problem}\n\\begin{equation}\\label{prob:sfm}\n\\min_{S\\subseteq V} f(S).\\tag{SFM}\n\\end{equation}\nThe base polytope of a submodular function is defined as\n\\begin{align*}\n  B(f) := \\{ x \\in \\mathbb{R}^V \\colon x(S) \\leq f(S) \\; \\forall S \\subseteq V, x(V) = f(V) \\}.\n\\end{align*}\nOne can optimize linear functions over $B(f)$ using the greedy algorithm.\nThe problem \\eqref{prob:sfm} can be reduced to finding the minimum norm point of the base polytope $B(f)$ \\cite{fujishige80}.\n\\begin{equation}\\label{prob:fuji}\n\\min\\left\\{ \\frac12 \\|y\\|_2^2\\colon y \\in B(f)\\right\\}.\\tag{Min-Norm}\n\\end{equation}\nThis reduction is the starting point of convex optimization approaches for \\eqref{prob:sfm}.\nWe refer the reader to Sections 44--45 in \\cite{Schrijver03} for concepts and results in submodular optimization, and  to \\cite{Bach-monograph} on machine learning applications.\n\nWe assume that $f$ is given in the decomposition \n\\[f(S) = \\sum_{i = 1}^r f_i(S), \\]\nwhere each $f_i: 2^{V} \\rightarrow \\mathbb{R}$ is a submodular function. Such functions are called \\emph{decomposable} or\n\\emph{Sum-of-Submodular (SoS)} in the literature. In this paper, we will use the abbreviation DSFM.\n\nFor each $i\\in [r]$, the function $f_i$ has an effective support $C_i$ such that  $f_i(S)=f_i(S\\cap C_i)$ for every $S\\subseteq V$. For each $i\\in [r]$, we assume that two oracles are provided: {\\em (i)} a value oracle that returns $f_i(S)$ for any set $S\\subseteq V$ in time $\\mathrm{EO}_i$; and {\\em (ii)} a quadratic minimization oracle ${\\cal O}_i(w)$.\nFor any input vector $w \\in \\mathbb{R}^n$, this oracle returns an optimal solution to \\eqref{prob:fuji} for the function $f_i+w$, or equivalently, an optimal solution to  $\\min_{y \\in B(f_i)} \\|y + w\\|^2_2$. We let $\\Theta_i$ denote the running time of a single call to the oracle ${\\cal O}_i$,  $\\Theta_{\\max}:=\\max_{i\\in[r]}\\Theta_i$ denote the maximum time of an oracle call, $\\Theta_{\\mathrm{avg}} := {1 \\over r} \\sum_{i \\in [r]} \\Theta_i$ denote the average time of an oracle call.\\footnote{For flow-type algorithms for DSFM, a slightly weaker oracle assumption suffices, returning a minimizer of $\\min_{S \\subseteq C_i} f_i(S) + w(S)$ for any given $w \\in \\mathbb{R}^{C_i}$. This oracle and the quadratic minimization oracle are reducible to each other:  the former reduces to a single call to the latter, and one can implement the latter using $O(|C_i|)$ calls to the former (see e.g. \\cite{Bach-monograph}).} We let $F_{i,\\max} := \\max_{S \\subseteq V} |f_i(S)|$, $F_{\\max}:= \\max_{S \\subseteq V} |f(S)|$ denote the maximum function values.\n\nDecomposable SFM thus requires algorithms on two levels. The \\emph{level-0} algorithms\nare the subroutines used to evaluate the oracles  ${\\cal O}_i$ for every $i\\in [r]$. The \\emph{level-1} algorithm minimizes the function $f$ using the level-0 algorithms as black boxes.\n\n\\subsection{Prior work}\nSFM has had a long history in combinatorial optimization since the early 1970s, following the influential work of Edmonds \\cite{Edmonds70}. The first polynomial-time algorithm was obtained via the ellipsoid method \\cite{Grotschel1981}; recent work presented substantial improvements using this approach \\cite{LeeSW15}.\nSubstantial work focused on designing strongly polynomial combinatorial algorithms \\cite{Schrijver00,iwata2001,fleischer03,Iwata03,Orlin09,IwataO09}. \nStill, designing practical algorithms for SFM that can be applied to large-scale problem instances remains an open problem.\n\nLet us now turn to decomposable SFM.\nPrevious work mainly focused on level-1 algorithms. These can be classified as \\emph{discrete} and \\emph{continuous} optimization methods. The discrete approach builds on techniques of classical discrete algorithms for  network flows and for submodular flows. Kolmogorov \\cite{kolmogorov12} showed that the problem can be reduced to submodular flow maximization, and also presented a more efficient augmenting path algorithm. Subsequent discrete approaches were given in \\cite{Arora12,Fix13,Fix14}.\nContinuous approaches start with convex programming formulation \\eqref{prob:fuji}. Gradient methods were \napplied for the decomposable setting in \\cite{StobbeK10,NishiharaJJ14,EneN15}. \n\nLess attention has been given to the level-0 algorithms. Some papers mainly focus on theoretical guarantees on the running time of level-1 algorithms, and treat the level-0 subroutines as black-boxes (e.g. \\cite{kolmogorov12,NishiharaJJ14,EneN15}). In other papers (e.g. \\cite{StobbeK10,JegelkaBS13}), the model is restricted to functions $f_i$ of a simple specific type that are easy to minimize. An alternative assumption is that all $C_i$'s are small, of size at most $k$; and thus these oracles can be evaluated by exhaustive search, in $2^k$ value oracle calls (e.g. \\cite{Arora12,Fix13}). \n\nShanu {\\em et al.}\\xspace \\cite{Shanu16} use a block coordinate descent method for level-1, and allow arbitrary functions $f_i$. The oracles are evaluated via the Fujishige-Wolfe minimum norm point algorithm \\cite{Fujishige11, Wolfe76} for level-0. \n\n\\subsection{Our contributions}\nOur paper establishes connections between discrete and continuous methods for decomposable SFM, as well as provides a systematic experimental comparison of these approaches.\nOur main theoretical contribution improves the worst-case complexity bound of the most recent continuous optimization methods \\cite{NishiharaJJ14,EneN15} by a factor of $r$, the number of functions in the decomposition. This is achieved by improving the bounds on the relevant condition numbers. Our proof exploits ideas from the discrete optimization approach. This provides not only better, but also considerably simpler arguments than the algebraic proof in \\cite{NishiharaJJ14}. \n\nThe guiding principle of our experimental work is the clean conceptual distinction between the level-0 and level-1 algorithms. Previous experimental studies considered the level-0 and level-1 algorithms as a single ``package''. For example, Shanu {\\em et al.}\\xspace \\cite{Shanu16} compare the performance of their \\emph{SoS Min-Norm algorithm} to the continuous approach of Jegelka {\\em et al.}\\xspace \\cite{JegelkaBS13} and the combinatorial approach of Arora {\\em et al.}\\xspace \\cite{Arora12}. However, these implementations are difficult to compare since they  use three different level-0 algorithms: Fujishige-Wolfe in SoS Min-Norm, a general QP solver for the algorithm of \n\\cite{JegelkaBS13}, and exhaustive search for \\cite{Arora12}. For potentials of large support, Fujishige-Wolfe outperforms these other level-0 subroutines, hence  the algorithms in \\cite{JegelkaBS13,Arora12} could have compared more favorably using the same Fujishige-Wolfe subroutine.\n\nIn our experimental setup, we compare level-1 algorithms by using the same level-0 subroutines. We compare the state-of-the-art continuous and discrete  algorithms: RCDM and ACDM from \\cite{EneN15} with Submodular IBFS from \\cite{Fix13}. We consider multiple options for the level-0 subroutines. For certain potential types, we use tailored subroutines exploiting the specific form of the problem. We also consider a variant of the Fujishige-Wolfe algorithm as a subroutine applicable for arbitrary potentials. \nOur experimental results reveal the following tradeoff. Discrete algorithms on level-1 require more calls to the level-0 oracle, but less overhead computation. Hence using algorithms such as IBFS on level-1 can be significantly faster than gradient descent as long as the potentials have fairly small supports. However, as the size of the potentials grow, or we do need to work with a generic level-0 algorithm, the better choice is using gradient methods. Gradient methods can perform better for larger potentials also due to weaker requirements on the level-0 subroutines: approximate level-0 subroutines suffice for them, whereas discrete algorithms require exact optimal solutions on level-0. \n\n{\\bf Paper outline.} The rest of the paper is structured as follows.\nSection~\\ref{sec:flow} describes the level-1 algorithmic framework for DSFM that is based on network flows, and outlines the IBFS algorithm. Section~\\ref{sec:conv-opt} describes the level-1 algorithmic framework for DSFM that is based on convex optimization, and outlines the gradient descent algorithms. Section~\\ref{sec:kappa} gives improved convergence guarantees for the gradient descent algorithms outlined in Section~\\ref{sec:conv-opt}. Section~\\ref{sec:level-0} discusses the different types of level-0 algorithms and how they can be used together with the level-1 frameworks. Section~\\ref{sec:experiments} presents our experimental results.\n\n\\section{Level-1 algorithms based on network flow}\n\\label{sec:flow}\n\nIn this section, we outline a level-1 algorithmic framework for DSFM that is based on a combinatorial framework first studied in \\cite{FujishigeZhang92}.\\footnote{The framework was introduced in a slightly different context, for the submodular intersection problem. The dual of this problem is minimizing a submodular function of the form $f=f_1+f_2$, with access to oracles minimizing $f_1$ and $f_2$.} \n\nFor a decomposable function $f$, every $x\\in B(f)$ can be written as $x = \\sum_{i = 1}^r x_i$, where\n$\\text{supp}(x_i)\\subseteq C_i$ and $x_i\\in B(f_i)$ (see e.g. Theorem 44.6 in \\cite{Schrijver03}). A natural algorithmic approach is to maintain an $x\\in B(f)$ in such a representation, and iteratively update it using the combinatorial framework described below. \nDSFM can be casted as a maximum network flow instance in a network that is suitably defined based on the current point $x$. This  can be viewed as an analogue of the residual graph in the maxflow\/mincut setting, and it is precisely the residual graph if the DSFM instance was a mincut instance.\n\n{\\bf The auxiliary graph.} For an $x\\in B(f)$ of the form $x=\\sum_{i = 1}^r x_i$, we construct the following directed auxiliary graph $G = (V, E)$, with $E = \\bigcup_{i = 1}^r E_i$ and capacities $c:E\\to {\\mathbb{R}}_+$. The arc sets $E_i$ are complete directed graphs (cliques) on $C_i$, and for an arc $(u,v)\\in E_i$, we define $c(u,v):=\\min\\{ f_i(S)-x_i(S)\\colon S\\subseteq C_i, u\\in S, v\\notin S\\}$. This is the maximum value $\\varepsilon$ such that $x'_i\\in B(f_i)$, where $x'_i(u)=x_i(u)+\\varepsilon$, $x'_i(v)=x_i(v)-\\varepsilon$, $x'_i(z)=x_i(z)$ for $z\\notin\\{u,v\\}$. \n\nLet \n $N := \\{ v \\in V \\colon x(v) < 0 \\}$ and $P := \\{ v \\in V \\colon x(v) > 0\\}$. \nThe algorithm aims to improve the current $x$ by updating along shortest directed paths from $N$ to $P$ with positive capacity; there are several ways to update the solution, and we discuss specific approaches later in the section. If there exists no such directed path, then we let $S$ denote the set reachable from $N$ on directed paths\nwith positive capacity; thus, $S\\cap P=\\emptyset$. It is easy to show that $S$ is a minimizer of the function $f$. \n\nUpdating along a shortest path ${\\cal Q}$ from $N$ to $P$ amounts to the following. Let $\\varepsilon$ denote the minimum capacity of an arc on $\\cal Q$. If $(u,v)\\in {\\cal Q}\\cap E_i$, then we increase $x_i(u)$ by $\\varepsilon$ and decrease $x_i(v)$ by $\\varepsilon$. The crucial technical claim \\cite{FujishigeZhang92} is the following. Let $d(u)$ denote the shortest path distance of positive capacity arcs from $u$ to the set $P$. Then, an update along a shortest directed path from $N$ to $P$ results in a feasible $x\\in B(f)$, and further, all distance labels $d(u)$ are non-decreasing.\n\n{\\bf Level-1 algorithms based on the network flow approach.} Using this auxiliary graph, and updating on shortest augmenting paths, one can generalize several maximum flow algorithms to a level-1 algorithm of DSFM.\nThese algorithms include: the Edmonds-Karp-Dinitz maximum flow algorithm, the preflow-push algorithm \\cite{GoldbergTarjan88}, the incremental breadth first search algorithm (IBFS) \\cite{goldberg11}, and the excesses incremental breadth first search algorithm \\cite{goldberg15}. Our experiments will use an implementation of IBFS, following~\\cite{Fix13}. \n\n{\\bf Submodular incremental breadth first search (IBFS).} Fix {\\em et al.}\\xspace\n\\cite{Fix13} adapt the IBFS algorithm to the above\ndescribed submodular framework  using the above\nmentioned claims by Fujishige \\& Zhang \\cite{FujishigeZhang92}. IBFS is an augmenting path algorithm\nfor the maximum flow problem. It identifies a shortest path from\nthe source set $N$ to the sink set $P$ via \ngrowing shortest path trees simultaneously forwards from $N$ and\nbackwards from $P$. \n\nThe submodular IBFS algorithm provides us with a level-1 algorithm for DSFM. Each step of the algorithm involves determining the capacity of an arc in the auxiliary graph; as we explain in Section~\\ref{sec:level-0}, each of these capacities can be computed using a single call to a level-0 subroutine ${\\cal O}_i$.\n\nBy combining the level-1 IBFS algorithm with appropriate level-0 subroutines, we obtain an algorithm for DSFM whose running time can be upper bounded as follows. On a directed graph with $n$ nodes and $m$ arcs, IBFS runs in time $O(n^2m)$. In the DSFM setting, we have $m=O(\\sum_{i\\in [r]} |C_i|^2 )$. Every step  involves determining an auxiliary capacity, which can be implemented using a single call to a level-0 subroutine ${\\cal O}_i$ (see Section~\\ref{sec:level-0}); the maximum time of such an oracle call is $\\Theta_{\\max}$. Hence, the running time bound for submodular IBFS can be given as $O(n^2\\Theta_{\\max} \\sum_{i\\in [r]}|C_i|^2)$. If all $C_i$'s are small, $O(1)$, then this gives $O(n^2r\\Theta_{\\max})$.\n\n\\section{Level-1 algorithms based on convex optimization}\n\\label{sec:conv-opt}\nIn this section, we outline the level-1 algorithms for DSFM that are based on gradient descent. \nRecall the convex quadratic program (\\ref{prob:fuji}) from the Introduction. This program has a unique optimal solution $s^*$; the set $S=\\{v \\in V \\colon s^*(v) < 0\\}$ is the unique smallest minimizer to \\eqref{prob:sfm}. We will refer to this optimal solution $s^*$ throughout the section. \n\nIn the DSFM setting, one can write (\\ref{prob:fuji}) in multiple equivalent forms \\cite{JegelkaBS13}. For the first formulation, we let $\\mathcal{P} := \\prod_{i = 1}^r B(f_i)\\subseteq {\\mathbb{R}}^{rn},$ and let $A\\in {\\mathbb{R}}^{n\\times (rn)}$ denote the following matrix:\n\t$$A := \\underbrace{[I_n I_n \\dots I_n]}_\\text{$r$ times}.$$\nNote that, for every $y \\in \\mathcal{P}$, $Ay = \\sum_{i = 1}^r y_i$, where $y_i$ is the $i$-th block of $y$, and thus $Ay \\in B(f)$. The problem \\eqref{prob:fuji} can be reformulated for DSFM as follow.\n\\begin{equation}\n\\label{Prox-DSFM}\n\t\\min\\left\\{ {1 \\over 2} \\left\\|Ay \\right\\|^2_2 \\colon \\tag{Prox-DSFM} y\\in \\mathcal{P}\\right\\}.\n\\end{equation}\nThe second formulation is the following. Let us define the subspace $\\mathcal{A} := \\{ a \\in \\mathbb{R}^{nr} \\colon Aa= 0\\}$, and minimize its distance from $\\mathcal{P}$:\n\\begin{equation}\n\\label{Best-Approx}\n\t\\min\\left\\{ \\|a - y\\|^2_2 \\tag{Best-Approx} \\colon a \\in \\mathcal{A}, y \\in \\mathcal{P}\\right\\}.\n\\end{equation}\nThe set of optimal solutions for both formulations (\\ref{Prox-DSFM}) and (\\ref{Best-Approx}) is the set \n$\\mathcal{E} := \\{y \\in \\mathcal{P} \\colon Ay = s^*\\}$,\n where $s^*$ is the optimum of (\\ref{prob:fuji}). We note that, even though the set of solutions to (\\ref{Best-Approx}) are pairs of points $(a, y) \\in \\mathcal{A} \\times \\mathcal{P}$, the optimal solutions are uniquely determined by $y \\in \\mathcal{P}$, since the corresponding $a$ is the projection of $y$ to   $\\mathcal{A}$.\n\n\\begin{lemma}[{\\cite{JegelkaBS13}, Lemma 2}]\n\\label{lem:decomp-opt-set}\n The set $\\mathcal{E}$ is non-empty and it coincides with the set of optimal solutions of (\\ref{Prox-DSFM}) and (\\ref{Best-Approx}).\n\\end{lemma}\n\n{\\bf Gradient methods.} The gradient descent algorithms of \\cite{NishiharaJJ14,EneN15} provide level-1 algorithms for DSFM. In the following, we provide a brief overview of these algorithms and we refer the reader to the respective papers for more details. \n\n{\\bf The alternating projections algorithm.} Nishihara {\\em et al.}\\xspace minimize (\\ref{Best-Approx}) using \\emph{alternating projections} \\cite{NishiharaJJ14}. The algorithm starts with a point $a_0 \\in \\mathcal{A}$ and it iteratively constructs a sequence $\\set{(a^{(k)}, x^{(k)})}_{k \\geq 0}$ by projecting onto $\\mathcal{A}$ and $\\mathcal{P}$: $x^{(k)} = {\\operatorname{argmin}}_{x \\in \\mathcal{P}}\\|a^{(k)} - x\\|_2$, $a^{(k + 1)} = {\\operatorname{argmin}}_{a \\in \\mathcal{A}}\\|a - x^{(k)}\\|_2$.\n\n{\\bf Random coordinate descent algorithms.} Ene and Nguyen minimize (\\ref{Prox-DSFM}) using \\emph{random coordinate descent} \\cite{EneN15}. The RCDM algorithm adapts the random coordinate descent algorithm of Nesterov \\cite{Nesterov10} to (\\ref{Prox-DSFM}). In each iteration, the algorithm samples a block $i \\in [r]$ uniformly at random and it updates $x_i$ via a standard gradient descent step for smooth functions. ACDM, the accelerated version of the algorithm, presents a further enhancement using techniques from  \\cite{FR13}.\n\n{\\bf Rate of convergence.} The algorithms mentioned above enjoy a \\emph{linear convergence rate} despite the fact that the objective functions of (\\ref{Best-Approx}) and (\\ref{Prox-DSFM}) are not strongly convex. Instead, these works show that there are certain parameters that one can associate with the objective functions such that the convergence is at the rate $(1 - \\alpha)^k$, where $\\alpha \\in (0, 1)$ is a quantity that depends on the appropriate parameter. Let us now precisely define these parameters and state the convergence guarantees as a function of these parameters.\n\n Let $\\mathcal{A}'$ be the affine subspace \n$\\mathcal{A}':= \\{ a \\in \\mathbb{R}^{nr} \\colon Aa = s^*\\}.$ \nNote that $\\mathcal{E} = \\mathcal{P} \\cap \\mathcal{A}'$.\nFor $y\\in {\\mathbb{R}}^{nr}$ and a closed set $K\\subseteq {\\mathbb{R}}^{nr}$, we let $d(y,K)=\\min\\set{\\|y-z\\|_2 \\colon z\\in K}$ denote the distance between $y$ and $K$.\n The relevant parameter for the Alternating Projections algorithm is defined as follows.\n\\begin{definition}[\\cite{NishiharaJJ14}]\nFor every $y\\in (\\mathcal{P} \\cup \\mathcal{A}') \\setminus {\\cal E}$, let\n\t\\begin{align*}\n\t\t\\kappa(y) &\\coloneqq \\frac{d(y, {\\cal E})}{\\max\\set{d(y, \\mathcal{P}), d(y, \\mathcal{A}')}},\\quad \\mbox{ and }\\\\\n\t\t\\kappa_* &\\coloneqq \\sup\\left\\{ \\kappa(y) \\colon y \\in (\\mathcal{P} \\cup \\mathcal{A}') \\setminus {\\cal E} \\right\\}.\n\t\\end{align*}\n\\end{definition}\n\nThe relevant parameter for the random coordinate descent algorithms is the following.\n\n\\begin{definition}[\\cite{EneN15}]\n\tFor every $y\\in\\mathcal{P}$, let $y^* \\coloneqq {\\operatorname{argmin}}_{p} \\{\\|p-y\\|_2 \\colon Ap=s^*\\}$ be the optimal solution to (\\ref{Prox-DSFM}) that is closest to $y$. We say that the objective function ${1 \\over 2} \\|Ay\\|^2_2$ of (\\ref{Prox-DSFM}) is \\emph{restricted $\\ell$-strongly convex} if, for all $y \\in \\mathcal{P}$, we have\n\t\\begin{align*}\n\t\t&\\|A(y - y^*)\\|^2_2 \\geq \\ell \\|y - y^*\\|^2_2,\\quad \\mbox{ and }\\\\\n\t\t\\ell_* \\coloneqq \\sup&\\left\\{\\ell \\colon {1 \\over 2} \\|Ay\\|^2_2 \\text{ is restricted $\\ell$-strongly convex}\\right\\}.\n\t\\end{align*}\n\\end{definition}\n\nThe running time dependence of the algorithms on these parameters is given in the following theorems.\n\\begin{theorem}[\\cite{NishiharaJJ14}]\n\tLet $(a^{(0)}, x^{(0)} =  {\\operatorname{argmin}}_{x \\in \\mathcal{P}}\\|a^{(0)} - x\\|_2)$ be the initial solution and let  $(a^*, x^*)$ be an optimal solution to (\\ref{Best-Approx}). The alternating projection algorithm produces in\n\t\t\\[k = \\Theta\\left(\\kappa_*^2  \\ln\\left( {\\|x^{(0)} - x^*\\|_2 \\over \\epsilon} \\right) \\right) \\]\n\titerations a pair of points $a^{(k)} \\in \\mathcal{A}$ and $x^{(k)} \\in \\mathcal{P}$ that is $\\epsilon$-optimal, i.e.,\n\t\t\\[\\|a^{(k)} - x^{(k)}\\|_2^2 \\leq \\|a^* - x^*\\|_2^2 + \\varepsilon. \\]\n\\end{theorem}\n\n\\begin{theorem}[\\cite{EneN15}]\n\tLet $x^{(0)} \\in \\mathcal{P}$ be the initial solution and let $x^*$ be an optimal solution to (\\ref{Prox-DSFM}) that minimizes $\\|x^{(0)} - x^*\\|_2$. The random coordinate descent algorithm produces in\n\t\\[ k =  \\Theta\\left(  {r \\over \\ell_*} \\ln\\left( {\\|x^{(0)}  - x^* \\|_2 \\over \\epsilon} \\right)\\right)\\]\n\titerations a solution $x^{(k)}$ that is $\\epsilon$-optimal in expectation, i.e., $\\mathbb{E}\\left[{1 \\over 2} \\|A x^{(k)} \\|^2_2 \\right] \\leq {1 \\over 2} \\|Ax^*\\|^2_2 + \\epsilon$.\n\t\n\tThe accelerated coordinate descent algorithm produces in \n\t\\[k = \\Theta \\left (r \\sqrt{{1 \\over \\ell_*}} \\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon} \\right) \\right)  \\]\n\titerations (specifically, $\\Theta\\left( \\ln\\left( { \\|x^{(0)} - x^*\\|_2 \\over \\epsilon} \\right) \\right)$ epochs with $\\Theta\\left(r \\sqrt{{1 \\over \\ell_*}}\\right)$ iterations in each epoch) a solution $x^{(k)}$ that is $\\epsilon$-optimal in expectation, i.e., $\\mathbb{E}\\left[{1 \\over 2} \\|A x^{(k)} \\|^2_2 \\right] \\leq {1 \\over 2} \\|Ax^*\\|^2_2 + \\epsilon$.\n\\end{theorem}\n\nNishihara {\\em et al.}\\xspace show that $\\kappa_* \\leq nr$, and a family of instances (in fact, minimum cut instances) is given for which $\\kappa_* \\geq \\Omega(n \\sqrt{r})$. Ene and Nguyen show that $\\ell_* \\geq {r \/\\kappa^2_*}$. In Theorem~\\ref{thm:kappa-bound}, we close the remaining gap and show that $\\kappa_* = \\Theta(n \\sqrt{r})$ and $\\ell_* = \\Theta(1 \/ n^2)$, and thus we obtain tight analyses for the running times of the above mentioned algorithms.\n\nBy combining the level-1 gradient descent algorithms with appropriate level-0 subroutines, we obtain algorithms for DSFM whose running times can be upper bounded as follows. Using our improved convergence guarantees, it follows that RCDM obtains in time $O\\left(n^2 r\\Theta_{\\mathrm{avg}}\\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon}\\right)\\right)$ a solution that is $\\varepsilon$-approximate in expectation. For ACDM, the improved time bound is $O\\left(nr \\Theta_{\\mathrm{avg}} \\ln\\left( { \\|x^{(0)} - x^* \\|_2 \\over \\epsilon} \\right) \\right)$. We can upper bound the diameter of the base polytope by $O(\\sqrt{n} F_{\\max})$ \\cite{Jegelka11}. For integer-valued functions, a $\\varepsilon$-approximate solution can be converted to an exact optimum if $\\varepsilon=O(1\/n)$ \\cite{Bach-monograph}.\n\n\\section{Tight convergence bounds for the continuous algorithms}\n\\label{sec:kappa}\n\nIn this section, we show that the combinatorial approach introduced in Section~\\ref{sec:flow} can be applied to obtain better bounds on the parameters $\\kappa_*$ and $\\ell_*$ defined in Section~\\ref{sec:conv-opt}. Besides giving a stronger bound, our proof is considerably simpler than the algebraic one using Cheeger's inequality in \\cite{NishiharaJJ14}.\nThe key is the following lemma.\n\\begin{lemma}\n\\label{lem:decompose}\n\tLet $y \\in \\mathcal{P}$ and $s^* \\in B(f)$. Then there exists a point $x \\in \\mathcal{P}$ such that $Ax = s^*$ and $\\|x - y\\|_2 \\leq \\frac{\\sqrt{n}}{2} \\|Ay - s^*\\|_1$.\n\\end{lemma}\nBefore proving this lemma, we show how it can be used to derive the bounds.\n\\begin{theorem}\\label{thm:kappa-bound}\n\tWe have $\\kappa_* \\leq n \\sqrt{r}\/2 + 1$ and $\\ell_* \\geq {4 \/ n^2}$.\n\\end{theorem}\n\\begin{proof}\n\tWe start with the bound on $\\kappa_*$. In order to bound $\\kappa_*$, we need to upper bound $\\kappa(y)$ for any $y\\in (\\mathcal{P}\\cup\\mathcal{A}')\\setminus \\mathcal{E}$. We distinguish between two cases: $y \\in \\mathcal{P} \\setminus\\mathcal{E}$ and $y \\in \\mathcal{A}' \\setminus\\mathcal{E}$.\n\n\t\\noindent{\\bf Case I: $y \\in \\mathcal{P}\\setminus\\mathcal{E}$.} The denominator in the definition of $\\kappa(y)$ is equal to $d(y, \\mathcal{A}') = {\\|Ay - s^*\\|_2}\/{\\sqrt{r}}$. This follows since the closest point $a=(a_1,\\ldots,a_r)$ to $y$ in $\\mathcal{A}'$ is to set $a_i=y_i+(s^*-Ay)\/r$ for each $i\\in [r]$. \nLemma~\\ref{lem:decompose} gives an $x \\in \\mathcal{P}$ such that $Ax = s^*$ and $\\|x - y\\|_2 \\leq \\frac{\\sqrt{n}}{2}  \\|Ay - s^*\\|_1\\le \\frac{n}2 \\|Ay - s^*\\|_2$. Since $Ax = s^*$, we have $x \\in {\\cal E}$ and thus the numerator of $\\kappa(y)$ is at most $\\|x - y\\|_2$. Thus $\\kappa(y) \\leq {\\|x - y\\|_2 \/(\\|Ay - s^*\\|_2\/\\sqrt{r})} \\leq n\\sqrt{r}\/2$.\n\n\t\\noindent{\\bf Case II: $y \\in \\mathcal{A}'\\setminus\\mathcal{E}$.}\nThis means that $Ay = s^*$. The denominator of $\\kappa(y)$ is equal to $d(y, \\mathcal{P})$. For each $i \\in [r]$, let $q_i \\in B(f_i)$ be the point that minimizes $\\|y_i - q_i\\|_2$. Let $q = (q_1, \\dots, q_r)\\in {\\cal P}$. Then $d(y, \\mathcal{P}) = \\|y - q\\|_2$. Lemma~\\ref{lem:decompose} with $q$ in place of $y$ gives a point $x\\in {\\cal E}$ such that $\\|q - x\\|_2 \\leq \\frac{\\sqrt{n}}2 \\|Aq - s^*\\|_1$. We have\n\t $\\|Aq - s^*\\|_1 = \\|Aq - Ay\\|_1 \\leq \\sum_{i = 1}^r \\|q_i - y_i\\|_1 = \\|q - y\\|_1 \\leq {\\sqrt{nr}} \\|q - y\\|_2.$\n\tThus $\\|q - x\\|_2 \\leq \\frac{{n \\sqrt{r}} }2\\|q - y\\|_2$. \n\tSince $x \\in {\\cal E}$, we have \n\t$d(y, {\\cal E}) \\leq \\|x - y\\|_2 \\leq \\|x - q\\|_2 + \\|q - y\\|_2 \\leq \\left(1 + \\frac{n \\sqrt{r}}2\\right) \\|q - y\\|_2 = \\left(1 + \\frac{n \\sqrt{r}}2\\right) d(y, \\mathcal{P}).$\n\tTherefore $\\kappa(p) \\leq 1 + \\frac{n \\sqrt{r}}2$, as desired.\n\nLet us now prove the bound on $\\ell_*$. Let $y \\in \\mathcal{P}$ and let  $y^* \\coloneqq {\\operatorname{argmin}}_{p} \\{\\|p-y\\|_2\\colon Ap=s^*\\}$. We need to verify that $\\|A(y - y^*)\\|^2_2 \\geq {4 \\over n^2} \\|y - y^*\\|^2_2$.\nAgain, we apply Lemma~\\ref{lem:decompose} to obtain a point $x\\in \\mathcal{P}$ such that $Ax = s^*$ and  \n$\\|x - y\\|_2^2 \\leq \\frac{n}4\\|Ax-Ay\\|_1^2 \\le \\frac{n^2}4 \\|Ax - Ay\\|_2^2$.\nSince $Ax= s^*$, the definition of $y^*$ gives $\\|y - y^*\\|_2^2 \\leq \\|x - y \\|_2^2$. Using that $Ax = Ay^* = s^*$, we have $\\|Ax - Ay\\|_2 =\\|Ay-Ay^*\\|_2$. The same calculation as in Case II above implies the required $\\|y - y^*\\|_2^2 \\leq \\frac{n^2}4 \\|A(y - y^*)\\|_2^2$.\n\\end{proof}\n\n\\begin{proofof}{Lemma~\\ref{lem:decompose}}\nWe give an algorithm that transforms $y$ to a vector $x\\in {\\cal P}$ as in the statement through a sequence of path augmentations in the auxiliary graph defined in Section~\\ref{sec:flow}.\nWe initialize $x=y$ and maintain $x\\in {\\cal P}$ (and thus $Ax\\in B(f)$) throughout. We now define the set of source and sink nodes as \n$N := \\{ v \\in V \\colon (Ax)(v) < s^*(v) \\}$ and $P := \\{ v \\in V \\colon (Ax)(v) > s^*(v)\\}$. Once $N=P=\\emptyset$, we have $Ax=s^*$ and terminate. Note that since $Ax,s^*\\in B(f)$, we have $\\sum_v (Ax)(v)=\\sum_v s^*(v)=f(V)$, and therefore $N=\\emptyset$ is equivalent to $P=\\emptyset$.\nThe blocks of $x$ are denoted as $x=(x_1,x_2,\\ldots,x_r)$, with $x_i\\in B(f_i)$.\n\\begin{claim} If $N\\neq\\emptyset$, then there exists a directed path of positive capacity in the auxiliary graph between the sets $N$ and $P$.\n\\end{claim}\n\\begin{proof}\nWe say that a set $T$ is $i$-tight, if $x_i(T)=f_i(T)$. It is a simple consequence of submodularity that the intersection and union of two $i$-tight sets are also $i$-tight sets.\nFor every $i\\in [r]$ and every $u\\in V$, we define $T_i(u)$ as the unique minimal $i$-tight set containing $u$. It is easy to see that for an arc $(u,v)\\in E_i$, \n$c(u,v)>0$ if and only if $v\\in T_i(u)$. We note that if $u\\notin C_i$, then $x(u)=f_i(\\{u\\})=0$ and thus $T_i(u)=\\{u\\}$.\n\nLet $S$ be the set of vertices reachable from $N$ on a directed path of positive capacity in the auxiliary graph. For a contradiction, assume $S\\cap P=\\emptyset$. \nBy the definition of $S$, we must have $T_i(u)\\subseteq S$ for every $u\\in S$ and every $i\\in[r]$. Since the union of $i$-tight sets is also $i$-tight, we see that  $S$ is $i$-tight for every $i\\in[r]$, and consequently, $x(S)=f(S)$. On the other hand, since $N \\subseteq S$, $S \\cap P = \\emptyset$, and $N \\neq \\emptyset$, we have $x(S) < s^*(S)$. Since $s^* \\in B(f)$, we have $x(S) < s^*(S) \\leq f(S)$, which is a contradiction. We conclude that $S \\cap P \\neq \\emptyset$.\n\\end{proof}\n\nIn every step of the algorithm, we take a shortest directed path ${\\cal Q}$ of positive capacity from $N$ to $P$, and update $x$ along this path. That is, \nif $(u,v)\\in {\\cal Q}\\cap E_i$, then we increase $x_i(u)$ by\n$\\varepsilon$ and decrease $x_i(v)$ by $\\varepsilon$, where\n$\\varepsilon$ is the minimum capacity of an arc on\n$\\cal Q$. Note that this is the same as running the Edmonds-Karp-Dinitz algorithm in the submodular auxiliary graph. Using the analysis in \\cite{FujishigeZhang92}, one can show that this change maintains $x\\in {\\cal P}$, and that the algorithm terminates in finite (in fact, strongly polynomial) time.\n\nIt remains to bound $\\|x-y\\|_2$. At every path update, the change in $\\ell_{\\infty}$-norm of $x$ is at most $\\varepsilon$, and the change in $\\ell_1$-norm is at most $n\\varepsilon$, since the length of the path is $\\le n$. At the same time, $\\sum_{v\\in N} (s^*(v)-(Ax)(v))$ decreases by $\\varepsilon$.\nThus, $\\|x-y\\|_\\infty\\le \\|Ay-s^*\\|_1 \/2$ and $\\|x-y\\|_1\\le n\\|Ay-s^*\\|_1 \/2$. Using the inequality $\\|p\\|_2\\le \\sqrt{\\|p\\|_1\\|p\\|_{\\infty}}$, we obtain $\\|x-y\\|_2\\le \\frac{\\sqrt{n}}2 \\|Ay-s^*\\|_1$, completing the proof.\n\\end{proofof}\n\n\\section{The level-0 algorithms}\n\\label{sec:level-0}\n\nIn this section, we briefly discuss the level-0 algorithms and the interface between the level-1 and level-0 algorithms.\n\n{\\bf Two-level frameworks via quadratic minimization oracles.}\nRecall from the Introduction the assumption on the subroutines ${\\cal O}_i(w)$ that finds the minimum norm point in $B(f_i+w)$ for the input vector $w\\in{\\mathbb{R}}^n$. \nThe continuous methods in Section~\\ref{sec:conv-opt} directly use the  subroutines ${\\cal O}_i(w)$ for the alternating projection or coordinate descent steps. For the flow-based algorithms in Section~\\ref{sec:flow}, the main oracle query is to find the auxiliary graph capacity $c(u,v)$ of an arc $(u,v)\\in E_i$ for some $i\\in[r]$. This can be easily formulated as minimizing the function $f_i+w$ for an appropriate $w$ with $\\text{supp}(w)\\subseteq C_i$; the details are given in Lemma~\\ref{lem:exchange-cap-oracle}. As explained at the beginning of Section~\\ref{sec:conv-opt}, an optimal solution to \\eqref{prob:fuji} immediately gives an optimal solution to \\eqref{prob:sfm} for the same submodular function. Hence, the auxiliary graph capacity queries can be implemented via the subroutines ${\\cal O}_i(w)$. \nLet us also remark that, while the functions $f_i$ are formally defined on the entire ground set $V$, their effective support is $C_i$, and thus it suffices to solve the quadratic minimization problems on the ground set $C_i$.\n\n\\begin{lemma}\n\\label{lem:exchange-cap-oracle}\n  The capacity $c(u, v) := \\min\\{f_i(S) - x_i(S) \\colon S \\subseteq\n  C_i, u \\in S, v \\notin S\\}$ can be computed as the minimum value of\n  $\\min_{S\\subseteq C_i} f_i(S)+w(S)$ for an appropriately chosen vector $w\\in{\\mathbb{R}}^n$,\n  $\\text{supp}(w)\\subseteq C_i$.\n\\end{lemma}\n\\begin{proof}\n  We define a weight vector $w \\in \\mathbb{R}^n$ as follows: $w(u)\n  = - (f_i(\\{u\\}) + 1)$; $w(v) = - (f_i(C_i) - f_i(C_i \\setminus\n  \\{v\\}) - 1)$; $w(a) = - x(a)$ for all $a \\in C_i \\setminus \\{u,\n  v\\}$, and $w(a)=0$ for all $a\\notin C_i$.\nLet $A\\subseteq C_i$ be a minimizer of $\\min_{S \\subseteq C_i} f_i(S)\n+ w(S)$.\nIt suffices to show that $u \\in A$ and $v \\notin A$. Note that\n$f_i(\\{u\\}) = f_i(\\{u\\}) - f(\\emptyset)$ is the maximum marginal value\nof $u$, i.e., $\\max_{S} (f_i(S \\cup \\{u\\}) - f_i(S))$. Moreover,\n$f_i(C_i) - f_i(C_i \\setminus \\{v\\})$ is the minimum marginal value of\n$v$. To show $u\\in A$, let us assume for a contradiction that $u\\notin A$.\n  \\begin{align*}\n    f_i(A \\cup \\{u\\}) + w(A \\cup \\{u\\}) &= (f_i(A) + w(A)) + (f_i(A \\cup \\{u\\}) - f_i(A)) + w(u)\\\\\n    &= (f_i(A) + w(A)) + (f_i(A \\cup \\{u\\}) - f_i(A)) - f_i(\\{u\\}) + 1\\\\\n    &\\leq f_i(A) + w(A) - 1.\n  \\end{align*}\n  Similarly, to show that $v\\notin A$, suppose for a contradiction that $v \\in A$, and consider the set $A \\setminus \\{v\\}$. Since $f_i(C_i) - f_i(C_i \\setminus \\{v\\}) \\leq f_i(A) - f_i(A \\setminus \\{v\\})$, we have\n  \\begin{align*}\n    f_i(A \\setminus \\{v\\}) + w(A \\setminus \\{v\\}) &= (f_i(A) + w(A)) - (f_i(A) - f_i(A \\setminus \\{v\\})) - w(v)\\\\\n    &= (f_i(A) + w(A)) - (f_i(A) - f_i(A \\setminus \\{v\\})) + (f_i(C_i) - f_i(C_i \\setminus \\{v\\})) - 1\\\\\n    &\\leq f_i(A) + w(A) - 1.\n  \\end{align*}\n  Therefore $u \\in A$ and $v \\notin A$, and hence $A \\in {\\operatorname{argmin}}\\{f_i(S) - x_i(S) \\colon u \\in S, v \\notin S\\}$. \n\\end{proof}\n\n\n\nWhereas discrete and continuous algorithms require the same type of oracles, there is an important difference between the two algorithms in terms of exactness for the oracle solutions. The discrete algorithms require exact values of the auxiliary graph capacities $c(u,v)$, as they must maintain $x_i\\in B(f_i)$ throughout. Thus, the oracle  must always return an optimal solution. The continuous algorithms are more robust, and return a solution with the required accuracy even if the oracle only returns an approximate solution. As discussed in Section~\\ref{sec:experiments}, this difference leads to the continuous methods being applicable in settings where the combinatorial algorithms are prohibitively slow.\n\n{\\bf Level-0 algorithms.}\nWe now discuss specific algorithms for quadratic minimization over the base polytopes of the functions $f_i$. Several functions that arise in applications are ``simple'', meaning that there is a function-specific quadratic minimization subroutine that is very efficient. If a function-specific subroutine is not available, one can use a general-purpose submodular minimization algorithm. The works \\cite{Arora12,Fix13} use a {\\em brute force search} as the subroutine for each each $f_i$, whose running time is $2^{|C_i|} \\mathrm{EO}_i$. However, this is applicable only for small $C_i$'s and is not suitable for our experiments where the maximum clique size is quite large. \nAs a general-purpose algorithm, we used the {\\em Fujishige-Wolfe minimum norm point algorithm} \\cite{Fujishige11, Wolfe76}. This provides an $\\varepsilon$-approximate solution in $O(|C_i| F^2_{i,\\max}\/\\varepsilon)$ iterations, with overall running time bound $O((|C_i|^4 + |C_i|^2 \\mathrm{EO}_i) F^2_{i, \\max} \/ \\varepsilon)$. \\cite{Chakrabarty14}. \nThe experimental running time of the Fujishige-Wolfe algorithm can be prohibitively large \\cite{jegelka11fast}. As we discuss in Section~\\ref{sec:experiments}, by warm-starting the algorithm and performing only a small number of iterations, we were able to use the algorithm in conjunction with the gradient descent level-1 algorithms. \n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\n\\begin{table}\n\\caption{Instance sizes}\n\\label{tb:instance-sizes}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# pixels} & {\\bf \\# edges} & {\\bf \\# squares}\\\\\n\\hline\nbee & 273280 & 1089921 & 68160 \\\\\n\\hline\noctopus & 273280 & 1089921 & 68160\\\\\n\\hline\npenguin & 154401 & 615200 & 38400\\\\\n\\hline\nplant & 273280 & 1089921 & 68160\\\\\n\\hline\nplane & 154401 & 615200 & 38400 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n{\\bf \\# regions} & \\multicolumn{3}{|c|}{{\\bf min, max, and average region size}}\\\\\n\\hline\n50 & 298 & 299 & 298.02\\\\\n\\hline\n49 & 7 & 299 & 237.306\\\\\n\\hline\n50 & 5 & 299 & 279.02\\\\\n\\hline\n50 & 8 & 298 & 275.22\\\\\n\\hline\n50 & 10 & 299 & 291.48\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Minimum cut experiments}\n\\label{tb:mincut}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1363201 & 1.70942\\\\\n\\hline\noctopus & 1363201 & 1.09101 \\\\\n\\hline\npenguin & 769601 & 0.684413 \\\\\n\\hline\nplant & 1363201 & 1.30977 \\\\\n\\hline\nplane & 769601 & 0.745521 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf UCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n0.951421 & 1.6234 & 13.4594 & 134.719 \\\\\n\\hline\n0.937317 & 1.6279 & 13.9887 & 137.969 \\\\\n\\hline\n0.492372 & 0.836147 & 7.1069 & 70.1742\\\\\n\\hline\n0.943306 & 1.63492 & 13.9559 & 137.865 \\\\\n\\hline\n0.521685 & 0.850145 & 7.31664 & 71.8874 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n1.3769 & 2.2696 & 18.4351 & 182.069 \\\\\n\\hline\n1.40884 & 2.33431 & 19.0471 & 188.887 \\\\\n\\hline\n0.757929 & 1.24094 & 9.99443 & 98.5717 \\\\\n\\hline\n1.39893 & 2.29446 & 18.6846 & 185.274 \\\\\n\\hline\n0.766455 & 1.26081 & 10.1244 & 99.0298 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Small cliques experiments}\n\\label{tb:smallcliques}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1431361 & 14.5125 \\\\\n\\hline\noctopus & 1431361 & 12.9877 \\\\\n\\hline\npenguin & 808001 & 7.58177 \\\\\n\\hline\nplant & 1431361 & 13.7403 \\\\\n\\hline\nplane & 808001 & 7.67518 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n4.14091 & 7.57959 & 66.0576 & 660.496 \\\\\n\\hline\n4.29358 & 7.80816 & 68.5862 & 675.23 \\\\\n\\hline\n2.16441 & 4.08777 & 37.8157 & 372.733 \\\\\n\\hline\n4.6404 & 8.21702 & 69.059 & 672.753 \\\\\n\\hline\n2.182 & 4.12521 & 37.8602 & 373.825 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n5.24474 & 10.0951 & 98.7737 & 932.954 \\\\\n\\hline\n5.5891 & 10.7124 & 99.4081 & 924.076 \\\\\n\\hline\n2.95226 & 5.71215 & 52.9766 & 512.665 \\\\\n\\hline\n5.8395 & 11.0806 & 102.023 & 900.979 \\\\\n\\hline\n2.95003 & 5.70771 & 53.7524 & 486.294 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Large cliques experiments with potential specific quadratic minimization for the region potentials. In order to be able to run IBFS, we used smaller regions: $50$ regions with an average size between $45$ and $50$.}\n\\label{tb:largecliques}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{\\bf image} & {\\bf \\# functions ($r$)} & {\\bf IBFS time (sec)}\\\\\n\\hline\nbee & 1431411 & 14.7271\\\\\n\\hline\noctopus & 1431411 & 12.698 \\\\\n\\hline\npenguin & 808051 & 7.51067 \\\\\n\\hline\nplant & 1431411 & 13.6282 \\\\\n\\hline\nplane & 808051 & 7.64527 \\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n 4.29954 & 7.87555 & 67.8876 & 664.816\\\\\n\\hline\n 4.18879 & 7.61576 & 66.7 & 656.71\\\\\n\\hline\n 2.132 & 4.01926 & 36.9896 & 364.694\\\\\n\\hline\n 4.55894 & 8.06429 & 67.72 & 659.685\\\\\n\\hline\n 2.16248 & 4.0713 & 37.1917 & 366.272\\\\\n\\hline\n\\end{tabular}\n\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n 5.34726 & 10.3231 & 100.24 & 912.477\\\\\n\\hline\n 5.44726 & 10.4446 & 96.2384 & 898.579\\\\\n\\hline\n 2.90223 & 5.60117 & 51.9775 & 500.083\\\\\n\\hline\n 5.72946 & 10.8512 & 99.6597 & 879.872\\\\\n\\hline\n 2.89726 & 5.61102 & 52.5439 & 475.967\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nWe evaluate the algorithms on energy minimization problems that arise\nin image segmentation problems.\n We follow the standard approach and model the image segmentation task of segmenting an object from the background as finding a minimum cost $0\/1$ labeling of the pixels. The total labeling cost is the sum of labeling costs corresponding to \\emph{cliques}, where a clique is a set of pixels. We refer to the labeling cost functions as clique potentials.\n\nThe main focus of our experimental analysis is to compare the running times of the decomposable submodular minimization algorithms. Therefore we have chosen to use the simple hand-tuned potentials that were used in previous work~\\cite{Shanu16,Arora12,StobbeK10}: the \\emph{edge-based costs} defined by \\cite{Arora12} and the \\emph{count-based costs} defined by \\cite{StobbeK10}. Specifically, we used the following clique potentials in our experiments, all of which are submodular:\n\\begin{compactitem}\n\\item {\\bf Unary potentials} for each pixel. The unary potentials are derived from Gaussian Mixture Models of color features \\cite{RotherKB04}.\n\\item {\\bf Pairwise potentials} for each edge of the $8$-neighbor grid graph. Each graph edge $(i, j)$ between pixels $i$ and $j$ is assigned a weight that is a function of $\\exp(- \\|v_i - v_j\\|^2)$, where $v_i$ is the RGB color vector of pixel $i$. The clique potential for the edge is the cut function of the edge: the cost of a labeling is equal to zero if the two pixels have the same label and it is equal to the weight of the edge otherwise.\n\\item {\\bf Square potentials} for each $2 \\times 2$ square of pixels. We view a $2 \\times 2$ square as a graph on $4$ nodes connected with $4$ edges (two horizontal and two vertical edges). The cost of a labeling is the square root of the number of edges of the square that have different labels. This is the basic edge-based potential defined by \\cite{Arora12}.\n\\item {\\bf Region potentials} for a set of regions of the image. We compute a set of regions of the image using the region growing algorithm suggested by \\cite{StobbeK10}. For each region $C_i$, we define a count-based clique potential as in \\cite{StobbeK10,Shanu16}: for each set $S \\subseteq C_i$ of pixels, $f_i(S) = |S| |C_i \\setminus S|$.\n\\end{compactitem}\nWe used five image segmentation instances to evaluate the algorithms\\footnote{The data is available at \\url{http:\/\/melodi.ee.washington.edu\/~jegelka\/cc\/index.html} and \\url{http:\/\/research.microsoft.com\/en-us\/um\/cambridge\/projects\/visionimagevideoediting\/segmentation\/grabcut.htm}}. Table~\\ref{tb:instance-sizes} provides the sizes of the resulting instances. The experiments were carried out on a single computer with a 3.3 GHz Intel Core i5 processor and 8 GB of memory. The reported times are averaged over 10 trials.\n\n\\begin{table}\n\\caption{Large cliques experiments with Fujishige-Wolfe quadratic minimization algorithm for the region potentials. The Fujishige-Wolfe algorithm was run for $10$ iterations starting from the current gradient descent solution. The region sizes are given in Table~\\ref{tb:instance-sizes}.}\n\\label{tb:largecliquesmnp}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf RCDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n4.4422 & 8.18077 & 69.0444 & 674.526 \\\\\n\\hline\n4.30835 & 7.86231 & 68.1428 & 665.57\\\\\n\\hline\n2.2724 & 4.28243 & 38.1329 & 366.549\\\\\n\\hline\n4.61008 & 8.20094 & 68.8351 & 660.469\\\\\n\\hline\n2.28484 & 4.30316 & 38.0435 & 366.825\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{{\\bf ACDM time (sec)}}\\\\\n\\hline\n{\\bf \\# iter = $5r$} & {\\bf \\# iter = $10r$} & {\\bf \\# iter = $100r$} & {\\bf \\# iter = $1000r$}\\\\\n\\hline\n5.29305 & 10.2853 & 103.452 & 936.613\\\\\n\\hline\n5.55511 & 10.6411 & 97.955 & 901.875\\\\\n\\hline\n2.95909 & 5.74585 & 54.3808 & 505.977\\\\\n\\hline\n5.71402 & 10.8467 & 99.6515 & 873.694\\\\\n\\hline\n2.9556 & 5.73271 & 54.0599 & 482.496\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n{\\bf Number of iterations for the coordinate methods.} We have run the coordinate descent algorithms for $1000 r$ iterations, where $r$ is the number of functions in the decomposition. Our choice is based on the empirical results of Jegelka {\\em et al.}\\xspace \\cite{JegelkaBS13} that showed that this number of iterations suffices to obtain good results.\n\n{\\bf Minimum cut experiments.} We evaluated the algorithms on instances containing only the unary potentials and the pairwise potentials. Table~\\ref{tb:mincut} gives the running times in seconds.\n\n{\\bf Small cliques experiments.} We evaluated the algorithms on instances containing the unary potentials, the pairwise potentials, and the square potentials. Table~\\ref{tb:smallcliques} gives the running times in seconds.\n\n\\iffalse\nFor the coordinate descent methods, we report the running times after a given number of iterations between $5r$ and $1000r$, where $r$ is the number of functions. \n\\fi\n\n{\\bf Large cliques experiments.} We evaluated the algorithms on instances containing all of the potentials: the unary potentials, the pairwise potentials, the square potentials, and the region potentials. For the region potentials, we used a potential-specific level-$0$ algorithm that performs quadratic minimization over the base polytope in time $O(|C_i| \\log(|C_i|) + |C_i| \\mathrm{EO}_i)$. Additionally, due to the slow running time of IBFS, we used smaller regions: $50$ regions with an average size between $45$ and $50$.\n\n{\\bf Large cliques experiments with Fujishige-Wolfe algorithm.} We also ran a version of the large cliques experiments with the Fujishige-Wolfe algorithm as the level-$0$ algorithm for the region potentials. The Fujishige-Wolfe algorithm was significantly slower than the potential-specific quadratic minimization algorithm and in our experiments it was prohibitive to run the Fujishige-Wolfe algorithm to near-convergence. Since the IBFS algorithm requires almost exact quadratic minimization in order to compute exchange capacities, it was prohibitive to run the IBFS algorithm with the Fujishige-Wolfe algorithm. In contrast, the coordinate descent methods can potentially make progress even if the level-$0$ solution is far from being converged.\n\nIn order to empirically evaluate this hypothesis, we made a simple but crucial change to the Fujishige-Wolfe algorithm: we \\emph{warm-started} the algorithm with the current solution. Recall that the coordinate descent algorithms maintain a solution $x_i \\in B(f_i)$ for each function $f_i$ in the decomposition. We warm-started the Fujishige-Wolfe algorithm with the current solution $x_i$, and we ran the algorithm for a small number of iterations. In our experiments, we ran the Fujishige-Wolfe algorithm for $10$ iterations. These changes made the level-$0$ running time considerably smaller, which made it possible to run the level-$1$ coordinate descent algorithms for as many as $1000r$ iterations. At the same time, performing $10$ iterations starting from the current solution seemed enough to provide an improvement over the current solution. Table~\\ref{tb:largecliquesmnp} gives the running times.\n\n{\\bf Conclusions.} The combinatorial level-$1$ algorithms such as IBFS are exact and can be significantly faster than the gradient descent algorithms provided that the sizes of the cliques are fairly small. For instances with larger cliques, the combinatorial algorithms are no longer suitable if the only choice for the level-$0$ algorithms are generic methods such as the Fujishige-Wolfe algorithm. The experimental results suggest that in such cases, the coordinate descent methods together with a suitably modified Fujishige-Wolfe algorithm provides an approach for obtaining an approximate solution.\n\n\n\\newpage \\clearpage\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe largest globular cluster associated with our Galaxy, $\\omega$\nCentauri, shows substantial ranges in abundance of all elements that\nare studied (e.g. \\citealt{nd95a, scl95, smi00, pan02}).  The iron\nabundance ranges from \\mbox{[Fe\/H]$\\approx$--2.0} up to\n\\mbox{[Fe\/H]$\\approx$--0.4}.  A peak at [Fe\/H]=--1.7 exists which accounts for\napproximately 70\\% of the population, and there is a long tail to\nhigher metallicities \\citep{nfm96, sk96, lee99, pan00, sta06a}.\n\n\nStudies of individual elements have also shown a range in their\nabundances within the cluster members.  Carbon, nitrogen and oxygen\nshow large scatters in their abundance ratios for a given\n[Fe\/H]~\\citep{per80, bw93, nd95a}.  The $\\alpha$ elements (Mg, Si, Ca\nand Ti) show a constant value of \\mbox{[$\\alpha$\/Fe]{$\\approx$}0.3} below\n\\mbox{[Fe\/H]$<$--1.0} \\citep{bw93, scl95, nd95a, smi00} but decrease\nto \\mbox{[$\\alpha$\/Fe]=0.0} at higher metallicities\n\\mbox{([Fe\/H]$>$--1.0)} \\citep{pan02}. The constancy with iron at lower\nabundances indicates they are produced by the same source, most likely\nsupernovae Type II.  At the higher metallicities, the [$\\alpha$\/Fe]\ndecreases, consistent with supernovae Type Ia contributions.\n\n\nSodium and aluminium abundance ratios are correlated, and both are\nanticorrelated with [O\/Fe] \\citep{bw93, nd95a, nd95b, smi00}.\n\\citet{smi00} also report [Al\/Fe] being anticorrelated with [Mg\/Fe].\n[Cu\/Fe] has been found to be constant for \\mbox{[Fe\/H]$<$--0.8}\n\\citep{smi00, cun02}, but \\citet{pan02} reported a trend of increasing\n[Cu\/Fe] as the metallicity increased from \\mbox{[Fe\/H]=--1.2} to~--0.5.\nStudies of the iron peak elements (Cr, Ni, Ti) and metals (Sc, V) have\nshown that there is no trend with respect to iron (up to\n\\mbox{[Fe\/H]$\\approx$--0.8)}, consistent with primordial enrichment from Type\nII supernovae.  The heavy neutron capture elements have been shown by\n\\citet{nd95a} and \\citet{scl95} to increase sharply as a function of\niron abundance.  These results are in contrast to normal globular\nclusters, and suggest that the stellar winds from AGB stars (sources\nof s-process elements) were involved with the enrichment of {$\\omega$ Cen}.\n\n\nThe cluster also has several members that are of distinct classes,\nsuch as CH stars, Ba stars, S stars \\citep{le83} and stars with strong\nCO \\citep{per80} to name a few.  Here we present another object that\nhas unusual abundance ratios, the formation of which is difficult to\nexplain.\n\n\nWe have obtained observations of a sample of main sequence and turnoff\n(MSTO) stars within the cluster for several purposes.  Firstly, the\npossibility of an age-metallicity relation within the cluster's member\nstars was investigated \\citep{sta06a}.  We found an age range of\n\\mbox{2--4} Gyrs exists between the most metal-poor and metal-rich\npopulations within the cluster, similar to that found by other studies\n\\citep{hil04, rey04, sol05}. Positions, photometry, metallicities and\nages of the stars in the catalogue can be found in the electronic\nversion of \\citet{sta06a}.  The second goal was to determine abundance\nratios of several elements --- carbon, nitrogen, strontium and barium\nas functions of [Fe\/H] \\citep{sta06b}. In that analysis a metal-rich\nmain sequence (MS) member, 2015448, was found that exhibited unusual\ns-process abundance ratios. The purpose of this Letter is to detail\nthe abundance analysis for star 2015448, and to propose possible\nformation scenarios.  \\S \\ref{OR_sect} briefly describes the\nobservation and reduction process for our sample of main sequence and\nturnoff members. The stellar parameters and analysis are detailed in\n\\S \\ref{SPA_sect}, while \\S \\ref{D_sect} discusses the possible\nformation scenarios for this object.\n\n\n\\section{Observations} \\label{OR_sect}\n\nThe observations and reduction of our data are described in detail in\n\\citet{sta06a} to which we refer the reader.  Briefly, photometry for\nthe cluster was obtained in the V and B bands, and samples were chosen\nwithin an annulus \\mbox{15'---25'} from the cluster center.  Two\nregions were defined near the MSTO, shown in Figure \\ref{cmd}, and\nspectra were obtained for objects within these CMD regions using the\nTwo Degree Field Spectrograph (2dF) on the Anglo-Australian Telescope\n\\citep{lew02}.  Figure \\ref{cmd} shows the color-magnitude diagram of\nall objects with no membership information as small dots, the sample\nof 420 radial velocity members as large dots, and the object that is\nthe topic of this Letter as a large star.  This object, 2015448, has\n$V=18.22$, and $B-V=0.69$.  This star immediately stood out in a\nvisual inspection of the spectra, due to the anomalously strong Sr\nlines, seen in a Figure \\ref{spec}.  Its position is\n\\mbox{RA=13$^h$24$^m$49.10$^s$} and\n\\mbox{Dec=--47$\\degr$40$'$34.7$''$} (J2000).  With a radial velocity\nof \\mbox{228$\\pm$11~kms$^{-1}$}, star 2015448 is likely to be a member of\nthe cluster as {$\\omega$ Cen} has a radial velocity of\n\\mbox{232$\\pm$0.7~kms$^{-1}$} \\citep{din99}\n\n\nThe observations were carried out in half hour exposures, and, as\nseveral such exposures were needed to obtain the required\nsignal-to-noise of $\\sim$30 for each star, there are several\nindividual observations for star 2015448.  Also, this object was observed\nin observing sessions in 1998, 1999 and 2002, in different fibres and\nspectrographs. This gives an independent check on the relative\nstrengths of the Sr and Ba features.  It was found that the two Sr\nfeatures were present in all observations, while Ba was not clearly\ndetected in any of our spectra.\n\nA spectroscopic abundance analysis of the full sample is detailed in\n\\citet{sta06b}.  Abundances relative to iron of C, N and Sr were\ndetermined using spectrum synthesis techniques.  The CH feature at\n$\\sim$4300{\\AA} was used to determine the [C\/Fe] abundance for each\nstar.  This [C\/Fe] was then adopted when the CN feature at 3883{\\AA}\nwas analyzed to find [N\/Fe].  The Sr abundance ratio was determined\nusing the two Sr lines at 4077.71{\\AA} and 4215.52{\\AA}.  When\npossible, [Ba\/Fe] was also determined from the feature at\n4554.03{\\AA}.\n\n\n\\section{Stellar Parameters and Analysis} \\label{SPA_sect}\n\nThe process for calculating metallicities is described in detail in\n\\citet{sta06a}.  The Sr-rich star has a metallicity of\n\\mbox{[Fe\/H]=--0.74$\\pm$0.23~dex}, determined solely from the\nAuto-Correlation Function method, and an age of\n\\mbox{14.5$\\pm$2.2~Gyrs}.  This age is consistent with the mean age of\nthe bulk of the stars in the cluster.  The ages for the individual\nstars in the sample were calculated by fitting an isochrone to each\nobject using the magnitude, color, metallicity and [$\\alpha$\/Fe]\nabundance (Y$^2$, \\citet{yi01}, Green Table).  This also enabled an\nindividual temperature and gravity to be assigned to each star.  A\ntemperature of 5820 (K), and \\mbox{log$g$=4.2~(cgs)} were adopted for\nthis object.  A reddening \\mbox{$E(B-V)$=0.11} \\citep{lub02} and\ndistance modulus \\mbox{(m--M)$_{V}$=14.10} were assumed.  Although the\nstar's color is redder than the bulk of the population in the cluster,\nit is also more metal-rich than that population. The hydrogen line\nstrengths in the observed spectrum are consistent with those expected\non the basis of the star's color.\n\nSynthetic spectra were obtained using the stellar models of Kurucz\n(1993), atomic line lists of Bell (2000, private communication) and\nmolecular line lists of Kurucz. The spectrum synthesis code was\ndeveloped by \\citet{cn78}.  More details of this process are given in\n\\citet{sta06b}.\n\n\n\n\\subsection{Abundance Ratios} \\label{A_sect}\n\nFigure \\ref{spec1} shows the synthetic spectrum fits to the CH, CN, Sr\nand Ba features.  The G band at $\\sim$4300{\\AA} was analyzed first,\nand gave a C abundance of \\mbox{[C\/Fe]=--0.5$\\pm$0.3~dex}.  This\nfeature loses its sensitivity for abundances less than\n\\mbox{[C\/Fe]=--0.5}, shown by the close proximity of the\n\\mbox{[C\/Fe]=--0.8} and --0.5~dex synthetic lines.  As we have no\ninformation on O, which affects the C abundance obtained from the CH\nfeature, a value of \\mbox{[O\/Fe]=0.18~dex} was assumed.  This star was\nassumed to have a \\mbox{[C\/Fe]=--0.5}, which was then used when\ndetermining [N\/Fe]. Synthesis of the CN feature at $\\sim$3883{\\AA} led\nto a N abundance ratio of \\mbox{[N\/Fe]$<$0.5~dex}.  A range of N\nabundance ratios are shown in the synthetic spectra, enabling an upper\nlimit on to be placed on the N abundance ratio.  The CN feature in\nFigure \\ref{spec1} shows that abundance ratios of \\mbox{[N\/Fe]=1.5}\nand 1.0 are too high.  However, this feature loses its sensitivity as\nsmaller N abundance ratios are considered and only an upper limit can\nbe determined.\n\n\n\nThe Sr and Ba lines were analyzed together.  There were two Sr lines\nused, Sr {\\sc} II at 4077.71{\\AA} and Sr {\\sc II} at 4215.52{\\AA}.\nThe Ba {\\sc II} 4554.03{\\AA} line was used to constrain the Ba\nabundance ratio.  The two Sr lines are in agreement and yield\n\\mbox{[Sr\/Fe]=1.6$\\pm$0.1~dex}.  It can be seen clearly that a solar\nSr abundance ratio does not fit the observed spectrum.  The CN\nbandhead at $\\sim$4216{\\AA} can affect the abundance ratio obtained\nfor the Sr 4215{\\AA} feature. However with an assumed solar N\nabundance ratio, there is little effect to the Sr line.  While the N\nabundance ratio could be as high as \\mbox{[N\/Fe]=0.5~dex}, spectrum\nsynthesis calculations show this does not affect on the Sr 4215{\\AA}\nline significantly, and does not alter the abundance ratio obtained.\nUsing the same enhancement for Ba that was found for Sr, one finds the\npredicted strength to be too high to fit the observed Ba line.  The\nsensitivity of the Ba feature is low up until abundance ratios of\n\\mbox{[Ba\/Fe]=0.6~dex}, and an upper limit can be placed at this\nvalue.\n\n\n\\subsection{Errors} \\label{E_sect}\n\nThe stellar parameters temperature, gravity and metallicity, for\nstar 2015448 were varied individually by their uncertainties to give an\nestimate of the error in our determined abundance ratios.  These were\nthen added in quadrature.  The error in temperature comes from\n$\\Delta(B-V)$ and the reddening, equating to $\\pm$100K uncertainty in\ntemperature.  This propagated to errors of $\\pm$0.2~dex in C and\n$\\pm$0.1~dex in Sr.  Gravity was changed by $\\pm$0.2~dex, and did not\nlead to any significant errors in the final abundance ratios.  The\nuncertainty in metallicity propagated to errors in abundance ratios\nfor \\mbox{$\\Delta$C$\\pm$0.2~dex} and \\mbox{$\\Delta$Sr$\\pm$0.1~dex}.\nThe CN feature used to determine N, adopting the previously determined\nC abundance ratio for 2015448, did not show any change with the above\nchanges in temperature, gravity, metallicity, or the determined error\nin the C abundance ratio, due to the low sensitivity of this feature.\nTo determine the effect of the assumed oxygen abundance,\n\\mbox{[O\/Fe]=0.0} and +0.3 were used and the CH feature reanalyzed.\nIt was found to have no noticeable effect on the [C\/Fe] abundance\ndetermined.\n\n\n\\section{Discussion} \\label{D_sect}\n\nThis peculiar star in {$\\omega$ Cen} shows \\mbox{[C\/Fe]=--0.5},\n\\mbox{[N\/Fe]$<$0.5}, enhanced \\mbox{[Sr\/Fe]=1.6}, and\n\\mbox{[Ba\/Fe]$<$0.6}.  The formation process that created it was\nunusual as there are no similar objects yet found within the cluster.\nWe do find MS objects with high Sr, but these also have similar\nenhancements in Ba.  This star was one out of 420 stars studied on the\nmain sequence turnoff.  Given that it is a metal-rich star, and we\nonly found 25 such objects, a more extensive search of other areas of\nthe center of the cluster may prove fruitful in determining if there\nare more of these objects.  On the RGB, at a metallicity of\n\\mbox{[Fe\/H]=--0.8}, most stars have \\mbox{[Ba\/Fe]$\\approx$+0.6}\n\\citep{nd95a}, which is consistent with the upper limit found for\nstar 2015448.  \\citet{nd95a} did not observe Sr, and a direct comparison\ncannot be made.  However, they did investigate other light s-process\nelements such as Y and Zr.  Both of these elements have abundance\nratios equal to or less than 0.6, and no stars on the RGB exhibit\ns-process abundances as high as is found here for [Sr\/Fe].\n\nThere are at least two possibilities that lead to the formation of\nthis star. Firstly, it may have been in a binary system with a\ncompanion that underwent unusual s-process enrichment and transfered\nmass to the star we see today.  At present we have no evidence that\nthis star is in a binary system: the individual velocities from the\n1998, 1999 and 2002 observations are consistent to within the\nmeasurement errors of \\mbox{$\\sim$10kms$^{-1}$}.  A second possibility\nis that the Sr-rich star formed out of already enriched material.\n\nAGB stars of low and intermediate mass dredge up $^{12}$C to the\nsurface via helium shell thermal pulses.  In general, if the star is\nmore massive than $\\sim$3M$_{\\odot}$, the $^{12}$C is processed to\n$^{14}$N via the CN cycle \\citep{vdm02}.  The main s-process is\nthought to occur within the $^{13}$C pocket where the neutron source\nis the \\mbox{$^{13}$C($\\alpha$,n)$^{16}$O} reaction\n(e.g. \\citealt{lk01}).  To generate the abundance ratio seen here it\nwould seem necessary that the s-processing occurs only for the light\ns-process elements, perhaps up to the peak at Zr, rather than\nprogressing through to heavy s-process elements which include Ba.\nThis was may be due to the neutron density not being high enough, as\nhigher neutron densities, assuming all other parameters being equal,\nwill produce larger amounts of heavier nuclei such as Ba or La,\nrelative to the lighter ones \\citep{smi05}.\n\nAnother source of s-processing, known as the weak s-process, occurs in\nmassive stars \\citep{phn90}. The \\mbox{$^{22}$Ne($\\alpha$,n)$^{25}$Mg}\nreaction is the source of the neutrons.  Most of the neutrons produced\nfrom this reaction are captured by the light elements, and only a\nsmall fraction are captured by the $^{56}$Fe seed nucleus, a process\nknown as ``self-poisoning''. This is the reason for the limited\nefficiency of the \\mbox{$^{22}$Ne($\\alpha$,n)$^{25}$Mg} source for\ns-process. It allows for the production of only light s-process nuclei\nwith mass numbers \\mbox{65$<$A$<$90} and Sr, with a mass number of 87,\nfalls into this range. Ba, on the other hand with mass number 137,\ndoes not and little of this element is produced by this process\n\\citep{phn90}. It is unclear whether massive stars could also produce\nthe necessary C and N abundance pattern found here, or whether star\n2015448 contains other elemental abundance patterns that may have\noriginated from these objects.\n\n\nThe r-process occurs in an environment that is rich in neutrons, and\nthe mean time between successive neutron captures in very short\ncompared with the time to undergo $\\beta$-decay.  This scenario as the\nenrichment source also requires the presence of SNe products such as\ncalcium and iron, or to have the Sr enriched material transferred but\nnot any SNe products.\n\nUsing the abundance yields for the r and (main) s-process (occurring\nin low-to-intermediate mass AGB stars) from \\citet{cam82}, and\nnormalizing to [Sr\/Fe] one finds a difference greater than one dex\nbetween predicted and observed abundances for Ba.  For the main\ns-process, the predicted abundance is \\mbox{[Ba\/Fe]=1.68}, while the\nr-process predicts \\mbox{[Ba\/Fe]=1.84}.  Both of these values are\nobviously too large, confirming the unusual nature of the source of\nthe abundance patterns found in star 2015448.  However, it should be\nnoted that AGB star yields do not explain the abundance anomalies of\nNa, Mg, Al, and O found in other stars of {$\\omega$ Cen} or globular clusters\n(see e.g. \\citealt{fen04}).  This may be due to the inadequacy of the\nmodels used to calculate yields for these objects.\n\nFrom weak s-process yields in massive stars \\citep{phn90}, using an\ninitial metallicity of \\mbox{Z\/Z$_{\\odot}$=10$^{-1}$} and mass of\n16M$_{\\odot}$, \\mbox{[Sr\/Fe]\\eq1.8} and \\mbox{[Ba\/Fe]\\eq0.7}, in good\nagreement with the Sr abundance and upper limit of the Ba one obtained\nhere.  This indicates the weak s-process is the more favorable option\nfor the source of enrichment in star 2015448.\n\n\nThis star is also highly unusual when compared to observations of\nfield stars.  Studies of n-capture elements in samples of field stars\nfor a range of metallicities show a spread in abundance ratios of Sr\nand Ba at low metallicities \\mbox{([Fe\/H]$<$-2.0)}, with most stars\nhaving [Sr\/Fe] or [Ba\/Fe] $\\leq$0.0~\\citep{mcw98, bur00, hon04}.  At\nhigher metallicities, the abundances are within $\\sim$0.5 of the solar\nabundance.\n\nA cool field giant, U Aquarii, has enhanced [Sr\/Fe] and [Y\/Fe]\nabundances, and low [Ba\/Fe] \\citep{bln79}.  This star is a faint R CrB\ntype variable star and shows no CH features, but strong $^{12}$C$_{2}$\nbands.  \\citet{bln79} concluded that U Aqr is a hydrogen deficient\ncarbon star with enhanced abundances of the light s-process elements\nSr and Y (by a factor of $\\sim$100) and little or no Ba.  It is now a\nHe-C core of an evolved star of $\\sim$1M$\\odot$ that ejected its H\nrich envelope at the He core flash. \\citet{bln79} postulated that a\nsingle neutron exposure occurred at the flash resulting in a brief\nneutron irradiation producing only the light s-process elements. A\nsimilar giant, known as Sakurai's Object \\citep{asp97} shows similarly\nenhanced C and light s-process elements, and is H deficient.  These\ntypes of stars may be responsible for the abundance pattern found in\nstar 2015448.  However, a significant difference is the C-rich nature of\nthe giants compared with the carbon depleted nature of star 2015448.\n\nOur results provide a challenging puzzle to determine the source of\nthe abundance patterns found for this star.  That said, the resolution\nof our data is inadequate to address this question.  Higher resolution\nspectra with high S\/N are needed to analyze as many elements as\npossible, in particular the s-process ones, to be able to obtain an\naccurate history for the evolution of star 2015448.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator $Q$ squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction. \n\nThis article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree $p$, we pay particular attention to $p=2$, i.e.~1-~and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g.~one as a particular kind of a vector-bundle twisted Courant algebroid. \n\\end{abstract}\n\\newpage\n\\tableofcontents\n\\section{Introduction and Results}\n\\subsection{Motivation}\nThese are basic and general considerations on higher gauge theories, i.e.~gauge theories where the standard connection 1-forms of Yang-Mills (YM) gauge theories are replaced by a whole tower of such gauge fields of different form degrees. Standard Yang-Mills theories are governed by some (finite-dimensional, quadratic) Lie algebra $\\g = \\Lie(G)$, where $G$ is the  structure group of the underlying principal bundle. On the other hand, when one attempts to construct an interacting theory of 1-form gauge fields, various consistency conditions would lead one to a constraint that has to be satisfied by the interaction constants $C^a_{bc}$, namely\\footnote{There is a second condition that one obtains: The constants $\\eta_{ab}$ entering the ``free'' kinetic term in a functional is also constrained; it has to correspond to an ad-invariant scalar product.}\n\\begin{equation} \\label{Jacobi}\nC^a_{eb} C^e_{cd} + {\\text{cycl.}}(bcd) = 0 \\, .\n\\end{equation}\nThis constraint is easily recognized as the Jacobi identity of a Lie algebra. Thus, also in the other direction, one is led by itself to a Lie algebra $\\g$ which, in a second step, then can be related to a principal bundle as mentioned before. In this paper we want to follow this strategy for the much more involved context of gauge fields of form degrees up to some $p \\in \\N$, including 0-forms for completeness. \n\nAt the place of \\eqref{Jacobi} one then obtains a much more involved set of equations for the unknown parameters introduced in an ansatz for the multiple interactions. Due to the fact that we permit also the 0-forms, these are even differential equations. It is one of the goals of this paper to identify this coupled system of structural equations with an underlying mathematical structure which it corresponds to, in generalization of the incomparably simpler relation of \\eqref{Jacobi} with a structural Lie algebra $\\g$, needed for the construction of an ordinary Yang-Mills theory. \n\nAnother goal is, however, to provide one mathematical formulation of what ``consistency conditions'', often imposed in more physically oriented constructions, can mean. In particular, we aim at isolating a basic or minimal set of such requirements, which we believe that the physics community would usually want to have and which we found to be fulfilled in all the examples we know or are aware of. \n\nTwo, three remarks are in place here in this context: First, there exists a very general and elegant approach to the \\emph{deformation} of free gauge theories to interacting ones, given by a BRST-BV-formalism, cf.~\\cite{BBH00} for a review. There one starts with just the kinetic terms of the fields that one wants to consider and the related, simple-to-find free gauge transformations. The result of the procedure provides the most general possible action that contains the original terms to lowest order and that preserves the ``number'' of the gauge symmetries, although not their form; e.g.~starting with $r$ 1-form fields $A^a$ and the free (abelian) action functional \n\\begin{equation}\\label{Skin}\nS[A] = \\int_\\Sigma \\eta_{ab} \\, \\ud A^a \\wedge *\\ud A^b \\, ,\n\\end{equation}\none is led to the most general Yang-Mills functional for a quadratic Lie algebra $\\g$ of dimension $r$ (cf.~the above discussion including the previous footnote), the abelian curvature or field strength $\\ud A^a$ being effectively replaced by its non-abelian generalization $F^a = \\ud A^a + \\frac{1}{2}C^a_{bc}A^b A^c$ in the above functional. There are several advantages of this procedure: except for possible problems of global nature or of convergence (one works in the setting of formal power series) the approach is completely systematic and extensive, providing the most general (formal) deformation moreover already up to trivial ones, that can be induced by changes of coordinates on the space of fields and that is captured by BRST-exact terms vanishing in the cohomological treatment; furthermore, the resulting theory already comes in its BV-form, thus ready for a subsequent quantization. There is, however, one decisive drawback of this formalism: The calculations are generically very technical and lengthy and the resulting constraints on the parameters of the deformation do not come with a mathematical interpretation. This does not pose a problem if they are of a comparatively simple form like in Equation \\eqref{Jacobi} (accompanied by a second condition expressing the ad-invariance of $\\eta$ in this case), where one retrieves a more mathematical interpretation rather easily. But it does so in a context like the present one, where the structural equations that one finds can easily fill a page (cf., e.g., results like in \\cite{Bizdadea09,Bizdadea09b,Bizdadea10}).\n\nSecond, what is also transparent from the above YM example already, in the case of blowing up the number of equations by adding another permitted form degree, it may be useful and will prove so to separate equations that replace the defining condition of the Lie algebra, Eq.~\\eqref{Jacobi}, from objects defined on it, like the ad-invariant tensor $\\eta$, that is needed for the functional  (cf.~the discussion above) and thus the dynamics of the physical fields, but not yet for the underlying geometry of a principal bundle. Note that a metric $h$ on $\\Sigma$ entered the functional (in terms of the Hodge duality operation $*$) and in more complicated theories the gauge transformations leaving invariant a functional can and will depend on $h$ also (cf.~\\cite{DSM} for an example), while some important part of the gauge transformations, to be specified below, will prove to be independent of this additional structure. Both these observations lead us to focus on generalizations of the curvature or field strength in a first step, where neither $\\eta$ nor $h$ enters. \n\nThis leads us also to postpone questions of the global bundle structure to a separate investigation: one may want to find the generalization of the structural Lie algebra, that governs the theory already on a local level, in a first instance. \nThis restricted focus permits one to work in a local chart on the spacetime or base manifold $\\Sigma$, to replace a connection by a set of 1-forms (in the ordinary YM case, but generalized to include higher form degrees here) and to deal with the question of how to represent an (adequately generalized) connection on a possibly non-trivial (higher) bundle in a second step only.\\footnote{In fact, this investigation appeared in the mean time already, at least on the arXiv, cf.~\\cite{KS07}. The present article thus can be seen, not solely but also, as a possible introduction or motivation from the perspective of physics to \\cite{KS07},\nwhich has an essentially mathematical focus. We will describe the resulting global picture in subsection \\ref{sec:phys} below, cf., e.g., Fig.~\\ref{fig1}.} \n\n\\subsection{Structure of the article}\nThe paper is organized as follows.  In the two subsections \\ref{sec:phys} and \\ref{sec:math} we summarize the main results of the present paper. \n\nIn Section~\\ref{s:Bian} we expand in detail on the first part of what is explained in subsections \\ref{sec:phys} below. In particular, we draw the necessary mathematical conclusions from the physical requirements, which we believe to be minimal or mandatory for any higher gauge theory. The main result of these considerations is summarized in Theorem~\\ref{thm:phys1} below. In Section~\\ref{s:examQ} we unravel the geometric structure of Q2-manifolds, as they appear as what replaces the structural Lie algebra of an ordinary YM gauge theory when considering a higher gauge theory with 0-form, 1-form, and 2-form gauge fields. They are in equivalence with Lie 2-algebroids or 2-term $L_\\infty$-algebroids. In Section~\\ref{s:examQ} we provide, in particular, a description generalizing the one of a Lie 2-algebra as a crossed module, in the Appendix~\\ref{s:Linf} we complement this by a descriptions in terms of multiple brackets as usual for $L_\\infty$-algebras (cf., also, subsection \\ref{sec:math} below).\n\nIn Section~\\ref{s:gauge} we take another step in the construction of a ``physically acceptable'' higher gauge theory, focusing on the gauge symmetries in this context. The main result of this investigation is summarized in Theorems~\\ref{thm:phys2} and \\ref{theo:Lie}) below. Having been led to regard derived brackets in this context, we take another view on Q2-manifolds or Lie 2-algebras from this perspective in the subsequent Section~\\ref{sec:derived}. This motivates the definition of what we call a V-twisted Courant algebroid. We derive its elementary properties in that section and provide sufficient conditions under which it is equivalent to a Lie 2-algebroid again. \n\nAppendix~\\ref{s:Linf} reviews briefly the relation between Q-structures and $L_\\infty$-algebras\/ algebroids in general.  \n\n\\subsection{Chronology of the results}\\label{s:hist\n\nSections \\ref{s:Bian}, \\ref{s:examQ}, and \\ref{s:gauge} were written already in 2005, Section  \\ref{sec:derived} between 2005 and 2008 (as visible also by several talks of TS and a partial distribution of preliminary versions). The Introduction and the Appendix, as well as some fine-tuning of the remainder, is from the end of 2013\/beginning of 2014, where we finally finished the article. For TS the results of this work as of 2005 were among the main motivations for starting his work with Alexei Kotov on characteristic classes and Q-bundles, the first results of which appeared as a preprint in 2007 on the arXiv and is now published right behind this article in the present journal \\cite{KS07}. We attempted to leave the Sections  \\ref{s:Bian}, \\ref{s:examQ}, \\ref{s:gauge}, and \\ref{sec:derived} as much as possible in their original form from the time of 2005 to 2008, including an updated perspective only in the two subsections to follow (to which we then also moved important formulas and statements from those older sections, like Equation \\eqref{ideal} and Theorem \\ref{thm:phys1}, respectively)\n\nFor what concerns our work summarized in section  \\ref{sec:derived}, we found out later that there were parallel developments arriving at a similar notion of a vector-bundle twisted Courant algebroid as the one given in the present paper. It is interesting to see that quite different motivations and starting points led to the same or similar objects of interest at about the same time.  In the following, we comment on the relation in more detail: Chen, Liu and Sheng have introduced $E$-Courant algebroids in \\cite{CLS08}.  An $E$-Courant algebroid as defined by them is a $V$-twisted Courant algebroid as defined in this paper after identifying their vector bundle $E$ with our $V$ iff in addition their inner product is surjective to this vector bundle.  Conversely a $V$-twisted Courant algebroid is a $V$-Courant algebroid in their sense iff the anchor $\\rho$ factors through $\\mathfrak{D}V$, the derived module of $V$.  Similarly, Li-Bland introduced  an $AV$-Courant algebroid in \\cite{LiB11}. This is a  $V$-twisted Courant algebroid in our sense and, conversely, every $V$-twisted Courant algebroid is an $AV$-Courant algebroid iff the anchor map factors through a Lie algebroid $A$ and the action of the twisted Courant algebroid on $V$ induces an action of $A$ on $V$.\n  \n\n\n\\subsection{Acknowledgments}\nWe thank the TPI in Jena and in particular A.~Wipf for the support of our work in 2005 and 2013, the ESI in Vienna for support in 2007, as well as the ICJ in Lyon for likewise support in 2008 and 2013. \nWe also thank the anonymous referee for helpful suggestions, such as a change of the subtitle. T.S.~thanks A.~Kotov for many valuable discussions during our stimulating parallel work. \n\n\\subsection{Mathematical framework for physical theories} \\label{sec:phys}\nLet us now describe our approach in some detail and summarize the main results of the paper. As mentioned above, we consider a tower of differential forms. Let us denote them collectively by $A^\\alpha$ and call them the ``gauge fields'' of the theory under investigation. So $A^\\alpha$ collects differential forms of various form degrees, in general at each level of a different number: $A^\\alpha = (X^i, A^a,B^D, \\ldots)$, where $(X^i)_{i=1}^n$ are $n$ 0-forms (or scalar fields), $(A^a)_{a=1}^r$ are $r$ 1-forms (called, in a somewhat misleading way, ``vector fields'' in the physics literature), $(B^C)_{C=1}^s$ are $s$ 2-forms etc., up to some highest form degree $p$. At this stage of the consideration we do not want to attach any geometrical interpretation to those differential forms yet, treating them in a most elementary way possible in the first step and believing in the power of ``physically oriented'' argumentation, supposed to lead us to a geometrically interesting picture more or less by itself in the end. \n\nNext we make a most general ansatz for the ``field strengths'' $F^\\alpha$ compatible with the form degrees and not using any additional structures (like a metric $h$ on $\\Sigma$), since we believe that these objects, which in the end should represent in one way or another some generalized curvature(s), should not depend on anything else but the data specified up to here. Most physicists would agree, moreover, that $F^\\alpha$ should start with the exterior derivative acting on the gauge fields, $F^\\alpha = \\ud A^\\alpha + \\ldots$, and that this should be complemented by terms, represented by  the dots, that are polynomial in the gauge fields of form degree 1 and higher, but, to stay as general as possible, with a priory unrestricted coefficient functions of the scalar fields $X^i$. This ansatz can be considered as still too restrictive, however. First, one may possibly want to replace the first term in $F^\\alpha$ by a coefficient matrix $M^\\alpha_\\beta$---depending on the $X^i$s---times $\\ud A^\\beta$; second, one may want to permit also $\\mathcal{A}^\\beta \\wedge \\ud A^\\gamma$-contributions to the remaining  terms indicated by the dots above. However, as we will argue in the main text, both of these apparent generalizations of our ansatz can be mapped in a first transformation to a situation of the simpler form above, provided only that the matrix $M^\\alpha_\\beta$ is invertible.\\footnote{Correspondingly, also theories that have field strengths containing Chern-Simons type of contributions can be tackled by the present formalism and, e.g., the Theorem \\ref{thm:phys1} below is applicable also in such situation. This is exemplified for instance in \\cite{SLS14}.} Thus, anticipating this argument, without loss of generality we can thus assume the field strengths to have the form\n\\begin{equation} \\label{curv}\nF^\\alpha = \\ud A^\\alpha + t^\\alpha_\\beta(X) A^\\beta + \\tfrac{1}{2} C^\\alpha_{\\beta \\gamma}(X) A^\\beta \\wedge A^\\gamma +\\tfrac{1}{6} H^\\alpha_{\\beta \\gamma\\delta}(X) A^\\beta \\wedge A^\\gamma \\wedge A^\\delta + \\ldots ,\n\\end{equation}\nwhere the coefficients or coefficient functions  $t^\\alpha_\\beta$,  $C^\\alpha_{\\beta \\gamma}$, $H^\\alpha_{\\beta \\gamma\\delta}$, $\\ldots$ are assumed to be fixed for defining the theory (on a local level and at this stage of constructing the theory). Their choice is now assumed to be constrained by some kind of ``physical consistency condition'', like one that would lead to \\eqref{Jacobi} in the case of only 1-form gauge fields $A^a$. It is thus now our task and declared goal to specify one or several possible consistency requirements, motivated by a more or less physical argument, to display the resulting constraints on the coefficients in some explicit form, and, in a final step, to give them some algebraic-geometrical meaning so as to unravel what replaces the structural Lie algebra $\\g$ of a standard YM theory in this much more general setting. \n\nBefore turning to these issues two more remarks on the nomenclature and the setting, however. We chose to include scalar fields (i.e.~0-forms) into that what we call gauge fields. In some sense this is unconventional, since they normally describe matter, while the conventional gauge fields (the 1-forms) represent the interaction forces. There are two reasons for this: First, it is suggested by the study of the Poisson sigma model (PSM) \\cite{Ikeda94, Str94}: As argued in  \\cite{Str04b} and \\cite{KS07}, the PSM can be seen as a ``Chern Simons (CS) theory''  of the Lie algebroid $T^*M$ associated to a Poisson manifold $M$; like the CS-theory corresponds to the Pontryagin class, also the PSM, and more generally all AKSZ sigma models \\cite{AKSZ}, correspond to characteristic classes (cf.~also \\cite{FRS13}). In this context, all the tower of differential forms correspond to a single map or section of an adequate higher bundle, and it in particular also includes the 0-forms. \n\nThe second reason for including the 0-forms is that even if one wants to consider the 0-forms $X^i$ as matter fields, it has advantages to include them right away.\\footnote{In this case the ``1-form field strength'' $F^i \\equiv \\ud  X^i - \\rho^i_a(X) A^a$ should be rather interpreted as a (generalized) covariant derivative. All the considerations concerning the ideal $\\I$ below will apply also to those.} Suppose one considers an ordinary YM-connection with scalar fields being a section of an associated (vector) bundle induced by a representation $\\rho$ of $\\g$\non a vector space or, more generally, by a $\\g$-action on a manifold $M$ that serves as a fiber of the associated bundle. In a local chart, the connection and the section correspond to a collection of 0-forms and 1-forms, $X^i$ and  $A^a$, respectively. On the other hand, $E:=\\g \\times M\\to M$ becomes a Lie algebroid in a canonical way and the data of the fields $X^i$ and $A^a$ are the same as a vector bundle morphism from $T\\Sigma \\to E$; moreover, the covariant derivative of the scalar fields and the curvature of the connection combine into a field strength map, cf.~\\cite{Str04b,Str09,KS07}. There are many more Lie algebroids\n(cf., e.g., \\cite{Mack87,CaWe99})\n than those coming from a Lie algebra action $\\rho$; if one intends to treat the 0-forms in a second step only, one misses all these additional possibilities. From a more physical perspective,  scalar fields are sometimes seen to deform the structural identities underlying the gauge fields of degrees at least one in a highly nontrivial way (cf, e.g., \\cite{Str04b} or some supergravity theories). Thus also from this perspective it seems reasonable to not exclude a geometrical explanation of such structures (in terms of some Lie algebroids for example) from the outset by the restriction to a two-step procedure.\n \n \n \n \nThe second remark concerns the nomenclature. We have chosen to call the $A^\\alpha$-fields \\emph{gauge fields} and the combinations $F^\\alpha$ as in \\eqref{curv} \\emph{field strengths}, following a ``physics oriented'' wording. We in particular avoid to call them generalized connections and curvatures, as it may seem more adequate from a (superficial) mathematical perspective. The reason for this is the following. As already obvious from the previously given example of the PSM, that what replaces the structural Lie algebra $\\g$ (as well as its integration $G$) of an ordinary principal bundle can and in general will be a bundle itself, in fact, even some kind of higher bundle. So, the fibers of what replace principal and associated bundles will be bundles themselves (and it may be useful  to choose auxiliary connections for technical reasons, that must be distinguished strictly from the ``dynamical'' gauge fields $A^\\alpha$). But even worse, a Poisson manifold for example does not always give rise to an integrating Lie groupoid, there can be obstructions in the integration process \\cite{CrF01}, while still the above mentioned characteristic classes as well as the PSM do exist as meaningful objects for any Poisson manifold $M$. To cover all these examples in a common mathematical framework it thus is indicated to work with bundles (over the base $\\Sigma$ or its tangent space $T\\Sigma$) where one does not need the analogue of the structural Lie group $G$, but it is sufficient to work with its Lie algebra $\\g$. In the ordinary setting this is the so-called Atiyah algebroid $A\\to T\\Sigma$ associated to a principal bundle $P\\to \\Sigma$, where $A$ can be identified with $TP\/G$ and, as a bundle over $T\\Sigma$, its fiber is isomorphic to $\\g$ only (cf.~\\cite{KS07} for further details). Now a connection in $P$, i.e.~an ordinary gauge field in physics language, corresponds to a \\emph{section} in $A$, to be distinguished from connections in $A$, that exist as well as on any bundle. (The generalization of $A$ will be what was called a $Q$-bundle in \\cite{KS07}.) These potential sources of a possible confusion with other natural connections lead us to avoid calling $A^\\alpha$ a generalized connection.\n\nWe now come back to our principal goal, finding conditions for the restriction of the coefficients appearing in \\eqref{curv}. The approach nearest at hand may seem to look at gauge transformations and to require some generalized form of invariance of the field strengths. This, however, has a decisive disadvantage: There is a lot of freedom in defining gauge transformations, and, as examples show (cf., e.g., the Dirac sigma model (DSM) \\cite{DSM}), they may depend on several additional structures not needed for the definition of the above field strengths. \n\n\nThe main idea for the departure is, therefore, to use a generalization of the Bianchi identities. Normally, they would contain a covariant derivative, that we have not yet defined as well, not yet knowing the adequate algebraic-geometric setting to use. Not wanting to use any additional new parameters\/coefficients, we thus will want to formulate a minimalistic version of Bianchi that at least most people in the field would agree with. Regarding the standard Bianchi identity for an ordinary YM theory, $\\ud F^a + C^a_{bc}A^b\\wedge F^c = 0$, we note that the application of the exterior derivative on the field strength becomes ``proportional'' to the field strength itself, albeit with a field dependent ``prefactor''. \n\nMathematically we express this as follows: We require that the exterior derivative $\\ud$ applied to any monomial containing a field strength $F^\\alpha$, i.e.~terms of the form $\\ldots \\wedge  A^\\alpha \\wedge F^\\beta \\wedge A^\\gamma \\wedge \\ldots$, yields expressions that again contain at least one field strength. In other words, we regard the ideal $\\I$ generated by expressions of the form \\eqref{curv} within the (graded commutative, associative) algebra $\\mathcal{A}$ generated by abstract elements $A^\\alpha$ and $\\ud A^\\alpha$. The condition we require is then simply\n\\beq \\boxed{\\ud \\I \\subset \\I } \\, ,\\label{ideal} \\eeq\ni.e.~$\\I$ has to be what is called a differential ideal. As innocent and weak as this condition may seem, it turns out to restrict the coefficient functions of \\eqref{curv} already very drastically, at least if the dimension $d$ of spacetime $\\Sigma$ exceeds the highest permitted form degree $p$ of $A^\\alpha$ by two, i.e.~$d \\geq p+2$. In that case, we will find that \\eqref{ideal} holds true iff the following vector field squares to zero:\n\\begin{equation} \\label{Q1}\nQ=  \\left(t^\\alpha_\\beta(x) q^\\beta + \\frac{1}{2} C^\\alpha_{\\beta \\gamma}(x) q^\\beta q^\\gamma +\\frac{1}{6} H^\\alpha_{\\beta \\gamma\\delta}(x) q^\\beta  q^\\gamma q^\\delta + \\ldots\\right) \\dfrac{\\partial}{\\partial q^\\alpha} .\n\\end{equation}\nHere we introduced for each gauge field $A^\\alpha$ an abstract coordinate $q^\\alpha$, carrying a degree that equals the form degree of  $A^\\alpha$ (for the degree zero variables we also use $q^i \\equiv x^i$ to distinguish them from the others, since they can appear non-polynomially).\nThese coordinates commute (anticommute) iff the respective differential forms commute (anticommute). Also, since by its definition, $F^\\alpha = \\ud A^\\alpha + \\ldots$ has a form degree that exceeds the one of $A^\\alpha$ by precisely one, the vector field $Q$ is an odd vector field of degree $+1$. In general, the condition\n\\beq\\label{Qsquare}\nQ^2 \\equiv \\tfrac{1}{2} [Q,Q] = 0\n\\eeq\nsubsummarizes a whole list of equations and in particular reduces precisely to \\eqref{Jacobi} in the case of coordinates of only degree one. \n\nThis observation turns out to be rather easy to prove, but so general that we consider it worth being called a fundamental theorem in the context of higher gauge theories:\n\\begin{thm} \\label{thm:phys1} If $\\dim \\Sigma \\geq p+2$ the generalized Bianchi identities \\eqref{ideal} hold true if and only if the vector field \\eqref{Q1} squares to zero, $Q^2=0$.\n\\end{thm}\nThus any higher gauge theory is associated to what is called a differential graded manifold or a Q-manifold \n$\\CM$. \n(In the case of an ordinary YM theory this corresponds to a reformulation of the Lie algebra $\\g$ in terms of its Chevalley-Eilenberg complex; in this case, simply $ (C^\\infty(\\CM), Q) \\cong (\\Lambda^\\cdot \\g^*, \\udCE)$). The most important part of the above theorem is not that a Q-manifold, as a generalization of a Lie algebra, gives rise to some generalized Bianchi identities, but the reverse direction: Even if Bianchi identities are formulated in as weak a version as (\\ref{bian}), one \\emph{necessarily}\ndeals with a Q-manifold on the target if one observes it or not. This is the strength of the above observation. \n\nTogether with \\cite{BKS,KS07} (and the considerations on the gauge symmetries to follow) this paves the way for an elegant, general, and simple picture of higher gauge theories. Their field content (and symmetries) is compactly described by a section $a$ in a Q-bundle (cf.~Fig.~\\ref{fig1} below):\n\\begin{figure}\n$$\\begin{xy}\/r0.15pc\/:\n  (-50,-70)*{}; (50,-70)*{} **\\crv{(0,-60)}+(0,-7)*\n  (T[1]\\Sigma,\\ud)};\n  (-50,0)*{}; (50,0)*{} **\\crv{(-20,20) & (20,-20)};  {\\ar_{\\textstyle a}(-10,-65)*{};(-10,2)*{}};\n  (-50,-50)*{}=\"A\"; (50,-50)*{} **\\crv{(0,-40)};    (72,30)*{(\\CM_{tot},Q_{tot})};\n  (50,-50)*{}; (50,50)*{} **\\crv{(40,-25)&(60,25)};\n  (50,50)*{}; (-50,50)*{} **\\crv{(0,60)};\n  \"A\"; (-50,50)*{} **\\crv{(-60,-25)&(-40,25)};\n  (-35,-47.5)*{};(-35,52.5)*{} **\\dir{-};\n  {\\ar(-60,10)*{(\\CM,Q)};(-35,20)*{}};\n\n\n\n  (-48,20)*{}; (52,20)*{} **\\crv{~*=<4pt>{.} (-20,30) & (20,10)};  {\\ar_{\\textstyle \\delta a}(0,0)*{};(0,20)*{}};\n\\end{xy}\n$$\n\\caption{\\label{fig1}  The resulting geometry governing all higher gauge theories: The generalization of a connection, called a \\emph{gauge field} in this work, corresponds to a section $a$ of a graded bundle $\\CM_{tot}$. The bundle is a Q-bundle and (the principal part of) gauge transformations correspond to vertical inner automorphisms. The section is a Q-morphism iff the generalization of curvature, called the \\emph{field strength} here, vanishes. In physical theories, however, it is almost exclusively those gauge fields which have a non-vanishing field strength that, up to gauge invariance, are of interest.}\n\\end{figure}\nThe tangent bundle $T\\Sigma$ of spacetime $\\Sigma$ can be viewed as a graded manifold when fiber-linear coordinates are regarded as carrying degree 1, functions on this graded manifold $T[1]\\Sigma$ correspond to differential forms, and the nilpotent, degree 1 de Rham differential $\\ud$ equips it with a Q-structure. The Q-manifold $(\\CM,Q)$ that we introduced above in the context of (\\ref{Q1}) can be viewed as a target if we interpret the tower of gauge fields $A^\\alpha$ as corresponding to a degree-preserving map from $T[1]\\Sigma$  to \n$\\CM$.\\footnote{We will explain all this in more detail in the main text, but we want to give an overview of the resulting picture already as this point as a guide for what we are going to obtain or derive as a final framework.} As a straightforward generalization of the considerations in \\cite{BKS} this can be reinterpreted as a section $a$ in a trivial Q-bundle, $(\\CM_{tot},Q_{tot}) := (\\CM \\times T[1]\\Sigma, Q+\\ud)$,  an observation that proves essential in the context of describing the gauge symmetries (in \\cite{BKS} for the PSM, here for general higher gauge theories). One of the observations fundamental to \\cite{KS07} is, on the other hand, that this picture generalizes also to the twisted case, i.e.~that a connection in an ordinary principal bundle corresponds just to a section of a (possibly non-trivial) Q-bundle. \n\nWe add a remark in parenthesis: The section $a \\colon T[1]\\Sigma \\to \\CM_{tot}$, corresponding to our gauge fields, is a section of the category of ($\\N_0$-)graded manifolds, and, not, in the sense of the category augmented by a Q-structure. It becomes an ``honest'' section of the category of Q-bundles, i.e.~a Q-morphism, iff all the field strengths $F^\\alpha$ vanish. As nice as this may seem from a purely mathematical perspective, as inadequate it is from the one of physics: It would exclude all the particles (like photons, gluons, etc) for which gauge fields were introduced in physics in the first place. In fact, even in topological models, such as the PSM, the DSM, or the AKSZ sigma models, the gauge fields become Q-morphisms only ``on-shell'', i.e.~on the level of the Euler-Lagrange equations of the corresponding action functionals, cf.~\\cite{BKS,DSM,KS08} for the proofs, respectively.\n\nLet us now address the gauge symmetries. Similarly to the general ansatz (\\ref{curv}), we make one for the gauge symmetries\n\\begin{equation}\\label{gaugedelta0}\n\\delta^{(0)}_{\\epsilon} A^\\alpha = \\ud \\epsilon^\\alpha +  \\left(s^\\alpha_\\beta(X)  + D^\\alpha_{\\gamma \\beta}(X) A^\\beta  +\\tfrac{1}{2} G^\\alpha_{ \\gamma\\delta\\beta}(X) A^\\gamma \\wedge A^\\delta + \\ldots \\right)\\wedge \\epsilon^\\beta,\n\\end{equation}\nfor some collection $\\epsilon^\\alpha$ of differential forms of degree up to  $p-1$, where we may allow the total gauge transformations to also \ncontain a part proportional to $F^\\alpha$ again:\n\\begin{equation} \\label{gaugedelta}\n\\delta_{\\epsilon} A^\\alpha = \\delta^{(0)}_\\epsilon A^\\alpha + O(F^\\alpha) \\, . \n\\end{equation} \nIt is important to remark that the terms $O(F^\\beta)$ have to be in the ideal $\\I$ for the following to be still true. This still includes higher gauge theories as they appear as a part of gauged supergravity for example, cf.~\\cite{SSW11,SLS14,SKS14}, and where they play an important role, but excludes terms containing e.g.~$*F^\\alpha$ (as they appear in the DSM \\cite{DSM}) or terms containing ``contractions'' of $F$ (with the inverse of the part of $A$ that may describe a vielbein) as they typically appear in the gravitational part of supergravity theories. In analogy to (\\ref{ideal}), we now require that also our gauge transformations do not leave the ideal \n$\\I$ generated by the field strengths,\n\\beq \\boxed{\\delta_\\varepsilon \\I \\subset \\I } \\, .\\label{deltaI} \\eeq\nThe result we obtain, illustrated and proven in detail in the main text, is the following (cf.~also Fig.~\\ref{fig1})\n\\begin{thm} \\label{thm:phys2}\nProvided $\\dim \\Sigma \\geq p+1$ and $Q^2=0$, infinitesimal gauge transformations of the form (\\ref{gaugedelta}) leave the field strength ideal $\\I$ invariant, Eq.~\\eqref{deltaI}, iff their principal part $\\delta^{(0)}_\\varepsilon$ corresponds to vertical inner automorphisms of the Q-bundle $(\\CM_{tot},Q_{tot})$. Moreover, if the infinitesimal gauge transformations are of the more restricted form (\\ref{gaugedelta0}), $\\delta_\\varepsilon \\equiv \\delta^{(0)}_\\varepsilon$, one does not need to require $Q^2=0$, but one can conclude it together with the generalized Bianchi identities (\\ref{bian}). \n\\end{thm}\nLet us briefly explain what inner automorphisms of a Q-manifold should be thought of to our mind. An automorphism should be degree preserving diffeomporphism that leaves invariant the vector field $Q$. Infinitesimally this corresponds to degree zero vector field annihilated by $\\ad_Q$. It is an immediate consequence of (\\ref{Qsquare}) that $\\ad_Q$ squares to zero as well. We call an inner automorphism of a Q-manifold one that is generated by the image of $\\ad_Q$, i.e.~by vector fields of the form  $\\ad_Q(\\epsilon)$, where $\\epsilon$ is a degree -1 on the Q-manifold. If the Q-manifold is an ordinary Lie algebra (shifted in degree by one) $\\g[1]$, this corresponds to the usual definition of inner automorphisms, i.e.~to  the adjoint action of the integrating group $G$ on $\\g=\\Lie(G)$ (generated infinitesimally by the left action on itself). For $(T[1]\\Sigma, \\ud)$ this definition of inner automorphism gives infinitesimally precisely the Lie derivatives of vector fields on $\\Sigma$, so they correspond to the canonical lift of a diffeomorphism of $\\Sigma$ to its tangent bundle. \n\nGauge transformations of an ordinary YM-theory are generated by vertical automorphisms of the underlying principal bundle $P$. They correspond precisely to inner vertical automorphisms in the above sense for the associated Q-bundle $T[1]P\/G$. The advantage of the present formulation is that like this it generalizes in a straightforward way to \\emph{all} higher gauge theories. \n\nSome further explanations are in place: First, additional structures entering a particular theory can certainly constrain the symmetries permitted in the respective case. For example, in the Poisson sigma model or, more generally, the AKSZ sigma models, the inner automorphisms also have to preserve the symplectic form defined on the fibers and that are compatible with the Q-structure on the fibers. For the PSM, for example, this implies that its gauge symmetries are generated infinitesimally by the Lie subalgebra $\\Omega^1_{closed}(M)$ of $\\Gamma(T^*M)$ ($T^*M$ viewed upon as the Lie algebroid of the Poisson manifold $M$ in this case). Second, there are cases where the gauge transformations including the additional terms of the form $O(F)$ in (\\ref{gaugedelta}) can be given a similarly good geometrical interpretation. For example, as will be shown in \\cite{SLS14}, for the higher gauge theory sector of the (1,0) superconformal model in six spacetime dimensions \\cite{SSW11}, the $O(F)$-part is essential for an off-shell closed algebra of gauge transformations on the one hand. On the other hand, in this case the gauge transformations (\\ref{gaugedelta}) correspond to nothing but inner automorphisms of another Q-bundle associated to the one in Fig.~\\ref{fig1}, where the typical fiber $(\\CM,Q)$ is replaced by $(T[1]\\CM, \\ud_{\\CM} \n+ \\L_Q)$ (cf.~also \\cite{KS07,SaS13}).\\footnote{Any Q-bundle $(\\CM_{tot},Q_{tot}) \\to (\\CM_1,Q_1)$ can be lifted to a tangent Q-bundle $(T[1]\\CM_{tot}, \\ud_{\\CM_{tot}} \n+ \\L_{Q_{tot}}) \\to (T[1]\\CM_1, \\ud_{\\CM_1} + \\L_{Q_1})$. Using the canonical 0-section $\\sigma \\colon \\CM_1\\to T[1]\\CM_1$, we can pull back $T[1]\\CM_{tot}$ to become a bundle over $\\CM_1$, which turns out to be compatible with the respective Q-structure\n\n}\n\n\n\nThird, it is important to be aware that $Q_{tot}$ projects to $\\ud$ on the base of the bundle, and it is like this that $\\ud \\varepsilon^\\alpha$ is generated in (\\ref{gaugedelta0}), (\\ref{gaugedelta}); this becomes possible only by going to the picture of a bundle, even in the case of a local treatment, where the bundle (at least when defined as in \\cite{KS07}) is trivial. \n\nIn the present description, the vector fields generating the gauge transformation always form a Lie algebra. They are vertical vector fields of degree zero of the form $\\ad_{Q_{tot}}(\\epsilon)$ and they close with respect to the usual commutator of vector fields. They are parameterized by vertical vector fields $\\epsilon$ of degree minus one \\emph{up to $\\ad_{Q_{tot}}$-closed} contributions. On the vector fields $\\epsilon$ one discovers that one has an induced bracket of the form \n\\begin{equation}\\label{derQtot}\n[\\epsilon,\\epsilon']_{Q_{tot}} \\equiv [\\ad_{Q_{tot}}(\\epsilon), \\epsilon'] \\, ,\n\\end{equation} called the ``derived bracket'' (following Cartan and \\cite{YKS03en}). For example, if one considers the Q-manifold $(\\g[1], \\udCE)$, the derived bracket just reproduces the Lie algebra bracket. While in general, however, this bracket is a Leibniz--Loday algebra only and not necessarily antisymmetric, it becomes so for the coexact elements, i.e.~for those $\\epsilon$ where one considers the quotient by closed contributions (which do not change the gauge transformation, as argued above). In this way one obtains the Theorem  \\ref{theo:Lie} in the main text.\\footnote{To simplify the notation in that section, we called the total Q-structure simply $Q$ and renamed the Q-structure on the fiber by $Q_2$.}\n\nIn a conventional treatment of gauge symmetries (i.e.~along the lines as described e.g.~in \\cite{HT}), they form a so-called \\emph{open} algebra for a generic higher gauge theory. This is not in contradiction to the present, to our mind more elegant formulation of its gauge symmetries. To resolve the apparent paradox, it is useful to consider the PSM---as a simpler toy-model, showing already all the essential features of this behavior. In \\cite{BKS} it is shown, in particular, that one can obtain the usual, only on-shell closed algebra of gauge symmetries of the PSM from a incomparably shorter calculation in the above-mentioned framework. This generalizes to the context of any higher gauge theory. For instance for a higher gauge theory containing 0-forms $X^i$, 1-forms $A^a$, and 2-forms $B^C$ one obtains in this way the result of Proposition \\ref{prop:onshellclosed} below. We also will comment further on this issue in \\cite{SLS14,GSS14}. \n \nWhile one can still discuss the usefulness or necessity  of using the exterior derivative $\\ud$ in vector analysis or $ \\udCE$ for Lie algebras, there will be no doubt in the case of $p \\ge 2$; already for $p=2$ there is, in general, a whole list of coupled differential equations satisfied by the structural quantities, cf.\\ Eqs.~\\eqref{B1}--\\eqref{B7} in the main text below, which are to be used in almost every calculation related to such a gauge theory and which can be replaced and subsummarized in a single operator $Q$ squaring to zero. The technical usefulness becomes particularly obvious in the above-mentioned context of the calculation of the commutator of gauge symmetries, but also already for the concrete form of the Bianchi identities (cf., e.g., the derivation of Eqs.~\\eqref{BFi}--\\eqref{BFB} in the main text below).\n\nLet us briefly come back to Fig.~\\ref{fig1} and the corresponding two theorems, Thm.~\\ref{thm:phys1} and \\ref{thm:phys2}. Not only the base of the Q-bundle is in fact a bundle in ordinary differential-geometric terms, but even worse so for the fibers: the Q-manifold $(\\CM,Q)$ isomorphic to the  typical fiber is generically a rather intricate (higher) bundle structure itself. We will come back to this issue in more detail in subsection \\ref{sec:math} below. What we consider important in this context is that the infinite-dimensional space of gauge fields is described as sections in \\emph{finite-dimensional} (graded) bundles. This gives additional mathematical control. The situation can be compared with the interest in Lie algebroids (for a definition cf., e.g., \\ref{def:Loid}): While for infinite-dimensional Lie algebras topological questions usually play an important role and are intricate to deal with, the situation simplifies greatly when this infinite-dimensional Lie algebra comes from the sections of a Lie algebroid. \n\nLikewise, the above operator $Q$ (or also $Q_{tot}$) must not be confused also with a BV- or BRST-operator. Although the BV-BRST operator $Q_{BV}$  also squares to zero, $Q_{BV}^2=0$,  and is of degree +1, it is defined over the \\emph{infinite-dimensional} graded manifold of (gauge) fields, ghosts, and their antifields, and, more importantly, it, in general, needs many more structures to be defined, namely those needed for an action functional that we have not yet addressed much up to here (and for which there are many options still open).\\footnote{Correspondingly, also the discovery of the role of the derived bracket in the context of the gauge symmetries, cf.~Thm.~\\ref{theo:Lie}, is not or at most vaguely related of the role of the derived bracket in the BV-context as observed in \\cite{YKS97}.} \n\n\nThere are very particular models of topological nature, like the Poisson sigma model or, more generally, the AKSZ-sigma models \\cite{AKSZ}, on the other hand, where there \\emph{is} a close relation of the two Q-structures $Q$ and $Q_{BV}$: $Q$ corresponds to a defining Q-structure on the target of the sigma model, which is also equipped with a compatible symplectic form $\\omega$, $\\ud$ is a Q-structure on its source $T[1]\\Sigma$, all very much like in Fig.~\\ref{fig1}. Then $a \\colon T[1]\\Sigma \\to \\CM$ is no more degree preserving (so as to capture ghosts and antifields), but a general supermap and $\\Omega_{BV}$ is, roughly speaking, the ``difference'' of $Q$ and $\\ud$ induced on this space of supermaps $\\CM_{BV}$. Using $\\omega$ and a canonical measure on $T[1]\\Sigma$, one then induces also a (weakly) symplectic form $\\omega_{BV}$ on $\\CM_{BV}$ and shows that $Q_{BV}$ is Hamiltonian, i.e.~$Q_{BV}=\\{ S_{BV}, \\cdot \\}_{BV}$, where $S_{BV}$ is the BV-action, extending the classical action of the PSM and AKSZ-sigma model, respectively (cf., e.g., \n\\cite{AKSZ,Catt01,Royt06}). \n\n\nThis changes quite drastically, however, in the case of non-topological, physically relevant theories; already for an ordinary YM gauge theory the BV operator is not constructed in such a simple way (unfortunately). Even for \\emph{relatively} simple generalizations like (for the PSM) the topological DSM \\cite{DSM} or (for YM) the non-topological Lie algebroid Yang-Mills theories \\cite{Str04b}, the BV operator is even not known at the present time; however, in both cases the target Q-structures are very well known, corresponding to Lie algebroids in both cases. And, again, the field content for any such a theory, and, more generally, for any higher gauge theory, its described compactly by sections in the finite-dimensional Q-bundle as sketched in Fig.~\\ref{fig1}. \n\nAlthough originally intended to do so, we will not address the construction of action functionals within this paper, leaving it for eventual later work (but cf.~also \\cite{Str04b,KS08,SLS14}). \n\n\n\\subsection{Further mathematical results} \\label{sec:math}\nHere we briefly summarize the ideas and results contained in the more mathematical sections \\ref{s:examQ} and \\ref{sec:derived}. It mainly consists in reformulating supergeometrically defined objects appearing in the Q-bundles as potential fibers in \nmore conventional algebraic-geometrical terms. \n\nOur graded or supermanifolds, on which the homological vector fields $Q$ are defined, always will be $\\N_0$ graded manifolds, being simultaneously supermanifolds by means of the induced $\\Z\/2\\Z$-grading. These are conventionally called simply\n N-manifolds and are defined as follows (for further details see also \\cite{Royt02,Gru09}): An N-manifold is a ringed second countable Hausdorff topological space $\\CM=(M,{C^\\infty})$ where the structure sheaf, which by abuse of notation we also denote as ${C^\\infty}$, is locally of the form smooth functions in tensor product with the exterior algebra in the odd coordinates tensor product with the polynomial algebra in the even coordinates not of degree 0. This implies that it \n consists of a tower of affine fibrations.  In degree 0 we just have a smooth manifold $M$.  In degree 1, also after truncation of a higher degree N-manifold, we have an odd vector bundle $E[1]$, i.e.\\ the fiber-linear coordinates of $E$ are declared as functions of degree 1.  Their parity is odd and thus the function algebra are the $E$-forms.  The topological structure of such an N-manifold comes from the smooth base only.  \n\nMorphisms of N-manifolds are morphisms of ringed spaces that preserve the $\\N_0$-grading of the structure sheaf.  Coordinate changes are particular local isomorphisms. Like this one finds e.g.~that the coordinates of degree 1 transform as the coordinates of a vector bundle so that any N-manifold $\\CM$  with degrees up to 1, i.e.~an N1-manifold,\\footnote{In general, we call an N$k$-manifold an N-manifold, where the highest coordinate degree is $k$. \\label{fn:Nk}} can be canonically identified with vector bundles $E\\to M$, $\\CM \\cong E[1]$.  \n\nIn degree 2, coordinate changes contain, however, also affine terms beside the terms one would expect for a vector bundle.  Globally an N2-manifold $\\CM$ is an affine bundle over the base $E[1]$ modeled after the pullback $\\pi_1^*V[2]$ of a vector bundle $V\\to M$ where $\\pi_1\\colon E[1]\\to M$.  Since the fibers $V_x$ for $x\\in M$ of this bundle are contractible, there exists a (non-canonical) global section (splitting) that identifies the affine bundle $\\CM$ with the graded vector bundle $E[1]\\oplus V[2]$ over the base $M$. In all of our theorems on Q2-manifolds, i.e.~N2-manifolds equipped with a degree +1 homological vector field, we will need such a splitting permitting us to identify the underlying graded manifold with the direct sum of two vector bundles $E\\to M$ and $V\\to M$, $\\CM \\cong E[1]\\oplus V[2]$. \n\nAs mentioned already in the previous subsection, the definition of an ordinary Lie algebra $\\g$ is equivalent to a Q1-manifold over a point, i.e.~with coordinates of degree 1 only. The corresponding N1-manifold gives the vector space (vector bundle $E$ over a point), $\\CM \\cong \\g[1]$ (as graded vector spaces), the functions $C^\\infty(\\CM)$ are identified with elements of $\\Lambda^\\cdot \\g^*$, and the degree +1 vector field $Q$ corresponds precisely to the Chevalley-Eilenberg differential, $Q\\cong \\udCE$, from which one certainly can also reconstruct the Lie bracket on the vector space. \n\nA general Q1-manifold can be seen to be equivalent to a Lie algebroid, as was first observed in \\cite{Vai97}. For completeness, we briefly provide a definition:\n\\begin{vdef} \\label{def:Loid} A \\emph{Lie algebroid} consists of a vector bundle $E\\to M$ together with a bundle map $\\rho \\colon E \\to TM$ (over the identity) and a Lie algebra $(\\Gamma(E),[\\cdot,\\cdot])$ on the sections of $E$ such that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(E)$ and $f \\in C^\\infty(M)$ one has the following Leibniz rule\n\\begin{equation} \\label{Leib}\n[\\psi_1, f \\psi_2] = f [\\psi_1,  \\psi_2] + \\rho(\\psi_1)f \\,\\cdot  \\psi_2 \\;.\n\\end{equation}\n\\end{vdef}\nThese data can be recovered from a homological vector field of degree 1 on an N1-manifold. In local coordinates $x^i$ and $\\xi^a$ of degree 0 and 1, respectively, it always is of the form \n\\beq \\label{Q0} Q = \\rho^i_a(x) \\xi^a \\frac{\\partial}{\\partial x^i}\n- \\frac12 C^a_{bc}(x) \\xi^b \\xi^c \\frac{\\partial}{\\partial \\xi^a} \\, . \\eeq\nThe axioms can be reconstructed\nby using $Q^2=0$ \\emph{and} the behavior of the coefficient functions $ \\rho^i_a$ and \n$C^a_{bc}$ with respect to (degree preserving) coordinate changes on $\\CM \\cong E[1]$ (so as to recover also \\eqref{Leib}). The details of this process can be found in the main text. It is also true, moreover, that the Q-morphisms (morphisms of N-manifolds such that their pullback intertwine the two Q-structures) correspond precisely to Lie algebroid morphisms in the sense of \\cite{MaH90} (cf.~\\cite{BKS} for an explicit proof). \nWe summarize this in \n\\begin{thm}[Vaintrob 97] \\label{Vain}\nA Q1-manifold is in 1:1-correspondence with a Lie algebroid. This is an equivalence of categories.\n\\end{thm}\nIn the present paper we treat the analogue question for one degree higher, i.e.~we address the description of a Q2-manifold. Instead of two coefficient functions $\\rho$ and $C$ satisfying two equations resulting from $Q^2=0$, one ends up with five coefficient functions satisfying seven coupled differential equations, cf.~Eqs.~\\eqref{Q} and \\eqref{B1}--\\eqref{B7}, respectively, below. For an orientation we first collect related results, corresponding to special cases.\n\nIf the Q2 manifold is one over a point, it corresponds to a (semistrict) Lie 2-algebra or, equivalently, to a 2-term $L_\\infty$-algebra. A special case of this are strict Lie 2-algebras, which, according to \\cite{Baez03vi}, are equivalent to Lie algebra crossed modules. We thus here define a Lie 2-algebra by means of this description:\n\\begin{vdef} A \\emph{strict Lie 2-algebra} \\label{strictL2}\nis a pair $(\\g,\\h)$ of Lie algebras \ntogether with a homomorphism $t \\colon \\h\\to \\g$ and a  representation\n$\\alpha \\colon \\g\\to\\Der(\\h)$ such that  $\\, t(\\alpha (x)v) =[x,t(v)]$   \nand $ \\alpha(t(v))w =[v,w]$  hold true for all $x \\in \\g$, $v,w \\in \\h$.  \n\\end{vdef}\nWe remark in parenthesis that this definition can be shortened:\n\\begin{lemma} A strict Lie 2-algebra $(\\g,\\h)$  is the same as an intertwiner, $t \\colon \\h \\to \\g$, from a $\\g$-representation $\\h$ to the adjoint representation on $\\g$ such that $t(v)\\cdot v = 0$ for all $v\\in \\h$.\n\\end{lemma}\nThe data of Definition \\ref{strictL2} can be retrieved easily with all their properties; e.g.~the Lie algebra structure on $\\h$ can then be \\emph{defined} by means of $[v,w]_\\h := t(v) \\cdot w$.\n\nThe definition of a semistrict Lie 2-algebra or simply \\emph{a Lie 2-algebra} is analogous, \nbut contains an additional map $H \\colon \\Lambda^3 \\g \\to \\h$, satisfying a ``coherence-type'' consistency relation. $H$ turns out to control the Jacobiators of brackets on $\\g$ and $\\h$ and enters a modification of the definition above in several places. For $H=0$ one recovers a strict Lie 2-algebra. (In fact, one may extract its precise definition from specializing the Definition \\ref{def:Lie2d} below to the case of $M$ being a point.)\n\n\nAlternatively, one can describe a Lie 2-algebra also by means of a 2-term $L_\\infty$-algebra. We recapitulate its definition in detail in the Appendix \\ref{s:Linf1} (cf.~also \\cite{SHLA,SHLA2} for the original definitions). Essentially this consists of a 2-term complex $\\h[-1] \\xrightarrow{t} \\g$,\\footnote{Note that in the context of N-manifolds, $\\h[-1]$ means that the linear coordinates are of degree -1, which implies that the \\emph{elements} of this vector space are of degree +1. The shifting is slightly changed with respect to the previous occurrence of $\\g$ and $\\h$, but this is due to different but equivalent conventions, cf.~also Appendix \\ref{s:Linf1}.} a unary bracket $[\\cdot]_1=t$ of degree -1, a binary bracket $[\\cdot,\\cdot]_2$ of degree 0 and a ternary bracket $[\\cdot,\\cdot,\\cdot]_3$ of degree 1 \nthat have to fulfill several quadratic relations which can be understood as generalized Jacobi identities. One of those is, e.g., for $x,y,z\\in\\g$\n\\begin{equation}\n [x,[y,z]_2]_2+[z,[x,y]_2]_2 +[y,[z,x]_2]_2 = [\\,[x,y,z]_3]_1 \\, ,\n\\end{equation}\nshowing that the 2-bracket forms a Lie algebra iff the image of the 3-bracket lie in the kernel of $t$. It is thus not surprising that this 3-bracket is the above mentioned map $H$. \nIt is thus again a non-vanishing $H$ that impedes the appearance of ordinary Lie algebras and that controls their deviation from being so on the other hand. In particular, $H$ itself has to fulfill a Jacobi-type quadratic relation. Further details on this perspective of a Q2-manifold over a point and $L_\\infty$-algebras in general are deferred to Appendix \\ref{s:Linf1}.\n\nFinally, concerning Q2-manifolds with an honest base manifold $M$, there is the following result (see also \\cite{Royt02}). \n\\begin{thm}[Roytenberg]$\\quad$\\\\ \\label{thm:Roy}\nA QP2-manifold is in 1:1-correspondence with a Courant algebroid.\n\\end{thm}\nSome comments are in order: A QP-manifold is a Q-manifold (in the above sense) equipped with a compatible symplectic form $\\omega$. QP2 signifies that the underlying N-manifold is an N2-manifold in our sense (cf.~footnote \\ref{fn:Nk}) or, equivalently, that the 2-form $\\omega$ has total degree $4=2+2$.\nA Courant algebroid can be defined as follows\n\\begin{vdef}\n\\label{def:Cour} A \\emph{Courant algebroid} consists of a vector bundle $E\\to M$ together with a bundle map $\\rho \\colon E \\to TM$ (over the identity), a Leibniz--Loday algebra $(\\Gamma(E),[\\cdot,\\cdot])$ on the sections of $E$, and fiber metric   $\\<.,.\\>$ on $E$\nsuch that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(E)$ one has the following rules\n\\begin{align} \\label{Coura}\n  \\< \\psi_1,[\\psi_2,\\psi_2]\\>&=  \\tfrac{1}{2}\\rho(\\psi_1)\\<\\psi_2,\\psi_2\\>  = \\<[\\psi_1,\\psi_2],\\psi_2\\> \\, .\n\\end{align}\n\\end{vdef}\nHere some further explanations may be in place: First, we call Leibniz--Loday algebra an algebra, where the left-multiplication is a derivation, i.e.\n\\begin{equation}\\label{JaCou}\n[\\psi_1,[\\psi_2,\\psi_3]] = [[\\psi_1,\\psi_2],\\psi_3] +[\\psi_2,[\\psi_1,\\psi_3]]\n\\end{equation}\n holds true $\\forall \\psi_k \\in \\Gamma(E)$. Moreover, one can deduce the Leibniz rule \\eqref{Leib} from the axioms. So, \\emph{if} the bracket $[\\cdot , \\cdot]$ were antisymmetric, this would be a Lie algebroid (with an additional fiber metric). However, the first equation of \\eqref{Coura} prohibits this: The inner product being non-degenerate, the symmetric part of the bracket can vanish only in the relatively uninteresting case of an identically vanishing map $\\rho$ \n(in which case one obtains a bundle of quadratic Lie algebras over $M$---ad-invariance following from the remainder of \\eqref{Coura} then). So, it is precisely the inner product (together with $\\rho$) that governs the symmetric part of the bracket. The second equation of \\eqref{Coura} expresses a (for non-vanishing $\\rho$ not just pointwise) ad-invariance of the fiber-metric.\n\nThis completes the explanations of the terms appearing in the above theorem. As an aside we mention that one may also rephrase the axioms of a Courant algebroid in terms of the anti-symmetrization of the above bracket, \n\\begin{equation}\\label{anti}\n[\\phi,\\psi]_2=\\tfrac12\\left([\\phi,\\psi] -[\\psi,\\phi]\\right) \\, .\n\\end{equation}\nIn that case, however, (\\ref{JaCou}) does not yield a Jacobi identity for this new bracket. It thus does not come as a surprise that this can be rephrased in terms of an (infinite-dimensional) 2-term $L_\\infty$-algebra $[.]_1\\:V_1\\to V_0$ \\cite{Royt98}: Here the vector spaces are $V_0=\\Gamma(E)$ and $V_1={C^\\infty}(M)$, respectively, the map $[.]_1$ is provided by means of the de Rham differential $\\ud$ followed by the transpose of the anchor map and a subsequent use of the inner product (to identify $E^*$ with $E$ and likewise so for the sections). Between two elements of $V_0$ the degree 0 binary bracket is given by \\eqref{anti}, while for $f\\in V_1$ and $\\phi \\in V_0$ one has $[\\phi,f]_2:=\\rho(\\phi)f$.\nThe degree 1 ternary bracket, finally, can be non-vanishing only on $[.,.,.]_3:\\Lambda^3V_0\\to V_1$, where it is given by $[\\psi_1,\\psi_2,\\psi_3]_3=\\tfrac13\\<[\\psi_1,\\psi_2]_2,\\psi_3\\>+{\\text{cycl.}}$  All higher brackets vanish for degree reasons.  \n\nWe now turn to describing our main result of section \\ref{s:examQ}. As in the previously mentioned results on Q1-manifolds, Lie 2-algebras, and QP2-manifolds, an essential ingredient of it is a definition (specifying to what precisely the Q2-manifold is equivalent). To obtain this definition, we proceeded as sketched above for Q1-manifolds, i.e.~analyzed the behavior of coefficient functions with respect to coordinate changes on the N2-manifold, which we identified with the sum of two vector bundles (in a non-canonical way, but different choices being related by an isomorphism). We first provide the generalization of Definition \\ref{strictL2} to the case of a general base $M$ of $\\CM$, trying to stay as close as possible to the known one for $M=\\mathrm{pt}$: \n\\begin{vdef}\\label{d:sLie2}\nA \\emph{strict Lie 2-algebroid} or a \\emph{crossed module of Lie algebroids} is a pair of \n $(E,V)$ of Lie algebroids over the same base $M$, where $\\rho_V \\colon V \\to TM$ vanishes,\ntogether with a homomorphism $t \\colon V\\to E$ and a representation ${{}^E{}\\nabla}$ of $E$ on $V$\nsuch that  $t\\left(\\Econn_\\psi v\\right) = [\\psi,t(v)]$\n and  $\\Econn_{t(v)}w = [v,w]$   hold true for all $\\phi,\\psi\\in\\Gamma(E)$, $v,w\\in\\Gamma(V)$.\n\\end{vdef}\nThe generalization of a representation of a Lie algebra $\\g$ on a vector space to a representation of a Lie algebroid $E$ on a vector bundle $V$ (over the same base) uses the notion of an $E$-covariant derivative: This is a map ${{}^E{}\\nabla}\\:\\Gamma(E)\\times\\Gamma(V)\\to\\Gamma(V)$ such that for all $\\psi\\in\\Gamma(E)$, $v\\in\\Gamma(V)$, and $f\\in{C^\\infty}(M)$\n\\[\\label{EConn}\n  {{}^E{}\\nabla}_{f\\psi} v = f {{}^E{}\\nabla}_\\psi v, \\quad\\quad \n  {{}^E{}\\nabla}_\\psi (f\\cdot v) = \\rho(\\psi)[f]\\cdot v +f{{}^E{}\\nabla}_\\psi v.\n\\]\nThis becomes a representation iff its $E$-curvature, defined by means of  ${{}^E\\!{}R}(\\phi,\\psi)v:=[\\Econn_\\phi,\\Econn_\\psi]v -\\Econn_{[\\phi,\\psi]}v$, vanishes,${{}^E\\!{}R}=0$.  This flatness condition implies also that $\\Econn_\\psi$ is a derivation of the (bundle of) Lie algebra(s) $(\\Gamma(V),[.,.])$.\n\n\n\\begin{vdef}\\label{def:Lie2d}  A (semi-strict) \\emph{Lie 2-algebroid} is defined as in Definition \\ref{d:sLie2} except that the (still antisymmetric) brackets on $\\Gamma(E)$ and $\\Gamma(V)$ are no more required to satisfy the Jacobi identity and\n ${{}^E{}\\nabla}$ is no more flat in general. Instead, all these deviations are governed by a single $E$-3-form $H\\in\\Omega^3(E,V)$ with values in $V$:  For all $\\psi_k\\in\\Gamma(E)$, $v,w\\in\\Gamma(V)$ and $f\\in{C^\\infty}(M)$ one has\n\\begin{align}\n  [\\psi_1,[\\psi_2,\\psi_3]] +{\\text{cycl.}} &= t\\left(H(\\psi_1,\\psi_2,\\psi_3)\\right), \\label{HJacobi} \\\\\n   {{}^E\\!{}R}(\\psi_1,\\psi_2)v &= H(\\psi_1,\\psi_2,t(v)), \\label{HRep}\\\\\n\n \n  {{}^E\\!{}\\mathrm{D}} H &= 0 \\, . \\label{DH}\n\\end{align}\n\\end{vdef}\nHere ${{}^E\\!{}\\mathrm{D}}$ is the exterior $E$-covariant derivative associated to ${{}^E{}\\nabla}$. We did not need to specify the Jacobiator of the bracket on $\\Gamma(V)$, since it follows at once from \\eqref{HRep} and $\\Econn_{t(v)}w = [v,w]$: \n\\begin{equation}\n[u,[v,w]] + {\\text{cycl.}} = H(t(u),t(v),t(w)) \\qquad \\forall u,v,w \\in \\Gamma(V) \\, .\n\\end{equation} \nIt is noteworthy that in this formulation, the coherence condition for $H$ takes the simple form \\eqref{DH}. \nIn this paper we will then prove the following:\n\\begin{thm}\\label{thm1.5} $\\quad$  \\\\ A Q2-manifold is in 1:1-correspondence with a (semi-strict) Lie 2-algebroid \\\\ up to isomorphism.\n\\end{thm}\nSimilarly to Theorem \\ref{Vain}, this statement can be turned also into an equivalence of categories.\n\n\nIn this context it is important to remark that the notion of a morphism of Lie 2-algebroids (as defined above in Definition \\ref{def:Lie2d}) is less restrictive than the one of  strict Lie 2-algebroids (Definition \\ref{d:sLie2}). This can be seen as follows: A definition of a morphism of a Lie 2-algebroid can be tailored from the corresponding Q-morphisms and the identification given in the above theorem  (such as one can do it for ordinary Lie algebroids and Q1-manifolds). On the one hand, the change of a splitting of the Q-manifold corresponds to a Q-morphism.\nOn the other hand, it corresponds to an element $B\\in\\Omega^2(E,V)$ and $H$ changes under this $B$ (in a slightly tricky way, cf.~Equation \\eqref{split3} below) and in particular this can be used to generate a non-zero $H$ from a vanishing one. \n\nA remark to the nomenclature seems important. One of the goals here is to give a definition of a mathematical structure equivalent to a Q2-manifold on the one hand and generalizing the definition of a Lie 2-algebra as given in \\cite{Baez03vi} on the other hand. In other words, we strive at a definition equivalent to the \\emph{super}-geometrical notion of a Q2-manifold in terms of notions used only in \\emph{ordinary} differential geometry. This can be viewed as the same philosophy underlying the Theorems \\ref{Vain} and \\ref{thm:Roy}---with the difference that in these two cases the definitions \\ref{def:Loid} and \\ref{def:Cour} were already established as a geometrical notion before the equivalence  (expressed in the two theorems mentioned above) to the (simpler) super- or graded-geometrical structure was observed. Some also call Q$k$-manifolds simply Lie $k$-algebroids \\emph{by definition}. To emphasize the non-tautological nature of Theorem \\ref{thm1.5}, it may thus be useful to add the specification ``semi-strict'' in the equivalence, with \\emph{semi-strict} Lie 2-algebroids being then \\emph{introduced} (to our knowledge for the first time) in \nDefinition \\ref{def:Lie2d} above. \n\nNote that as a simple corollary of the above theorem, we find that any Q2-manifold gives rise to an infinite-dimensional Lie 2-algebra or also 2-term $L_\\infty$-algebra, $\\Gamma(V) \\xrightarrow{t} \\Gamma(E)$. In fact, a Lie 2-algebroid provides the additional control over such an infinite-dimensional Lie 2-algebra in the same way as a Lie algebroid provides one for its infinite-dimensional Lie algebra of sections. \n\nCertainly, a QP2-manifold is also a Q2-manifold. The infinite-dimensional Lie 2-algebra found in the above way is, however, \\emph{not} the same one as the one constructed in \\cite{Royt98}: There the  2-term complex is $\\Omega^0(M) \\xrightarrow{t} \\Gamma(E)$ and, containing  the de Rham differential  $\\ud$, neither $t$ nor the Jacobiator are  $C^\\infty(M)$-linear, \nwhereas in our construction one has $\\Omega^1(M) \\xrightarrow{t} \\Gamma(E)$ rendering  the map $t$ and the Jacobiator $C^\\infty(M)$-linear.\n\nFurther perspectives on Q$2$-manifolds in terms of particular  $L_\\infty$-algebras or algebroids are deferred to Appendix \\ref{s:Linf}).\n\n\nThere is also another way of obtaining the axioms for a Lie algebroid from studying a Q1-manifold than the one we sketched above (by a study of components of $Q$ with respect to changes of coordinates). This is provided by regarding the Q-derived bracket on the space of degree $-1$ vector fields on the Q-manifold $\\CM$: Let $\\psi$, $\\psi'$ be two such vector fields, then\n\\begin{equation}\n[\\psi,\\psi']_Q := [[\\psi,Q],\\psi']] \\label{QderLie}\n\\end{equation}\nis another vector field of degree -1 on $\\CM$. While in the case of a Q1-manifold one easily identifies vector fields of degree -1 on $\\CM = E[1]$ with sections in the Lie algebroid $E$ and the bracket \\eqref{QderLie} with the Lie bracket between them, so that at the end one rediscovers also by this procedure the usual axioms of a Lie algebroid, for a Q2-manifold one obtains a \\emph{different} picture. First of all, the derived bracket does not need to be antisymmetric anymore, since $[\\psi,\\psi]$ can be non-zero now as, in contrast to a Q1-manifold, a Q2-manifold carries non-trivial degree -2 vector fields (for more details and explanations we refer to section \\ref{sec:derived}). The derived bracket, however, always gives a Leibniz--Loday algebra, satisfying a condition of the form \\eqref{JaCou}. These observations, which are motivated moreover by our study of the gauge transformations (cf., in particular, Equation \\eqref{derQtot})\\footnote{Note that $\\ad_Q(\\psi) \\equiv [Q,\\psi] = [\\psi,Q]$ since both vector fields are odd.}, lead to an attempt of finding another description of a Q2-manifold $\\CM$. In fact, since $[\\cdot,\\cdot]$, restricted to degree -1 (and thus odd) vector fields, yields a $C^\\infty(M)$-linear symmetric map from the degree -1 vector fields on $\\CM$ to the degree -2 vector fields, and  since vector fields of negative degrees on an N-manifold can always be identified with sections of vector bundles, one swiftly extracts from the above an axiomatic formulation very close to the one of a Courant algebroid, with the difference that now the inner product can take values in another bundle. \n\n\\begin{vdef}\n\\label{def:VCour} A \\emph{$V$-twisted Courant algebroid} consists of a vector bundle $W\\to M$, a bundle map $\\rho \\colon W \\to TM$, a Leibniz--Loday algebra $(\\Gamma(W),[\\cdot,\\cdot])$, a non-degenerate surjective symmetric product $\\<.,.\\>$ taking values in another vector bundle $V \\to M$, and  a $W$-connection  ${{}^W\\nabla}$ on $V$\nsuch that for all $\\psi_1$, $\\psi_2 \\in \\Gamma(W)$ one has the following axioms\n\\begin{align} \\label{VCoura}\n  \\< \\psi_1,[\\psi_2,\\psi_2]\\>&=  \\tfrac{1}{2}{{}^W\\nabla}_{\\psi_1}\\<\\psi_2,\\psi_2\\>  = \\<[\\psi_1,\\psi_2],\\psi_2\\> \\, .\n\\end{align}\n\\end{vdef}\nIt is easy to see that this reduces to an ordinary Courant algebroid if $V=M \\times \\Real$ and ${{}^W\\nabla}(1)=0$ (in this case of the trivialized bundle we can identify its sections with functions and if the unit section is parallel, ${{}^W\\nabla}$ reduces to $\\rho$ on these functions). More generally, we can say that a $\\rk V=1$, $V$-twisted Courant algebroid admitting a non-vanishing, $W$-constant section $v\\in\\Gamma(V)$ (i.e.~$v(x)\\neq 0 \\; \\forall x\\in M$ and $\\Wconn_\\psi v=0 \\; \\forall \\psi \\in \\Gamma(W)$) is in a canonical correspondence with a Courant algebroid. However, line-bundle twisted Courant algebroids are a strictly more general notion than the one of a Courant algebroid.\n\nA nontrivial example of a $V$-twisted Courant algebroid can be provided as follows: Let $A$ be a Lie algebroid together with a flat $A$-connection ${}^A\\nabla$ on a vector bundle $V$. Then the couple $W$, $V$, becomes a $V$-twisted Courant algebroid if one takes $W:=A\\oplus \\Hom(A,V) \\cong A\\oplus A^*\\otimes V$ and equips it with the bracket\n\\begin{equation}\\label{ACourant}\n[X\\oplus\\alpha,Y\\oplus\\beta] = [X,Y] \\oplus [\\imath_X,\\uD]\\beta -\\imath_Y\\uD\\alpha +H(X,Y,\\bullet)\n\\end{equation}\nwhere $\\uD\\:\\Omega^k(A,V)\\to\\Omega^{k+1}(A,V)$ is the exterior covariant derivative constructed from ${}^A\\nabla$, $[.,.]$ the Lie algebroid bracket on $A$, and $H\\in\\Omega^3(A,V)$ is any $A$-3-form with values in $V$ satisfying $\\uD H=0$.  The other maps are obvious, for example $\\Wconn_{X\\oplus\\alpha} v={}^A\\conn_X v$.  Certainly, this reduces to the standard Courant algebroid $W\\cong TM \\oplus T^*M$ twisted by a closed 3-form $H$ if $A$ is chosen be the standard Lie algebroid $TM$ (and $V$ and ${{}^W\\nabla}$ as mentioned above), in which case \\eqref{ACourant} reduces to the known, $H$-twisted Courant-Dorfman bracket \n\\begin{equation} \\label{standardtwistCour}\n[X\\oplus\\alpha,Y\\oplus\\beta] = [X,Y] \\oplus \\L_X \\beta -\\imath_Y\\ud\\alpha + \\imath_Y \\imath_X H \\, .\n\\end{equation}\nWhile an ordinary Courant algebroid reduces to a quadratic Lie algebra when $M$ is a point, the ``decoupling'' of $\\rho$ and ${{}^W\\nabla}$ has the advantage that now we can also host (non-Lie) Leibniz--Loday algebras in this framework. For $M=\\mathrm{pt}$, the definition reduces to a Leibniz--Loday algebra together with an invariant inner product with values in a vector space that has a representation.  Also, conversely, one can promote \\emph{any} Leibniz--Loday algebra $W$ to a $V$-twisted Courant algebra $W$ in a \\emph{canonical} way, cf.~Proposition \\ref{canonical} below.\n\n\n\nThe main result of section \\ref{sec:derived} is then the following (for further details, cf.~Proposition~\\ref{p:FWder} and Theorem~\\ref{thm:der})\n\\begin{thm}$\\,$ \\\\ Up to isomorphism and  provided that $\\rk V>1$, there is a 1:1 correspondence between Lie 2-algebroids $V \\xrightarrow{t} E$ and $V$-twisted Courant algebroids $W$ fitting into the following exact sequence \\begin{equation} \\label{VEW}\n0\\longrightarrow\\Hom(E,V) \\longrightarrow W\\xrightarrow{\\;\\;\\pi\\; \\;} E\\longrightarrow 0\n\\end{equation} with $\\rho_W = \\rho_E \\circ \\pi$. In particular, if $\\rk V >1$ and after the choice of a splitting of \\eqref{VEW},\nthe bracket on a $V$-twisted Courant algebroid $W$ as above has the form\n\\begin{equation}\\label{VCourant2}\n[X\\oplus a,Y\\oplus b]_W = [X,Y]_E - t(a(Y)) \\oplus [\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a +\\imath_Y \\imath_X H+ b \\circ t \\circ a -  a\\circ t \\circ b  ,\n\\end{equation}\nwhere all the operations on the r.h.s.~are the induced ones on the corresponding Lie 2-algebroid.\\label{thm1.6}\n\\end{thm}\nIn the above formula, $X$, $Y \\in \\Gamma(E)$ and $a$, $b$ are either viewed as elements of $\\Omega^1(E,V)$ (in the first two terms following the $\\oplus$) or as homomorphisms from $E$ to $V$ so that they can be composed with the fixed map $t \\in \\Hom(V,E)$ of the Lie 2-algebroid. \n\nThe similarity of formula \\eqref{VCourant2} with Equations \\eqref{standardtwistCour} and \\eqref{ACourant} is striking, in particular in view of the fact that, in contrast to those examples, now the (anti-symmetric) Lie 2-algebroid bracket $[ \\cdot , \\cdot ]_E$ does no more satisfy a Jacobi condition in general, while still \\eqref{VCourant2} obeys the left-derivation property \\eqref{JaCou}. Similarly, ${{}^E\\!{}\\mathrm{D}}$ does no more square to zero in general (such as $\\uD$ and $\\ud$ in the analogous formulas \\eqref{ACourant} and \\eqref{standardtwistCour}, respectively),\nbut rather gives the $E$-curvature \\eqref{HRep}. It is here where the simple $t$-contributions to the new bracket become essential.\n\n\n\nTheorems \\ref{thm1.5} and \\ref{thm1.6} certainly also induce a 1:1 correspondence between the above mentioned ``exact V-twisted Courant algebroids'' (with $\\rk V >1$) and  Q2-manifolds $\\CM$ (with at least two independent generators of $C^\\infty(\\CM)$ of degree 2). Exact $V$-twisted Courant algebroids thus provide an alternative, ordinary differential geometric description of the graded-geometrical notion of a Q2-manifold---with respective restrictions provided in the parenthesis of the previous sentence.\n\nThe condition of $\\rk V \\neq 1$ seems necessary, since there exist examples of line bundle twisted Courant algebroids not corresponding to Q2-manifolds (in the above way). In fact, one to our mind interesting problem left open in the analysis of the present paper is a classification of line bundle twisted Courant algebroids, i.e.~$V$-twisted Courant algebroids $W$ with $\\rk V = 1$. \n\nOn the technical level this restriction on the rank can be traced back to the following simple fact of linear algebra (proven in Lemma \\ref{p:Jh=0}): Let $V_1$ and $V_2$ be two vector spaces. Then any linear operator $\\Delta \\colon \\Hom(V_1,V_2) \\to V_1$ satisfying $a(\\Delta(a))=0$ for any $a \\in \\Hom(V_1,V_2)$ has to necessarily vanish \\emph{iff} $\\dim V_2 >1$, while, e.g., for $V_2 = \\Real$ one may identify $\\Delta$ with an element of $V^1 \\otimes V^1$ and then any $\\Delta \\in \\Lambda^2 V_1$ obviously satisfies the condition for symmetry reasons. \n\nIn view of the different descriptions of QP2 manifolds as infinite dimensional Lie 2-algebras sketched before, it is at this point no more astonishing that if the QP2 manifold is regarded as a Q2 manifold (by the forgetful functor), one does not obtain the usual Courant algebroid in the above fashion. Instead, as before, $M \\times \\Real$ is replaced by $T^*M$, so one obtains a particular $T^*M$-twisted Courant algebroid in the above identification (when applied to a QP2 manifold viewed upon as a Q2 manifold). \n\nThe bracket \\eqref{standardtwistCour} is also well-defined for sections of $TM \\oplus \\Lambda^k T^*M$ if $H \\in \\Omega^{k+2}_{closed}(M)$, (without the $H$-twist, this bracket goes back to Vinogradov, cf.~\\cite{Vino90}). One obtains similar properties to those of the standard Courant algebroid, but, for example, the symmetrized contraction of its elements provides an inner product which now takes values in $V=\\Lambda^{k-1} T^*M$. In section \\ref{sec:derived} we propose an axiomatization of these properties:  Very much as \\cite{Xu97} does it for the case of $k=1$, leading  (together with its later reformulations\/simplifications) to the Definition \\ref{def:Cour} above, we propose the definition of a \\emph{Vinogradov algebroid} (cf.~Definition \\ref{def:Vino} below). We also show that $V$-twisted Courant algebroids are particular Vinogradov algebroids. \n\nSection \\ref{sec:derived} is a relatively long section and contains further potentially interesting subjects (like the axiomatization of the bracket \\eqref{standardtwistCour} when applied to higher form degrees into a Vinogradov algebroid and a simple non-standard Q-description of $H$-twisted standard Courant and Vinogradov algebroids). We do not want to summarize all this here, so as to keep some tension for the interested reader. \n\n\n\n\\vskip\\newpassage\n\n\nFinally, in the Appendix \\ref{s:Linf} we provide a further reformulation of Q$k$-manifolds in terms of $k$-term $L_\\infty$-algebroids, which we also define there so as to generalize the more standard relation between Q-manifolds over a point and $L_\\infty$-algebras \\cite{SHLA}.\n\n\\newpage\n\\section{From Bianchi identities to Q-structures} \\label{s:Bian}\nWe want to construct a gauge theory for some $p$-form gauge field. As examples\nshow, at least if we want to go beyond a trivial abelian version of such a theory with $p\\ge 1$, \nthis necessarily involves a tower of gauge fields, where the highest form\ndegree is $p$. For definiteness we take $p=2$, but the generalization to\narbitrary $p$ is immediate. We thus consider a collection of 2-forms $B^D$,\ntogether with 1-form fields $A^a$, and in general also 0-form fields\n$X^i$. The scalar fields are permitted to take values in some internal\nmanifold $M$, which is not necessarily a vector space---so in general the\ntheory is a sigma model. Denoting spacetime by $\\Sigma$, one has\n $X \\colon \\Sigma \\to M$, and the particular collection of scalar fields\n $X^i=X^i(\\x)$ ($\\x \\in \\Sigma$) results after a (local) choice of\n coordinates $x^i$ \non $M$, $X^i = X^*(x^i)$.\\footnote{It is\ntempting to exchange the notation of $M$ and\n$\\Sigma$; then $M$ would be spacetime, and if $\\Sigma$ is nontrivial, one has a\nsigma model. However, here we conform to previous publications.}  In\nmany examples there are no scalar fields and the tower of gauge fields starts\nwith 1-forms only. This just corresponds to the choice $M$ a point, in which\ncase one may drop the map $X$ as it contains no information. However, even in this case, scalar fields would then enter the theory in a second step representing matter; intending to stay as general as possible, we permit their presence thus right from the beginning. \n\nOne of the main ingredients in a gauge theory are the field strengths. We\nalso know how they should start, namely with the exterior derivative acting\non the respective gauge field, possibly then modified by adding wedge\nproducts of gauge fields to this expression. We thus make the most\ngeneral ansatz of this kind:\n\\begin{eqnarray}  F^i &:=& \\ud X^i - \\rho^i_a A^a \\, , \\label{Fi}\\\\\n        F^a&:=& \\ud A^a + \\frac12 C^a_{bc} A^b A^c - t^a_D B^D\\, , \\label{Fa}\\\\\n        F^D&:=& \\ud B^D + \\Gamma^D_{aC} A^a B^C - \\frac{1}{6} \n        H^D_{abc} A^a A^b A^c \\, ,\n\\label{FB}\n\\end{eqnarray}\nwhere wedge products between differential forms are understood \nand\nall the yet unrestricted coefficients can be functions of  $X$.\nFor some purposes it will prove convenient to use a more condensed\nnotation: We then denote the field strengths \ncollectively by $F^\\alpha$ and likewise \nthe gauge fields by $A^\\alpha$ (so $A^i \\equiv X^i$, $A^D \\equiv B^D$).\n\n\nWe now want to require a  very mild form of  Bianchi identities. In\nordinary YM-gauge theories Bianchi identities tell us that $\\ud\nF^a\\equiv -C^a_{bc} A^b F^c$. We want to relax\/generalize this\ncondition on the complete set of field strengths $F^\\alpha$ by requiring\nthat the exterior derivative of them produces terms that all \n\\emph{contain} field strengths again, $\\ud F^\\alpha = \\lambda^\\alpha_\\beta F^\\beta$,\nwhere $\\lambda^\\alpha_\\beta$ may be  arbitrary, field dependent factors.\nInterpreted in this sense, we require that one has:\n\\beq \\boxed{\\ud F^\\alpha |_{F^\\beta=0} \\equiv 0}\\, . \\label{bian}\\end{equation}\nThis equation has to be understood in such a way that for \\emph{every} choice of the gauge fields $A^\\alpha$ satisfying $F^\\alpha =0$ one has $\\ud F^\\alpha =0$.\nWe remark right away that this is satisfied also for generalizations\nsuch as for \nhigher YM theories as discussed in \\cite{Baez02jn}\nor the Algebroid YM theories of \\cite{Str04b}. Moreover, there are\noccasions where\nthe $X$-fields above, considered as part of the gauge field here, can\nbe identified with some Higgs type scalar fields  and $F^i$ then\ncorresponds to the covariant\nderivative of this scalar field $X^i$. In standard\nYM-theory e.g.~we may use the fact that the square of the\nexterior covariant derivative\nis proportional to the curvature to show\nthat also in this enlarged setting \n\\eqref{bian} is satisfied.\n\nLet us rephrase the\nrequirement \\eqref{bian}\nin more mathematical terms: Within\nthe differential graded commutative algebra $(\\mathcal{A},\\ud)$\ngenerated by abstract elements $A^\\alpha$ and $\\ud A^\\alpha$ there is an ideal $\\I$\ngenerated by $F^\\alpha$ as \ndefined in Eqs.~\\eqref{Fi}--\\eqref{FB}, i.e.~by the choice of\nstructural functions $\\rho^i_a(X)$, $\\ldots$, $H^D_{abc}(X)$. The\ngeneralized Bianchi identities hold iff (by definition) this ideal is\nan invariant subspace with respect to the exterior derivative, i.e.~iff Eq.~(\\ref{ideal}) holds true.\n\nThere are some further possible reformulations of the above\nconditions.\nOne may rephrase them as saying e.g.~that the\nstructural functions must be such that the quotient\nalgebra $\\mathcal{A}\/\\I$ forms a so-called free differential algebra\n(cf.~\\cite{Sull77b} or, for a physical\nexample in low dimensions, e.g.~\\cite{Izq99}). Since this quotient\ncorresponds to looking at vanishing field strengths only, which will\n\\emph{not} be our field equations when interested in constructing\na (non-topological) generalization of Yang-Mills theories, we avoid\nthis perspective, such as that we avoid the notation $\\ud F^\\alpha\n\\approx 0$ instead of \\eqref{bian} since for non-topological\ntheories the ``on-shell'' equality would not coincide with the on-shell\nnotion dictated by field equations. Finally, Eq.~\\eqref{bian} may\nalso be viewed as an integrability condition in the following sense:\nFor all fields $A^\\alpha$ which satisfy $F^\\alpha(\\sigma)=0$ for some point $\\sigma\n\\in \\Sigma$, it follows automatically (by the choice of structural\nfunctions) that also $\\left(\\ud F^\\alpha\\right)(\\sigma)=0$. \n\n \n\n\nThe main purpose of this section is twofold: First, we want to show\nthat the simple ``physical'' and rather general requirement \n\\eqref{bian}\n\nautomatically and naturally leads to\nQ-structures.\\footnote{By definition (but cf.~also the more detailed\n explanation below), Q-structures, introduced in the \n context of topological field theories in \\cite{AKSZ}\\label{ct:Scha}\n  are homological vector fields of degree one on a\nsupermanifold.}\nSecondly, to obtain the explicit form\nof the generalized Bianchi identities (cf., e.g.,\nEqs.~\\eqref{BFi}--\\eqref{BFB}\nbelow), the use of this formalism shortens calculations extremely. \n\n\nFor this purpose we first unify the gauge fields into a single map.\nThe scalar fields $X ^i(\\x)$ describe some map  $a_0 \\colon \\Sigma \\to\nM$ ($a_0 \\equiv X$ in previous notation); as already mentioned, they\nare the pullback\nof coordinate functions \non $M$ to functions on $\\Sigma$  by precisely this map (possibly\nrestricted to local charts). To imitate\nthis for the other components $A^a, \nB^D, \\ldots$ of the gauge \nfield, we introduce auxiliary spaces spanned by coordinates \n$\\xi^a, b^D, \\ldots$, taking these spaces to be linear and thus \ndefined by the respective linear coordinates: $W = \\langle \\xi_a   \\rangle$,\n$\\V = \\langle b_I   \\rangle$, where $\\xi_a$ is a basis \nof  vectors \ndual to $\\xi^a$ (linear coordinates on a vector space are a basis in the\ndual vector space) etc. \n\nSince every 1-form on $\\Sigma$ may be viewed as a (fiberwisely linear)\nfunction on $T\\Sigma$, we may thus\nregard $A^a$ as the pullback of the coordinate function $\\xi^a$ by a map from\n$T\\Sigma$ to $W$. A local coordinate system on $T\\Sigma$ can be chosen in the form\n$\\x^\\mu, \\theta^\\mu$, where $\\theta^\\mu=\\ud \\x^\\mu$. Since the natural\nmultiplication of 1-forms is the antisymmetric wedge product, it is near at\nhand to declare the fiber coordinates $\\theta^\\mu$ of $T\\Sigma$ as anti-commuting;\nthis is taken into account by the notation $T[1]\\Sigma$: the coordinates $\\x^\\mu$\nof the base are of degree zero (and thus even and commuting), while\n$\\theta^\\mu$ is of degree one (and in particular odd). In this way we may interpret $A^a$ as\nthe pullback of $\\xi^a$ to functions on $T[1]\\Sigma$ (i.e.~differential forms) by\na map $a_1 \\colon T[1]\\Sigma \\to W[1]$, $A^a = (a_1)^* \\xi^a$; here we declared\nthe coordinates $\\xi^a$ to be of degree one, too, so that the map $a_1$ becomes\ndegree preserving, and $(a_1)^*$ a morphism of algebras.  Likewise,\n $B^D = (a_2)^* b^D$, where $a_2 \\: T[1]\\Sigma \\to \\V[2]$, and $b^D$ are declared to\nbe of degree two, since $B^D$ is a 2-form (that is a function on $T[1]\\Sigma$ of\ndegree two) and we want $a_2$ to be degree preserving again.\\footnote{If one considers supersymmetric or supergravity theories, the form degree does not determine if a homogeneous element is commuting or anti-commuting. If one has bosonic and fermionic 0-forms, for example, then already $M$ will be a corresponding supermanifold. Likewise, there may be anticommuting and commuting 1-forms in a supergravity theory; to describe such objects one would introduce a bidegree on the target, so that the first degree corresponds to the form degree and the other one to the commutativity property. We will not go into further details about this possible generalization here; as a simple prototype of such a theory one may look at supergravity theories reformulated as Poisson sigma models with a super-Poisson structure on the target as given in \\cite{Strobl:1999zz}.}\n\nIf $X^i$ denote local coordinates on $U \\subset M$, we can assemble the above\nmaps into a single map $a$ from (open domains of) $T[1]\\Sigma$ to $U \\times W[1] \\times\n\\V[2]$. It is tempting to require this direct product form of the target\nmanifold only on local charts $U$. Globally we may permit a twist (nontrivial\ntransition functions) in changing the vector spaces when changing charts in\n$M$.  In fact, what we permit is a super-manifold $\\CM_2$ of degree 2: Locally\nit is described by degree 0 coordinates $x^i$, degree 1 coordinates $\\xi^a$,\nand degree 2 coordinates $b^D$.\\footnote{\\label{fn:pos} \nIn this and the following section we will \nconsider only non-negatively graded supermanifolds. (We will often call an integer graded manifold with its induced superalgebra of functions determined by the parity of the degrees simply a supermanifold, understanding implicitly from the context that we consider the full $\\Z$-grading (and not just modulo 2)). Speaking of the degree of it we will \nthen always mean the maximal degree of ``local coordinate functions'' (local generators). \nLater on also supermanifolds with (coordinate) functions of negative degrees will play \nsome role, cf in particular Section~\\ref{sec:derived}.} \nThe transition functions need to be degree\npreserving (by definition). This implies that forgetting about the degree two\ncoordinates one has a vector bundle $E \\to M$ with typical fiber\n$W$. Similarly, $x^i, b^D$ define a vector bundle $V \\to M$ with typical fiber\n$\\V$.  (Since the transition functions of $b^D$s may also contain a part\nquadratic in the degree one coordinates $\\xi^a$, the total space\n$\\CM_2$ does not correspond to the direct sum vector bundle $E \\oplus\nV$ canonically, but does so only after an additional choice of\nsection; we will come back to this subtlety in the subsequent section).\n\nThus in sum we obtain a degree preserving map $a \\colon \\CM_1 \\to \\CM_2$,\nwhere $\\CM_1 = T[1]\\Sigma$ is a degree one super-manifold.\\footnote{More generally,\nwe may also permit a nontrivial fibration over $\\CM_1$, such that $\\CM_2$ is\nonly a typical fiber, leading to the notion of ``$Q$-bundles''. This\nis dealt with in \\cite{KS07}, while in this paper we restrict to\ntrivial bundles (except for the target bundle, encoded in the\nsupermanifold $\\CM_2$). Still, the present \nformulas remain correct also in the more general setting if they are\nreinterpreted as local ones.}\nChoosing local coordinates $q^\\alpha : = (x^i, \\xi^a, b^D)$\non $\\CM_2$ this gives rise to \nfunctions on $\\CM_1$ by means of  the pullback $a^*$, which then \njust yields the set of  local 0-forms, 1-forms, and 2-forms,\nrespectively, with which we started this section, $A^\\alpha = a^*(q^\\alpha)$. \n\nThe de Rham differential $\\ud$ on $\\Sigma$ corresponds to a degree one\nvector field \non $T[1]\\Sigma \\equiv \\CM_1$, namely $\\Qo = \\theta^\\mu \n\\partial\/\\partial \\x^\\mu$. It squares to zero, $(\\Qo)^2 \\equiv 0$, and thus defines \nwhat is called a $Q$-structure on $\\CM_1$. We now define a likewise vector field on $\\CM_2$ \nusing the above data. It is sufficient to know its action on coordinates $q^\\alpha$: \n\\begin{eqnarray} {Q_2} x^i &:=& \\rho^i_a \\xi^a \\, , \\quad \\label{Qi}\\\\\n     {Q_2} \\xi^a& := &- \\frac12 C^a_{bc} \\xi^b \\xi^c +t^a_D b^D\\, , \n     \\label{Qa}\\\\\n        {Q_2} b^D&:= & -\\Gamma^D_{aC} \\xi^a b^C + \\frac{1}{6} \n        H^D_{abc} \\xi^a \\xi^b \\xi^c \\, .\\label{QB}\n\\end{eqnarray}\nThis vector field is obviously also of degree one, i.e.~it raises the degree\nof the respective coordinate by precisely one unit. Furthermore we\nintroduce (cf.~also \\cite{BKS})\n\\beq \\CF := \\Qo \\circ a^* - a^* \\circ {Q_2} \\qquad \\Rightarrow \\qquad \\CF (q^\\alpha) \n\\equiv F^\\alpha \\, . \\label{CF}\\end{equation}\n\nUp to here we only rewrote the initial data contained in\nEqs.~\\eqref{Fi} --  \\eqref{FB}. We now come to analyze\nCondition~\\eqref{bian}. Denoting by $\\ev_\\x$ evaluation of a\nfunction on $\\CM_1=T[1]\\Sigma$ at a point $\\x \\in \\Sigma$, in the present\nlanguage the generalized Bianchi identities (in their formulation as\nintegrability conditions for $F^\\alpha=0$ equations) can be translated into the \nrequirement that for every $\\x \\in \\Sigma$\nthe composed operator $\\ev_\\x \\circ \\Qo \\circ\n\\CF$ vanishes whenever \n$a\\colon\\CM_1 \\to \\CM_2$ is such that the operator $\\ev_\\x\n\\circ \\CF$ vanishes.\n\nNow, using  \n$(\\Qo)^2 \\equiv 0$, we find the identity\n\\begin{equation}\n\\ev_\\x \\circ \\Qo \\circ \\CF =- \\ev_\\x \\circ \\left( \\CF \\circ {Q_2}\n  +a^* \\circ ({Q_2})^2\\right). \\label{preBian}\n\\end{equation}\nThus, for \\eqref{bian} to hold true\nit is obviously sufficient that the vector field ${Q_2}$ \nsquares to zero. But also the converse is true, provided only that the\ndimension of spacetime $\\Sigma$ is at least $p+2$ ($p$ is the degree of\nthe supermanifold $\\CM_2$, in our discussion above $p=2$): Assume that\n$({Q_2})^2 \\neq 0$, that $a\\colon \\CM_1 \\to \\CM_2$ is such that\n$\\ev_\\x \\circ \\CF \\equiv 0$, and that the integrability\ncondition holds true. Then there exists at least one\nbasic coordinate function $q^{\\alpha_0} \\in C^\\infty(\\CM_2)$ for which\n$({Q_2})^2 q^{\\alpha_0}=: \\varphi$ is a non-vanishing function on $\\CM_2$.\nNote that this function has at most degree $p+2$, so $a^* \\varphi$ is a\ndifferential form on $\\Sigma$ of at most degree $p+2$. \nApplying \\eqref{preBian} to $q^{\\alpha_0}$ under the above assumptions,\nwe obtain $\\ev_\\x a^* \\varphi =0$, valid for every $\\sigma \\in \\Sigma$. \nIf we are able to show that\nthere always exists some $a$ fulfilling $(F^\\alpha)_\\x=0$ for which this\nequation is violated, we obtained a contradiction, proving that ${Q_2}$\nmust square to zero. For this to be the case, we need the condition\n$\\dim \\Sigma \\ge p+2$, because, otherwise, \\emph{any} violation\n$\\varphi$ of $({Q_2})^2=0$ in a degree higher than $\\dim \\Sigma$ will not contribute\nto $\\ev_\\x a^* \\varphi$. On the other hand, it is precisely the\nintegrability of \n the equations $F^\\alpha =0$ that  ensure that for \\emph{every} choice of\n $\\ev_\\x \\circ a$ (i.e.~for every choice of $(A^\\alpha)_\\x$)\nwe can \nfind a map $a$, defined at least in a neighborhood of $\\x \\in \\Sigma$, such\nthat also $(F^\\alpha)_\\x=0$. This concludes the proof. \n\n\n\n\nWe thus managed to translate the relatively weak form of Bianchi identities \n\\eqref{bian}, $\\ud F^\\alpha \\sim \\sum F^\\beta$ (where the sum contains\narbitrary field dependent coefficients), \ninto a $Q$-structure on a target super manifold $\\CM_2$. This gives the first part of Theorem \\ref{thm:phys1} mentioned in the introduction, or, in the\nparticular case $p=2$, it means in detail:\n\\begin{thm} \\label{theo:nilpot}\nFor a spacetime dimension of at least 4, the nilpotency of the\nsuper vector field ${Q_2}$\ndefined in Eqs.~\\eqref{Qi} -- \\eqref{QB} is necessary and\nsufficient for the validity of \\eqref{bian}. \n\\end{thm}\n\nThe relation $({Q_2})^2=0$ contains \\emph{all} the conditions we\nwant to place on the \nstructural functions in \\eqref{Fi} -- \\eqref{FB}. They contain the \ngeneralization of the Jacobi identity of the structure constants in ordinary \nYang-Mills gauge theories. In the subsequent section we will display all \nthe conditions (in the example $p=2$), cf.~Eqs.~\\eqref{B1} --\n\\eqref{B7} below, discuss\ntheir general structure and present examples of it\nsuch as Lie 2-algebras or Lie algebroids (an alternative perspective, containing Courant \nalgebroids and their generalizations will be discussed separately in Section~\\ref{sec:derived}).\n\nIn the present section we still want to display the explicit form of\nthe Bianchi identities that we obtain when \\emph{using} the nilpotency \nof ${Q_2}$. One finds\n\\begin{align}  \\ud F^i &\\equiv \\rho^i_{a,j}A^a F^j - \\rho^i_a F^a \\, ,\n        \\label{BFi}\\\\\n      \\ud   F^a&\\equiv \\frac12C^a_{bc,i}A^b A^c F^i -t^a_{D,i}B^D F^i\n         -C^a_{bc} A^b F^c - t^a_D F^D\\, ,  \\label{BFa}\\\\\n      \\ud   F^D&\\equiv -\\Gamma^D_{aC,i} A^aB^C F^i +\\Gamma^D_{aC} B^C F^a\n        +\\frac16H^D_{abc,i} A^aA^bA^c F^i  \\nonumber\\\\\n      &\\quad  -\\frac12H^D_{abc} A^aA^b F^c -\\Gamma^D_{aC} A^a F^C \\, .\n      \\label{BFB}\n\\end{align}\nHere the power of the employed super language becomes particularly transparent: \nWhereas in every explicit calculation one needs to make heavily use \nof the Eqs.~\\eqref{B1} -- \\eqref{B7} below, one may derive the above \nconditions within some lines by use of the following obvious two relations\n(which\nfollow directly from the defining Equation~\\eqref{CF} and the fact\nthat $\\Qo$, ${Q_2}$ are nilpotent vector fields, cf.~also \\cite{BKS,Str04b})\n\\begin{eqnarray}  \\Qo  \\circ \\CF &=& - \\CF \\circ  {Q_2} \\label{dF} \\\\\n   \\CF(\\phi) &=& F^\\alpha \\, a^*(\\partial_\\alpha \\phi) \\label{FLeibniz} \n\\end{eqnarray}\nwhere the second equation holds for all functions $\\phi$ on the target\nand $\\partial_\\alpha$ denotes a left derivation w.r.t.~$q^\\alpha$. \nWe illustrate this for $\\alpha=i$: \nApplying the left hand side of \\eqref{dF} to $x^i$, we get $\\ud F^i$. On the \nother hand $\\CF ({Q_2} x^i) = \\CF(\\rho^i_a\\xi^a) = \\CF(\\rho^i_a) a^*(\\xi^a)+\na^*(\\rho^i_a) \\CF(\\xi^a) = \\rho^i_{a,j} \\CF(x^j) A^a + \\rho^i_a F^a$.\n(Following \nusual physics notation,\nwe did not explicitly display $a^*$ when acting on \nfunctions over $M$ such as  $\\rho^i_{a,j}\\equiv \\partial_j \\rho^i_a$,\nwhich are then understood as\nfunctions over $\\Sigma$ via the ``scalar fields'' $X^i$).\n\n\n   \nThe result above is readily specialized to particular cases of interest. \nE.g., for $M$ a point, we merely drop all terms containing $F^i$. If there is \nno $B$-field (i.e.~there are no degree two generators on $\\CM_2$), we \nput $F^D$ and $B^D$ to zero wherever they occur. Combining both steps\nwe regain \nthe Bianchi identities of ordinary Yang-Mills theory. \n\n\\vskip\\newpassage\n\nBefore continuing we briefly comment on some further\ngeneralization. In the ansatz \\eqref{Fi}--\\eqref{FB} we could\nhave permitted also terms proportional to $\\ud A^\\alpha$, adding e.g.~a\nterm $\\nu^a_i \\ud X^i$ to the r.h.s.~of \\eqref{Fa} etc. Since\nsuch terms enter field strengths $F^\\beta$ only for\nvalues of $\\beta$ corresponding to a higher form degree than the one\nof $A^\\alpha$, it is always possible to replace $\\ud A^\\alpha$ by $F^\\alpha$ by\nmerely renaming coefficient functions in $F^\\beta$. Thus, in the above\nexample of Eq.~\\eqref{Fa}, we may instead equally well add $\\nu^a_i\nF^i$ to the ansatz, merely by\nreplacing $C^a_{bc}$ with new structure functions. Constructing\nout of these new structural functions---except for the additional\ncoefficients $\\nu^\\beta_\\alpha$, which we may just ignore at this point---the \nvector field $Q_2$ as before, Eqs.~\\eqref{Qi}--\\ref{QB}, \nTheorem~\\ref{theo:nilpot}\nstill holds true: While the ideal $\\I$ generated by field strengths changes with \nthe change of structural functions in the first step of the rewriting, it does not change by adding or dropping the terms with lower degree field strengths on the right hand sides in their definition and thus $\\ud \\I \\subset \\I$ gives the\nsame conditions.\n\nMore generally, the ideal $\\I$ remains unaltered by every\nredefinition of the form $\\widetilde F^\\alpha = M^\\alpha_\\beta F^\\beta$ if the field\ndependent matrix $M^\\alpha_\\beta$ is invertible. If the form degree\nof $A^\\alpha$ is denoted by $|\\alpha|$, then $M^\\alpha_\\beta$ has form\ndegree $|\\alpha| - |\\beta|\\ge 0$. It thus is a lower triangular matrix, where\nthe non-vanishing off-diagonal components correspond to $\\nu^\\alpha_\\beta$\nintroduced above. The (block) diagonal pieces, those where $|\\alpha| =\n|\\beta|$, correspond to $X$-dependent coefficient matrices in front of \n$\\ud A^\\alpha$ in the definition of the new field strength $\\widetilde\nF^\\alpha$---the matrix $M^\\alpha_\\beta$ is thus invertible, if and only if each of its \ndiagonal components is nonzero everywhere. For example, instead of \n\\eqref{Fi} we would consider\n$\\widetilde F^i = M^i_j \\ud X^j - \\widetilde \\rho^i_a A^a$. Now, if the\nmatrix $M^i_j(X)$ is invertible (for every point in the target $M$),\nwe may introduce $\\rho^i_a := (M^{-1})^i_j \\widetilde \\rho^j_a$, and\nthis is the first component of the vector field $Q_2$ of Theorem\n\\ref{theo:nilpot}, \nand so on.  Only\nredefinitions of the above form with a non maximal and possibly even\nvarying rank of the\ndiagonal pieces of $M^\\alpha_\\beta$ would require a new and in general more\nintricate analysis. \n\nAlso coordinate transformations on $M$, or, more\ngenerally, on the supermanifold $\\CM_2$, will induce\nredefinitions of field strengths. In an active interpretation of coordinate transformations, i.e.~viewing them as (local expressions of) diffeomorphisms instead of a \nchange of a chart (passive interpretation in physics terminology), the ideal $\\I$ changes in general, but in a covariant way. In this active interpretation $\\I$ remains unchanged only under very particular super-diffeomorphisms, namely those that leave the $Q$-structure invariant; we will come back to them in the context of gauge symmetries in section \\ref{s:gauge} below. \n\nIf $M$ is not a point, \nthe field strength components $F^\\alpha$ themselves do, however, \\emph{not} transform covariantly, except for the lowest ones with $|\\alpha| =0$. (Only the complete set\nof the whole ideal $\\I$ does). \n This can be corrected by means of the choice of connections in\nthe vector bundles $E$ and\n $V$.\\footnote{The bundles $E$ and $V$ and these connections are not to\nbe confused with the (total) bundle of the gauge theory and the gauge fields,\nrespectively. The first one, the total bundle, is assumed to be trivial\nwithin this paper---for a generalization cf.~\\cite{KS07}---and we\non purpose did not call $A^\\alpha$ a connection since this would require\nfurther geometrical justification, and in some way is even the wrong\nnotion. The choice of $M$, $E$, $V$, and possibly connections on the\nlatter two, are fixed background data; they replace the choice of a\nLie algebra in ordinary Yang-Mills theories. Only the gauge fields $A^\\alpha$ \nare dynamical.} One way of seeing this is to first determine the\ntransformation behavior of all the structural coefficient functions \nunder a change of coordinates on ${\\CM_2}$---we will do this within the\nsubsequent section---and then to correct for the overall behavior of\n$F^a$ and $F^D$ by appropriate additions. There is a much simpler route,\nhowever, using the fact\nthat $\\CF$ is a (graded) Leibniz-type operator\nover $a^*$, cf.~Eq.~\\eqref{FLeibniz}:\nA change of local frame in the vector bundle $E$, e.g.,\ncorresponds to a change of coordinates $\\xi^a \\mapsto\n\\widetilde{\\xi^a} = M^{\\tilde a}_b \\xi^b$ on ${\\CM_2}$, where $M^{\\tilde a}_b$ are\nfunctions on the base\\footnote{which is not to be confused with\nspacetime $\\Sigma$---both $M$ and $E$ are data of the target.}\n $M$ of $E$. With \\eqref{FLeibniz} this implies $F^{\\widetilde{a}}= M^{\\tilde\n a}_b(X) F^b +  M^{\\tilde a}_{b,i}(X) F^i  A^b$, \n where $M^{\\tilde a}_{b,i} \\equiv \\partial\n M^{\\tilde a}_{b}\/\\partial x^i$. Only the first term of the two\n corresponds to a tensorial behavior. From this it is obvious\n that if $\\Gamma^a_{ib}$ and\n $\\Gamma^D_{iC}$ denote connection coefficients in local frames in $E$\n and $V$, respectively, the redefined field strengths\n \\begin{eqnarray} F^a_{(\\Gamma)} &:=& F^a + \\Gamma^a_{ib} F^i  A^b \\label{FaG}\n \\\\\n     F^D_{(\\Gamma)} &:=& F^D + \\Gamma^D_{iC} F^i  B^C\n  \\label{FBG} \\end{eqnarray}\n(together with the unaltered $F^i$) are tensorial. In some gauge invariant \naction functionals $S$  there may be\na contribution of the form $\\Lambda_i F^i$, where $\\Lambda_i$ is a\nLagrange multiplier field;\\footnote{Cf, e.g., \\cite{Str04b,KS08}.} in such a case one may replace $F^a_{(\\Gamma)},\nF^D_{(\\Gamma)}$ by the simpler expressions\n$F^a, F^D$ in the remaining part of the action $S$---at\nthe mere expense that the new, redefined Lagrange multipliers loose\ntheir tensorial behavior. For other functionals these contributions\nmay be mandatory, however. \n\nFinally, an interesting alternative to the use of $\\CF$ or the auxiliary connections needed for a tensorial behavior of the field strengths like in Eqs.~\\eqref{FaG} and \\eqref{FBG} is to introduce a map $f \\colon \\CM_1 \\to T[1]\\CM_2$ defined by means of $f^*(q^\\alpha)=a^*(q^\\alpha)$ and $f^*(\\ud q^\\alpha)=\\CF (q^\\alpha)$;\\footnote{This was found later in the paper \\cite{KS07}.} \n equipping $T[1]\\CM_2$ with an appropriate canonical Q-structure, $f$ becomes a Q-morphism for \\emph{every} map $a \\colon \\CM_1 \\to \\CM_2$ and generalizes the Chern-Weil map in the case of non-trivial Q-bundles as is useful for constructing corresponding characteristic classes. In particular, $f^* \\colon {C^\\infty}(T[1]\\CM_2 \\to \\Omega^\\bullet(\\Sigma))$ is an algebra morphism then, $f^*(x^i\\xi^a \\ud \\xi^D) = X^i A^a \\wedge F^D$ etc. \n\n\\newpage\n\n\\section{Q-structures for $p=2$ --- Lie 2-algebroids}\n\\label{s:examQ}\nIn this section we study general vector fields of the form\n\\eqref{Qi}--\\eqref{QB}. Dropping the index 2 for notational\nconvenience within \\emph{this} section,\n\\beq \\label{Q} Q = \\rho^i_a \\xi^a \\frac{\\partial}{\\partial x^i}\n- \\frac12 C^a_{bc} \\xi^b \\xi^c \\frac{\\partial}{\\partial \\xi^a} +t^a_D b^D\n\\frac{\\partial}{\\partial \\xi^a}  -\\Gamma^D_{aC} \\xi^a b^C\n\\frac{\\partial}{\\partial b^D}\n+ \\frac{1}{6} \n        H^D_{abc} \\xi^a \\xi^b \\xi^c \\frac{\\partial}{\\partial b^D} \\eeq\nis the most general degree one vector field on a degree two\nsupermanifold (cf.~also footnote \\ref{fn:pos}). \nThe range of indices is $i = 1, \\ldots, n$, $a = 1,\n\\ldots r$, $D=1 \\ldots s$, respectively,\nwhere, in previous notation,  $n = \\dim M$, $r = \\dim V$, and $s  = \\dim W$.\nNote that $C^c_{ab}$ and $H^D_{abc}$ are completely\nantisymmetric in their lower indices by construction. \nRequiring $Q$ to be homological, that is to square to\nzero and thus to define a Q-structure, one obtains the following identities to\nhold true: \n\\begin{align}\n  \\rho^j_{[a}\\rho^i_{b],j} -\\frac12\\rho^i_c C^c_{ab}  &= 0  \\label{B1}\\\\\n  \\rho^i_a t^a_D &= 0  \\label{B2}\\\\\n  3C^e_{[ab} C^d_{c]e} +3\\rho^i_{[c} C^d_{ab],i} - t^d_D H^D_{abc} &= 0\n    \\label{B3}\\\\\n  \\rho^i_c t^a_{D,i} -\\Gamma^F_{cD} t^a_F -C^a_{bc}t^b_D &= 0  \\label{B4}\\\\\n  \\rho^i_{[b}\\Gamma^D_{a]F,i}  +\\frac12 \\Gamma^D_{eF}C^e_{ab}\n    -\\Gamma^D_{[aC}\\Gamma^C_{b]F} +\\frac12H^D_{abc}t^c_F &= 0  \\label{B5}\\\\\n  t^a_{(A} \\Gamma^D_{aC)} &= 0  \\label{B6}\\\\\n  \\Gamma^D_{[aF}H^F_{bcd]} +\\rho^i_{[a}H^D_{bcd],i} -\\frac32H^D_{e[ab}C^e_{cd]}  &= 0  \\label{B7}\n\\end{align}\nThe first two equations result upon twofold application of $Q$ to\ncoordinates $X^i$, the next three to $\\xi^a$, and the last two when\n$Q^2$ is applied to $b^D$ (the result being put to zero\nin all cases). Round\/square brackets indicate symmetrization\/antisymmetrization\nof enclosed\nindices of equal type (in Eq.~\\eqref{B5}, e.g., this concerns the indices $a$\nand $b$ in the respective two terms, and only them).\n\nWe now discuss the meaning of these equations in more standard,\nindex-free terms. We start with $M =\\mathrm{pt}$ and $s=0$, i.e.~there are no $X^i$\nand $b^D$ coordinates. Correspondingly,  in \\eqref{Q} only the term\ncontaining $C^c_{ab}$  survives. From the above set of equations, the\nonly nontrivial one is Eq.~\\eqref{B3}, reducing to $C^e_{[ab}\nC^d_{c]e}=0$, which is the Jacobi identity for the coefficients\n$C^a_{bc}$. Thus in this simple case, in which the supermanifold\n$\\CM$ has a\ntrivial base and is actually only of degree one,\none finds the Q-manifold\n$(\\CM,Q)= (\\g[1],\\ud_{CE})$, where $\\g$ is some Lie algebra and\n$\\ud_{CE}$ the corresponding Chevalley-Eilenberg\ndifferential. Super vector spaces of degree one\nwith Q-structure are equivalent to\nLie algebras. \n\nLet us in a next step drop the restriction that $M$ is a point,\ni.e.~we deal with a general degree one supermanifold (which,\ncertainly, may also be viewed as a particular degree two\nsupermanifold) with Q-structure. This leaves the first two\nterms in \\eqref{Q}. The corresponding equations, \\eqref{B1} and\n\\eqref{B3} with $t^d_D \\equiv 0$, are recognized as the\nstructural equations for a Lie algebroid: A Lie algebroid is a vector\nbundle $E$ over $M$ together with a bundle map $\\rho \\colon E \\to TM$,\ncalled the anchor map, \nand a Lie algebra defined on the sections of $E$ satisfying the\nLeibniz rule $[\\psi, f\n\\varphi] = f [\\psi, \\varphi] + \\rho(\\psi) f \\, \\varphi$.\\footnote{For\nlater use we mention that we call $E$ an \\emph{almost Lie algebroid} if it\nis defined as above with the mere difference that the bracket $[\n\\cdot, \\cdot ]$ needs to\nbe antisymmetric only, but not necessarily to satisfy the Jacobi\nidentity. If $\\rho \\equiv 0$ in an (almost) Lie algebroid, the bracket\ndefines a fiberwise\nproduct and $E$ becomes a bundle of (almost) Lie algebras.}\n{} From these\ndata one can infer that $\\rho$ is also a morphism of Lie algebras, $\\rho([\\psi, \n\\varphi]) =  [\\rho(\\psi), \\rho(\\varphi)]$. Choosing a local frame\n$\\xi_a$ in $E$ and local coordinates $x^i$ on $M$, the anchor gives\nrise to structural functions $\\rho^i_a=\\rho^i_a(x)$ \nvia $\\rho(\\xi_a) = \\rho^i_a \\partial_i$. Likewise, the bracket between\ntwo sections from the basis induces $C^c_{ab}=C^c_{ab}(x)$:\n$[\\xi_a,\\xi_b]= C^c_{ab} \\xi_c$. Now it is easy to see that the\nmorphism property of $\\rho$ implies Eq.~\\eqref{B1} and the Jacobi\nidentity for the Lie bracket on $E$, together with the Leibniz rule,\nyields (the respective remainder of) Eq.~\\eqref{B3}. \n\nFor the reverse direction, i.e.~to recover the index free geometrical\nstructure from the above formulas, one needs to address coordinate\nchanges on the Q-manifold $\\CM$. By definition, coordinate changes on\na graded manifold need to be degree preserving; therefore, \n$\\widetilde{x^i}= \\widetilde{x^i}(x)$,\n$\\widetilde{\\xi^a} = M^{\\tilde a}_b(x) \\xi^b$. Thus one deals with a\nvector bundle $E$ over the base $M$ of $\\CM$ and $\\CM = E[1]$ (by\ndefinition, for a vector bundle the shift in degree\nconcerns only the fiber coordinates). Implementing this coordinate\nchange on $Q$, we see that $\\rho^i_a$ transforms like a tensor, but\n$C_{ab}^c$ does not. Indeed,\n$\\partial_i=\\frac{\\partial \\widetilde{x^j}}{\\partial x^i}\n\\widetilde{\\partial_j} + \\frac{\\partial \\widetilde{\\xi^a}}{\\partial\nx^i} \\widetilde{\\partial_a}$, where \n$\\partial_i = \\partial\/\\partial x^i$, $\\partial_a = \\partial\/\\partial \\xi^a$\netc. So, there is a contribution proportional to $\\rho^i_a$ and the\n$x$-derivative of $M^{\\tilde a}_b$ to the transformation law of\n$C_{ab}^c$. Since $C_{ab}^c \\xi_c = [\\xi_a, \\xi_b]$ is to hold true\nin every frame---after all we do not want to have frame-dependent\ndefinitions of a bracket---this results in the Leibniz property of the\nbracket. The rest is then obvious.\n\nSummarizing, a degree one supermanifold with Q-structure, \ncalled Q1-manifold for simplicity,  is \\emph{tantamount}\nto a Lie algebroid, $(\\CM,Q)=(E[1],{}^E\\ud)$.\\footnote{This was observed first by Vaintrob \\cite{Vai97}, including also the compact formulation of Lie algebroid morphisms (a proof of equivalence of this definition with the standard, more complicated one of \\cite{MaH93} can be found in \\cite{BKS}). --- Recall \\label{f:8}  that we \nrestricted to non-negatively graded supermanifolds; Q-manifolds with \nthis restriction are also called NQ-manifolds in the literature, cf., e.g., \n\\cite{Sch93,AKSZ}.} Here ${}^E\\ud$ is the\ncanonical differential that generalizes $\\ud_{CE}$ from above for\n$M=\\mathrm{pt}$ and the de Rham differential for $E=TM$, $\\rho = \\id$,\nthe standard Lie algebroid (cf., e.g., \\cite{CaWe99} for\n${}^E\\ud$  or just\nuse the above formulas for $Q$, reinterpreted correspondingly, as a\npossible definition). \n \nIn order to reveal the differential geometry contained in a general\n``Q2-manifold'', i.e.~a degree two graded manifold with Q-structure, \nEq.~\\eqref{Q} together\nwith Eqs.~\\eqref{B1}--\\eqref{B7}, we follow the same strategy: We\nfirst look at degree preserving coordinate changes on $\\CM$,\n\\beq \\widetilde{x^i}= \\widetilde{x^i}(x), \\qquad \n\\widetilde{\\xi^a} = M^{\\tilde a}_b(x) \\xi^b , \\qquad\n \\widetilde{b^D}  = N^{\\tilde D}_C(x)  b^C +\\frac12 L^{\\tilde D}_\n{ab}(x)\\xi^a\\xi^b  \\, . \\label{trafo} \\eeq\n If there were no $\\xi^a$ coordinates,\ni.e.~$r=0$, similarly to before we would obtain $\\CM = V[2]$ for some\nvector bundle $V$ over $M$. The above $L$-terms imply that in general\n$\\CM$ is not just the direct sum of two vector bundles $E$ and $V$,\nbut instead one has the sequence of supermanifolds\n\\beq V[2] \\to \\CM \\to  E[1] \\, . \\label{sequence} \\eeq\n(The first map is an embedding, characterized \nby $\\xi^a =0$, and the second map a (surjective)\nprojection, dropping the $b^D$ coordinates). \nOnly after the choice of a\nsection $\\imath \\colon E[1] \\to \\CM$, characterized locally by $b^D = L^D_{ab}(x) \\xi^a \\xi^b$, \nthis gives rise to $\\CM = E[1] \\oplus V[2]$. The \\emph{difference} between two \nsections is a \nsection $B$ of $\\Lambda^2 E^* \\otimes V$---similarly to the fact that the\ndifference between two connections in a vector bundle\ncorresponds to a tensorial object.\nIn what follows the choice of a section \nshall be understood, so that in terms of ordinary differential geometry \nthere is a total space corresponding to the Whitney sum  \n$E \\oplus V$ of two vector bundles over $M$. However, let us mention that when the Q-manifold under discussion carries \nadditional structures, like a compatible symplectic form, this may not be the best adapted description.\n\n\n\n\nSo, under the above assumption, a (non-negatively) graded manifold of (maximal) degree two gives rise to\na vector bundle $E \\oplus V$ \nover $M$. Next we\ndetermine the transformation properties of the coefficients in\n\\eqref{Q}. For later purposes we keep also $L$-contributions,\ndisregarding them only afterwards within the rest of this\nsection. With \\eqref{trafo} one\nobtains:\n\\begin{eqnarray}\n  \\widetilde{\\rho^i_a} &=& \\frac{\\partial\\tilde x^\\imath}{\\partial\nx^j} \\rho^j_b (M^{-1}) ^b_{\\tilde a}  \\label{trafo1} \\\\\n  \\widetilde{C_{ab}^c} M^{\\tilde a}_d M^{\\tilde b}_e    \n   &=& M^{\\tilde c}_f C_{de}^f\n    -2M^{\\tilde c}_{d],i}\\rho^i_{[e}\n    +M^{\\tilde c}_f t^f_D (N^{-1})^D_{\\tilde E} L^{\\tilde E}_{de}\n   \\label{trafo2} \\\\\n  \\widetilde {t^a_D} &=& M^{\\tilde a}_b t^b_C  (N^{-1})^C_{\\tilde D}\n  \\label{trafo3}\n  \\\\\n  \\widetilde {\\Gamma^C_{aE}} M^{\\tilde a}_b N^{\\tilde E}_D \n   &=&  N^{\\tilde C}_E \\Gamma^E_{bD}\n    -N^{\\tilde C}_{D,i} \\rho^i_b\n    -L^{\\tilde C}_{db} t^d_D  \\label{trafo4} \\\\\n  \\widetilde {H^D_{def} } M^{\\tilde d}_a M^{\\tilde e}_b M^{\\tilde f}_c\n       \n   &=& N^{\\tilde D}_C H^C_{abc} \n    +3N^{\\tilde D}_F \\Gamma_{a]C}^F (N^{-1})^C_{\\tilde E} L^{\\tilde E}_{[bc}\n    +3\\rho^i_{[a} L^{\\tilde D}_{bc],i}\n    -3L^{\\tilde D}_{d[a}C^d_{bc]}\n  \\nonumber\\\\  &&\n    -3N^{\\tilde D}_{C,i}\\rho^i_{[a} \\big(N^{-1}\\big)^C_{\\tilde E}\n    L^{\\tilde E}_{bc]}\n    -3L^{\\tilde D}_{d[a} t^d_C (N^{-1})^C_{\\tilde E} L^{\\tilde E}_{bc]}\n  \\label{trafo5}\n\\end{eqnarray}\nNote that $C_{ab}^c$, $H^D_{abc}$ and $\\Gamma_{aD}^C$ are affected by the affine \n$L$-contributions. Only the two quantities $\\rho^i_a$ and $t_D^a$ are\nnot; they both correspond to bundle maps, namely $t \\colon V\n\\to E$ and $\\rho \\colon E \\to TM$ or, equivalently, to sections of\n$V^* \\otimes E$ and $E^* \\otimes TM$, respectively.\n If $L$-terms are suppressed, $H$ is likewise just a \nsection of $\\Lambda^3 E^* \\otimes V$. But the nature of all three\nobjects, $H$,\nthe bracket on $E$ induced by $C^c_{ab}$, and the object corresponding\nto $\\Gamma_{aD}^C$ discussed further below\ncan change if $L^D_{ab}$ is not forced to vanish. \nFor the next steps we will first restrict to vanishing $L$-contributions, i.e.~to\na fixed choice of a section of \\eqref{sequence}\n\nEven under this assumption, the nature of $\\Gamma_{aD}^C$ is more\nintricate than that of $H$. \nAs anticipated by the notation,  $\\Gamma_{aD}^C$ are\nthe coefficients of a kind of connection. More precisely, \nintroduce an\n$E$-connection ${{}^E{}\\nabla}$ in $V$, i.e.\\ for every $\\psi \\in \\Gamma(E)$ an\noperator $\\Econn_\\psi \\colon \\Gamma(V)\n\\to \\Gamma(V)$ satisfying the Leibniz rule \\mbox{\\( \\Econn_\\psi (fv) =\n\\rho(\\psi)f \\cdot v +f \\Econn_\\psi v \\)} such that the assignment $\\psi\n\\mapsto \\Econn_\\psi$ is $C^\\infty(M)$-linear in\n$\\psi$.\\footnote{More abstractly, $\\Econn_\\cdot$ corresponds to a map from $E$\n  to $A(V)$, the Atiyah algebroid of $V$. $A(V)$ is a vector bundle over $M$\n  itself, namely the Lie algebroid\n  corresponding to the canonical Lie groupoid of automorphisms of the\n  vector bundle $V$.}\n This definition reduces to the one of an ordinary covariant\nderivative in the case of $E=TM$. Then $\\Gamma_{aD}^C$ are just the\nE-connection coefficients in a local frame, $\\Econn_{\\xi_a} b_D =\n\\Gamma_{aD}^C b_C$; now Eq.~\\eqref{trafo4}  follows from\n\\eqref{trafo} (both without the respective\n$L$-contribution, \\eqref{trafo} interpreted as a change of frames in\nthe two vector bundles) by means of the properties of ${{}^E{}\\nabla}$ and vice versa.     \n\nFinally the coefficients $C_{ab}^c$ induce a\nbracket $[ \\cdot , \\cdot ]$ on sections of $E$  via $[\\xi_a,\\xi_b] :=\nC_{ab}^c \\xi_c$, which is antisymmetric by construction and, \n\\emph{according to} \\eqref{trafo2}\n without $L$-terms, satisfies a Leibniz\nrule. \n\nNow, using the above data, we are to reinterpret the conditions\nensuring $Q^2=0$. In this way we arrive at:\n\\begin{prop} \\label{prop:double}\nA Q-structure on a supermanifold of degree two (a Q2-manifold)  \nwith section of the sequence \\eqref{sequence} as described above \nis equivalent to a Lie 2-algebroid.\n\\end{prop}\n\\begin{vdef} \\label{Lie2def} A \\emph{Lie 2-algebroid} is a complex of vector bundles\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq   \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(1297,885)(601,-361)\n\\thicklines\n{\\put(1351,314){\\vector( 1, 0){375}}\n}%\n{\\put(1876,164){\\vector(-2,-1){540}}\n}%\n{\\put(751,314){\\vector( 1, 0){375}}\n}%\n{\\put(1276,164){\\vector( 0,-1){300}}\n}%\n{\\put(676,164){\\vector( 2,-1){540}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{V}%\n}}}\n\\put(1201,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{E}%\n}}}\n\\put(1801,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{TM}%\n}}}\n\\put(1201,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(826,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n\\put(1426,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n\\end{picture}\n\n \\label{compl} \n\\eeq \ntogether with an antisymmetric bracket $[ \\cdot , \\cdot ]$ on $E$, an $E$-connection ${{}^E{}\\nabla}$ \non $V$, and an ``anomaly'' $H \\in \\Gamma(\\Lambda^3 E^* \\otimes V)$ satisfying\nthe following compatibility conditions: \n\\begin{align} [\\psi_1,f\\psi_2] &= \\rho(\\psi_1)f \\, \\psi_2 + \nf[\\psi_1,\\psi_2] \\, ,  \\label{Leibn}\\\\\n  {}[\\psi_1,[\\psi_2,\\psi_3]\\,] +\\textup{cycl(123)} &= t \\big(H(\\psi_1,\n  \\psi_2, \\psi_3)\\big) \\, , \n    \\label{Jac}\\\\{}\n[\\Econn_{\\psi_1},\\Econn_{\\psi_2}]v - \\Econn_{[\\psi_1,\\psi_2]}v &= \nH\\big(\\psi_1,\\psi_2,t(v)\\big) \\, ,  \\label{repr1}\n\\end{align}\nas well as  $t(\\Econn_\\psi v) = [\\psi, t(v)]$ and $\\Econn_{t(v)} w = -\n\\Econn_{t(w)} v$. In addition, $H$ has to satisfy\n\\beq\n  \\Econn_{\\psi_1} H(\\psi_2,\n  \\psi_3, \\psi_4) -\\frac32 H([\\psi_1,\\psi_2],\\psi_3,\\psi_4)\n  +\\textup{Alt}(1234)= 0  \\label{strange} \\, .\n\\eeq\n\\end{vdef}\nWe will now make further remarks on the statement of the above\nproposition and some parts of the above definition. In this way we\nwill arrive also at another, somewhat simpler form of the description of Q2-manifolds (or of what we just called Lie 2-algebroids), provided by Theorem\n\\ref{theo1} below.\\footnote{In this part of the paper we provide a \nclassical differential geometry description of a Q2-manifold. The relatively simple notion of a Q2 manifold in graded geometry encodes relatively involved data in the classical geometry setting. These are, however, chosen so as to mimic the standard definition of a Lie algebroid (corresponding to Q1-manifolds) and, simultaneously,  generalize existing definitions of strict and semi-strict Lie 2-algebras. It may be reasonable to call the above or any equivalent data in \\emph{classical} differential geometry \\emph{semi-strict} Lie 2-algebroids, so as to clearly distinguish from definitions like the ones given in the Appendix or increasingly popular conventions where Q$k$-manifolds are considered as a defintion of Lie $k$-algebroids (cf also the corresponding discussion in the Introduction following Theorem \\ref{thm1.5}).}\n\n\nIn the above, $\\psi_i \\in \\Gamma(E)$, $f \\in\nC^\\infty(M)$, and $v,w \\in \\Gamma(V)$. \\eqref{compl} defining a\ncomplex just means that $ \\rho \\circ t = 0$, as corresponds to\nEq.~\\eqref{B2}. The three projections going to one and the same\ncopy of the base $M$ means that the bundle maps $t$ and $\\rho$ cover\nthe identity map on $M$. Eqs.~\\eqref{Jac}, \\eqref{repr1}, and\n\\eqref{strange} correspond to Eqs.~\\eqref{B3}, \\eqref{B5},\nand \\eqref{B7}, respectively; the two equations in the text are\n\\eqref{B4} and \\eqref{B6}. From Eqs.~\\eqref{Leibn} and\n\\eqref{Jac} together with $ \\rho \\circ t = 0$ one may show that\n$\\rho$ is a morphism of brackets:\n\\beq   \\rho\\big([\\psi_1,\\psi_2]\\big) = [\\rho (\\psi_1), \\rho (\\psi_2)] \\, , \n\\label{rhomor}  \\eeq\nwhere the bracket on the r.h.s.~denotes the commutator of \nvector fields (i.e.~the bracket in the standard Lie algebroid $TM$). \nThis equation, finally, reduces to \\eqref{B1} in components. \n\n\n\n\nCondition \\eqref{Leibn} is the Leibniz rule for the bracket on\n$E$, $t \\circ H$ defining its Jacobiator according to\n\\eqref{Jac}. So, for $H=0$, or even for $H$ living only in the\nkernel of $t$, $E$ becomes a Lie algebroid. As follows from\n\\eqref{repr1}, $H$ also governs the $E$-curvature ${{}^E\\!{}R}$ of ${{}^E{}\\nabla}$; indeed,\n$$ {{}^E\\!{}R}(\\psi_1,\\psi_2)  \\equiv [\\Econn_{\\psi_1},\\Econn_{\\psi_2}] -\n\\Econn_{[\\psi_1,\\psi_2]} \\quad , \\qquad \n{{}^E\\!{}R} \\in \\Gamma(\\Lambda^2E^* \\otimes \\End(V)) \\, , \n$$  where $[\\Econn_{\\psi_1},\\Econn_{\\psi_2}]$ denotes the ordinary commutator of derivations, \nis nothing but the generalization of\ncurvature to the context of $E$-connections in $V$ for the \nalmost Lie algebroid $E$. Denoting by ${{}^E{}\\Omega}^p(M,V)$ the sections of\n$\\Lambda^p E^* \\otimes V$, the $E$-$p$-forms with values in $V$, where $V$ is some\nvector bundle over the base $M$ of $E$, ${{}^E\\!{}R}$ becomes an $E$-2-form\nwith values in the endomorphisms of $V$. Note that we used\nEq.~\\eqref{rhomor} to show \nthat the operator ${{}^E\\!{}R}(\\psi_1,\\psi_2)$ is\n$C^\\infty(M)$-linear and thus indeed defines an \nendomorphism of $V$. $H=0$ implies a flat E-connection, $ {{}^E\\!{}R}=0$, which \nis a \\emph{representation} of $E$ on $V$.  The condition\n\\eqref{strange} implies that the Jacobi condition for $[ \\cdot ,\n\\cdot ]$  and the\nrepresentation property of ${{}^E{}\\nabla}$ are violated in a relatively mild,\ncontrolled way.\n\nEq.~\\eqref{strange} can be rewritten in a more compact form.\nIntroduce the generalization of an exterior covariant derivative\n${{}^E\\!{}\\mathrm{D}} \\colon {{}^E{}\\Omega}^\\cdot (M,V) \\to {{}^E{}\\Omega}^{\\cdot \\, +1} (M,V)$ \nby means of a generalized Cartan\nformula\n\\begin{align} \\label{genCar} {{}^E\\!{}\\mathrm{D}} \\omega (\\psi_0, \\ldots , \\psi_p) = \n  &  \\sum_i (-1)^i \\Econn_{\\psi_i}\n  \\omega(\\psi_0,\\ldots,\\widehat{\\psi_i},\\ldots,\\psi_p)  \\nonumber \\\\\n  & +\\sum_{i<j} (-1)^{i+j} \\omega\\big( [\\psi_i,\\psi_j], \\psi_0, \\ldots,\\widehat{\\psi_i},\\ldots\\widehat{\\psi_j},\\ldots,\\psi_p \\big)\n  \\, .  \n\\end{align}\nNote that this defines a map in the indicated manner only because\n${{}^E{}\\nabla}$ and the bracket satisfy a Leibniz rule---with the same anchor\n$\\rho$. \nThen the condition to be satisfied by $H$ is nothing but simply\n${{}^E\\!{}\\mathrm{D}} H=0$. \n\nAt this point it is worthwhile to return to the question of changes \nof the section of \\eqref{sequence}. As\nalready mentioned above it corresponds to an $E$-2-form $B$ with\nvalues in $V$. For this purpose we thus need to interpret\nEqs.~\\eqref{trafo1} to \\eqref{trafo5} for the case that\nboth $M^{\\tilde a}_b$ and\n$N^{\\tilde D}_C$ are the identity and $L$ equals $B$.\\footnote{Note\nthat this is\n\\emph{not} the only option one has. Indeed, $L$ may be a function of\n$M$ and $N$ and then changes of the frame in $E$ and $V$ induced by\n$M$ and $N$ result in a different nature of e.g.~the bracket, which\nthen will in general \\emph{not} satisfy a Leibniz rule as before. This\nis what happens in the case of the Courant algebroid, which we will\ndiscuss briefly in a later section; here additional structures, present\nonly in that case, give rise to a natural unique relation of $L$ to\nchanges of frames.} This induces the following changes of the bracket,\nthe $E$-connection, and the anomaly:\n\\begin{eqnarray} [\\psi_1,\\psi_2] &\\mapsto&\\widetilde{ [\\psi_1,\\psi_2]}\n\\equiv\n [\\psi_1,\\psi_2] + t(B(\\psi_1,\\psi_2)) \\label{split1}\n\\\\ \\Econn_\\psi v &\\mapsto& ^E{}\\widetilde{\\nabla}_{\\!\\psi\\,} v \\equiv  \\Econn_\\psi v +\nB(\\psi,t(v))\\label{split2} \n\\\\ H &\\mapsto& H +{{}^E\\!{}\\mathrm{D}} B -C \\, ,  \\label{split3}\n\\end{eqnarray}\nwhere $C(\\psi_1,\\psi_2,\\psi_3) \\equiv B(t(B(\\psi_1,\\psi_2)),\\psi_3) +{\\text{cycl.}}$\nNote in this context that ${{}^E\\!{}\\mathrm{D}}$ does not define a differential for two\nreasons: first, ${{}^E{}\\nabla}$ has a non-zero curvature, and, secondly, the\nbracket on $E$ does not satisfy the Jacobi condition---and these two\nfacts do \\emph{not} cancel against one another. Correspondingly, the\nanomaly $H$ up to changes of the section is not just an element of a\ncohomology. The new and the old $H$ are closed only w.r.t.~the\nnew and the old  $E$-derivatives, ${{}^E\\!{}\\mathrm{D}}$ and  \n$^E\\!\\!{}\\widetilde{D}$, respectively.\n\n\n\n\n\n\nFinally,\n${{}^E{}\\nabla}$ and $t$ may be used to define naturally a bracket on $V$ according to \n\\([v,w]_V:= \\Econn_{t (v)} w \\). According to Prop.~\\ref{prop:double}, \nthis bracket is antisymmetric. Moreover, $\\rho_V := \\rho \\circ t = 0$\ndefines a canonical \nanchor map from $V$ to $TM$ with respect to\nwhich $[ \\cdot , \\cdot ]_V$ may be seen to \nsatisfy a Leibniz rule, i.e.~just $[v, fw]_V = f[v,w]_V$. \nIn this way also $V$ becomes an \nalmost Lie algebroid, in fact a bundle of almost Lie algebras,\nwith Jacobiator determined by $H$ via\n \\beq [v_1,[v_2,v_3]_V]_V +\n{\\text{cycl.}}(123) = H(tv_1,tv_2, tv_3) \\, ,\\label{Jac2} \\eeq\nas can be derived from the equations in Prop.~\\ref{prop:double}.  \nAlso, $t \\colon V \\to E$ may be seen to be a morphism of brackets, \n$t\\big([v,w]_V\\big) = [t (v), t(w)]$. Finally, it remains to check in what sense $\\Econn_\\psi$ \nrespects the bracket on $V$; one obtains\n\\beq \\Econn_\\psi ([v,w]_V) = [ \\Econn_\\psi (v),w]_V + [v, \\Econn_\\psi (w)]_V + \nH(\\psi, t(v),t(w)) \\, . \\label{der} \\eeq\nIn summary, \n\\begin{thm} \\label{theo1}\nA Q2-manifold with section or a Lie 2-algebroid is equivalent to an\nalmost Lie algebroid $(E,[ \\cdot , \\cdot ])$ and a bundle of almost\nLie algebras $(V,[ \\cdot , \\cdot ]_V)$ together with a morphism $t\n\\colon V \\to E$, covering the identity on $M$, an $E$-connection on\n$V$ such that \\beq \\Econn_{t(v)} w= [v,w]_V \\quad , \\qquad t(\\Econn_{\n\\psi} v) = [\\psi, t(v)] \\, , \\label{repr2} \\eeq as well as an\nanomaly $H\\in {{}^E{}\\Omega}^3(M,V)$ such that the Jacobiators on $E$ and $V$ are\n$t \\circ H$ and $H \\circ t$, respectively, and the representation and\nderivation property of ${{}^E{}\\nabla}$ are violated according to the\n$E$-curvature ${{}^E\\!{}R} \\cdot = H( \\, , \\, , t(\\cdot))$ and\nEq.~\\eqref{der}, respectively.  $H$ fulfills the consistency\ncondition ${{}^E\\!{}\\mathrm{D}} H=0$.\n\nA change of section of \\eqref{sequence} corresponds to $L \\in\n{{}^E{}\\Omega}^2(M,V)$ and induces Eqs.~\\eqref{split1} to \\eqref{split3} as\nwell as $[v,w]_V \\mapsto [v,w]_V + L(t(v),t(w))$, with all remaining\nstructures unmodified. \n\\end{thm}\nInstead of the first three lines we could have said also \n``A Q2-manifold with section is equivalent to a complex of almost \nLie algebroids\n\\begingroup\\makeatletter\n\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(2120,885)(601,-361)\n\\thicklines\n{\\put(1276,314){\\vector( 1, 0){300}}\n}%\n\\put(1351,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n{\\put(2251,314){\\vector( 1, 0){300}}\n}%\n\\put(2251,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n{\\put(1726,164){\\vector( 0,-1){300}}\n}%\n{\\put(906,172){\\vector( 2,-1){749.200}}\n}%\n{\\put(2699,169){\\vector(-2,-1){870}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(V,[.,.])}%\n}}}\n\\put(1651,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(2626,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(TM,[.,.])}%\n}}}\n\\put(1651,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{(E,[.,.])}%\n}}}\n\\end{picture}\n\n  \\label{compl2} \n\\eeq\ntogether with an $E$-connection on $V$ such that \\ldots''. \nIn this sequence $TM$ is regarded as \nthe standard Lie algebroid over \n$M$ and arrows are morphisms. Since $t$ and $\\rho$ cover the identity\nmap, the \nmorphism of almost Lie algebroids then implies that\n$\\rho_V = \\rho \\circ t$ and this has to  vanish due to the complex \nproperty. Note also that we did not try to formulate minimal \nconditions: e.g.~Eqs.\\ \\eqref{Jac2} and  \\eqref{der} \nmay be established from the rest as obvious from our derivation. \n\nOf particular interest is certainly the special case of vanishing \nanomaly, where the situation simplifies considerably: \n\\begin{prop} \n A Q2-manifold with a section such that $H =0$ is equivalent to a \ncomplex of Lie algebroids\n\\begingroup\\makeatletter\n\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n  \\reset@font\\fontsize{#1}{#2pt}%\n  \\fontfamily{#3}\\fontseries{#4}\\fontshape{#5}%\n  \\selectfont}%\n\\fi\\endgroup%\n\\beq \\mbox\n\\setlength{\\unitlength}{3947sp}%\n\\begin{picture}(1297,885)(601,-361)\n\\thicklines\n{\\put(1351,314){\\vector( 1, 0){375}}\n}%\n{\\put(1876,164){\\vector(-2,-1){540}}\n}%\n{\\put(751,314){\\vector( 1, 0){375}}\n}%\n{\\put(1276,164){\\vector( 0,-1){300}}\n}%\n{\\put(676,164){\\vector( 2,-1){540}}\n}%\n\\put(601,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{V}%\n}}}\n\\put(1201,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{E}%\n}}}\n\\put(1801,239){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{TM}%\n}}}\n\\put(1201,-361){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{M}%\n}}}\n\\put(826,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$t$}%\n}}}\n\\put(1426,389){\\makebox(0,0)[lb]{\\smash{\\SetFigFont{12}{14.4}{\\rmdefault}{\\mddefault}{\\updefault}{$\\rho$}%\n}}}\n\\end{picture}\n} \n  \\label{compl1} \n\\eeq \ntogether with a representation ${{}^E{}\\nabla}$ of \n$E$ on $V$ satisfying the compatibility conditions \\eqref{repr2}. \n\\label{cor:Liecomp}\n\\end{prop}\nIn the notion of a representation of one Lie algebroid on another one we \nmay include $\\Econn_\\psi ([v,w]_V) = [ \\Econn_\\psi (v),w]_V + [v,\n\\Econn_\\psi (w)]_V$, or otherwise we may even derive \nthis condition from the other assumptions in the proposition.\n\nIt remains to specialize to the case of $M=\\mathrm{pt}$, \ni.e.~to ``Q-vector spaces''. E.g.~then Prop.~\\ref{cor:Liecomp} reduces to \n\\begin{prop} A Q2-vector space with section such that $H=0$ is \nequivalent to a Lie 2-algebra. \n\\end{prop}\n\\begin{vdef} A Lie-2-algebra (a differential crossed module) \nis a pair $(\\g,\\h)$ of Lie algebras \ntogether with a homomorphism $t \\colon \\h\\to \\g$ and a  representation\n$\\alpha \\colon \\g\\to\\Der(\\h)$ such that $\\, t((\\alpha x)v) =[x,t(v)]$   \nand $ \\alpha(t(v))w =[v,w]_\\h$  for all $x \\in \\g$, $v,w \\in \\h$ hold true.  \n\\end{vdef}\nHere we adapted to the notation of \\cite{Baez02jn}. The relation is \nestablished easily via $\\g =E$, $\\h = V$ and $\\alpha(x) \\equiv \\Econn_x$ for \n$x \\in \\g \\cong \\Gamma(\\g)$, $\\alpha_{aD}^C \\equiv \\Gamma_{aD}^C$.\\footnote{$\\alpha$ \nis not an arbitrary representation on $\\h$, \n$\\alpha \\in \\End(\\h)$ and $[\\alpha(x),\\alpha(y)] = \\alpha([x,y])$---the flatness property of ${{}^E{}\\nabla}$, \nbut the fact that $\\h$ is a Lie algebra itself is taken into account \nby restricting to $\\Der(\\h) \\subset  \\End(\\h)$: \n$\\alpha(x) \\in \\Der(\\h) \\Leftrightarrow \\alpha(x) ([v,w]) =  [\\alpha(x)(v),w]\n+[v,\\alpha(x)(w)]$---which follows from our Condition~\\eqref{der} with $H=0$.} \nThe case of a Q2-vector space with non-vanishing $H$ (and fixed\nsection $\\imath$) corresponds to a \nsemistrict Lie 2-algebra in the nomenclature of Baez, cf.~e.g.~\\cite{Baez03vi}. \n\n  \n\n\n\\newpage\n\n\\section{Gauge transformations} \\label{s:gauge}\nHere we introduce a notion of infinitesimal gauge\ntransformations. The present section is intended to\nparallel Section~\\ref{s:Bian}, where we introduced an as general as possible\nnotion of field strength of gauge fields, imposing conditions that a\nphysicist would generically consider mandatory---including certainly\nordinary Yang--Mills gauge theories as particular example, but also\nknown generalizations like those of \\cite{Str04b}, \\cite{SaWei:2005bp,SaWit:2005hv}, \\cite{BM05}, \\cite{ACJ05} or \\cite{BS05}. \nSuch as we required the field strengths $F^\\alpha$ to start with $\\ud\nA^\\alpha$, likewise now we want to require that the gauge\ntransformations\nfor $A^\\alpha$ start with $\\ud \\epsilon^\\alpha$ (except those for the zero\nforms $X^i$ certainly). Again this is satisfied in all the \nexamples we just mentioned---at least after choosing appropriate parameters. Correspondingly, infinitesimal gauge\ntransformations are\nparameterized by quantities $(\\epsilon^\\alpha) :\\equiv\n(\\epsilon^a,\\mu^D)$ which are differential\nforms on $\\Sigma$ of form degree one less\nthan the respective $A^\\alpha$ (so $\\mu^D$ are 1-forms); due to the index\n$\\alpha$, in general $\\epsilon^\\alpha$ should be considered as functions of\n$X^i(\\sigma)$ as well---otherwise formulas are at least frame\ndependent---but we postpone this issue to a more careful\ndiscussion below. We thus arrive at the following rather general ansatz: \n\\begin{align}  \\delta_\\varepsilon X^i &= \\bar\\rho^i_a  \\varepsilon^a  \\nonumber\\\\\n  \\delta_\\varepsilon A^a &= \\ud\\varepsilon^a +\\bar C^a_{bc}A^b \\varepsilon^c -\\bar t^a_D \\mu^D +O(F^i) \\nonumber\\\\\n  \\delta_\\varepsilon B^D &= \\ud\\mu^D -\\bar\\Gamma^D_{aC} \\varepsilon^a B^C +\\bar\\Gamma^{\\prime\\,D}_{aC}\n    A^a \\mu^C +\\frac12 \\bar H^D_{abc} A^a  A^b \\varepsilon^c\\,\n    +O(F^i,F^a) \\,.  \\label{gsymm0}\n\\end{align}\nHere $O(F^\\alpha)$ is supposed to denote contributions that vanish for\n(the respective)\n$F^\\alpha=0$. They are required already to obtain frame-independence of\nformulas, while there exist several options to achieve this \n(cf.~also the discussion in \\cite{BKS}). Moreover, there may be\nadditional, model dependent contributions necessary to render a\nparticular action functional $S$ invariant w.r.t.~gauge transformations.\nSuch contributions may use additional data present in $S$,\nincluding even a metric $h$ on $\\Sigma$; an illustration is provided by\nthe Dirac sigma model \\cite{DSM}, where the $A^a$ variations contain\ncontributions proportional to $\\ast F^i$, where $\\ast$ denotes the\nHodge duality operation induced by $h$ on $\\Sigma$, which is 2-dimensional\nin this case (moreover, such a contribution makes sense only in this\ndimension).\nIn this way one\ncould even consider completely general\n$F^a$ contributions to $\\delta A^a$, or likewise\ncontributions to $\\delta X^i$. We excluded them in the above (thus\nonly ``rather general'') ansatz, where we assume that the additions live in the ideal $\\I$ and are homogeneous of the correct form-degree (without any additional structures); that is why, e.g., only $F^i$ is permitted to enter $\\delta A^a$ in \\eqref{gsymm0}. \n\nThe coefficients in the ansatz~\\eqref{gsymm0} are arbitrary\nfunctions of $X(\\sigma)$ at this point. The notation chosen anticipates\nthe result one obtains, where one simply drops the bars\n(and also the prime in the third contribution to the last formula).\n\n\n\n\n\nIn analogy to the Bianchi identities \\eqref{bian}, \\eqref{ideal}\nwe require that gauge transformations of field strengths result into\ncombinations that \\emph{contain} field strengths again, i.e.~we\nrequire: $\\delta_\\varepsilon F^\\alpha = \\L_\\beta^\\alpha\nF^\\gamma$ for all possible choices of gauge parameters \n$\\varepsilon^\\alpha$ and field configurations $A^\\alpha$---where  the coefficients\n$\\L_\\beta^\\alpha$ depend \nlinearly on $\\varepsilon^\\alpha$ (and, in principle, its first derivatives) and are\npermitted to be also field dependent---or,  in other\nwords, we require\n\\beq  \\boxed{\\delta_\\varepsilon F^\\alpha|_{F^\\beta=0} \\equiv 0 \\quad \\textrm{or}  \\quad \\delta_\\varepsilon\n\\I \\subset \\I } \\; .\n\\label{gauge}\\eeq\nSo the ideal $\\I$, defined in Section~\\ref{s:Bian}, should not only\nbe invariant w.r.t.~the operator $\\ud$ but also w.r.t.~the operator\n$\\delta_\\varepsilon$ (for arbitrary choices of the gauge transformation\nparameters $\\varepsilon^a$ and $\\mu^D$). \n\nSuch relations are certainly always satisfied in all the examples\nmentioned above. In particular, even if some of the $X^i$ fields are\nconsidered as scalar fields (and not as part of the gauge field) and,\ncorrespondingly, the respective $F^i = \\ud X^i - \\rho^i_a A^a$ then\nhas the interpretation of a covariant derivative of $X^i$, again it is\na minimal requirement on what one would usually like to\ncall a covariant derivative.\n \nFollowing the same strategy as in the Section~\\ref{s:Bian}, the\nCondition~\\eqref{gauge} is now used to restrict the coefficient\nfunctions in the ansatz~\\eqref{gsymm0}. There we obtained \nTheorem \\ref{theo:nilpot}. Consequently, \nwe now assume that $Q_2$\nsquares to zero in the definition of field strengths.\\footnote{At the\nend of the following proof we will\nmake a brief remark on what still can be said if we drop this\nrequirement.}\nWe obtain\n\\begin{thm} \\label{theo:gauge1}  \nFor a spacetime dimension of at least 3, equality of the coefficients\nin \\eqref{gsymm0} with the respective unbared (and unprimed)\ncoefficient in the\nfield strengths \\eqref{Fi}--\\eqref{FB}---or, likewise, \nin the (nilpotent) super vector field ${Q_2}$\nin Eqs.~\\eqref{Qi}--\\eqref{QB}---is necessary and\nsufficient for the validity of \\eqref{gauge}.\n\\end{thm}\nAgain, a likewise statement holds true for every choice of $p$, provided\nonly the dimension of $\\Sigma$ exceeds $p$ by at least one (otherwise one\nstill has sufficiency, but necessity is maybe the more striking part). \nNote that we assumed the Bianchi\nidentities to be satisfied, so \\emph{any}\ncontribution of the form $O(F^\\alpha)$, i.e.~lying inside the ideal $\\I$, to \\eqref{gsymm0} does not\nmodify the result!\n\nTo prove the statement we will content ourselves with a halfway\nQ-language at this point, not yet\nintroducing any further abstractness beyond what we already did\nso far, while still avoiding any lengthy direct calculation in this manner.\nIn a condensed notation we may summarize the ansatz~\\eqref{gsymm0}\nas follows:\n\\beq \\delta^{(0)}_\\varepsilon A^\\alpha \\equiv\n\\ud \\varepsilon^\\alpha + \\varepsilon^\\beta a^*V^\\alpha_\\beta \\label{gauge0}\n\\, ,\n\\eeq\nwhere $V^\\alpha_\\beta$ is a function on $\\CM_2$ (of degree $|\\alpha| - |\\beta| +1\\ge\n0$, vanishing otherwise as well as for $|\\beta|=0$),\ncollecting all the yet undetermined\ncoefficients of the ansatz. The superscript $(0)$\nwas used, moreover, so as to indicate that we dropped the\nterms $O(F^\\alpha)$, which are irrelevant for the present\npurposes. We assume for simplicity, moreover,\nthat $\\varepsilon^\\alpha$\ndepends on $\\sigma$ only (and not also\non fields).\\footnote{Both steps, dropping $O(F^\\alpha)$ terms and having $\\varepsilon^\\alpha$\ndepend on $\\sigma$ only, imply, among others, that we fixed a frame or target space coordinates for\nthe calculation. But this is legitimate at this point.} We also kept\nthe slightly more suggestive notation $\\ud$ instead of $Q_1$ here.\n\nNow we are to calculate $ \\delta^{(0)}_\\varepsilon F^\\alpha$, where, according to\n\\eqref{CF}, $F^\\alpha= \\ud A^\\alpha -a^* Q_2^\\alpha$ with  $Q_2^\\alpha$ denoting \n the (left) components of the super vector field\n$Q_2$ in a basis $\\partial_\\alpha \\equiv \\partial \/\\partial q^\\alpha$. A\nrather simple calculation (using at this point only $\\ud^2=0$) yields\n\\begin{eqnarray}\\delta^{(0)}_\\varepsilon F^\\alpha  &= \n&  \\ud \\varepsilon^\\beta \\, a^* (V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha) \\nonumber \\\\ \n  & & - \\varepsilon^\\beta \\, a^* \\left[ \\partial_\\beta (Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha) + O(V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha) \\right] \n\\nonumber \\\\ \n  & &- (-1)^{|\\beta|} \\varepsilon^\\beta \\CF(V^\\alpha_\\beta) \\, , \\label{Fchangeold} \n\\end{eqnarray}\nwhere $ O(V^\\alpha_\\beta - \\partial_\\beta\nQ_2^\\alpha)$ denotes terms vanishing with $V^\\alpha_\\beta = \\partial_\\beta\nQ_2^\\alpha$ and, as before, $\\CF \\equiv \\ud a^* - a^* Q_2$.\nSo, obviously the sum of the first two lines has to vanish.\nFor every point $\\sigma_0 \\in \\Sigma$ we may choose $\\varepsilon^\\alpha$ such that $\\ud \\varepsilon^\\alpha(\\sigma_0)$\nis non-zero and $\\varepsilon^\\alpha(\\sigma_0)$ vanishes and vice versa. Also the map $a \\colon\nT[1]\\Sigma \\to \\CM_2$ is at our disposal. Correspondingly,\nif the dimension of $\\Sigma$ exceeds $p$, this requires\n\\beq V^\\alpha_\\beta = \\partial_\\beta\nQ_2^\\alpha \\qquad \\textrm{and} \\qquad  \\partial_\\beta (Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha) = 0 \\, \\qquad \\textrm{for} \\quad\n   |\\beta| \\ge 1 \\,\n   . \\label{consistency}\n   \\eeq\nThe second equation is fulfilled identically for $(Q_2)^2=0$, while\nthe first one gives the claimed necessity. \n\nOn the other hand, if we restrict ourselves to gauge transformations of the form \\eqref{gauge0} (and in general only then!), one may even\n\\emph{conclude} $(Q_2)^2=0$ from \\eqref{gauge}. In this case one needs\nonly $\\dim \\Sigma \\ge p+1$. The reason is that according to the second Equation~\\eqref{consistency} $Q_2^\\gamma \\partial_\\gamma\n  Q_2^\\alpha \\equiv (Q_2 \\circ Q_2)^\\alpha$ can only be a function of degree zero, which contradicts with the fact that it must have a strictly positive degree by knowing that $Q_2$ has degree plus one. This gives the result summarized in Theorem \\ref{thm:phys2}.\n\n\n\nFor convenience of the reader, we display the resulting\nformulas\nexplicitly: \n\\begin{align}  \\delta^{(0)}_\\varepsilon X^i &= \\rho^i_a  \\varepsilon^a \\, ,  \\label{g1a}\\\\\n  \\delta_\\varepsilon^{(0)} A^a &= \\ud\\varepsilon^a + C^a_{bc}A^b \\varepsilon^c +t^a_D \\mu^D \\, , \n    \\label{g2a}\\\\\n  \\delta_\\varepsilon^{(0)} B^D &= \\ud\\mu^D -\\Gamma^D_{aC} \\varepsilon^a B^C +\\Gamma^D_{aC}\n    A^a \\mu^C +\\frac12 H^D_{abc} A^a A^b \\varepsilon^c \\, ,  \\label{g3a}\n\\end{align}\nwhile, according to \\eqref{gsymm0} and the above result, the full\ntransformation then takes the form\n\\begin{align}\n  \\delta_\\varepsilon X^i &= \\delta^{(0)}_\\varepsilon X^i  \\label{g1b}\\\\\n  \\delta_\\varepsilon A^a &= \\delta^{(0)}_\\varepsilon A^a +F^i \\lambda^a_i  \\label{g2b} \\\\\n  \\delta_\\varepsilon B^D &= \\delta^{(0)}_\\varepsilon B^D +F^i \\lambda^D_i \n    +F^a \\lambda^D_a   \\,, \\label{g3b}\n\\end{align}\nfor some coefficients $\\lambda^\\alpha_\\beta$ depending on $\\varepsilon$ (and the gauge fields). We will\nspecify two covariant options for a choice of $\\lambda$s below. Again we\nmention that this is not yet the most general possible form; there may\nbe further model- and\/or dimension-dependent contributions of the more\ngeneral form that they just vanish with vanishing $F^\\alpha$\\footnote{If, for example, we add a term containing $*F^\\alpha$ linearly to, say, $\\delta A^\\beta$, $*$ denoting the Hodge dual with respect to some metric on $\\Sigma$, then the gauge transformation of $F^\\beta = \\ud A^\\beta + \\ldots$ produces a term linear in $\\ud *F^\\alpha$ from which, in general, we cannot say that it belongs to the ideal $\\I$ (while it still would vanish together with $F^\\alpha=0$).}---but in any case the above $\\delta_\\varepsilon$ (as\nwell as $\\delta^{(0)}_\\varepsilon$) is always an important part of such gauge\ntransformations as well.\n\n\n\\vskip\\newpassage\n\nIn the following we will now refine the Q-language so as to also be\nable to cover the gauge transformations of the form \\eqref{g1a}--\\eqref{g3a} or even \\eqref{g1b}--\\eqref{g3b}. We restrict ourselves\nto infinitesimal ones and we will not strive to present their\ngeometrical meaning here (this is rather subject of a more\nmathematical accompanying paper \\cite{KS07}); the main goal here\nis to provide a technical tool, which e.g.~permits one to calculate\nthe commutator of two gauge transformations in an efficient manner.\nAgain we want to be model independent here and thus we aim at\ncalculating the generic part of such a commutator,\n$[\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha$;\nevery explicit calculation, making use of \\eqref{B1}--\\eqref{B7}, \nwould be rather tedious and lengthy---the method presented\nbelow\\footnote{Cf.~also \\cite{BKS} for the same idea, illustrated\nthere at the example of the\nPoisson sigma model.} shortens the calculation considerably; also it\nprovides a relation to so-called derived brackets \\cite{YKS96},\nwhich were studied in the mathematical literature and which we will\nthen develop further for our particular case in a subsequent section. \n\nIn Section~\\ref{s:Bian} we found that $A^\\alpha = a^* (q^\\alpha)$, the\nmap $a \\colon {\\CM_1} \\to {\\CM_2}$ containing all the information about our\ngauge fields. Likewise, we now could attempt\nto cast gauge transformations into the form\n$\\delta_\\varepsilon A^\\alpha = (\\delta_\\varepsilon a)^* (q^\\alpha)$ with $(\\delta_\\varepsilon\na)^* = a^* \\circ  V_\\varepsilon$  for some degree preserving vector field $V_\\varepsilon$ on\n${\\CM_2}$ (parameterized somehow by $\\varepsilon$). Observing moreover that $\\varepsilon^\\alpha =\n(0,\\varepsilon^a,\\mu^D)$ naturally carries an upper index of ${\\CM_2}$, it is natural\nto regard these coefficients as components of a vector field\non ${\\CM_2}$, $\\varepsilon := \\varepsilon^\\alpha \\partial_\\alpha$. There are two obvious problems\nwith this ansatz: First, $\\varepsilon$ should depend on $\\sigma$ not just via the\nmap $X$ and, second, in this manner we would not be able to have $\\ud = Q_1$ enter respective formulas. \n\nBoth issues are cured simultaneously by the following construction\n\\cite{BKS,KS07}: we can trivially extend the map $a$ from before\nto a map \\beq a \\colon {\\CM_1} \\to\n\\CM := {\\CM_1} \\times {\\CM_2} \\, , \\label{aext}\n\\eeq\nwhich, by definition, is the identity on the\nfirst factor. For notational simplicity we do not introduce a new\nsymbol for this extended map, henceforth $a$ always corresponds to\nthis extended map; as before $A^\\alpha = a^* (q^\\alpha)$, since (local)\nfunctions $q^\\alpha$ on ${\\CM_2}$ can be regarded also as functions on $\\CM$\n(together with coordinate functions $\\sigma^\\mu$ on ${\\CM_1}$ they provide\ncoordinate functions on the product supermanifold $\\CM$). Obviously,\n$\\CM$ naturally becomes a Q-manifold itself; just add the two nilpotent\nvector fields to obtain\n\\beq Q = Q_1 + Q_2 \\, . \\eeq\nThis is again a nilpotent degree one vector field since by definition\n$Q_1$ and $Q_2$ supercommute (i.e.~they anticommute). In other words,\ngiven the two {complexes} $(C^\\infty({\\CM_1}), Q_1)$ and  $(C^\\infty({\\CM_2}),\nQ_2)$, one forms a double complex in the standard way,  \n$C^\\infty({\\CM_1}) \\otimes C^\\infty({\\CM_2}) \\cong C^\\infty(\\CM)$. In this\nextended setting, the operator $\\CF$ of\nEquation~\\eqref{CF} is replaced by\n\\beq \\CF := \\Qo \\circ a^* - a^* \\circ Q \\qquad \\Rightarrow \\qquad \\CF (q^\\alpha) \n\\equiv F^\\alpha \\, , \\label{CFext}\\end{equation}\nsince essentially all remains the same only that the target\n$({\\CM_2},{Q_2})$ is replaced by $(\\CM,Q)$---merely with an additional restriction on \n\\eqref{aext} that $a$ is the identity on the first factor.\\footnote{In other\nwords, $a$ is a section of the bundle $\\CM \\to {\\CM_1}$. Cf.~\\cite{KS07}\nfor the obvious generalization to nontrivial (total) bundles, which,\ndespite its simplicity,\nmay be seen to generalize the notion of an ordinary\nconnection in a nontrivial principal bundle (while certainly\nin this case a connection is not just\na globally defined Lie algebra valued 1-form on the base of that bundle).} Note that\nthis extended $\\CF$ vanishes identically on functions coming from\n${\\CM_1}$ (i.e.~constant along ${\\CM_2}$). \n\n\nThe space of vector fields $\\X_\\bullet(\\CM)= \\Gamma(T\\CM)$ of every Q-manifold $\\CM$ carries canonically the structure of a complex. If $\\CM$ is a supermanifold of degree $p$\n(cf.~Section~\\ref{s:examQ}), the space $\\X_\\bullet(\\CM)$ has also elements of\nnegative degree, bounded from below by $-p$. The differential is\nprovided by the graded commutator with $Q$, $\\ud_Q \\colon\nV \\mapsto [Q,V]$, since the bracket\n$[ \\cdot , \\cdot ]$ maps derivations of $C^\\infty(\\CM)$ into itself,\n$\\ud_Q \\equiv [Q,\\cdot]$ is of degree one since $Q$ is, and\n$(\\ud_Q)^2=0$ due to $Q^2 = \\frac{1}{2} [Q,Q]=0$ as one shows easily by means of\nthe (graded) Jacobi identity of the bracket.\n\nFor the purpose of gauge symmetries, we will be interested in a\nsubcomplex $({\\X_\\bullet^{\\mathrm{vert}}(\\CM)},\\ud_Q)\\subset ({\\X_\\bullet(\\CM)},\\ud_Q)$,  spanned by ``vertical''\nvector fields, i.e.~vector fields parallel to ${\\CM_2}$.\\footnote{Here we made\nuse of the fact that $\\ud_{Q_1}$ maps ${\\X_\\bullet^{\\mathrm{vert}}(\\CM)}$ into itself: $\\epsilon =\\epsilon^\\alpha\n\\partial_\\alpha \\in {\\X_\\bullet^{\\mathrm{vert}}(\\CM)}\n\\Rightarrow [Q_1,\\epsilon] = Q_1(\\epsilon^\\alpha)\n\\partial_\\alpha \\in {\\X_\\bullet^{\\mathrm{vert}}(\\CM)}$.}\n This complex is\nisomorphic to $(C^\\infty({\\CM_1}),Q_1) \\otimes (\\X_\\bullet({\\CM_2}),\\ud_{Q_2})$,\nas one verifies easily (cf.~also Eq.~\\eqref{complex}  below for\nmore details). In this form we recognize that both of the potential problems of the ansatz \nof symmetries generated by vector fields are cured at the same time: while still dealing essentially with vector fields on the target, the coefficient functions now depend also on ${\\CM_1}$ and there is $Q_1\\cong \\ud$ in the game as well. \n\nLet $V \\in \\X_0^{\\mathrm{vert}}(\\CM)$, a vertical vector field of degree zero,\nand set $\\delta A^\\alpha = (a^* \\circ V) (q^\\alpha)$. Together with\n\\eqref{CFext} this implies\n\\beq \\delta F^\\alpha = ( \\Qo \\circ a^*\\circ V - a^*\\circ V \\circ Q)(q^\\alpha)\n\\equiv \\CF ( V(q^\\alpha)) + (a^* \\circ \\ud_Q(V))(q^\\alpha) \\, . \\label{Ftrafo1}\n\\eeq\nThus if $\\dim \\Sigma \\ge p+1$, $\\delta \\I \\subset \\I$ is tantamount \nto requiring $V$ to be $\\ud_Q$-closed. The solution found before in\nTheorem \\ref{theo:gauge1} corresponds to an exact $V$, i.e.~$V=\\ud_Q\n\\epsilon$, where $\\epsilon$ is\na degree minus one vertical vector field,\n$\\epsilon \\in \\X_{-1}^{\\mathrm{vert}}(\\CM)$. We will now make this more explicit (and thereafter\nadd a remark about the $\\ud_Q$-cohomology). \n\n\nThe degree $k$ elements of\n$C^\\infty({\\CM_1})$ may be identified with $\\Omega^k(\\Sigma)$ and $Q_1$ is the\nde Rham differential $\\ud$ on $\\Sigma$. Correspondingly,\n\\begin{align}\n (\\X_{-1}^{\\mathrm{vert}}(\\CM),\\ud_Q)\n\\cong & \\; (\\Omega^0(\\Sigma),\\ud) \\otimes (\\X_{-1}({\\CM_2}),\\ud_{Q_2})\n  \\nonumber \\\\ &  \\oplus\n (\\Omega^1(\\Sigma),\\ud) \\otimes\n(\\X_{-2}({\\CM_2}),\\ud_{Q_2})  \\; \\oplus \\ldots \\nonumber \\\\\n  &  \\oplus  (\\Omega^{p-1}(\\Sigma),\\ud) \\otimes\n(\\X_{-p}({\\CM_2}),\\ud_{Q_2})\n  \\, .  \\label{complex}\n\\end{align}\nThus, for $p=2$, the most general $\\epsilon \\in \\X_{-1}^{\\mathrm{vert}}(\\CM)$ takes the form\n\\beq \\epsilon = \\epsilon^a \\frac{\\partial}{\\partial \\xi^a} + \\epsilon^D\n\\frac{\\partial}{\\partial b^D} \n\\eeq\nwith $\\epsilon^a \\in C^\\infty(\\Sigma) \\otimes C^\\infty(M)\\cong\nC^\\infty(\\Sigma\\times M)$ and  $\\epsilon^D = (\\epsilon^D)_{\\mathrm{main}} + \\xi^a\n\\epsilon^D_a$, where $(\\epsilon^D)_{\\mathrm{main}}\n\\in \\Omega^1(\\Sigma) \\otimes C^\\infty(M)$ and $\\epsilon^D_a \\in\nC^\\infty(\\Sigma\\times M)$. Here we used that the degree zero\npart of $C^\\infty({\\CM_2})$ is $C^\\infty(M)$ and that $\\xi^a\n\\frac{\\partial}{\\partial b^D} \\in \\X_{-1}({\\CM_2})$; \neverything is understood as\ndefined only locally over the target base $M$ certainly. We now\nintroduce\n\\beq \\varepsilon^a := a^* \\epsilon^a \\; , \\quad  \\mu^D \\equiv \\varepsilon^D:= a^* \\epsilon^D \\, ;\n\\eeq\nthus obviously, $\\varepsilon^a \\in  C^\\infty(\\Sigma)$ and $\\mu^D \\in\n\\Omega^1(\\Sigma)$.\\footnote{Note: $\\varepsilon^a(\\sigma) = \\epsilon^a(\\sigma,X(\\sigma))$---it was,\namong others, this extra \\emph{explicit}\n$\\sigma$-dependence that made it necessary to extend the target as\nin \\eqref{aext}: For a fixed choice of $X \\colon \\Sigma \\to M$ that is\nan embedding this would not have been necessary, but, on the other\nextreme side---which, however, is \\emph{always} realized in ordinary YM\ngauge theories---if the\nimage of $X$ is just a point in $M$, one now\nstill has the possibility for a nontrivial\n$\\sigma$-dependence of $\\varepsilon^a$. The situation with $\\mu^D$ is similar, but\na bit more involved: The coefficient $\\epsilon^D_a$ corresponds to a part\nof $\\mu^D$ that is not only $X$, but also $A^a$--dependent. We mostly\nwill consider only the case with $\\epsilon^D_a=0$.}\nNow, to prove the above claim, we just\nperform the computation:\n\\begin{align}\n\\left(a^* \\circ \\ud_Q(\\epsilon)\\right)(q^\\alpha)& =\n  a^* \\left(\\epsilon(Q_2^\\alpha)\\right)  + \\left(a^* \\circ Q\\right) (\\epsilon^\\alpha) \n  \\nonumber \\\\ \n& =  a^*(\\epsilon^\\beta\\partial_\\beta Q_2^\\alpha) - \\CF (\\epsilon^\\alpha) + \\left(Q_1 \\circ\na^*\\right)  (\\epsilon^\\alpha) \\nonumber \\\\\n& = \\ud \\varepsilon^\\alpha + \\varepsilon^\\beta \\, a^*(\\partial_\\beta Q_2^\\alpha) - F^\\beta \\, a^*\n( \\partial_\\beta \\epsilon^\\alpha ) \\, .\n  \\end{align}\nComparison with Eqs.~\\eqref{gauge0}, \\eqref{consistency}, and \n\\eqref{g1b}--\\eqref{g3b} shows that indeed\n\\beq \\boxed{\\delta_\\varepsilon A^\\alpha = (a^* \\circ [Q,\\epsilon] )\\, (q^\\alpha)}\n\\label{gaugebox} \\eeq\nwith particular coefficients $\\lambda^\\alpha_\\beta$ in\n\\eqref{g1b}--\\eqref{g3b}, namely\n\\beq \\lambda^\\alpha_\\beta = \n-a^*(\\partial_\\beta \\epsilon^\\alpha)\\, , \\label{lambdas1} \\eeq\ni.e., in\ndetail, $\\lambda^a_i = \n-a^*(\\partial_i \\epsilon^a)$, $\\lambda^D_i = \n-a^*(\\partial_i \\epsilon^D) \\equiv -\na^*\\left(\\partial_i (\\epsilon^D)_{\\mathrm{main}}\\right) -\n A^a a^*(\\partial_i \\epsilon^D_a)$, and $\\lambda^D_a = \n-a^*(\\epsilon^D_a)$. For the particular case of prime interest that\n$\\epsilon^D =(\\epsilon^D)_{\\mathrm{main}}$, this simplifies to\n\\beq \\lambda^a_i = \n-a^*(\\partial_i \\epsilon^a) \\, , \\quad  \\lambda^D_i = \n-a^*(\\partial_i \\epsilon^D) \\, , \\quad \\lambda^D_a =0 \\, ,   \\label{lambdas2}\n\\eeq\nwhere $\\lambda^D_i$ depends on the choice of $X$ but not also on\n$A^a$.\n\nNote that, given a fixed map $a$, i.e.~a\nfixed choice of $A^\\alpha$, we can always adapt for an arbitrary choice of\n$\\varepsilon^\\alpha$ \\emph{and}  $\\lambda^\\alpha_\\beta$ in \\eqref{g1b}--\\eqref{g3b}\nby an appropriate choice of\n$\\epsilon^\\alpha$. On the other hand, from the point of view of\nEq.~\\eqref{gaugebox}, gauge transformations are governed by\n``vertical vector fields'' on the target, which in turn determine, for\n\\emph{every} independent choice of $A^\\alpha$, both $\\varepsilon^\\alpha$ and\n$\\lambda^\\alpha_\\beta$. \n\nWe use the opportunity for a side remark about \\emph{one} alternative covariant\nchoice of $\\lambda$s in  \\eqref{g1b}--\\eqref{g3b}. At the\nexpense of introducing a connection in both  $E$ and $V$, say those\nused to define the field strengths \\eqref{FaG} and \\eqref{FBG},\none may choose\n\\beq \\lambda^a_i=\\varepsilon^b\\Gamma^a_{ib} \\qquad\n\\lambda^D_i=\\mu^C\\Gamma^D_{iC} \\label{lambdas3}\n\\eeq \nbesides $\\lambda^D_a =0$. This  has the advantage that now already $\\varepsilon^\\alpha \\equiv a^*(\\epsilon^\\alpha)$\ndetermines $\\lambda^\\alpha_\\beta$ uniquely (in \\eqref{lambdas1} also the first germ of $\\epsilon^\\alpha$ at the section $a$ is needed). To see that this choice is indeed\ncovariant, one can argue by comparison with Eqs.~\\eqref{lambdas1},\n\\eqref{lambdas2}: By construction, Eq.~\\eqref{gaugebox}\ngives a covariant gauge transformation---note that here covariance\nmeans that $\\delta_\\varepsilon$ is to act as a derivation, but the r.h.s.~of\n\\eqref{gaugebox} obviously does so since $[Q,\\varepsilon]$ is a\nvector field.\\footnote{Cf.~also \\cite{BKS} for a  more detailed\ndiscussion of this issue.} In detail this implies that when we change\nlocal frames in $E$ and $V$ according to \\(\\tilde{\\xi^a}=M^a_b \\xi^b\\),\n\\(\\tilde{b^D}=N^D_C b^C\\), every transformation of $\\lambda$s such as\nthose of Eqs.~\\eqref{lambdas1}, \\eqref{lambdas2}, namely\n\\begin{align}  \\widetilde{\\lambda^a_i} &=\n    M^{\\tilde a}_b \\lambda^b_i - M^{\\tilde a}_{b,i}\\varepsilon^b\n  \\label{lambdatrafo1} \\\\\n  \\widetilde{\\lambda^D_a} &=\n    N^{\\tilde D}_F \\lambda^F_c\n   (M^{-1})^c_{\\tilde a}  \\label{lambdatrafo2} \\\\\n  \\widetilde{\\lambda^D_i} &=\n    N^{\\tilde D}_C\\lambda^C_i - N^{\\tilde D}_{C,i}\\mu^C\n   \n    \\label{lambdatrafo3}\n  \\, ,\n\\end{align}\ngive a covariant gauge transformation in \n\\eqref{g1b}--\\eqref{g3b}.\n(We suppressed writing pullback by $X$ here: E.g., $M^{\\tilde a}_b$ in\n\\eqref{lambdatrafo1} is rather $X^*(M^{\\tilde a}_b) \\equiv M^{tilde a}_b(X)$ and likewise\n$M^{\\tilde a}_{b,i}$ denotes $X^*(\\partial_i M^{\\tilde a}_b)$.)\nThe above choice of $\\lambda$s using background connections\nobviously fulfills the transformation requirements\n\\eqref{lambdatrafo1}--\\eqref{lambdatrafo3}. Note that the\nmethod of obtaining these transformation properties is\nincomparably simpler than an elementary one where, in a first step, one would \ncompute all changes in \\eqref{g1a}--\\eqref{g3a} using \n\\eqref{trafo1}--\\eqref{trafo5}. \n\nIn the considerations around Eq.~\\eqref{Ftrafo1} we obtained only that\n$V$ needs to be $\\ud_Q$-closed, but not necessarily $\\ud_Q$-exact. This\nreconciles with the result of Theorem \\ref{theo:gauge1} since \nunderlying \\eqref{gauge} is a particular form of\ngauge transformations, namely the one specified in\nEqs.~\\eqref{gsymm0}---starting with exact forms on $\\Sigma$.\nThis already gives the restriction\nto trivial elements of the otherwise nontrivial cohomology of\n$\\ud_Q$. It may be interesting to relax this condition, however.\n\nNote that the $\\ud_Q$--cohomology on vertical vector fields\ncan be computed by means of an algebraic\nversion of the K\\\"unneth formula, in detail:  $H_{\\ud_Q}^0(\\CM) =\n\\bigoplus_{k=0}^2 \\left(H_{\\textrm{deRham}}^k(\\Sigma) \\otimes\nH_{\\ud_{Q_2}}^{-k}({\\CM_2})\\right)$. E.g.~a change of coordinates on the\nsupermanifold ${\\CM_2}$ induces a symmetry only if it leaves the Q-structure invariant, i.e.~if its (degree zero) generator $V=V^\\alpha \\partial_\\alpha$ commutes with $Q_2$. \nThus such a generator corresponds to an element with $k=0$\nin the previous sum. \n\n\n\\vskip\\newpassage\n\nWe now come to the announced application of the above framework for\ngauge transformations. As a warm-up exercise we return to\nEq.~\\eqref{Ftrafo1} and use it to\ncompute the explicit form of the change of field strengths $F^\\alpha$\nunder the generic part $\\delta^{(0)}_\\varepsilon$ of gauge transformations of\nour gauge fields. Needless to say that again every explicit\ncalculation would be very lengthy, making heavy use of\n\\eqref{B1}--\\eqref{B7}. We first observe that locally and for every\nfixed choice of $(\\varepsilon^a,\\mu^D)$ we can choose frames in $E$ and $V$\nsuch that $\\epsilon^a$ and $\\epsilon^D$ are constant along ${\\CM_2}$---correspondingly,\nin \\emph{this} particular frame, $\\delta_\\varepsilon A^\\alpha = \\delta^{(0)}_\\varepsilon\nA^\\alpha$ and we may make use of \\eqref{Ftrafo1} with $V=\\ud_{Q}\\epsilon$:\\footnote{Alternatively, here we may equally well\nuse \\eqref{Fchangeold} together with Eqs.~\\eqref{consistency}.}\n\\begin{eqnarray}\n  \\delta^{(0)}_\\varepsilon F^i &=& \\varepsilon^a \\, \\rho^i_{a,j} \\, F^j \\, , \\label{deltaFi} \\\\\n  \\delta^{(0)}_\\varepsilon F^a &=&-C^a_{bc}\\, \\varepsilon^b F^c + \\left(C^a_{cb,i}\\, \\varepsilon^bA^c\n  -t^a_{D,i}\\, \\mu^D \\right)F^i \\, , \\label{deltaFa} \\\\\n  \\delta^{(0)}_\\varepsilon F^D &=& -\\Gamma^D_{aC} \\, \\varepsilon^a F^C + \\left(\\Gamma^D_{aC}\n  \\, \\mu^C  - H^D_{abc} \\,\n  \\varepsilon^b A^c \\right)  F^a \\nonumber \\\\\n  && + \\left[\\tfrac{1}{2} H^D_{abc,i}\\, \\varepsilon^a A^bA^c -\\Gamma^D_{aC,i}\n  \\left(\\varepsilon^a B^C + \\mu^CA^a \\right) \\right] F^i \\, . \\label{deltaFB}\n\\end{eqnarray}\nFor example, the first line was obtained as follows:\n$\\delta_\\varepsilon F^i=\\CF([Q,\\epsilon]x^i)=\\CF(\\rho^i_a\\epsilon^a) =\nX^*\\left(\\partial_j(\\rho^i_a\\epsilon^a)\\right) F^j$,\nwhere in the last step we  used \\eqref{FLeibniz}. \n\n\n\n  \n\n\nWe now come to the commutator of gauge transformations. For this, we first make\nthe following simple but important observation\n\\begin{thm} \\label{theo:Lie}\nInfinitesimal gauge transformations, written in form of\n\\eqref{gaugebox},\nform a Lie algebra isomorphic to $(\\X_{-1}^{\\mathrm{vert}}(\\CM)\/\\ker\\ud_Q,[\\cdot ,\\cdot]_Q)$,\nwith\nthe derived bracket:\n\\beq [\\epsilon,\\bar \\epsilon]_Q = [\\ud_Q(\\epsilon),\\bar \\epsilon] \\equiv  [[\\epsilon,Q],\\bar\n\\epsilon] \\, . \\label{induced}\n\\eeq \n\\end{thm}\nTo prove this we observe that infinitesimal gauge transformations\n\\eqref{gaugebox} are generated by  degree minus one vertical\nvector fields \\emph{modulo} elements that are annihilated by\n$\\ud_Q \\equiv [Q, \\cdot]$. Such elements are called \\emph{coexact} in\n\\cite{YKS96}, in generalization of coexact differential forms. \nIt is now easy to convince oneself that the commutator of two gauge\ntransformations \\eqref{gaugebox} gives rise to a new bracket\nbetween vector fields:\n\\beq \n [\\delta_\\epsilon , \\delta_{\\bar \\epsilon} ] A^\\alpha = \\left(a^* \\circ\n [\\ud_Q(\\epsilon), \\ud_Q(\\bar \\epsilon)]\\right) (q^\\alpha) =\n \\left( a^* \\circ \\ud_Q ( [\\ud_Q(\\epsilon),\\bar \\epsilon])\\right) (q^\\alpha) \\, ,\n\\label{comm}\n \\eeq\nwhere we made use of the (graded) Leibniz property of $\\ud_Q$ \nw.r.t.~$[ \\cdot , \\cdot ]$---noting that\n$\\ud_Q(\\epsilon)$ is of degree zero---and of \n$(\\ud_Q)^2=0$. This proves \\eqref{induced}. The induced or\n\\emph{derived} bracket\n$[\\epsilon,\\bar \\epsilon]_Q = [\\ud_Q(\\epsilon),\\bar \\epsilon]$ on $\\X_{-1}^{\\mathrm{vert}}(\\CM)$ is \\emph{not}\na Lie bracket; however, as a result of a general construction,\nrecalled in the subsequent section, on the\nquotient by the kernel of the differential $\\ud_Q$ it is,\ncf.~Proposition \\ref{p:YKSder} below.\n\nIt is possible to characterize Q2-manifolds \nalso by means of its derived bracket---as an alternative to the\ndescription found in Section~\\ref{s:examQ}. Since\nwe were now naturally led to this bracket, and since this formulation\nis well-suited also for comparison with the usual formulation of what\nis a Courant algebroid, we will reanalyze\nQ2-manifolds from this perspective in the subsequent section.\n\n\nThe above theorem\nimplies in particular that if gauge transformations are\ninterpreted---in generalization\nof \\cite{BKS}---in the sense advocated in the present section, they close\noff-shell---and this also in theories which in the \\emph{usual}\nterminology have an\n\\emph{open algebra} of gauge transformations. In fact,\nas a kind of corollary of Theorem \\ref{theo:Lie}, we will now verify\n\\begin{prop}  For the\ngauge transformations \\eqref{g1a}--\\eqref{g3a} one finds\n\\begin{eqnarray} [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] X^i &=&\n\\delta^{(0)}_{\\widetilde \\varepsilon}\nX^i  \\label{gcom1}\\\\{}\n  [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^a &=&\n  \\delta^{(0)}_{\\widetilde \\varepsilon}\n \n  A^a -C^a_{bc,i} \\varepsilon^b\n    \\bar \\varepsilon^{c} F^i  \\label{gcom2}\\\\{}\n  [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] B^D &=&  \n  \\delta^{(0)}_{\\widetilde \\varepsilon}\n\n B^D \n     +H^D_{abc} \\varepsilon^b  \\bar \\varepsilon^{c} F^a \\nonumber \\\\ \n  & & +\\,\\Gamma^D_{aC,i}\n     (\\varepsilon^a\\bar \\mu^{C}-\\bar \\varepsilon^{a}\\mu^C) F^i\n  - H^D_{abc,i} \\varepsilon^{a}\\bar\\varepsilon^b A^c F^i  \\label{gcom3}\n\\end{eqnarray}\n   \nwhere $\\,{\\widetilde \\varepsilon}^a\n\\equiv C^a_{bc}   \\varepsilon^b \\bar \\varepsilon^c$\nand  \n \n \n \n   \n    $\\,{\\widetilde \\varepsilon}^D\n   \n    \\equiv  \\Gamma^D_{aC}(\n \\varepsilon^a \\bar \\varepsilon^{C}-\\bar \\varepsilon^{a}\\varepsilon^C ) \n -  H^D_{abc}\\varepsilon^a\\bar \\varepsilon^{ b}\n  A^c \\, .  $\n\\label{prop:onshellclosed}  \n\\end{prop}\nThe idea is simple: As when computing the variation of\nfield strengths w.r.t.~$\\delta^{(0)}_\\varepsilon$, for a given choice of \n$\\varepsilon^\\alpha \\in C^\\infty({\\CM_1})$ \nand given a particular coordinate system $q^\\alpha$ on ${\\CM_2}$, we may\ndefine $\\epsilon := \\varepsilon^\\alpha \\partial_\\alpha$, which now is a vertical vector\nfield. In that coordinate system, we then have\n$\\delta^{(0)}_\\varepsilon A^\\alpha = \\delta_\\epsilon A^\\alpha$ and likewise so for the\nbarred variations.\\footnote{Here we\nwrote  $\\delta_\\epsilon A^\\alpha$ instead of  $\\delta_\\varepsilon A^\\alpha$\nas in \\eqref{gaugebox} or \\eqref{g1b}--\\eqref{g3b}, so as to\nmake clear that the gauge transformation \\eqref{gaugebox} is\nparameterized not only by $\\varepsilon^\\alpha(\\sigma)=\\epsilon^\\alpha(\\sigma,A^\\beta(\\sigma))$, but also by the\nfirst derivative of $\\epsilon^\\alpha$\nw.r.t.~$q^\\beta$---cf.~Eq.~\\eqref{lambdas1}. This is in contrast to the\nsituation when choosing \\eqref{lambdas3} in\n\\eqref{g1b}--\\eqref{g3b}.} \nThus also for\nthe commutators we have $[\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha\n= [\\delta_\\epsilon,\\delta_{\\bar \\epsilon}] A^\\alpha$. Now we can apply\n\\eqref{comm} and Theorem \\ref{theo:Lie}. This contains the ``heavy\ncalculation'', so to say (which, in the present formulation, is almost a\ntriviality). We are left with splitting  $\\delta_{[\\epsilon,\\bar \\epsilon]_Q}\nA^\\alpha$ into the part $\\delta^{(0)}_{[\\epsilon,\\bar \\epsilon]_Q} A^\\alpha$ and, using\n\\eqref{lambdas1}, the remaining \nterms containing $F^\\alpha$s according to \\eqref{g2b} and \\eqref{g3b}.\nThe main point is that even if $\\epsilon$ and $\\bar \\epsilon$ are constant along\n${\\CM_2}$, their induced bracket is not! Let us be explicit about this:\n\\beq [ [\\varepsilon^\\alpha \\partial_\\alpha , Q_1 + Q_2 ], \\bar \\varepsilon^\\beta \\partial_\\beta ] =\n\\bar \\varepsilon^\\beta \\varepsilon^\\alpha  [[\\partial_\\alpha ,Q_2 ],\\partial_\\beta ]\n\\eeq\nsince $\\varepsilon^\\alpha$ and $\\bar \\varepsilon^\\beta$ are constant along ${\\CM_2}$ and\n$[Q_1(\\varepsilon^\\alpha) \\partial_\\alpha, \\bar \\varepsilon^\\beta \\partial_\\beta ] =0$. Thus we need to\ncompute $[\\partial_\\alpha,\\partial_\\beta ]_{{Q_2}}$, which is easily found to\nequal\n$(-1)^{|\\beta|+1}(\\partial_\\alpha \\partial_\\beta {Q_2}^\\gamma) \\partial_\\gamma$. So,\n\\begin{eqnarray}\n\\widetilde \\epsilon := [\\varepsilon^\\alpha \\partial_\\alpha,  \\bar \\varepsilon^\\beta \\partial_\\beta ]_Q \n  &=& -(-1)^{|\\beta|}\\bar \\varepsilon^\\beta \\varepsilon^\\alpha\n(\\partial_\\alpha \\partial_\\beta {Q_2}^\\gamma) \\, \\partial_\\gamma\n\\label{derivedsymm} \\\\\n&\\equiv&   \\varepsilon^a \\bar \\varepsilon^b C^c_{ab} \\partial_{c}\n\\,+ (\\varepsilon^a \\bar \\mu^D- \\bar \\varepsilon^a \\mu^D ) \\, \\Gamma^C_{aD}\n \\partial_{C} \n \\, - \\varepsilon^a\\bar \\varepsilon^{ b}H^D_{abc}\n  \\xi^c \\partial_{D}\n \\, , \\nonumber\n \\end{eqnarray}\nwhere we used the general form \\eqref{Q} for ${Q_2}$ and $\\varepsilon^D \\equiv\n\\mu^D$. Despite the fact that $\\epsilon$ and  $\\bar \\epsilon$\nwere vertical vector fields ``constant along ${\\CM_2}$'' (in the chosen\nframe $\\partial_\\alpha$), the vector field $\\widetilde \\epsilon = \\widetilde\n\\epsilon^\\alpha \\partial_\\alpha$ has coefficients which \\emph{do} depend\nnon-trivially on ${\\CM_2}$. By comparison of coefficients it is now easy to\nconclude the above statement from\n\\beq [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}] A^\\alpha\n= [\\delta_\\epsilon,\\delta_{\\bar \\epsilon}] A^\\alpha = \\delta_{\\widetilde \\epsilon} A^\\alpha =\n\\delta^{(0)}_{\\widetilde \\varepsilon} A^\\alpha + F^\\beta \\widetilde \\lambda_\\beta^\\alpha\n\\, , \n\\eeq\nwith $ \\widetilde \\varepsilon^\\alpha = a^* \\widetilde \\epsilon^\\alpha$\nand $\\widetilde \\lambda_\\beta^\\alpha = - a^*(\\partial_\\beta \\widetilde\n\\epsilon^\\alpha)$.\nIt is seen also that the nontrivial\n$\\xi^a$--dependence of $\\widetilde \\epsilon^D$ is responsible for both, the\n$A^a$-dependence of $\\widetilde \\varepsilon^D$ and the $F^a$-contribution to \n\\eqref{gcom3}. \n\nWe conclude this section by illustrating how the previous considerations can be  used to also compute the commutator of symmetries of the form  \n\\begin{align}  \\delta^{}_\\varepsilon X^i &= \\rho^i_a  \\varepsilon^a \\, ,  \\label{g1an}\\\\\n  \\delta_\\varepsilon^{} A^a &= \\ud\\varepsilon^a + C^a_{bc}A^b \\varepsilon^c +t^a_D \\varepsilon^D +\\Gamma^a_{ib}\\varepsilon^bF^i  \\label{g2bn} \\\\\n  \\delta_\\varepsilon B^D  &= \\ud\\varepsilon^D -\\Gamma^D_{aC} \\varepsilon^a B^C +\\Gamma^D_{aC}\n    A^a \\wedge \\varepsilon^C +\\frac12 H^D_{abc} A^a \\wedge A^b \\varepsilon^c  -  \\Gamma^D_{iC}  \\epsilon^C \\wedge F^i \\,, \\label{g3bn}\n\\end{align}\ninduced by (ordinary, ``non-dynamical'') connections on the (target) bundles $E$ and $V$\nand which result from implementing \\eqref{lambdas3} (together with $\\lambda_a^D=0$) into the general symmetries \\eqref{g1b}--\\eqref{g3b}. The symmetries on the 0-forms $X^i$ coincide with the ones in \\eqref{g1a} and we thus can take recourse to \\eqref{gcom1} for the corresponding commutator. Changes occur when calculating the commutator on the 1-form fields $A^a$ and the 2-form fields $B^D$, where it is useful to split the infinitesimal symmetries according to $\\delta_\\varepsilon= \\delta^{(0)}_\\varepsilon+ \\delta^{(1)}_\\varepsilon$, where $\\delta^{(0)}_\\varepsilon$ is given by \\eqref{g2a} and \\eqref{g3a}. In the following we restrict ourselves to the 1-form fields, leaving the rest as an exercise to the reader. Evidently \\begin{equation}\n\\delta^{(1)}_{\\varepsilon} A^a = \\Gamma^a_{ib}\\varepsilon^b F^i \\, .\n\\end{equation}\nMoreover, \n\\begin{equation}\n[\\delta^{}_\\varepsilon,\\delta^{}_{\\bar \\varepsilon}] A^a \n= [\\delta^{(0)}_\\varepsilon,\\delta^{(0)}_{\\bar \\varepsilon}]A^a \n + [\\delta^{(1)}_\\varepsilon,\\delta^{(1)}_{\\bar \\varepsilon}]A^a \n + \\left(\\delta^{(0)}_\\varepsilon \\delta^{(1)}_{\\bar \\varepsilon}A^a \n    +\\delta^{(1)}_\\varepsilon \\delta^{(0)}_{\\bar \\varepsilon}A^a  \n\t- (\\varepsilon \\leftrightarrow \\bar{\\varepsilon})\\right)\\, .\n\\end{equation}\nThe first commutator is given by \\eqref{gcom2}. For the second commutator, one observes    $\\delta^{(1)}X^i=0$ such that  $\\delta_{\\varepsilon}^{(1)}\\left(\\Gamma^a_{ib}\\bar{\\varepsilon}^b F^i\\right)=\\Gamma^a_{ib}\\bar{\\varepsilon}^b\\delta_{\\varepsilon}^{(1)}\\!F^i= \n-\\Gamma^a_{ib}\\bar{\\varepsilon}^b\\rho^i_c \\Gamma^c_{jd}\\varepsilon^d F^j$. Since we also know how $F^i$ transforms with respect to $\\delta^{(0)}$, cf.~Eq.~\\eqref{deltaFi}, we are now left with merely collecting all terms so as to arrive at\\footnote{There were some later developments about such commutators in the mean time. First, in \\cite{Str09} the terms violating closeness were given a geometric interpretation in the case that $E$ is a Lie algebroid: An ordinary connection on $E$ induces an $E$-connection generalizing the adjoint representation of a Lie algebra \\cite{Str04b}; its $E$-curvature factorizes over the anchor map  $\\rho$,  ${{}^E\\!{}R}^a{}_{bcd}=\\rho_b^i \\, S^a{}_{icd}$. With this one obtains $ [\\delta_\\epsilon,\\delta_{\\bar{\\epsilon}}]A^a = \\delta_{\\widetilde{\\epsilon}}^{}A^a+S^a_{ibc}\\epsilon^b\\bar{\\epsilon}^{c}F^i$.\nSecond, the methods used here were extended so as to perform calculations about such commutators more directly, replacing $a^*$ by $f^*$---cf., e.g., \\cite{SaS13,SLS14}.}\n\\begin{align}\n [\\delta_\\epsilon,\\delta_{\\bar{\\epsilon}}]A^a =& \\delta_{\\widetilde{\\epsilon}}^{}A^a\n -(C^a_{bc,i}+\\Gamma_{id}^aC^d_{bc} ) \\epsilon^b\\bar{\\epsilon}^{c}F^i\n  \\\\\n & +\\left(C^a_{db}\\Gamma^d_{ic} -\\Gamma^a_{jb}\\rho^j_{c,i}-\\Gamma^a_{ib,j}\\rho^j_c\n +\\Gamma^a_{jb}\\rho^j_d\\Gamma^d_{ic} \\right)( \\epsilon^b\\bar{\\epsilon}^{c}-\\bar{\\epsilon}^{b}\\epsilon^c)F^i , \\nonumber\n\\end{align}\nwhere the new parameter $\\widetilde{\\epsilon}$ is given by \nProposition \\ref{prop:onshellclosed}. Note that the part of the symmetries on the $A^a$ fields parameterized by the 1-forms $\\varepsilon^D\\equiv \\mu^D$, cf.~Eq.~\\eqref{g2bn}, do commute\nevidently. Still, these symmetries do not commute with those parameterized by the 0-forms $\\epsilon^a$. This explains why on the r.h.s.~the parameter $\\widetilde{\\epsilon}^D$ is non-vanishing in general---in fact, as we learn from Proposition \\ref{prop:onshellclosed}, I 1-form part of the symmetries is even induced if the initial two symmetries $\\epsilon$ and $\\bar{\\epsilon}$ have vanishing 1-form parts (like in a central extension).  \n\n\n\n\\newpage\n\n\\section{Derived bracket and extended Courant algebroids} \\label{sec:derived}\nIn Section~\\ref{s:examQ}\nwe studied Q-manifolds $\\CM$ in terms of ordinary differential\ngeometric notions by analyzing the components of the vector field $Q$\nin some chosen (graded) coordinate system, taking into account their\ntransformation properties. There is, however, also an alternative\nprocedure using the space of vector fields on $\\CM$ with its graded\ncommutator bracket as well as its derived one, as is inspired by the\nprevious section on gauge symmetries. While in the case of\nQ1-manifolds or Lie algebroids this provides an alternative route\nleading to an \\emph{evidently} equivalent picture, for Q2-manifolds\none obtains an alternative description of Lie 2-algebroids, closer to\nthe description of Courant algebroids (they will be a particular case\nof what we will call vector bundle twisted Courant algebroids), with a\nnontrivial translation of one picture into the other one,\ncf.~definition \\ref{def:VCourant} and, e.g., Theorem \\ref{thm:der}\nbelow (or, equivalently, Theorem \\ref{thm1.6} in the summary).\n\n\nAs mentioned, the bracket between gauge symmetries \\eqref{induced} is\na so called derived bracket \\cite{YKS03en}. In Subsection~\\ref{sec:Der}\nwe first recall some generic features of derived brackets, starting\nfrom a general differential graded Lie algebra (DGLA). We extend the\nconsiderations of the literature to the context of algebroids by\nintroducing an anchor map; vector fields on a Q$p$-manifold equipped\nwith the differential $\\ud_Q$ are an example of this construction for\narbitrary $p \\in \\N$. We show that applying this perspective to\n$p=1$ reproduces the respective considerations of section\n\\ref{s:examQ} (i.e.~it provides the usual definition of a Lie \nalgebroid) in a rather straightforward way. We also provide the\ndefinition of Courant algebroids in this subsection as another example\nof the derived bracket construction. \n\nInspired by the definition of Courant algebroids and the derived\nbracket construction applied to vector fields on Q$p$-manifolds for\narbitrary $p$ we are led to define Vinogradov algebroids and vector\nbundle twisted Courant algebroids in Subsection~\\ref{subsec:Vino}. The\nformer ones are also inspired by the Vinogradov bracket (which\ngeneralizes the Courant or Dorfman bracket to higher form degrees). \nIn vector bundle twisted\nCourant algebroids, which\nare special Vinogradov algebroids, the inner product in the definition of an\nordinary Courant algebroid is replaced by a bilinear form taking\nvalues in a second vector bundle $V$ over the same base. Although in part\ninspired by Q-manifolds, it is not clear that all Vinogradov and V-twisted\nCourant algebroids have a description in terms of Q-manifolds. \n\nIn Subsection~\\ref{subsec:Lie2} we show that particular V-twisted\nCourant algebroids are in one-to-one correspondence to (split)\nQ2-manifolds, and, consequently (cf.~Theorem \\ref{theo1}) to Lie\n2-algebroids. This equivalence, cf.~Theorem \\ref{thm:der} below, was\nsuggested by the changed perspective on Q-manifolds as stemming from\nthe gauge transformations in our context and would not be that obvious\nto invent without such a motivation. The subsection also contains a\nclassification of these type of V-twisted Courant algebroids, which is\nsomewhat analogous to the one of exact Courant algebroids.\n\n\n\\subsection{Derived brackets and anchored DGLAs}\n\\label{sec:Der}\nThe prototype of a derived bracket goes back to E.~Cartan. Consider the space of differential \nforms $\\Omega^\\cdot(M)$. Besides the de Rham operator $\\ud$ we can also consider contractions \n$\\imath_v$ with a vector field $v$ as an operator. The Lie derivative  $\\L_v$ \nthen results as the graded commutator $\\L_v = [\\imath_v, \\ud]$. On the other hand, certainly, \n$[\\L_v, \\imath_w] = \\imath_{\\L_v(w)}$, by the Leibniz rule of the Lie derivative. Using \n$\\L_v(w)=[v,w]$ and the above formula for the Lie derivative, we see that the bracket $[v,w]$ \nof two vector fields can be viewed as arising from the left hand side of the following \nequation\n\\beq[[\\imath_v,\\ud], \\imath_w] = \\imath_{[v,w]}\\,.\\label{derbasic}\\eeq\nNote the similarity of the left hand side with Eq.~\\eqref{induced}. In fact, the latter \nequation can be almost literally  translated into that setting: Consider the\nsupermanifold $T[1]M$ with the canonical Q-structure $\\ud$ on it. Every vertical vector\nfield on $T[1]M$ corresponds to some $\\imath_v$ for $v \\in \\Gamma(TM)$. It is a nontrivial\nbut easy-to-check fact that the result of the l.h.s.~of \\eqref{derbasic} is a vertical\nvector field again; therefore it comes from a vector field on $M$, defining a product\n$[v,w]$. \n\nThe above construction can be easily extended to include also exterior multiplication \nwith differential forms. Let $\\alpha,\\beta \\in \\Omega^1(M)$ and compute\\footnote{A.~Alekseev,\nprivate communication 1999.} \n\\beq [[\\imath_v+\\alpha\\wedge,\\,\\ud],\\,\\imath_w+\\beta \\wedge]\n  =\\imath_{[v+\\alpha,w+\\beta ]_W} \\, ; \\label{Anton} \\eeq\nIn this way one obtains the  (non-skewsymmetric) Courant, i.e.~the Dorfman bracket \\cite{Cour90,Dor87,Dor93},\n\\beq [v\\oplus \\alpha,w\\oplus \\beta]_W = [v,w]\\oplus \\L_v\\beta -\\imath_w\\ud\\alpha \\, . \\label{Cour}\n\\eeq\nAlso note that the mere bracket,\n$[\\imath_v+\\alpha\\wedge,\\imath_w+\\beta \\wedge]$ induces a natural\nbilinear, symmetric and non-degenerate pairing: when acting on\n$\\Omega^\\cdot(M)$ this operator reduces to multiplication with $\\alpha(w)\n+ \\beta(v)$. Viewing $v \\oplus \\alpha$ as a section of $W=TM\\oplus T^*M$, one\ncan axiomatize \\cite{Xu97,SevLett} rules that can be derived for these\noperations  so as to end up with the definition\n \n \\newpage\n\\begin{vdef} \\label{def:Cour2} A {\\rm Courant algebroid} is a vector bundle $W \\to M$ together with a bundle map\n$\\rho \\colon W \\to TM$, a pairing of its sections $[ \\cdot , \\cdot ]_W \\colon \\Gamma(W)\n\\otimes \\Gamma(W) \\to \\Gamma(W)$ as well as a non-degenerate fiber metric $\\< \\cdot, \\cdot\\>$\nsuch that the following three identities hold true \n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{cJacobi}  \\\\\n  [\\psi,\\psi]_W &= \\frac12\\uD\\<\\psi,\\psi\\>  \\label{cnSkew}\\\\\n  \\rho(\\psi)\\<\\phi_1,\\phi_2\\> &=\n    \\<[\\psi,\\phi_1]_W,\\phi_2\\> + \\<\\phi_1,[\\psi,\\phi_2]_W\\> \n\\, ,  \\label{cInvar}\n\\end{align}\nwhere $\\uD$ is induced by means of $\\rho$ and the inner product via $\\<\\uD f , \\phi\\> =\n\\rho(\\phi) f$.  \n \\end{vdef}\nThe first axiom is the (left-)Leibniz rule of the bracket with respect\nto itself. From \\eqref{cInvar} one can also conclude \\cite{LiB11} a\nLeibniz rule w.r.t.~multiplication of sections by functions\n\\begin{equation} [\\psi,f\\phi]_W = \\rho(\\psi)f\\phi +f[\\psi,\\phi]_W\n\\label{cLeibniz} \\, . \\end{equation}\nThe properties \\eqref{cJacobi} and \\eqref{cLeibniz} render  \n$(W,\\rho,[ \\cdot , \\cdot ]_W)$\nwhat we want to call a Leibniz--Loday algebroid (cf.~also \\cite{KS08})\\footnote{In fact, in \\cite{KS08} we called it Loday algebroid. We finally decided for the present paper to call an algebra with the property \\eqref{cJacobi} a Leibniz-Loday algebra and, if it is an algebra of sections and the condition \\eqref{cLeibniz} is satisfied, a Leibniz-Loday algebroid.}. \nThe second line brings in the fiber metric so as\nto control the violation of the antisymmetry of the bracket, while the last line is the\nad-invariance of the inner product.\nEqs.~\\eqref{cJacobi} and\n\\eqref{cLeibniz} permit to conclude that $\\rho$ is a morphism of\nbrackets, cf.~e.g.~the corresponding generalization in\nProp.~\\ref{p:Vinogrelem} below; together with \\eqref{cLeibniz} this\ngives the five axioms often demanded in the literature for the\ndefinition of a Courant algebroid. \n\nFor a general Courant algebroid one always has the complex\n\\beq 0 \\to T^*M \\to W \\to TM \\to 0 \\, , \\label{seq}\\eeq\ninduced by $\\rho$ and its adjoint. Exact Courant algebroids, which, by\ndefinition, means that the above sequence is exact, are classified by\nan element $[H] \\in H^3_{{\\mathrm{dR}}}(M)$ \\cite{SevLett}; up to a\ncontribution $H(v,w,\\cdot)$  added to the r.h.s.~of\n\\eqref{Cour}, this reproduces the explicit formulas on $W=TM\\oplus\nT^*M$ introduced above (with $\\rho$ being projection to the first\nfactor and a change of splitting in \\eqref{seq} corresponding to a\nchange $H \\mapsto H + \\ud B$).\n\nThe construction in \\eqref{Anton} can be translated easily into\nsuper-language. As before we may use $\\CM=T[1]M$ and consider\n$\\imath_v$ as a vertical vector field and $\\alpha \\wedge$ as\nmultiplication with a function of degree one. If we want to have\n$\\alpha$'s enter as vector fields as well, getting a model for what\nhappens in the previous section, we can simply extend $T[1]M$ by one\n(graded) copy of $\\Real$. Denoting this coordinate by $b$, and\nchoosing local $x^i$ on $M$ with its induced odd coordinates $\\xi^i =\n\\ud x^i$ on $T[1]M$, we then associate to every $\\psi = v \\oplus \\alpha$\nthe vector field\n\\begin{equation} \\psi = v^i \\frac{\\partial}{\\partial \\xi^i} + \\alpha_i \\xi^i  \\frac{\\partial}{\\partial b} \\, ; \\end{equation}\ndeclaring $b$ to have degree two, this becomes a homogeneous vector\nfield on the graded manifold $\\CM = T[1]M \\times \\Real [2]$ of degree\nminus one (and, moreover, it has the form of the most general vector\nfield of this degree). The bracket \\eqref{Cour} now follows as a\nderived bracket of (graded) vector fields on $\\CM$, $[ v \\oplus \\alpha , w\n\\oplus \\beta]_W = [\\psi,\\psi']_Q$ where the latter is defined by \n\\begin{equation} [ \\psi, \\psi']_Q := [[ \\psi , Q ], \\psi'] , \\label{Cour2} \\end{equation}\nwith $Q$ being the\ncanonical vector field on $T[1]M$ corresponding to the de Rham\ndifferential on $M$ extended trivially to the product and the bracket\ndenoting the graded Lie bracket of vector fields.\n\nObviously, this\ncan be ``twisted'' by a closed 3-form $H$ by replacing the above $Q$\nby \\begin{equation} Q = \\xi^i  \\frac{\\partial}{\\partial x^i} + \\frac{1}{3!} H_{ijk}\n\\xi^i \\xi^j \\xi^k  \\frac{\\partial}{\\partial b} \\label{Qtwist1} \\end{equation}\nwhich again yields a Q-structure on $\\CM$. This reproduces the general\nsituation of an exact Courant algebroid mentioned above. Note also\nthat in this language the inner product on $W=TM \\oplus T^*M$ results\nfrom the (graded) bracket of the corresponding vector fields on\n$\\CM$: \n\\begin{equation} [\\psi,\\psi'] = \\<\\psi,\\psi'\\>   \\frac{\\partial}{\\partial b} \\, .\n\\label{inner2}\\end{equation} \n\n\nTo have vector fields $v$ and 1-forms $\\alpha$ enter more symmetrically,\none may alternatively lift the graded construction on $\\CM = T[1]M$\ncanonically to the (super)symplectic manifold $T^*[2]\\CM$: The local\ncoordinates $x^i$ and $\\xi^i$ of degree 0 and 1 on $\\CM$ are then\naccompanied by canonically conjugate momenta $p_i$ and $\\xi_i$ of\ndegree 2 and 1, respectively. $\\psi= v + \\alpha = v^i \\xi_i + \\alpha_i \\xi^i$\nthen describes the most general function of (total) degree\none\\footnote{A vector field on $\\CM$ is always a fiber linear function\non its cotangent bundle and a function can be pulled back canonically\nby the projection. The shift in grading by two has been chosen so as\nto have them both of the same degree.}, and the canonical Hamiltonian $\\tilde Q$ for\nlifting the vector field $Q=\\xi^i \\partial\/\\partial x^i$ to the\ncotangent bundle, $\\tilde Q = \\xi^i p_i$, then reproduces\n\\eqref{Anton} and the inner product by means of the (graded)\nPoisson brackets\n\\begin{align}\n\\{ \\, \\{ \\psi , \\tilde Q \\} , \\psi' \\} &= [ \\psi ,  \\psi' ]_W \n\\label{dBrack1}\\\\\n\\{  \\psi , \\psi' \\} &= \\< \\psi , \\psi' \\> \\, .\n\\label{dInner}\\end{align}\nIn fact, such a construction can be extended to the general setting of\na Courant algebroid as defined above. One then finds that Courant\nalgebroids are in {\\em one-to-one} correspondence to {\\it symplectic}\nQ2-manifolds \\cite{Royt02} (where, by definition, the symplectic form\nis required to be of degree two and to also be compatible with the\nQ-structure, i.e.\\ to be preserved by the vector field $Q$).\n\nOne observes the similarity of the formulas \\eqref{Cour2},\n\\eqref{inner2} with \\eqref{dBrack1}, \\eqref{dInner}. In both\ncases the original bracket is a (graded) Lie bracket, in the second\ncase of degree minus two, in the first case of degree\nzero.\\footnote{\\label{f:24} In fact, the second bracket is even a\n(graded) Poisson bracket, i.e.~there exists a second compatible graded\ncommutative multiplication $\\cdot$, usually taken to be of degree zero. (Here\ncompatibility means a graded Leibniz condition: $\\{F,G \\cdot H \\} = \\{F,G\n\\} \\cdot H + (-1)^{(|F|+d)|G|} G \\cdot \\{F, H \\}$, where $d$ is the degree of the bracket, cf.~also the subsequent footnote).  If one\nprefers, also the bracket of vector fields can be embedded into a\nPoisson algebra, by viewing vector fields as fiber linear functions on\nthe cotangent bundle and thus replacing $\\CM$ by the graded manifold\n$T^*\\CM$, if one prefers, shifted also by some degree (by degree one\nif the canonical Poisson bracket is to reproduce the\nSchouten--Nijenhuis  bracket of (graded)\nvector fields on $\\CM$). We will come back to this perspective below.}\nIn the previous section, moreover, we viewed the left-adjoint action\nof $Q$ as a (compatible) differential $\\ud_Q$. Thus, to cover all\nrelevant cases simultaneously, one is led to regard the derived\nbracket construction of a differential graded Lie algebra (DGLA),\ni.e.\\ a graded Lie algebra with a differential (nilpotent odd\noperator) compatible with the bracket:\\footnote{We call\n$(A^\\bullet,[.,.])$ a (graded) \\emph{Leibniz--Loday algebra} of degree $d$, if\n$[A^k,A^l]\\subset A^{k+l+d}$ and $[a,[b,c]]=[[a,b],c]\n+(-1)^{(|a|+d)(|b|+d)}[b,[a,c]]$. It is a \\emph{differential graded\nLeibniz--Loday algebra} (DGLoA) if in addition there is an operation $D \\colon\nA^\\bullet \\to A^{\\bullet+1}$ squaring to zero and satisfying\n$D([a,b])=[Da,b] + (-1)^{(|a|+d)} [a,Db]$. If the bracket $[.,.]$ is\nalso graded antisymmetric, i.e.~$[b,a]=-(-1)^{(|a|+d)(|b|+d)}[a,b]$, we\ncan replace ``Leibniz--Loday'' by ``Lie'' in these definitions.}\n\n\\begin{prop}[Kosmann-Schwarzbach \\cite{YKS96}]\\label{p:YKSder} Let\n$(A^\\bullet,[.,.],\\uD)$ be a DGLA (with bracket\nof degree $d$) and define its derived bracket as:\n\\begin{equation}\\label{Dder} [\\phi,\\psi]_D = (-1)^{|\\phi|+d+1}[\\uD\\phi,\\psi] \\;,\n\\end{equation}\nwhere $|\\phi|$ denotes the degree of $\\phi$ (assumed to be\nhomogeneous).  \n\n\\noindent Then the following facts are true: \n\\begin{enumerate}\n\\item This new bracket has degree $d+1$ and is a graded\nLeibniz-Loday bracket,\n\\begin{equation}\\label{Loday} [\\phi,[\\psi_1,\\psi_2]_D]_D = [[\\phi,\\psi_1]_D,\\psi_2]_D\n  +(-1)^{(|\\phi|+d+1)(|\\psi_1|+d+1)}[\\psi_1,[\\phi,\\psi_2]_D]_D \\;.\n\\end{equation}\n\\item If $|\\phi|$, $|\\psi_!|$, and $|\\psi_2|$ all have the opposite parity of $d$, one has \n\\begin{eqnarray}\\label{dSkew} [\\phi,\\phi]_D &=&\\frac12 \\uD[\\phi,\\phi] \\;, \\\\\n             {}[\\phi,[\\psi_1,\\psi_2]]_D &=& [[\\phi,\\psi_1]_D,\\psi_2]+ \n[[\\psi_1,[\\phi,\\psi_2]_D]\\label{Dcomp} \\; .  \\label{adInvar}\n\\end{eqnarray}\n\\item The quotient of  $(A^\\bullet,[.,.]_D)$ by $\\ker D$ is a graded Lie\nalgebra. \n\\end{enumerate} \n\\end{prop}\nThese facts are easy to verify and recommended to the reader as an\nexercise. (To see that $A^\\bullet\/\\ker D$ is well-defined one observes\nthat $[\\phi,\\psi]_D$ equals $[\\phi,D\\psi]$ modulo a $D$-exact term,\nwhich can also be used to establish the graded antisymmetry of the\ninduced bracket). \n\nIn fact, the Leibniz-Loday Property~\\eqref{Loday} also follows in the more\ngeneral context of $(A^\\bullet,[.,.],\\uD)$ being a DGLoA only, cf.~the\npreceding footnote and Lemma \\ref{p:der2} below. The choice of the\nsign in \\eqref{Dder} is essential and can be motivated also from\nexamples where $\\uD$ is an inner derivation, $\\uD=[Q,.]$ with\n$[Q,Q]=0$, and the brackets arise as in \\eqref{Cour2}. This\ndefinition also agrees with (the first part of) Eq.~\\eqref{induced}\nwith the identification $\\uD=\\ud_Q$, since there the bracket has\ndegree zero and the elements are odd.\n\n\\bigskip\n\nWe now turn to the alternative description of Q-manifolds when\nstudying the vector fields on them and their derived bracket. To\nexplain this procedure, we start with a Q1-manifold, $\\CM = E[1]$:\nSections $\\psi$ in $E$ can be identified with degree minus one vector\nfields on $\\CM$, $\\psi = \\psi^a\n\\partial\/\\partial \\xi^a$ in our notation.  Replacing $\\ud$ in Cartan's\nformula \\eqref{derbasic} with the differential of the Lie algebroid (acting\n on ${{}^E{}\\Omega}^\\bullet(M)$) every Lie algebroid bracket occurs naturally as a\nderived bracket. In super-language this means explicitly that the\nLie algebroid bracket on $E$, which for clarity we now denote by $[\n\\cdot , \\cdot ]_E$ is a derived bracket w.r.t.~$Q$,\n\\begin{equation} [\\phi,\\psi]_E = [[\\phi,Q],\\psi] \\, , \\label{Eder}\n\\end{equation}\nwhere, for notational simplicity, we did not introduce a symbol for\nthe map identifying sections in $E$ (as they appear on the l.h.s.~of\nthis equation) with the respective degree minus one vector field (as\nunderstood on the r.h.s.)---in \\eqref{derbasic} the symbol\n$\\imath_\\cdot$ could be interpreted as such an isomorphism. We will\nkeep such a notational convention also further on. From the previous\ngeneral considerations about DGLAs and their derived brackets, we can\nnow at once conclude about two of the three defining properties for\n$[\\cdot , \\cdot ]_E$ to be a Lie algebroid bracket: Since there are\n\\emph{no} degree minus two vector fields on $\\CM = E[1]$, the right\nhand side of \\eqref{dSkew} vanishes identically. With the fact that\n$\\phi$, $\\psi$ have odd degree and the natural bracket of super vector\nfields degree $d=0$, Eqs.~\\eqref{dSkew} and \\eqref{Loday} yield\n$[\\cdot , \\cdot ]_E$ to be (ungraded) antisymmetric and to satisfy an\n(ungraded) Jacobi identity, respectively. To retrieve the Leibniz \nproperty w.r.t.~an or the anchor map of $E$,\nwe also need to make use of the module structure of vector fields\nw.r.t.~functions on $\\CM$, where we may restrict to degree zero\nfunctions $C^\\infty_0(\\CM) \\cong C^\\infty(M)$ so as to preserve the\ndegree of our vector fields. This will be provided by\nEq.~\\eqref{eq:Leibniz} below, then establishing that we obtain a Lie\nalgebroid structure on $E$ by this procedure. \n\nSince we will need such type of anchor maps also in the general\ncontext of higher degree $Q$-manifolds and their compatibility with\nthe derived bracket construction, we will extend the\nconsiderations on DGL(o)As to this setting. A rather minimalistic way of\ndoing so is \n\\begin{vdef}\nWe call $(A^\\bullet,[.,.],\\uD, R, \\rho_0)$ a  \\emph{weakly anchored} DGL(o)A\nover the ring $R$ (commutative, with unity) if $(A^\\bullet,[.,.],\\uD)$\nis a DGL(o)A of degree $d$ and $\\cdot \\colon R \\times A^k\n\\to A^k$ and $\\rho_0 \\colon A^{-d} \\to \\End_{\\Real} (R)$\nare maps such that \n\\begin{equation} [\\phi, f \\cdot \\psi] = \\left(\\rho_0(\\phi)\\right)(f) \\cdot \\psi + f\n\\cdot [\\phi, \\psi] \\label{ring}\n\\end{equation}\nholds true for every $\\phi \\in  A^{-d}$, $\\psi \\in A$, and\n$f\\in R$.\n\nWe call it merely \\emph{anchored} iff in addition $\\rho_0$ is $R$-linear and\nso is its composition with $\\uD$, i.e.~for all $\\phi \\in  A^{-d-1}$\none has\n\\[ \\label{rhoBund} \\rho_0(\\uD(f\\phi))=f\\rho_0(\\uD\\phi) \\,. \\]\nWe call it \\emph{strongly anchored} if also ($\\forall f,g \\in R, \\psi \\in\nA$) \n\\[  \\uD(fg\\psi) +fg\\uD \\psi= f\\uD(g\\psi) +g\\uD(f\\psi) \\,. \\label{1stder} \\]\n\\end{vdef} \n\nIn fact, \\eqref{1stder} is \\emph{equivalent} to the\nexistence of an operator  $\\CD \\colon R\\to \\Hom_R(A^\\bullet,A^{\\bullet+1})$ such that\n\\begin{align}\n  \\uD(f\\phi) &= (\\CD f)(\\phi) +f\\uD \\phi  \\label{DLeib} \\,, \\\\\n  \\CD(fg) &= f\\CD g +g\\CD f \\label{D0Leib}\n\\end{align}  holds true. \\eqref{D0Leib} implies that  $\\CD$ is a first \norder differential operator and \\eqref{DLeib} that likewise is $\\uD$\nw.r.t.~$R$. In fact Eq.~\\eqref{DLeib} also shows constructively how to\ndefine $\\CD$ once \\eqref{1stder} is fulfilled. We have chosen the\nformulation as in \\eqref{1stder} since it is intrinsic and does not\nneed the introduction of new operations such as the above map $\\CD$. \n\nA graded Lie algebroid over an ungraded manifold $M$ gives an example\nof a strongly anchored DGLA with $\\uD=0$; here $A^\\bullet$ are the\nsections in the graded vector bundle and $R=C^\\infty(M)$, as an\n$R$-module $A^\\bullet$ is projective in this example. Obviously the\nabove definition can be easily extended to graded commutative rings\n$R$ in which case general graded Lie algebroids can be covered as\nwell; this is however not needed for the applications we are having in\nmind here (cf.~Proposition \\ref{p:tangentLie} below). An example with\nnontrivial $\\uD$ can be constructed from Lie bialgebroids satisfying\nEq.~\\eqref{rhoBund}; here $A^\\bullet$ consists of the sections of the\nexterior powers of the (ungraded) vector bundle. But more important\nfor our context is the following observation:\\footnote{Cf.~also\nfootnote\n\\ref{f:8} concerning our simplified nomenclature and footnote \\ref{f:24} for\na possible relation of the two parts of the proposition. A differential\ngraded Poisson algebra $(A^\\bullet, \\{ . , . \\}, \\cdot , \\uD)$ of degree $d$ is\na DGLA $(A^\\bullet, \\{ . , . \\}, \\uD)$ and a graded Poisson algebra\n$(A^\\bullet, \\{ . , . \\}, \\cdot)$, both of degree $d$, such that $\\uD(F \\cdot\nG) = \\uD F \\cdot G + (-1)^{|F|} F \\cdot \\uD G$ for all $F\n\\in A^\\bullet$, $G \\in A$.}\n\\begin{prop}\\label{p:tangentLie}\n$\\,$\n\\begin{enumerate}\n\\item $(A^\\bullet,[.,.])=\\X(\\CM)$ with $\\uD=\\ud_Q$\nof a Q-manifold is a strongly anchored DGLA with bracket of degree zero \nw.r.t.\\ $R=C^\\infty_0(\\CM)\\cong{C^\\infty}(M)$, where $\\rho_0(\\phi)[f]=\\phi(f)$.\n\\item \nA differential graded Poisson algebra (DGPA) on $C^\\infty(\\CM)$ is a weakly\nanchored DGLA w.r.t.\\ $R=C^\\infty_0(\\CM)$ and $\\rho_0(\\phi)f=\\{\\phi,f\\}$. It\nis strongly anchored iff $\\{f,g\\}=0=\\{\\phi \\cdot \\uD f,g\\}$ for all $f,g\\in\nR$ and $\\phi \\in C^\\infty_{-d-1}(\\CM)$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof} For the case of vector fields use $\\uD \\phi = [Q,\\phi]$ and find\n\\eqref{DLeib} with \n$(Df)(\\phi)$ being the multiplication of $\\phi$ by $Q(f)$. $Q$ being a vector\nfield, this proves also \\eqref{D0Leib} and thus \\eqref{1stder}. From its\ndefinition it is immediate that $\\rho_0$ is $R$-linear in $\\phi\n\\in \\X_0(\\CM)$. Moreover using \\eqref{DLeib} and observing that $\\rho_0\n\\left(Q(f) \\phi\\right)=0$, which follows since here $\\phi$ is degree minus\none and vanishes when applied to degree zero functions, also \\eqref{rhoBund}\nfollows. \n \n\nIn the Poisson case the two additional conditions are found by directly\nchecking the desired conditions using the definition of\n$\\rho_0$. E.g.~$\\{f,g\\}=0$ follows immediately from $R$-linearity of\n$\\rho_0$. \n\\end{proof}\n\n\\pagebreak[2]These additional structures are compatible with the\nderived bracket construction in the following sense:\\nopagebreak[3]\n\\begin{lemma}\\label{p:der2}\n  Let $(A^\\bullet,[.,.],\\uD,R,\\rho_0)$ be a (weakly\/strongly) anchored DGLoA, \n  then $(A^\\bullet,[.,.]_\\uD,\\uD,R,\\rho)$ is also a (weakly\/strongly)\n  anchored DGLoA with \n\\begin{equation} \\rho(\\phi) = \\rho_0(\\uD\\phi)  \\;. \\label{eq:rho1}\n\\end{equation}\n\\end{lemma}\nThis can be proven by simple straightforward calculations. The Lemma contains\nthree different statements. \n\nWe now apply these considerations first by returning to Q1-manifolds. As a\nconsequence of Proposition \\ref{p:tangentLie} and the preceding lemma,\nwe notice that the derived bracket \\eqref{Eder} satisfies a Leibniz\nrule \\begin{equation} [\\phi, f \\cdot \\psi]_E =\\rho(\\phi)(f) \\cdot \\psi + f\n\\cdot [\\phi, \\psi]_E \\label{eq:Leibniz} \n\\end{equation} with an $R$- or $C^\\infty(M)$-linear anchor map \n$\\rho$ given by $\\rho(\\phi)f = [Q,\\phi] f$. $C^\\infty(M)$-linearity\nimplies  that $\\rho$ indeed comes from a vector bundle map $E\\to TM$. \n\n\n\nThus, indeed, the bracket on $\\Gamma(E)$ defined via the derived\nbracket construction\n\\eqref{Eder} induces a Lie algebroid structure on $E$. Moreover, this is the \n\\emph{same} Lie algebroid structure as found before in Section~\\ref{s:examQ}, \nwhich one easily verifies on a basis: Using the above formulas as definitions \nfor the bracket and the anchor, one verifies easily that $[\\partial_a,\n\\partial_b]_E = C^c_{ab} \\partial_c$ and $\\rho(\\partial_a)=\\rho^i_a \n\\partial_i$ (with $\\partial_a \\equiv \\frac{\\partial}{\\partial \\xi^a}$ and \n$\\partial_i \\equiv \\frac{\\partial}{\\partial x^i}$ for the parameterization used \nin \\eqref{Q}).\n\nWe now want to imitate this second procedure, illustrated at the\nexample of Q1-manifolds and in spirit close to the considerations of\nthe previous section on gauge symmetries, so as to find an alternative\ncharacterization of degree two Q-manifolds. In fact, we want to\nperform some of the considerations common to all choices of degree $p$\nfirst and later specialize the discussion to $p=2$. Courant algebroids\nshould be particular examples of the latter case, moreover, and we\nwill find a description of Q2-manifolds or Lie-2-algebroids\ncomplementary to the one given in Section~\\ref{s:examQ} and much closer to\nthe usual formulation of Courant algebroids.\n\n\n\\subsection{Vinogradov and $V$-twisted Courant algebroids}\n\\label{subsec:Vino}\nTo obtain a prototype of a bracket for higher $p$ than two, we\nmay look at a generalization of the Courant-Dorfman bracket, the\nso-called Vinogradov\nbracket. It is in fact the bracket \\eqref{Cour} with $\\alpha,\\beta\n\\in \\Omega^k(M)$ for some arbitrary $k \\in \\N$, and it can be \nobtained precisely as for the case $k=1$. Accordingly, we again have\nat least two options translating this scenario into super-language. In\nthe $\\CM=T[1]M \\times \\Real[p]$ picture\\footnote{We have found this description in 2005. In the mean time it appeared also in \\cite{Uri13}}, sections $v\\oplus\\alpha$ correspond\nto \\begin{equation}  \\label{psi}\n\\psi = \\imath_v + \\alpha \\frac{\\partial}{\\partial b} \\, , \\end{equation} \nwhere $b$ obviously needs to have degree $p=k+1$ for $\\psi$ to be\nhomogeneous of degree minus one. The Vinogradov bracket then follows\nagain as a derived bracket \\eqref{Cour2} with $Q$ being the de Rham\ndifferential. In generalization of Eq.~\\eqref{Qtwist1} this may\ncertainly again be twisted, \\begin{equation} Q = \\ud + H \\frac{\\partial}{\\partial\nb} \\, , \\label{Qtwist2} \\end{equation} where now $H$ needs to be a closed\n($k+2$)-form obviously. Like in the Courant algebroid, also here the\noriginal (graded) Lie bracket plays an important role,\ncf.~Eq.~\\eqref{inner2} (or \\eqref{dInner}), generating a\n$C^\\infty(M)$ bilinear pairing; it comes from the contraction of the\nvector fields with the $k$-forms, as one easily verifies using\n\\eqref{psi}. Thus, this time the pairing maps two elements in\n$W=TM \\oplus \\Lambda^k T^*M$ to an element of another bundle, namely\nof $V=\\Lambda^{k-1} T^*M$. This, the appearance of a second bundle $V$\nof relevance, will be one feature we want to keep also for the more\ngeneral setting in what follows.\n\nBefore axiomatizing these data, in analogy to the step from\n\\eqref{Cour} to Definition \\ref{def:Cour}, into what one may want\nto call a Vinogradov algebroid, we briefly translate the above picture\nalso into a graded Poisson language. We therefore regard \n$\\CM=T^*[p]T[1]M$ with its canonical Poisson bracket $\\{ \\cdot , \\cdot\n\\}$ of degree $-p$. Local coordinates $x^i$ and $\\xi^i$ of degree 0\nand 1, respectively, are then accompanied by momenta $p_i$ and $\\xi_i$\nof degree $p$ and $p-1$, respectively. Sections of $W$ correspond to\nfunctions on this $\\CM$ of degree $k=p-1$, $\\psi= v^i \\xi_i +\n\\frac{1}{k!}\\alpha_{i_1\n\\ldots i_k} \\xi^{i_1} \\ldots \\xi^{i_k}$, and the bracket and pairing \non $W$ results from formulas \\eqref{dBrack1} and \\eqref{dInner},\nrespectively, where $\\tilde{Q}=\\xi^i p_i +H$. \n\n\nWe now return to the task of axiomatization. In particular, one may\nwant to check properties of the bracket and pairing, finding\nappropriate generalizations of those written in definition\n\\ref{def:Cour}. There are natural candidates for the analogues of the\nobjects appearing in \\eqref{cLeibniz} and \\eqref{cnSkew}, which will\nalso turn out to be realized for what one may call the Standard\nVinogradov algebroid ($W=TM \\oplus \\Lambda^k T^*M$, $V=\\Lambda^{k-1}\nT^*M$), possibly twisted by some $H\\in \\Omega_{cl}^{k+2}(M)$. The\nanchor map $\\rho$ will be projection to $TM$, for $\\uD$ one can choose\nthe de Rham differential (succeeded by the natural embedding of\n$k$-forms into sections of $W$). For the ad-invariance of the pairing\n(cf.~Eq.~\\eqref{cInvar}), however, we can no more use solely the\nanchor map, since the pairing lands in sections of $V$. The operator\non the l.h.s.~of\n\\eqref{cInvar} will then be found to be replaced by the Lie derivative\nw.r.t.~the anchor of the section $\\psi$. Note that this is for $k>1$\n(or, likewise, $p>2$) no more $C^\\infty(M)$-linear in $\\psi$. So, in\nthe axioms there will be besides $\\rho$ and $\\uD$ one more new object,\nwhich we will call $\\rho_D$, not necessarily coming from a bundle map,\nbut being restricted appropriately otherwise. Also, the relation\nbetween $\\rho$ and $\\uD$ being adjoint to one another does not make\nany sense anymore.\\footnote{Only the generalization that $\\rho_D$\nwould be adjoint of $\\uD$ w.r.t.~the pairing can be meaningful. In\nfact, although it is not satisfied in the Standard Vinogradov\nalgebroid for $k>1$, posing this additional condition will be studied\nin detail the context of $p=2$ below, cf., e.g., definition\n\\ref{def:VCourant} and Proposition \\ref{p:FW}.}  $\\uD$ needs to be\nrestricted then by something like Equation~\\eqref{1stder} so as to\nensure that it is a first order differential operator. Likewise for\n$\\rho_D$ we require it to take values in the module $\\Gamma(\\CDO)(V)\\equiv \\Gamma(\\CDO(V))$ of covariant differential operators  on $V$: Keeping in\nmind that the $\\Real$-linear first order differential operators on a\nvector bundle $V\\to M$ are themselves a projective module over $M$, we\ncall the generating bundle $\\CDO(V)$, the bundle of covariant\ndifferential operators on $V$; elements in  $\\Gamma(\\CDO)(V)$ are then sections of this bundle. There is a canonical map from $\\CDO(V)$\nto $TM$ which we may use to require $\\rho_D$ to cover the\nmap from $\\Gamma(W)$ to $\\Gamma(TM)$ induced by $\\rho$ (and\nconventionally denoted by the same letter), i.e.~in formulas we require \n\\begin{equation} \n  \\rho_D(\\phi)[fv] = \\rho(\\phi)[f]v +f\\rho_D(\\phi)[v] \\, .\n\\label{vLeibn3} \n\\end{equation} where $\\phi \\in \\Gamma(W)$, $f \\in C^\\infty(M)$, and $v \\in \\Gamma(V)$.\nWe thus are led to the following\n\n\\newpage\n\\begin{vdef} \\label{def:Vino} A \\emph{Vinogradov algebroid} are two vector bundles $W$ and $V$\ntogether with a bracket $[.,.]_W:\\Gamma(W)\\times\\Gamma(W)\\to \\Gamma(W)$,\na map $\\rho\\:W\\to TM$, a non-degenerate\\footnote{A (skew)-symmetric bilinear form $\\<.,.\\>$ is non-degenerate if for every $\\phi\\in E_x$, $\\phi\\ne0$ there is a $\\psi\\in E_x$ such that $\\<\\phi,\\psi\\>\\ne0$.} surjective inner product\n$\\<.,.\\>\\: W\\otimes_M W\\to V$, an $\\Real$-linear map\n$\\uD:\\Gamma(V)\\to \\Gamma(W)$,\nand a map $\\rho_D: \\Gamma(W)\\to \\Gamma(\\CDO)(V)$ covering the map\n$\\rho \\colon  \\Gamma(W)\\to \\Gamma(TM)$ subject to the\nfollowing axioms.\n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +\n[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{vJacobi}  \\\\\n  [\\psi,\\psi]_W &= \\frac12\\uD\\<\\psi,\\psi\\>  \\label{vSkew}\\\\\n  \\rho_D(\\psi)\\<\\phi,\\phi\\> &=\n  2\\<[\\psi,\\phi]_W,\\phi\\>  \\label{vInvar}\\\\\n  \\uD(fgv) +fg\\uD(v) &= f\\uD(gv) +g\\uD(fv)  \\label{vLeibn2}\n\\end{align} where $\\phi_i,\\psi \\in \\Gamma(W)$, $f,g \\in {C^\\infty}(M)$,\nand $v \\in \\Gamma(V)$.\n\\end{vdef}\nIt is easy to see that \\eqref{vInvar} and \\eqref{vLeibn3} imply \n\\begin{equation}\n[\\psi,f\\phi]_W = \\rho(\\psi)f\\phi +f[\\psi,\\phi]_W \\label{vLeibniz} \\;\n\\end{equation} and, conversely,\none may also conclude \\eqref{vLeibn3} from \\eqref{vInvar}\nand \\eqref{vLeibniz}.  One may consider adding $[\\uD v,\\phi]_W =\n0\\label{brwExact}$ to the axioms as this rule is suggested from\nseveral perspectives (e.g.~it is satisfied for the Vinogradov\nbracket). For the case of\n $\\rho_D(\\phi)[v]=\\<\\phi,\\uD v\\>$, on the other hand,\nthis follows automatically, cf.~Proposition \\ref{p:Royt} below.\n\nTo require surjectivity of the pairing goes almost without loss of\ngenerality since $V$ enters the axioms only via the image of  $\\<\n\\cdot , \\cdot \\>$; only if this image changes its rank, one cannot\nconsistently restrict to it as a vector bundle. On the other side, it\nis suggested from some perspectives to drop the condition of\nnon-degeneracy,\nwhich, however, then would go with a partially\ndifferent formulation of the axioms and we\nwill not pursue this idea in the present article. Let us mention that\nin the application to Q-manifolds the inner product is automatically\nsurjective and non-degenerate, cf.~Proposition \\ref{p:genStdVinogr}\nbelow.\n\n\nNote that the Equation~\\eqref{vInvar} is nothing but the invariance of the\ninner product under sections of $W$, being equivalent to\n\\begin{align*}\n  \\rho_D(\\psi)\\<\\phi_1,\\phi_2\\> &=\n    \\<[\\psi,\\phi_1]_W,\\phi_2\\> + \\<\\phi_1,[\\psi,\\phi_2]_W\\> \\,\n\\end{align*}\nby a standard polarization argument. It is an instructive exercise to\ncompute the analog of the Leibniz rule (Eq.~\\eqref{vLeibniz}) for\nthe left hand side of the bracket, which shows that the bracket\nis also a first order linear partial differential operator in the l.h.s.\nAlso one has the following \n\\begin{prop}\\label{p:Vinogrelem}  The maps $\\rho$ and $\\rho_D$ are morphisms of\nbrackets.\\footnote{In the case of Courant algebroids it was apparently\nfirst K. Uchino \\cite{Uchi02} who observed that the morphism property\nof $\\rho$ can be deduced from the other axioms.} Furthermore, their\ncomposition with $\\uD$ vanishes ($\\rho\\circ\\uD =0=\\rho_D\\circ\\uD$).\n\\end{prop}\n\n\\begin{proof}\n \nThe morphism property of $\\rho$ can be concluded easily from\n\\eqref{vJacobi} and \\eqref{vLeibniz}, cf., e.g., \\cite{HS08}. \nFor the morphism property of $\\rho_D$ write a section of\n$V$ locally as scalar product of two sections of $W$.  Then use \\eqref{vInvar}\nand \\eqref{vJacobi} to proceed in a similar manner.\nThe last two properties follow from \\eqref{vSkew} and the first or second property, respectively.\n\\end{proof}\n\n\n\n\nLet us now check that the axioms are indeed all fulfilled for the\n(twisted standard) Vinogradov bracket described above.  Since we\nalready introduced some super language for the bracket, let us start\nfrom there:\n\\begin{prop}\\label{p:derVino}  Given a strongly anchored DGLA structure\n  $(A^\\bullet,[.,.],\\uD,\\rho_0)$ with bracket of degree $d$ over\n  $R={C^\\infty}(M)$ of a smooth manifold $M$, where $A^{-d-1}$ and\n  $A^{-d-2}$ are projective modules over $M$, then there exist vector\n  bundles $W\\to M$ and $V\\to M$ such that $A^{-d-1}$ is (canonically\n  isomorphic to) the space of sections of $W$ denoted\n  $\\phi,\\psi,\\ldots$ in what follows and $A^{-d-2}$ is the space of\n  sections of $V$ a typical element of which is denoted by $v$ below.\n  Thus $\\uD \\colon \\Gamma(V)\\to \\Gamma(W)$. Now the following operations\n  can be defined by their right hand sides:\n\\begin{align}\n  [.,.]_W &: \\Gamma(W)\\otimes\\Gamma(W)\\to \\Gamma(W),&  [\\phi,\\psi]_W&:=[\\uD\\phi,\\psi] \\label{dBrack2}\\\\\n  \\rho &: W\\to TM,&\n \\rho(\\phi)[f]&:= \\rho_0(\\uD\\phi)[f]  \\label{dAnchor}\\\\\n  \\<.,.\\> &: \\Gamma(W)\\otimes\\Gamma(W)\\to \\Gamma(V),&\n    \\<\\phi,\\psi\\>&:=[\\phi,\\psi] \\label{dInner2}\\\\\n  \\rho_D &: \\Gamma(W)\\to \\Gamma(\\CDO)(V), \\;& \\rho_D(\\phi)[v]&:= [\\uD\\phi,v] \\,. \\label{rhoQ}\n\\end{align}\n  $\\<.,.\\>$ is ${C^\\infty}(M)$-linear iff $[\\psi,f\\phi]=f[\\psi,\\phi]$ for\n  all functions $f$ on $M$. Requiring this and that the induced inner product\n  $\\<.,.\\>\\:W\\otimes_M W\\to V$ is non-degenerate and surjective, these\n  operations form a Vinogradov algebroid.\n\\end{prop}\n\\begin{proof}  Follows from Proposition \\ref{p:YKSder}\n  and Lemma \\ref{p:der2} with the interpretation that \\eqref{vLeibn3} is just\n  the Leibniz rule for the derived bracket restricted to $A^{-d-2}$ in the\n  right-hand argument.\n\\end{proof}\n\nDue to Proposition \\ref{p:tangentLie} the vector fields on our super manifold\n$\\CM=T[1]M\\times\\Real[k+1]$ together with the above $Q$-structure fulfill the\nfirst condition.  ${C^\\infty}(M)$-linearity of the inner product is\nautomatically fulfilled for Q-manifolds (since $d=0$ and thus $\\Gamma(W)$\nis identified with degree minus one vector fields in this case). \nNon-degeneracy follows by direct inspection, noting that  \n$W \\cong TM\\oplus \\Lambda^kT^*M$ for\n$1\\le k\\le\\dim M$. This shows that the (twisted) Vinogradov\nbracket and the canonical pairing indeed provide an example of a\nVinogradov algebroid; we will call this the ($H$-twisted) standard \nVinogradov algebroid.\n\nAn immediate generalization of this is to replace the tangent bundle\nin the above construction by a Lie algebroid $A$.  Starting with a Lie\nalgebroid $(A,[.,.]_A,\\rho_A)$ and a positive integer $k\\le\\rk A$ we\nconsider the bundle $W:=A\\oplus\\Lambda^kA^*$ with the canonical\nsymmetric (non-degenerate) pairing $\\<X\\oplus\\alpha,Y\\oplus\\beta\\>\n:=\\imath_{X}\\alpha+\\imath_{Y}\\beta$.  The choice for $V$ is therefore\n$\\Lambda^{k-1}A^*$.  The choice of the super manifold $\\CM$ is analog\nto the tangent case $\\CM=T^*[k+1]A[1]$.  $A[1]$ also has a canonical\n$Q$-structure, the differential $\\ud_A$ of the algebroid $A$.  Its\nHamiltonian lift to $\\CM$ is the $Q$-structure we use.  Since the\nHamiltonian is itself of degree $k+2$, we can add an $A$-$(k+2)$-form\n$H$, satisfying $\\ud_AH=0$ so as to render the total charge further on\nnilpotent.  This leads us to\n\\begin{prop}\\label{p:genStdVinogr}  Let $(A,[.,.]_A,\\rho_A)$ be a Lie algebroid over\n$M$. Let $W=A\\oplus\\Lambda^kA^*$ and $V= \\Lambda^{k-1}A^*$. Then  \n\\begin{align}\n  (X\\oplus\\alpha,Y\\oplus\\beta) &:= \\imath_X\\beta+\\imath_Y\\alpha \\\\\n  [X\\oplus\\alpha,Y\\oplus\\beta]_W &:= [X,Y]_A\\oplus \\L_X\\beta\n    -\\imath_Y\\ud_A\\alpha +\\imath_X\\imath_Y H  \\label{stdVbrack}\\\\\n  \\rho(X\\oplus\\alpha) &:= \\rho_A(X)  \\\\\n  \\uD(v) &:= 0\\oplus\\ud_Av  \\label{stdVder}\\\\\n  \\rho_D(X\\oplus\\alpha) &:= \\L_X  \\, ,  \\label{stdVrhoQ}\n\\end{align}  where $X,Y \\in \\Gamma(A)$, $\\alpha,\\beta \\in\n\\Gamma(\\Lambda^kA^*)$,  and $v \\in \\Gamma(V)$, equips $(W,V)$ with the\nstructure of a Vinogradov algebroi\n.\n\\end{prop}\n\\begin{proof}  Follows from Proposition \\ref{p:derVino} with the\nrealization as a Q-structure. \n\\end{proof}\nWe will call this example an ($H$-twisted) generalized standard\nVinogradov algebroid, reducing to an ($H$-twisted) standard Vinogradov\nalgebroid for the choice of $A$ being a standard Lie algebroid, $A\n=TM$. Note that for every generalized standard Vinogradov algebroid we\nhave two quite distinct super-geometric descriptions, one as described\nabove where $\\CM = T^*[k+1]A[1]$ and another one on the supermanifold\n$\\CM' = A[1] \\times\n\\Real[k+1]$  with $Q'=\\ud_A + H \\partial\/\\partial b$, $b$ being the\ncanonical coordinate on $\\Real[k+1]$.\\footnote{Also these two pictures\nare not just related by the Hamiltonian reformulation of $(\\CM',Q')$\naccording to footnote \\ref{f:24}, $T^*[\\bullet]\\CM'$ does not agree\nwith $\\CM$.}\n\nAn important class of Vinogradov algebroids is provided by those where\nthe map $\\rho_D$ is $C^\\infty(M)$-linear (note that for a generalized\nstandard Vinogradov algebroid this is the case only for $\\rk\nV=1$). A special case of this is provided by the following definition\n(cf.~Prop.~\\ref{p:FW} below):\n\\begin{vdef}[{$V$}-twisted Courant algebroid]\\label{def:VCourant}\nLet $V,W$ be vector bundles over $M$, $W$ equipped with an anchor map $\\rho\n\\colon W\\to TM$, a bracket $[ . , . ]_W$ on its sections, and a non-degenerate\nsurjective symmetric product $\\<.,.\\>$ taking values in $V$. Let\n ${{}^W\\nabla}$ be a $W$-connection on $V$. We call this a vector bundle twisted (or\nV-twisted) Courant algebroid if these data are subject to the\nfollowing axioms\n\\begin{align}\n  [\\psi,[\\phi_1,\\phi_2]_W]_W &= [[\\psi,\\phi_1]_W,\\phi_2]_W +[\\phi_1,[\\psi,\\phi_2]_W]_W\n    \\label{fwJacobi}  \\\\\n  \\<[\\psi,\\phi]_W,\\phi\\> &=\\frac12\\Wconn_\\psi \\<\\phi,\\phi\\> =  \\<\\psi,[\\phi,\\phi]_W\\>\n  \\label{fwInvar} \\;.\n\\end{align}\n\\end{vdef}\n\nWe remark that the usual Leibniz rule \\eqref{vLeibniz} follows from \\eqref{fwInvar} after polarization, $\\<[\\psi,\\phi_1],\\phi_2\\>+\\<\\phi_1,[\\psi,\\phi_2]\\> = \\Wconn_\\psi\\<\\phi_1,\\phi_2\\> $, from the Leibniz rule for ${{}^W\\nabla}$.\n\n\n\\begin{prop} \\label{p:FW} A V-twisted Courant algebroid\n  is a Vinogradov algebroid with   $\\rho_D(\\phi):=\\Wconn_\\phi$ and $\\uD$\nits adjoint, i.e.~$\\uD$ being defined via\n\\begin{align} \n  \\label{FW}  \\<\\phi,\\uD v\\> =\\Wconn_\\phi v \\;.\n\\end{align}\nConversely, a Vinogradov algebroid is a V-twisted Courant algebroid if\n\\begin{align} \n  \\label{FW'}  \\<\\phi,\\uD v\\> =\\rho_D(\\phi) v \\;.\n\\end{align}\n\\end{prop}\n\\begin{proof} Given \\eqref{FW} then \\eqref{fwInvar} is equivalent to \n  \\eqref{vSkew}--\\eqref{vInvar}.\n  It remains to check that $\\uD$ fulfills the Leibniz rule \\eqref{vLeibn2},\n  but this follows since \\eqref{vLeibn3} shows that the adjoint\n  ${{}^W\\nabla}$ of $\\uD$ is a first order linear PDO. For the other direction note that \\eqref{FW'} replaces \\eqref{FW} thus \\eqref{vSkew}--\\eqref{vInvar} are fulfilled.  Further ${{}^W\\nabla}=\\rho_D$ the adjoint of $\\uD$ implies that it is ${C^\\infty}$-linear in $\\phi$.\n\\end{proof}\n\nNote that the l.h.s.~ of \\eqref{FW} also permits the interpretation that\n$\\uD\\:\\Gamma(V)\\to{}^W{}\\Omega^1(M,V)$ and this 1-form is then applied to a vector\n$\\phi\\in W$, i.e.\\ $\\uD$ is an exterior $W$-covariant derivative. From\nProposition \\ref{p:Vinogrelem} we learn that ${{}^W\\nabla}$ is a\nmorphism of brackets,\ni.e.~$$\\Wconn_{[\\phi,\\psi]_W}=[\\Wconn_\\phi,\\Wconn_\\psi] \\, .$$  In other words,\nthe $W$-connection of a V-twisted Courant algebroid is always\nflat and thus also $\\uD^2=0$. Furthermore, one has \n\\begin{prop}\\label{p:Royt}  In a twisted generalized standard Vinogradov \n  algebroid as well as in a V-twisted Courant algebroid\n   brackets for $\\uD$-exact sections satisfy \\\\(for all $v,v' \\in \\Gamma(V)$ and\n$\\phi \\in \\Gamma(W)$):\n\\begin{align*} \\<\\uD v,\\uD v'\\>&=0 \\\\ [\\uD v,\\phi]_W &=0\\\\\n  [\\phi,\\uD v]_W &= \\uD\\<\\phi,\\uD v\\>  \n\\end{align*}\n\\end{prop}\n\\begin{proof} The first equality follows from $ \\<\\uD v,\\uD\nv'\\>=\\Wconn_{\\uD v}v' $ and Proposition \\ref{p:Vinogrelem}. For the\nremaining two equations one can use the explicit formula\n\\eqref{stdVbrack} for the derived bracket \nin the case of twisted generalized standard Vinogradov algebroids and\ngeneralize the proof in \\cite[p.20, lemma 2.6.2]{Royt99} to\n$V$-twisted Courant algebroids in a straightforward way, respectively.\n\\end{proof}\n\nAs the terminology suggests, ordinary Courant algebroids\n$(W,[.,.],\\rho,\\<.,.\\>)$ provide examples of $V$-twisted ones: one just\ntakes $V$ to be  the trivial $\\Real$ bundle over the base $M$ so that its\nsections can be identified with functions on $M$ and identify ${{}^W\\nabla}$\nwith $\\rho$ in this case. For the reverse direction we need an\nadditional condition: Consider a V-twisted Courant algebroid with $V$\nof rank one. Suppose we can find a section $v$ of $V$ which vanishes\nnowhere (consequently $V$ has to be trivial) and which is annihilated\nby $\\uD$, $\\uD v=0$. Then we can define a non-degenerate symmetric\nbilinear form $\\<.,.\\>_C$ on $W$ by means of\n$\\<\\phi,\\psi\\>=\\<\\phi,\\psi\\>_Cv$.  Moreover, $\\<\\uD (fv),\\phi\\>=\n\\left(\\rho(\\phi) f\\right) v$. It now is easy to see that $([.,.]_W,\n\\rho, \\<.,.\\>_C)$ defines a Courant algebroid.\n\nLet us remark, however, that a line-bundle twisted Courant algebroid,i.e.~a $V$-twisted Courant algebroid with $\\rk V = 1$, is a strictly more general notion than the one of an ordinary Courant algebroid.  We intend to come back to this elsewhere.\n\n\nOrdinary Courant algebroids over a point reduce to quadratic Lie\nalgebras. This is no more the case for the above $V$-twisted\ngeneralizations, where even the bracket need not be antisymmetric. In\nthis case $W$ is what one calls a Leibniz--Loday algebra and $V$ becomes a\n$W$-module by means of $\\Wconn_\\cdot \\colon W \\to \\End(V)$. Whereas the\nr.h.s.~of \\eqref{cnSkew} vanishes identically for $M$ being a point,\nthe analogous second equality in \\eqref{fwInvar} does \\emph{not} lead\nto $[\\phi,\\phi]_W=0$ for any representation $\\Wconn_\\cdot$. An explicit\nexample with a bracket having a symmetric part is given by the\nfollowing construction which was motivated by the example in \\cite[chap.\\ 1]{San07}\n\n\\begin{prop}  Let $W:=\\End(\\Real^2)$, $V:=\\Real^3$\n\\begin{align*}\n P\\: W&\\to W: (x_{ij})\\mapsto\\pmatrix{cc}{x_{11}&0\\\\0&0} \\\\\n [X,Y]_W &:= P(X)Y -YP(X) \\\\\n \\<X,Y\\> &:= (x_{11}y_{12}+x_{12}y_{11}, -x_{11}y_{21}-x_{21}y_{12},x_{22}y_{22}) \\\\\n \\Wconn_Y(p,q,r) &:= (2y_{11}p,2y_{11}q,0)\n\\end{align*}  Then these 5 structures form a V-twisted Courant algebroid over a point.\n\\end{prop}\n\\begin{proof}  Straightforward calculations show that\n $$P(P(X)Y)=P(X)P(Y)=P(XP(Y)) \\;. $$  Due to \\cite{San07} $(W,[.,.]_W)$ forms a Leibniz--Loday algebra.  By inspection the given inner product is non degenerate.  It is now a straightforward calculation that the given connection fulfills \\eqref{fwInvar}.  Thus the given structure is a V-twisted Courant algebroid.\n\\end{proof}\n\nIn fact, even the following statement is true: \n\\begin{prop} \\label{canonical}\n\\emph{Any} Leibniz--Loday algebra $(W,[ \\cdot , \\cdot ]_W)$ becomes a $V$-twisted Courant algebroid over a point in a \\emph{canonical} way. \n\\end{prop}\n\\begin{proof}  For this purpose one takes $V$ to be the two-sided ideal generated by quadratic elements, $V=\\langle [w,w]_W| w \\in W \\rangle$, and the inner product to be nothing but the symmetrization of the original bracket, $(w_1, w_2) := \\frac{1}{2} [w_1 , w_2 ]_W + \\frac{1}{2}[w_2 , w_1 ]_W$, mapping elements from $S^2W$ into $V\\subset W$. With the evident choice $\\Wconn_{w_1} (w_2) := [w_1,w_2]_W$, the conditions \\eqref{fwInvar} follow directly from \\eqref{fwJacobi}. \n\\end{proof}\n\nIn the present context the most important examples of $V$-twisted Courant algebroids are, however, provided by Q2-manifolds. This is the subject of the\nfollowing subsection.\n\n\n\\subsection{Lie-2-algebroids and  V-twisted Courant algebroids}\n\\label{subsec:Lie2}\nEvery Q2-manifold gives rise to a V-twisted Courant algebroid as follows:\n\\begin{prop}\\label{p:derVcour} For the strongly anchored DGLA $\\X_\\bullet (\\CM)$ of\na Q2-manifold (cf.~Prop.~\\ref{p:tangentLie}) the conditions in \nProp.~\\ref{p:derVino} are satisfied and the resulting Vinogradov\nalgebroid is even a V-twisted Courant algebroid.\n\\end{prop}\n\n\\begin{proof}  First we recall that $d=0$ and thus vector fields of\ndegree -2 are sections of $V\\to M$ while those of degree -1 are\nsections of the vector bundle $W\\to M$\n(cf.~Prop..~\\ref{p:tangentLie}). We have \n $[\\psi,f\\phi]=f[\\psi,\\phi]$ for all $\\psi, \\phi \\in \\Gamma(W) \\cong \\X_{-1}(\\CM)$ and\n $f \\in {C^\\infty}(M)\\cong C^\\infty_0(\\CM)$ since there are no negative\ndegree functions on $\\CM$. In every  (local) splitting of\n\\eqref{sequence} we find $W=E\\oplus E^*\\otimes V$ and  the bracket \nbetween tangent vector fields of degree -1 is just the canonical pairing\non this direct sum \\begin{equation} \\<X\\oplus a,Y\\oplus b\\> = a(Y)+b(X)\n\\label{pair} \\, ,  \\end{equation}\nwhere $X,Y \\in \\Gamma(E)$ and $a,b \\in \\Gamma(E^* \\otimes V)\\cong {}^E\\Omega^1(M,V)$.\nThis is\nobviously non-degenerate and surjective. Thus by Prop.~\\ref{p:derVino}\nwe conclude to have a Vinogradov algebroid. According to\nProp.~\\ref{p:FW} it now suffices to verify Eq.~\\eqref{FW'}: Indeed,\nthis follows from the compatibility of the Lie bracket with $\\uD$ and\nthe fact that $[\\psi,v]\\equiv 0$ $\\forall \\psi \\in \\Gamma(W) \\cong \\X_{-1}(\\CM)$ and $v \\in \n\\Gamma(V) \\cong \\X_{-2}(\\CM)$, since there are no vector fields of\ndegree -3.\\end{proof} In Section~\\ref{s:examQ} we found an\ninterpretation of the components of the vector field $Q$,\nEq.~\\eqref{Q}, on a (split) Q2-manifold in terms of a Lie\n2-algebroid. Using the same $Q$ to calculate explicitly the derived\nbracket given in the previous proposition, one arrives at the\nfollowing\n\\begin{prop}\\label{p:FWder} Let $(E,V, [.,.]_E,\\rho_E,t,{{}^E{}\\nabla},H)$ be a Lie\n2-algebroid as defined in Def.~\\ref{Lie2def\n. Then $W:=E\\oplus E^*\\otimes V$ with \n\\begin{align} \\rho_W(X\\oplus a):=&\\; \\rho_E(X) \\, , \\label{Rho}\\\\\n  \\Wconn_{X\\oplus a}v := &\\Econn_Xv -\\imath_{t(v)}a \\label{rhoQCont} \\\\\n  \\<X\\oplus a,Y\\oplus b\\> :=&\\; \\imath_Y a + \\imath_X b \\, ,  \\label{Paar}\\\\\n  [X\\oplus a, Y\\oplus b]_W :=&\\; \\left([X,Y]_E  \\oplus\n    [\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a  + \\imath_X \\imath_Y H  \\right) \\nonumber\\\\\n  &-\\left(t(a(Y))\\oplus T(a,b)\\right)\\, , \\label{brCont}\n\\end{align} where $X,Y \\in \\Gamma(E)$, $a,b \\in \n{{}^E{}\\Omega}^1(M,V)$, and $v \\in \\Gamma(V)$, is a $V$-twisted Courant\nalgebroid. Here ${{}^E\\!{}\\mathrm{D}}$ has been defined in Eq.~\\eqref{genCar} and the\noperator $T\\colon (E^*\\otimes V)^{\\otimes2} \\to E^*\\otimes V$ is induced by $t$ according to the composition of maps \n\\begin{equation} \nT(a,b)\\equiv a\\circ t\\circ b - b\\circ t\\circ a \\, . \\label{Tdefn}\n\\end{equation}\n \n\\end{prop}\n\\begin{proof}  Instead of checking all axioms separately one shows that the given algebroid is the V-twisted Courant algebroid from Proposition \\ref{p:derVcour}.\n\nThe Lie 2-algebroid induces a Q-structure via Theorem \\ref{theo1}.  To prove the above formulas one can e.g.\\ use local coordinates on the Q-manifold.\n\\end{proof}\n\nIt turns out that under certain circumstances Proposition \\ref{p:FWder} can be reversed. For that purpose we now consider a generalization of exact Courant algebroids.\nLet $W$ be an ordinary Courant algebroid. Then the anchor map $\\rho\\colon W\\to TM$ together with the inner product induce a map $\\rho^* \\colon T^*M\\to W$, where we used the inner product to identify $W^*$ with $W$.  Due to Proposition \\ref{p:Vinogrelem} the composition $\\rho\\circ\\rho^*$ vanishes. Thus for \\emph{any} Courant algebroid $W$ one has the complex\n \\begin{equation}\n 0\\to T^*M\\xrightarrow{\\rho^*} W\\xrightarrow{\\rho} TM\\to 0 \\, ; \\label{Wexact}\n\\end{equation}   \n$W$ is said to be exact iff the above short sequence is exact.\nExamples of exact Courant algebroids occur on $W=TM\\oplus T^*M$ via the choice of a closed 3-form $H$ with the bracket \\eqref{Cour}.  Conversely given an exact Courant algebroid we can choose an isotropic splitting $j\\:TM\\to E$\\footnote{We will be more explicit on this in the more general context below.} and define a 3-form $H$ as\n $$ H(X,Y,Z):= \\<[j(X),j(Y)],j(Z)\\> \\;. $$  \\eqref{cnSkew} shows that $H$ is skew-symmetric in the arguments $X$ and $Y$, while \\eqref{cInvar} implies that it is invariant under cyclic permutations of $X$, $Y$ and $Z$. Thus $H$ is a 3-form. The remaining  identity on the bracket implies that this 3-form has to be closed under the de Rham differential.  Finally, a change of splitting, which is necessarily induced by a skew-symmetric 2-form $B$, changes $H$ by an exact term, $H \\mapsto H+\\ud B$.  Thus, exact Courant algebroids are in 1:1 correspondence to classes of the third real cohomology on $M$, an observation going back to \\v{S}evera \\cite{SevLett}.\n\nWe wish to generalize this result to certain V-twisted Courant algebroids $W$.  Since the inner product on $W$ has values in $V$, the map $\\rho \\equiv \\rho_W \\colon W \\to TM$ induces a map $\\rho^* \\colon T^*M \\otimes V\\to W$ by means of $( \\rho^*(\\alpha \\otimes v), \\psi ) = \\alpha(\\rho(\\psi))$ for all $\\psi \\in W$. Thus, \\emph{any} $V$-twisted Courant algebroid $W$ gives rise to the complex\n\\begin{equation}\n 0\\to T^*M\\otimes V \\xrightarrow{\\rho^*} W\\xrightarrow{\\rho} TM\\to 0 \\, . \\label{VWexact}\n\\end{equation}\nLet us call a $V$-twisted Courant algebroid exact if this complex is acyclic, in generalization of ordinary exact Courant algebroids (which fit into this definition since for $V=M\\times \\Real$ \\eqref{VWexact} reduces to \\eqref{Wexact}). More generally, consider a map $\\pi\\colon W\\to E$. Together with the V-valued inner product it induces a map $\\pi^*\\colon E^*\\otimes V\\to W$.  Here $E^*\\otimes V$ is the $V$-dual of E and $W$ the $V$-dual of itself. We are thus led to \n\\begin{vdef}\nSuppose that in a V-twisted \nCourant algebroid $(V,W)$ the anchor map $\\rho_W \\colon W \\to TM$ factors through a bundle map $\\pi\\colon W\\to E$ for some vector bundle $E$. If the then induced complex \n\\[  0 \\rightarrow E^*\\otimes V \\xrightarrow{\\pi^*} W \\xrightarrow{\\pi} E \\to 0\n   \\label{Vexact}\\]\nis acyclic, we call $(V,W)$ an \\emph{exact $V$-twisted (or $V$-exact) Courant algebroid over $E$} and for $E=TM$, $\\pi=\\rho_W$ simply a \\emph{$V$-exact Courant algebroid}.\n\\end{vdef}\nThere then is the following remarkable fact:\n\\begin{thm}\\label{thm:der} Let $(V,W)$ be a $V$-exact Courant algebroid over some $E$ with $\\rk V>1$ and let $j\\: E\\to W$ be an isotropic \nsplitting  of \\eqref{Vexact}.\n\nThen $E$ and $V$ carry canonically the structure of a semistrict Lie 2-algebroid, $(E,V,[\\cdot, \\cdot]_E,\\rho,t,{{}^E{}\\nabla},H)$, and the structural data of the $V$-twisted Courant algebroid $W$ are necessarily of the form  of Proposition \\ref{p:FWder} above. \n\nA change of splitting $j(X)\\mapsto j(X)-B^\\#(X)$ results into a change of the underlying data $[.,.]_E$, $H$, and ${{}^E{}\\nabla}$ according to \\eqref{split1}--\\eqref{split3} furthermore.\n\\end{thm}\n\nWe first remark that an isotropic splitting of \\eqref{Vexact} always exists. This is seen as follows: First, as for any exact sequence of vector bundles, we can choose \\emph{some} splitting $j_0\\:E\\to W$ of the short exact sequence \\eqref{Vexact}.  This permits us to identify $E\\oplus E^*\\otimes V\\xrightarrow\\sim W$ via $(X,a)\\mapsto j_0(X)+\\pi^*(a)$ for $X,Y\\in E_x$, $a,b\\in E^*\\otimes V_x$ and $x\\in M$. Computing the symmetric 2-form $L(X,Y):=\\<j_0(X),j_0(Y)\\>$, $L\\:E\\otimes E\\to V$, it may or may not vanish.  If it does, $j_0$ provides an isotropic splitting.  If it does not, it induces a map $L^\\#\\:E\\to E^*\\otimes V,\\, X\\mapsto L(X,\\bullet)$.  If we change $j_0$ by $-\\pi^*\\circ L^\\#$, we obtain an isotropic splitting $W\\cong E\\oplus E^*\\otimes V$ and the inner product on $W$ is identified with the standard product on $E\\oplus E^*\\otimes V$, $\\<j(X)+\\pi^*(a),j(Y)+\\pi^*(b)\\>=a(Y)+b(X)$.\n\nWe henceforth do no more write $j$ and $\\pi^*$ explicitly, but use the identification  $W\\cong E\\oplus E^*\\otimes V$ induced by the isotropic splitting to write elements of $W$ in the form $X\\oplus a$ (instead of $j(X)+\\pi^*(a)$) with $X \\in E$ and $a \\in \\Hom(E,V)$.\n\nWe also remark that the condition on the rank of $V$ is necessary. The 1:1 relation between $V$-exact Courant algebroids over $E$ and semi-strict Lie 2-algebroids $(E,V)$ breaks down explicitly in the case of $\\rk V=1$. While due to Proposition \\ref{p:FWder} also for $V$ being a line bundle, one obtains a $V$-exact Courant algebroid over $E$ from any semi-strict Lie 2-algebroid, not any such a generalized Courant algebroid is of this form, even if the line bundle is trivial, $V=M \\times \\Real$.\nThis will become clear within the proof below, which will also permit us to provide an explicit example of this fact right after the proof.\n\\begin{proof}\nOur task now is to retrieve the data\n$[.,.]_E$, $\\rho_E$, $t$, ${{}^E{}\\nabla}$, and $H$ from the data of the $V$-twisted exact Courant algebroid over $E$ and to show that \nthey satisfy all the identities they have to so as to form a Lie 2-algebroid.\n\nWe start with Equation \\eqref{rhoQCont} and \\emph{define} ${{}^E{}\\nabla}$ and $t$ implicitly by means of\n\\[\n  \\Wconn_{X\\oplus a} v =: \\Econn_Xv -a(t(v)) \\, , \\label{naTdefn}\n\\]\nwhich has to hold true for all $v \\in \\Gamma(V)$ and $X \\oplus a \\in \\Gamma(E) \\oplus \\Gamma(\\Hom(E,V)) \\cong \\Gamma(W)$. We also know by assumption that $\\rho_W = \\rho_E \\circ \\pi$ for some $\\rho_E \\colon E \\to TM$. Noting that after the choice of a splitting, $\\pi$ just becomes projection to the first factor, $\\pi(X \\oplus a)=X$, the Leibniz property of ${{}^W\\nabla}$ and Definition \\eqref{naTdefn} (for simplicity one may set $a=0$) ensure that ${{}^E{}\\nabla}$ is an $E$-covariant derivative on $V$ with the anchor map $\\rho_E$, as anticipated by the notation chosen.  Returning with this information to \\eqref{naTdefn}, it implies that the map $t$ is $C^\\infty(M)$-linear, and thus comes from a bundle map $t\\colon V\\to E$.\n\nNext, anticipating what we want to achieve in \\eqref{brCont}, we \\emph{define} $[.,.]_E$ and some  ${H}$ from $[X,Y]_W$ by means of\n$$  [X\\oplus0,Y\\oplus0]_W =: [X,Y]_E\\oplus -{H}(X,Y) \\, .$$\nFirst we can conclude from the known properties of the left-hand-side that $[X,Y]_E$ satisfies a Leibniz property with respect to the second argument for the anchor map $\\rho_E$. Likewise we also follow that $H(\\cdot, \\cdot)$ is $C^\\infty$-linear in the second argument. Moreover, as a consequence of the isotropy of the chosen splitting, $(X\\oplus0,Y\\oplus0)=0$, and the second equation of \\eqref{fwInvar} yields antisymmetry of the $E$-bracket and $H$ in $X$ and $Y$. By definition, $H(X,Y) \\in \\Gamma(E^* \\otimes V)$. The first equation of \\eqref{fwInvar} then implies that $H(X,Y,Z) := \\imath_Z H(X,Y)$ \nis also antisymmetric in $Y$ and $Z$. Thus it is antisymmetric in all three arguments and the $C^\\infty$-linearity extends to all these arguments. All together this implies that  $H\\in{{}^E{}\\Omega}^3(M,V)$.\n\nWe now have all of the ingredients $[.,.]_E$, $\\rho_E$, $t$, ${{}^E{}\\nabla}$, and $H$ necessary to define the bracket \\eqref{brCont}. Let us denote the r.h.s.~of this equation by $[X\\oplus a,Y\\oplus b]_W^0$ and parameterize the difference to this form of the bracket by several bilinear maps $\\delta_i$ as follows:\n\\begin{align}\\label{ansatzbr}\n  [X\\oplus a,Y\\oplus b] =: &\\;[X\\oplus a,Y\\oplus b]_W^0 +\\\\ &+\\delta_1(a,Y)+\\delta_2(b,X)+\\delta_3(a,b)  \n    \\oplus  \\delta_4(X,b)+\\delta_5 (Y,a) +\\delta_6(a,b) \\, . \\nonumber\n\\end{align}\nThese maps are certainly $\\Real$-bilinear, but due to the property of $[X\\oplus a,Y\\oplus b]_W^0$ they are even $C^\\infty(M)$-bilinear (due to the already established features of the quantities entering this bracket).\n\nWe now want to show that for $\\rk V\\ne1$, the Axiom~\\eqref{fwInvar} implies that $\\delta_1 = \\ldots = \\delta_6=0$, while the Leibniz--Loday property \\eqref{fwJacobi} implies the three Equations \\eqref{HJacobi}, \\eqref{HRep}, and \\eqref{DH} needed to endow $(E,V)$ with the structure of a semi-strict Lie 2-algebroid. \n\nThe term in the center of  \\eqref{fwInvar} does not depend on the bracket at all and these terms are precisely balanced by the terms coming from $[X\\oplus a,Y\\oplus b]_W^0$ in the right equation. This implies that for equal arguments, $X=Y$ and $a=b$, all the delta-contributions have to vanish in \\eqref{ansatzbr}. This is the case iff\n\\begin{equation}\n\\delta_1=-\\delta_2 \\; , \\quad \\delta_4 = -\\delta_5 \\; , \\label{delta4} \n\\end{equation}\nand  $\\delta_3$ and $\\delta_6$ are antisymmetric in their two arguments. We now turn to the l.h.s.~of \\eqref{fwInvar}. The left Equation \\eqref{fwInvar} then gives the following additional constraints:\n\\begin{eqnarray}\n\\delta_5 &=& 0 \\, ,\\label{delta5}\\\\\nb(\\delta_2(b,X)) &=& 0 \\, ,\\label{delta2}\\\\\nb(\\delta_3(a,b)) &=& 0\\, , \\label{delta3}\\\\\n\\iota_Y\\delta_6(a,b)+ b(\\delta_1(a,Y)) &=& 0 \\, ,\\label{delta6} \n\\end{eqnarray}\nto hold true for all choices of arguments. To exploit Equations \\eqref{delta2} and \\eqref{delta3}, we prove the following\n\\begin{lemma}\\label{p:Jh=0}\n Let $V_1$ and $V_2$ be two vector spaces and let $\\Delta \\in \\left(\\Hom(V_1, V_2)\\right)^* \\otimes V_1 \\cong (V_1)^{\\otimes 2} \\otimes V_2^*$ satisfying \\begin{equation} \\label{aa}\n  a(\\Delta(a))=0\n\\end{equation} for all $a \\in \\Hom(V_1, V_2)$. Then, for $\\dim V_2 >1$, necessarily $\\Delta =0$, while for $V_2 \\cong \\Real$ any $\\Delta \\in \\Lambda^2 V_1$ has this property.\n\\end{lemma}\n\\begin{proof}[{ (of Lemma \\ref{p:Jh=0})}] \nLet us first consider the case of a factorizing $a$: $a = \\alpha \\otimes v$ with $\\alpha \\in V_1^*$ and $v \\in V_2$. Then \\eqref{aa} holds true iff $\\Delta \\in \\Lambda^2(V_1) \\otimes V_2^*$. It remains to check \\eqref{aa} for $a = a_1 + a_2$ with both summands such products. This is satisfied iff the contraction of $\\Delta$ in the $V_2$-slot with the first factor of any term of the form $v_1 \\wedge v_2 \\in \\Lambda^2 V_2$ has to vanish. If $\\dim V_2 = 1$, this condition is empty and the statement follows from $\\Lambda^2(V_1) \\otimes \\Real^* \\cong \\Lambda^2(V_1)$. In the case that $V_2$ has at least two directions, on the other hand, evidently one is forced to a vanishing $\\Delta$.\n\\end{proof}\nBy $C^\\infty(M)$-linearity, the Equations \\eqref{delta2} and \\eqref{delta3} are pointwise in $M$ and for $\\rk V>1$ we can conclude the vanishing of $\\delta_2$ and $\\delta_3$ by setting $V_1 := E|_x$, $V_2 :=V|_x$ for any fixed $x \\in M$ and $\\delta(\\cdot)$ equal to \n$\\delta_2(\\cdot , X)|_x$ and $\\delta_3(a,\\cdot)|_x$, respectively (again for any fixed $X$ and $a$, respectively). \n\nThus, if the rank condition on $V$ is satisfied, Equations \\eqref{delta4}---\\eqref{delta6} imply the vanishing of $\\delta_i$, $i=1,\\ldots,6$. It now remains to exploit Equation \\eqref{fwJacobi}. \n\nIt is a straightforward calculation to check that  \\eqref{fwJacobi}---together with the already established form of the bracket $[ \\cdot , \\cdot ]_W = [ \\cdot , \\cdot]_W^0$, i.e.~\\eqref{brCont} holding true,---implies the remaining identities \\eqref{HJacobi}, \\eqref{HRep}, and \\eqref{DH}. For example, regarding the $\\Gamma(E)$-part of  \\eqref{fwJacobi} yields \n\\begin{gather}\n [[X,Y]_E,Z]_E -[t(\\imath_Y a),Z]_E-t\\left( \\imath_Z\\left([\\imath_X,{{}^E\\!{}\\mathrm{D}}]b -\\imath_Y{{}^E\\!{}\\mathrm{D}} a  + \\imath_X \\imath_Y H  -T(a,b)\\right)\\right)= \\nonumber \\\\\n  = [X,[Y,Z]_E]_E -[X,t(\\imath_Zb)]_E -t\\left( a\\left([Y,Z]_E-t(\\imath_Z b)\\right)\\right) - \\left(X \\leftrightarrow Y \\,\\,\\text{and}\\,\\, a \\leftrightarrow b\\right)\\, ,\n\\end{gather}\nwhich, after the dust clears and using the definitions \\eqref{genCar} and \\eqref{Tdefn} for ${{}^E\\!{}\\mathrm{D}}$ and $T$, respectively, yields the the first of the three equations \nand the right hand equation of \\eqref{repr2}. Establishing the three quadratic equations on the structural data for $E$ and $V$ shows that they form a semi-strict Lie 2-algebroid.\n\nAlternatively one may also proceed as follows in the last step: Given the above data, we can define a vector field of the form \\eqref{Q} on  $E[1]\\oplus V[2]$. Vector fields of degree -1 can then be identified with sections of $W$. Equation~\\eqref{vJacobi} then translates into $0=\\frac12[[[[Q,Q],\\psi],\\phi_1],\\phi_2]$. \n It remains to check that\nthis equation contains the three Equations \\eqref{B3}, \\eqref{B5}, and \\eqref{B7}. In fact, it contains even all seven components \\eqref{B1}--\\eqref{B7} of\n$Q^2=0$ as one may convince oneself.\n\nA change of splitting of the degree 2 Q-manifold used above corresponds precisely to a change of splitting of the sequence \\eqref{Vexact} if one regards the vector fields of degree -1 on that graded manifold.  This then proves the  theorem.\n\\end{proof}\n\nWe conclude with some related remarks. First, we saw that in the case $\\rk V =1$ the discussion of $V$-exact Courant algebroids over some $E$ is at least more involved. Let us restrict to a trivial bundle $V=M \\times \\Real$ so that we can identify $\\Hom(E,V)$ with $E^*$. Then, according to the above considerations, the data appearing in Proposition \\ref{p:FWder} can be extended by a modification of the bracket by means of two tensors: $\\delta_1 = - \\delta_2 \\sim \\delta_6 \\in \\Gamma(\\Lambda^2E \\otimes E^*)$ and $\\delta_3 \\in \\Gamma(\\Lambda^3 E)$. While the general analysis of this special case gets somewhat involved---one still needs to analyze the consequences of \\eqref{fwJacobi}---, we just make two remarks at this point here: First, \\emph{every} Lie 2-algebra gives rise to a $V$-exact Courant algebroid over $E$ according to Proposition \\ref{p:FWder}, also in the case of $\\rk V =1$. Second, one simple example of a $V$-exact Courant algebroid over $E$ with $\\rk V = 1$ \\emph{not} covered by this parameterization may be provided by the following construction: Choose the Lie 2-algebroid to have all structural data vanish and take, in the above parameterization also $\\delta_1=0$ but $\\delta_3\\equiv h \\in \\Gamma(\\Lambda^3 E)$ arbitrary.  \nThen for arbitrary sections in $E \\oplus E^*\\otimes V \\cong E \\oplus E^*$ one has \n\\begin{equation}\n[X \\oplus \\alpha , Y \\oplus \\beta ]_W := h(\\alpha, \\beta) \\oplus 0 \\, . \n\\end{equation} \nIt is evident that double brackets always vanish with this definition, and thus Equation \\eqref{fwJacobi} is satisfied trivially. But also each of the three terms in \\eqref{fwInvar} vanish by the antisymmetry of $h$ in all three arguments and the vanishing of ${{}^W\\nabla}$, so that also these two equations are satisfied. \n\nWe intend to come back to a more systematic investigation of line-bundle-twisted exact Courant algebroids over some $E$ elsewhere. \n\n\n\n\n \n \n \n \n     \n \n \n \n \n \n \n\n\n\n\n\n\n \n \n \n \n \n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main  aims of research in nuclear physics is to make reliable predictions for the ground state nuclear properties of all nuclei in the periodic table with one nuclear model. For this purpose, several approaches have been developed and they can be classified into three categories: The first one called\nthe macroscopic models such as the Bethe-Weizs\u00e4cker mass formula~\\cite{samanta2002}. The second one is the macroscopic-microscopic models such as the Finite Range Droplet Model\n(FRDM)~\\cite{moller1995}. The last one consists of the microscopic models such as the conventional Hartree Fock method~\\cite{skyrme1956,decharg1980,bassem2015,bassem2016,antonov2017} with effective density-dependent interactions and its relativistic analog, the relativistic mean field (RMF) theory~\\cite{walecka1974,reinhard1989}; the former is based on the nonrelativistic kinematics, in which the density-dependent and the spin-orbit interactions are important ingredients, while the latter is based on the relativistic kinematics, \nwhere the ingredients are the nucleons, the mesons and their interactions.\nTherefore, the spin appears automatically and the interplay of the scalar and the vector potentials leads naturally to a proper spin orbit interaction and appropriate shell structure. The RMF theory  has received wide attention due to its successful description of lots of properties of nuclei not only in the valley of stability but also far away from it.~\\cite{meng1998,zhou2003,meng1999,geng2004,ring1996,meng1996,ginocchio1997}.\n\n\nDensity functional theories (DFT's) are extremely useful in understanding nuclear many-body dynamics. Among different nuclear DFTs, the covariant density functional theory (CDFT)~\\cite{niki2014,lalazissis2005,rocamaza2011,niki2002} based on the energy density functionals (EDFs) is one of the most attractive and is very successful in  describing very well the ground and excited states throughout the\nchart of nuclei~\\cite{abusara2012,agbemava2015,agbemava2017} as well as in the nuclear structure analysis~\\cite{meng2015,matev2007,afanasjev2008}.\nIn Ref.~\\refcite{Agbemava2014}, the global performance of some covariant\nenergy density functionals on some nuclear observables was analyzed.\n\nIn this work, we are interested in calculating and analyzing some ground-state properties of even-even Pt isotopes, N=82-160, within the framework of the covariant density functional theory by using two of the state-of-the-art functionals which provide a complete and an accurate description of different ground states and excited states over the whole nucleic chart~\\cite{karim2015,afanasjev2010,elbassem2019}, namely : The density-dependent point-coupling DD-PC1~\\cite{niki2008} and the density-dependent meson-exchange DD-ME2~\\cite{DD-ME2}.\n\nThe paper is organized in the following way: The covariant density functional theory and details of the numerical calculations are presented in section~\\ref{Theoretical Framework}. \nSection~\\ref{Results and Discussion} is devoted to present our results and discussion.\nFinally, the conclusions of this study are presented in section~\\ref{Conclusion}.\n\n\\section{Covariant density functional theory}\n\\label{Theoretical Framework}\nTwo classes of covariant density functional models are used throughout this paper: the density-dependent point-coupling (DD-PC) model and the density-dependent meson-exchange (DD-ME) model. \nThe first has a finite interaction range and has been fitted \nto binding energies and radii of spherical nuclei; while the latter uses a zero-range interaction and has been fitted to nuclear matter data and for finite nuclei only to binding energies  of a large range of deformed nuclei.\\\\\n\n\nIn the meson-exchange model, the nucleus is considered as a system of Dirac nucleons which interact via the exchange of mesons with finite masses leading to finite-range interactions~\\cite{typel1999,lalazissis2009}.\nThe standard Lagrangian density with medium dependence vertices that defines the meson-exchange model~\\cite{gambhir1990} is given by:\n\\begin{eqnarray}\n\\mathcal{L}  &  =\\bar{\\psi}\\left[\n\\gamma(i\\partial-g_{\\omega}\\omega-g_{\\rho\n}\\vec{\\rho}\\,\\vec{\\tau}-eA)-m-g_{\\sigma}\\sigma\\right]  \\psi\\nonumber  +\\frac{1}{2}(\\partial\\sigma)^{2}-\\frac{1}{2}m_{\\sigma}^{2}\\sigma^{2}\\\\\n&\n-\\frac{1}{4}\\Omega_{\\mu\\nu}\\Omega^{\\mu\\nu}+\\frac{1}{2}m_{\\omega}^{2}\\omega\n^{2}\\label{lagrangian}  -\\frac{1}{4}{\\vec{R}}_{\\mu\\nu}{\\vec{R}}^{\\mu\\nu}+\\frac{1}{2}m_{\\rho}%\n^{2}\\vec{\\rho}^{\\,2}-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n\\end{eqnarray}\nwhere $m$ is the bare nucleon mass and $\\psi$ denotes the Dirac spinors. $m_\\rho$, $m_\\sigma$ and $m_\\omega$ are the masses of  $\\rho$ meson,  $\\sigma$ meson and  $\\omega$ meson, \nwith the corresponding coupling constants for the mesons to the nucleons as $g_\\rho$, $g_\\sigma$ and $g_\\omega$ respectively, and $e$ is the charge of the proton.\\\\\n\nThe point-coupling model represents  an alternative formulation of the self-consistent RMF framework~\\cite{manakos1988,rusnak1997,brvenich2002,zhao2010}. The Lagrangian for the DD-PC model~\\cite{niki2008,nikolaus1992} is given by:\n\\begin{eqnarray}\n&\\mathcal{L}    =\\bar{\\psi}\\left(i\\gamma \\cdot \\partial-m\\right)  \\psi\n-\\frac{1}{2}\\alpha_S(\\hat\\rho)\\left(\\bar{\\psi}\\psi\\right)\\left(\\bar{\\psi}\\psi\\right)-\\frac{1}{2}\\alpha_V(\\hat\\rho)\\left(\\bar{\\psi}\\gamma^{\\mu}\\psi\\right)\\left(\\bar{\\psi}\\gamma_{\\mu}\\psi\\right)\\label{Lag-pc}\\\\\n&\n-\\frac{1}{2}\\alpha_{TV}(\\hat\\rho)\\left(\\bar{\\psi}\\vec\\tau\\gamma^{\\mu}\\psi\\right)\\left(\\bar{\\psi}\\vec\\tau\\gamma_{\\mu}\\psi\\right)-\\frac{1}{2}\\delta_S\\left(\\partial_v\\bar{\\psi}\\psi\\right)\\left(\\partial^v\\bar{\\psi}\\psi\\right) - e\\bar\\psi\\gamma \\cdot A\n\\frac{(1 - \\tau_3)}{2}\\psi .\\nonumber\n\\end{eqnarray}\nThe Eq.~(\\ref{Lag-pc}) contains the free-nucleon Lagrangian, the point coupling interaction terms, and the coupling of the proton to the electromagnetic field, the derivative terms account for the leading effects of finite-range interaction\nwhich are important in nuclei. \n\nIn the present work, we have used the recently developed density-dependent meson-exchange relativistic energy functional DD-ME2~\\cite{DD-ME2} and the very successful density-dependent point-coupling interaction DD-PC1~\\cite{niki2008} with the parameter sets listed in Tables~\\ref{tab1}~and~\\ref{tab2} respectively. \n\n\n\\begin{table}[!ht\n\t\\caption{ The parameters of the  DD-ME2 functional. \n\t\t\\label{tab1}}\n\t\\begin{center}%\n\t\t\\begin{tabular}\n\t\t\t[c]\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\t\\hline Parameter &  DD-ME2 \\\\\\hline\n\t\t\t$m$   & 939  \\\\\n\t\t\t$m_{\\sigma}$  & 550.1238 \\\\\n\t\t\t$m_{\\omega}$  & 783.000  \\\\\n\t\t\t$m_{\\rho}$    & 763.000 \\\\\n\t\t\t$g_{\\sigma}$  &  10.5396\\\\\n\t\t\t$g_{\\omega}$  & 13.0189\\\\\n\t\t\t$g_{\\rho}$    & 3.6836  \\\\\n\t\t\t$a_{\\sigma}$  & 1.3881 \\\\\n\t\t\t$b_{\\sigma}$  & 1.0943 \\\\\n\t\t\t$c_{\\sigma}$  & 1.7057 \\\\\n\t\t\t$d_{\\sigma}$  & 0.4421 \\\\\n\t\t\t$a_{\\omega}$  & 1.3892 \\\\\n\t\t\t$b_{\\omega}$  & 0.9240 \\\\\n\t\t\t$c_{\\omega}$  & 1.4620 \\\\\n\t\t\t$d_{\\omega}$  & 0.4775 \\\\\n\t\t\t$a_{\\rho}$    & 0.5647 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{table}[!ht]\n\t\\caption{The parameters of the DD-PC1 functional.}%\n\t\\label{tab2}\n\t\\begin{center}\n\t\t\\begin{tabular}\n\t\t\t[c]\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\t\\hline Parameter &  DD-PC1 \\\\\\hline\n\t\t\t$m$ &   939\\\\\n\t\t\t$a_{\\sigma}$ & -10.04616\\\\\n\t\t\t$b_{\\sigma}$ & -9.15042\\\\\n\t\t\t$c_{\\sigma}$ & -6.42729\\\\\n\t\t\t$d_{\\sigma}$ &  1.37235\\\\\n\t\t\t$a_{\\omega}$ &  5.91946\\\\\n\t\t\t$b_{\\omega}$ &  8.86370\\\\\n\t\t\t$d_{\\omega}$ &  0.65835\\\\\n\t\t\t$b_{\\rho}$   &  1.83595\\\\\n\t\t\t$d_{\\rho}$   &  0.64025\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\n\n\\section{Results and Discussion}\n\\label{Results and Discussion}\n\nIn this section, we present the ground-state properties of $^{160-238}$Pt nuclei obtained in the framework of the CDFT by using the interactions DD-ME2~\\cite{DD-ME2} and DD-PC1~\\cite{niki2008}. \nOur results are compared with the available experimental data, the predictions of the RMF model with NL3~\\cite{lalazissis1999} functional and with the results of HFB theory with SLy4 Skyrme force calculated by using the computer code HFBTHO v2.00d~\\cite{Stoitsov2013} as in Ref~\\refcite{bassem2017}. \n\n\\subsection{Binding energy}\n\nBinding energy (BE) is directly related to the stability of nuclei and is a very important quantity in nuclear physics. \nThe binding energies per nucleon (BE\/A) of ground states for platinum isotopes, $^{160-238}$Pt, are presented in Fig.~\\ref{BEexp} as a function of the neutron number N. The available\nexperimental data~\\cite{wang2012} as well as the predictions of the RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4) theories are also shown for comparison.\n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.5]{BE_Pt.eps}\n\t\\caption{(Color online) The binding energies per nucleon for even-even $^{166-238}$Pt isotopes.}\n\t\\label{BEexp}\n\\end{figure}\t\t\n\nIt can be clearly seen from Fig. \\ref{BEexp} that the theoretical predictions reproduce the experimental data accurately and, qualitatively, all curves show a similar behavior.\n\nIn order to provide a further check of the accuracy of our results, the differences between the experimental total BE for even-even platinum isotopes and the calculated results obtained in this work are presented in Fig.~\\ref{Delta_BE} as a function of the neutron number N. \n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.5]{deltaBE_Pt.eps}\n\t\\caption{(Color online) Differences between theoretical total binding energies and experimental values for even-even Pt isotopes.}\n\t\\label{Delta_BE}\n\\end{figure}\n\nThe comparison of the available experimental total binding energies of the ground state for $^{166-238}$Pt with the present calculations and with the other theoretical models, is also done by the root mean square (rms) deviation tabulated in Table~\\ref{rms}. \n\n\n\\begin{table}[!htb]\n\t\\caption{The rms deviations of the total binding energies of platinum isotopes.\\label{rms}}\n\t\\centering\n\t\\begin{tabular}\n\t\t{@{\\hspace{0pt}}c@{\\hspace{24pt}}c@{\\hspace{24pt}}c@{\\hspace{0pt}}@{\\hspace{24pt}}c@{\\hspace{0pt}}}\n\t\t\\hline\\noalign{\\smallskip}\n\t\tDD-ME2 \t&   DD-PC1\t&  NL3\t\t & SLy4\\\\ \n\t\t\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\t\t1.864 \t&\t1.880\t&\t2.755\t & \t4.019\t\\\\  \n\t\t\\noalign{\\smallskip}\\hline\n\t\\end{tabular}\n\\end{table}\n\nAs we can see from Fig.~\\ref{Delta_BE} and Table~\\ref{rms}, DD-ME2 exceeds in accuracy the other nuclear models DD-PC1, NL3 and SLy4.\n\n\\subsection{Two neutron separation energy ($S_{2n}$) and shell gap ($\\delta_{2n}$)}\n\nIn the present work, we have calculated two-neutron separation energies, \n$S_{2n}(N,Z)$ = BE$(N,Z)$ - BE$(N - 2,Z)$, for Pt isotopes by using the density-dependent effective interactions DD-PC1 and DD-ME2. \n\nIn Fig.~\\ref{S2n}, we display the calculated S$_{2n}$ of  even-even platinum isotopes, as a function of the neutron number N, in comparison with the available experimental data~\\cite{wang2012} and the predictions of RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4).\n\n\\begin{figure}[ht]\n\t\\centering \\includegraphics[scale=0.52]{S2n_Pt.eps}\n\t\\caption{(Color online) The two-neutron separation energies, S$_{2n}$, for Pt\n\t\tisotopes.}\n\t\\label{S2n}\n\\end{figure}\n\nAs one can see from Fig.~\\ref{S2n}, the results of the two density-dependent models DD-ME2 and DD-PC1 as well as those of NL3 and SLy4 reproduce the experimental data quite well except some small discrepancies which are mainly due to the missing beyond mean field corrections~\\cite{Litvinova2006}. S$_{2n}$ gradually decreases with N, and a sharp drop is distinctly seen at N = 126 in both experimental and theoretical curves, except in SLy4 curve, which corresponds to the closed shell at this magic neutron number. \nThe discontinuity of S$_{2n}$ at this nucleus indicates that it corresponds to a critical point of first order phase transition to prolate shape.\n\nA more sensitive observable for locating the shell closure is the two-neutron shell gap $\\delta_{2n}=S_{2n}(N,Z)-S_{2n}(N+2,Z)$, also known as the S$_{2n}$~differential. \n$\\delta_{2n}$ is shown in Fig.~\\ref{ShellGap} as a function of the neutron number N. The strong peaking in the two-neutron shell gap ($\\delta_{2n}$) clearly seen at N~=~126 further supports the shell closure at this neutron magic number as shown in Fig.~\\ref{S2n} by two-neutron separation energy ($S_{2n}$). Here again, only the SLy4 curve does not show a strong peak at the neutron magic number N~=~126. \n\n\n\\begin{figure}[!ht]\n\t\\centering \\includegraphics[scale=0.55]{ShellGap_Pt.eps}\n\t\\caption{(Color online) The two-neutron shell gap $\\delta_{2n}$ for even-even $^{94-168}$Pt isotopes.}\n\t\\label{ShellGap}\n\\end{figure}\n\n\n\n\\subsection{Quadrupole deformation} \n\nThe quadrupole deformation is also an important property for describing the\nstructure and shape of the nucleus. \nIn Fig.~\\ref{23beta}, we show for every Pt isotope (covering the mass\ninterval $160 \\leqslant A \\leqslant 204$) the energy curves along\nthe axial symmetry axis, as a function of the deformation\nparameter, $\\beta$, obtained within CDFT framework by using the density-dependent effective interactions DD-ME2 and DD-PC1.\n\n\n\\begin{figure}[!htb]\n\t\\centering \\includegraphics[scale=0.6]{23beta_Pt.eps}\n\t\\caption{(Color online) The total energy curves for $^{160-204}$Pt as a function of the axial quadrupole deformation parameter $\\beta_2$.}\n\t\\label{23beta}\n\\end{figure}\n\nAs we can see from Fig.~\\ref{23beta}, the interaction DD-PC1 provides potential\nenergy curves which are extremely similar to the ones obtained with DD-ME2. The deformations of the oblate and prolate minima are practically independent of the force. \n\nThe lightest isotopes, $^{160-162}$Pt, exhibit spherical shape. The next isotope, $^{164}$Pt, starts to develop two shallow degenerate minima, oblate\nand prolate, that correspond to a small value of $\\beta$. The next isotope, $^{166}$Pt, starts to develop a more pronounced prolate minimum. The $^{168-186}$Pt isotopes show a similar structure, with a well-deformed prolate minimum, $\\beta \\approx 0.3$, and an oblate local minimum. \n\nA transition from prolate to oblate shapes occurs smoothly between $^{188}$Pt(prolate) and $^{190}$Pt(oblate). In $^{190-200}$Pt two minima appear, with the opposite situation\noccurring in $^{168-186}$Pt. As the mass number increases, the two well-deformed minima gradually disappear and we get a flat potential energy curve at A = 202. At A = 204, we get a sharp single minimum, which confirms the spherical shape at the magic neutron number N = 126. \n\nThese results are in good agreement with recent works \\cite{Rodriguez2010,Garcia2014,Nomura2011} in terms of predicting $^{188}$Pt as critical. However, other calculations have different results that are not in agreement with ours, such as: Ref.~\\refcite{Yao2013} which predicts that the shape transition in Pt isotopes within a beyond-mean-field approach with the Skyrme SLy6 occurs at A = 186 to 188 instead of A = 188 to 190 in our calculations. In the same line, \nconstrained Hartree-Fock+ BCS calculations with the Skyrme forces Sk3, SGII, and SLy4 suggest a prolate to oblate shape transition at $^{182}$Pt~\\cite{Boillos2015}. Furthermore, triaxial D1M-Gogny calculations predict a smooth shape transition at A = 184 to 186~\\cite{Nomura2013}.\n\nThese differences between theoretical methods in predicting the exact location of the shape transition are due, firstly, to the difference between the models used and, secondly, to the fact that the shape transition is very sensitive to small details of the calculation, since the shape transition occurs exactly around the region where the energy of the competing shapes are practically degenerate.\n\nIn Fig.~\\ref{PESs} we display the triaxial contour plots of $^{186-190}$Pt\nisotopes in the ($\\beta$, $\\gamma$) plane. To study the dependency on $\\gamma$,  systematic constrained triaxial calculations have been done for mapping the quadrupole deformation space defined by $\\beta_2$ and $\\gamma$ using the effective interaction DD-ME2. \nThe constrained calculations are performed by imposing constraints on both axial and triaxial mass quadrupole moments. The potential energy surface\n(PES) study as a function of the quadrupole deformation parameter is performed by the method of quadratic constraint~\\cite{Ring1980}  (see Ref.~\\refcite{Abusara2017} for more details).\nEnergies are normalized with respect to the binding energy of the global minimum such that the ground state has a zero MeV energy.\n\n\n\\begin{figure}[!htb]\n\t\\minipage{0.5\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{Pt186.eps}\n\t\\endminipage\\hfill\n\t\\minipage{0.5\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{Pt188.eps}\n\t\\endminipage\\hfill\n\t\\centering \\includegraphics[scale=0.4]{Pt190.eps}\n\t\\caption{(Color online) Potential energy surfaces for  $^{186-190}$Pt in the ($\\beta$, $\\gamma$) plane, obtained from a CDFT calculations with the DD-ME2 parameter set. The color scale shown at the right has the unit of MeV, and scaled such that the ground state has a zero MeV energy.}\n\t\\label{PESs}\n\\end{figure}\n\nFrom this figure, we can notice that the location of the ground state minimum moves from near prolate shape at $^{186}$Pt to near oblate shape at $^{190}$Pt. $^{188}$Pt is slightly triaxial with its global minimum at (0.25, 10\\si{\\degree}). Thus, the shape transition is smooth, and there are no sudden changes in the nuclear shape. These  results confirm those seen previously in Fig.~\\ref{23beta} and are in full agreement with the results shown in Fig. 5 of Ref.~\\cite{Garcia2014} obtained with Hartree-Fock-Bogoliubov based on Gogny-D1S interaction. \n\n\n\\subsection{Neutron, Proton and Charge radii}\n\nThe charge radii calculated within the framework of the covariant density functional theory by using the functionals DD-ME2 and DD-PC1 are compared with the available experimental data~\\cite{angeli2004} and with the predictions of RMF(NL3)~\\cite{lalazissis1999} and HFB(SLy4). \nGood agreement between theory and experiment can be clearly seen in Fig.~\\ref{Rc}, except in the region $100 \\le N \\le 108$ where a small difference between the models and experiment is seen. This discrepancy is mainly due to the deformation effect since the experimental $\\beta_2$ values are extracted from experimental $B(E2)$$\\uparrow$ values (the reduced probability of the transition from the ground state of the nucleus to the first excited $2^+$ state) by using the Bohr model, which is valid only in the case of well-deformed nuclei, as it has been explained in Ref.~\\refcite{elbassem2019}.\\\\\n\n\\begin{figure}[ht]\t\t\t\t\t\t\t\t   \n\t\\centering \\includegraphics[scale=0.55]{Rc_Pt.eps}\n\t\\caption{(Color online) The charge radii of Pt isotopes}\n\t\\label{Rc}\n\\end{figure}\t\t\t\t\t\t\t\t   \n\nWe display in Fig. \\ref{RnRp} the calculated neutron and proton radii, $R_n$ and $R_p$, of platinum isotopes obtained by using the two functionals DD-ME2 and DD-PC1. The results of HFB theory with SLy4 Skyrme force as well as those of RMF model with NL3 functional are also shown for comparison.  \n\n\n\\begin{figure}[!htb]\n\t\\minipage{0.48\\textwidth}\n\t\\centering \\includegraphics[scale=0.4]{R_Pt.eps}\n\t\\endminipage\\hfill\n\t\\minipage{0.48\\textwidth}\n\t\\centering \\includegraphics[scale=0.41]{Rn-Rp_Pt.eps}\n\t\\endminipage\\hfill\n\t\\caption{(Color online) The neutron and proton radii of Pt isotopes (left panel) and the neutron skin thicknesses ($\\Delta R$ = $R_n$ \u2212 $R_p$)(right panel).}\n\t\\label{RnRp}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{RnRp}, we see that the neutron rms radii obtained by using the density-dependent\neffective interactions DD-ME2 and DD-PC1 and by HFB(SLy4) are almost equal\nthroughout the isotopic chain, but the NL3 results are overestimated. The origin of this difference is the missing density-dependence in the iso-vector channel of NL3~\\cite{lalazissis2005}. The proton rms radii obtained by all the formalisms, CDFT(DD-ME2 and DD-PC1), HFB(SLy4) and RMF(NL3) are almost similar.\n\nWe also notice from Fig.~\\ref{RnRp} (right panel) that the difference between the rms radii of neutrons and protons ($\\Delta R = R_n - R_p$) increases by increasing the neutron number in favor of developing a neutron skin. \n$\\Delta R$ attains its maximum for $^{238}$Pt in which it reaches 0.41~fm in the case of DD-ME2 and DD-PC1 formalisms, 0.38~fm in HFB(SLy4) and 0.55~fm in RMF(NL3) calculations.\n\nThere are also some abnormalities in the charge and proton radii of Pt isotopes as we can see in Figs.~\\ref{Rc} and~\\ref{RnRp} (left panel), these little jumps in $R_c$ and $R_p$ in the two regions $98 \\leqslant N \\leqslant 108$ and $140 \\leqslant N \\leqslant 160$ are due to the deformation effect~\\cite{elbassem2019}.\\\\\n\n\\section{Conclusion}\n\\label{Conclusion}\nIn the present work, we have studied the ground state properties of even-even\nplatinum isotopes, $^{160-238}$Pt, from the proton-rich side up to the neutron-rich one within the framework of the covariant density functional theory, by using two of the most recent functionals: The density-dependent point-coupling DD-PC1 and the density-dependent meson-exchange DD-ME2.\nThe bulk ground state properties are quite well reproduced in our calculations and are in good agreement with the experimental data.\nA strong shell closure is clearly seen at N=126. The neutron skin in this study reaches 0.41 fm for $^{238}$Pt.  \nThe total energy curves for $^{160-204}$Pt obtained in this work suggest a smooth\nprolate to oblate shape transition at $^{188}$Pt.\\\\\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe sampling Kantorovich operators have been introduced to approximate and reconstruct not necessarily continuous signals. In \\cite{BABUSTVI}, the authors introduced this operators starting from the well-known generalized sampling operators (see e.g. \\cite{BUFIST,BUST2,BAVI2,VI1,BABUSTVI2}) and replacing, in their definition, the sample values $f(k\/w)$ with $w \\int_{k\/w}^{(k+1)\/w}f(u)\\,du$. Clearly, this is the most natural mathematical modification to obtain operators which can be well-defined also for general measurable, locally integrable functions, not necessarily continuous. Moreover, this situation very often occur in Signal Processing, when one cannot match exactly the sample at the point $k\/w$: this represents the so-called \"time-jitter'' error. The theory of sampling Kantorovich operators allow us to reduces the time-jitter error, calculating the information in a neighborhood of $k\/w$ rather that exactly in the node $k\/w$. These operators, as the generalized sampling operators, represent an approximate version of classical sampling series, based on the Whittaker-Kotelnikov-Shannon sampling theorem (see e.g. \\cite{ANVI}).\n\n   Subsequently, the sampling Kantorovich operators have been studied in various settings. In \\cite{COVI,COVI2} the multivariate version of these operators were introduced. Results concerning the order of approximation are shown in \\cite{COVI3}. Extensions to more general contexts are presented in \\cite{VIZA1,VIZA2,VIZA3,BAMA2}.\n\n   The multivariate sampling Kantorovich operators considered in this paper are of the form:\n$$\n(S_w f)(\\underline{x})\\ :=\\ \\sum_{\\underline{k} \\in \\Z^n} \\chi(w\\underline{x}-t_{\\underline{k}})\\left[\\frac{w^n}{A_{\\underline{k}}} \\int_{R_{\\underline{k}}^w}f(\\underline{u})\\ d\\underline{u}\\right], \\hskip0.7cm (\\underline{x} \\in \\R^n), \\hskip0.8cm \\mbox{(I)}\n$$\nwhere $f: \\R^n \\to \\R$ is a locally integrable function such that the above series is convergent for every $\\underline{x} \\in \\R^n$. The symbol $t_{\\underline{k}}=\\left(t_{k_1},...,t_{k_n}\\right)$ denotes vectors where each $(t_{k_i})_{k_i \\in \\Z}$, $i=1,...,n$ is a certain strictly increasing sequence of real numbers with $\\Delta_{k_i}=t_{k_{i+1}}-t_{k_i}>0$.\nNote that, the sequences $(t_{k_i})_{k_i \\in \\Z}$ are not necessary equally spaced. We denote by $R_{\\underline{k}}^w$ the sets:\n$$\nR_{\\underline{k}}^w\\ :=\\ \\left[\\frac{t_{k_1}}{w},\\frac{t_{k_1+1}}{w}\\right]\\times\\left[\\frac{t_{k_2}}{w},\\frac{t_{k_2+1}}{w}\\right]\\times...\\times\\left[\\frac{t_{k_n}}{w},\\frac{t_{k_n+1}}{w}\\right], \\hskip2.2cm \\mbox{(II)}\n$$\n$w>0$ and $A_{\\underline{k}} = \\Delta_{k_1} \\cdot \\Delta_{k_2} \\cdot...\\cdot \\Delta_{k_n}$, $\\underline{k} \\in \\Z^n$. Moreover, the function $\\chi:\\R^n \\to \\R$ is a kernel satisfying suitable assumptions. \n\n   For the operators in (I) we recall some convergence results. We have that the family $(S_wf)_{w>0}$ converges pointwise to $f$, when $f$ is continuous and bounded and $(S_w f)_{w>0}$ converges uniformly to $f$, when $f$ is uniformly continuous and bounded. Moreover, to cover the case of not necessarily continuous signal, we study our operators in the general setting of Orlicz spaces $L^{\\varphi}(\\R^n)$. For functions $f$ belonging to $L^{\\varphi}(\\R^n)$ and generated by the convex $\\varphi$-function $\\varphi$, the family of sampling Kantorovich operators is \"modularly'' convergent to $f$, being the latter the natural concept of convergence in this setting.  \n\n   The latter result, allow us to apply the theory of the sampling Kantorovich operators to approximate and reconstruct images. In fact, static gray scale images are characterized by jumps of gray levels mainly concentrated in their contours or edges and this can be translated, from a mathematical point of view, by discontinuities (see e.g. \\cite{GOWO}).\n\n  Here, we introduce and analyze in detail some practical applications of the sampling Kantorovich algorithm to thermographic images, very useful for the analysis of buildings in seismic engineering. The thermography is a remote sensing technique, performed by the image acquisition in the infrared. Thermographic images are widely used to make non-invasive investigations of structures, to analyze the story of the building wall, to make diagnosis and monitoring buildings, and to make structural measurements. A further important use, is the application of the texture algorithm for the separation between the bricks and the mortar in masonries images. Through this procedure the civil engineers becomes able to determine the mechanical parameters of the structure under investigation. Unfortunately, the direct application of the texture algorithm to the thermographic images, can produce errors, as an incorrect separation between the bricks and the mortar. \n\n  Then, we use the sampling Kantorovich operators to process the thermographic images before to apply the texture algorithm. In this way, the result produced by the texture becomes more refined and therefore we can apply structural analysis after the calculation of the various parameters involved. In order to show the feasibility of our applications, we present in detail a real-world case-study.\n\n\n\\section{Preliminaries}\n\nIn this section we recall some preliminaries, notations and definitions.\n\n  We denote by $C(\\R^n)$ (resp. $C^0(\\R^n)$) the space of all uniformly continuous and bounded (resp. continuous and bounded) functions $f:\\R^n\\to \\R$ endowed with the usual sup-norm $\\|f\\|_{\\infty} := \\sup_{\\uu \\in \\R^n}\\left|f(\\uu)\\right|$, $\\uu = (u_1,\\, ...,\\, u_n)$, and by $C_c(\\R^n) \\subset C(\\R^n)$ the subspace of the elements having compact support. Moreover, $M\\left(\\R^n\\right)$ will denote the linear space of all (Lebesgue) measurable real functions defined on $\\R^n$.\n\n   We now recall some basic fact concerning Orlicz spaces, see e.g. \\cite{MUORL,MU1,RAO1,BAVI_1,BAMUVI}.  \n\n\\noindent   The function $\\varphi: \\R^+_0 \\to \\R^+_0$ is said to be a $\\varphi$-function if it satisfies the following assumptions: $(i)$ $\\varphi \\left(0\\right)=0$, and $\\varphi \\left(u\\right)>0$ for every $u>0$; $(ii)$ $\\varphi$ is continuous and non decreasing on $\\R^+_0$; $(iii)$\n$\\lim_{u\\to \\infty}\\varphi(u)\\ =\\ + \\infty$.\n\n  The functional $I^{\\varphi} : M(\\R^n)\\to [0,+\\infty]$ (see e.g. \\cite{MU1,BAMUVI}) defined by\n$$\nI^{\\varphi} \\left[f\\right] := \\int_{\\R^n} \\varphi(\\left| f(\\underline{x}) \\right|)\\ d\\underline{x},\\ \\hskip0.5cm \\left(f \\in M(\\R^n)\\right),\n$$\nis a modular in $M(\\R^n)$ . The Orlicz space generated by $\\varphi$ is given by\n$$\nL^{\\varphi}(\\R^n) := \\left\\{f \\in M\\left(\\R^n\\right):\\ I^{\\varphi} [\\lambda f]<+\\infty,\\ \\mbox{for\\ some}\\ \\lambda>0\\right\\}.\n$$\nThe space $L^{\\varphi}(\\R^n)$ is a vector space and an important subspace is given by\n$$\nE^{\\varphi}(\\R^n)  := \\left\\{f \\in M\\left(\\R^n\\right):\\ I^{\\varphi} [\\lambda f]<+\\infty,\\ \\mbox{for\\ every}\\ \\lambda>0\\right\\}.\n$$\n$E^{\\varphi}(\\R^n)$ is called the space of all finite elements of $L^{\\varphi}(\\R^n)$. It is easy to see that the following inclusions hold:\n$\nC_c(\\R^n) \\subset E^{\\varphi}(\\R^n) \\subset L^{\\varphi}(\\R^n).\n$\nClearly, functions belonging to $E^{\\varphi}(\\R^n)$ and $L^{\\varphi}(\\R^n)$ are not necessarily continuous. A norm on $L^{\\varphi}(\\R^n)$, called Luxemburg norm, can be defined by\n$$\n\\left\\|f\\right\\|_{\\varphi}\\ :=\\ \\inf \\left\\{\\lambda>0:\\ I^{\\varphi}[f\/\\lambda] \\miu \\lambda\\right\\}, \\hskip0.7cm (f \\in L^{\\varphi}(\\R^n)).\n$$\n\n   We will say that a family of functions $(f_w)_{w>0} \\subset L^{\\varphi}(\\R^n)$ is norm convergent to a function $f \\in L^{\\varphi}(\\R^n)$, i.e., $\\left\\|f_w-f\\right\\|_{\\varphi}\\rightarrow 0$ for $w\\to +\\infty$, if and only if \n$\\lim_{w \\to +\\infty} I^{\\varphi}\\left[\\lambda(f_w-f)\\right]\\ =\\ 0$, for every $\\lambda>0$.\nMoreover, an additional concept of convergence can be studied in Orlicz spaces: the \"modular convergence\". The latter induces a topology (modular topology) on the space $L^{\\varphi}(\\R^n)$ (\\cite{MU1,BAMUVI}). \n\n  We will say that a family of functions $(f_w)_{w>0} \\subset L^{\\varphi}(\\R^n)$ is modularly convergent to a function $f \\in L^{\\varphi}(\\R^n)$ if\n$\n\\lim_{w \\to +\\infty} I^{\\varphi}\\left[\\lambda(f_w-f)\\right]\\ =\\ 0$,\nfor some $\\lambda>0$. Obviously, norm convergence implies modular convergence, while the converse implication does not hold in general. The modular and norm convergence are equivalent if and only if the $\\varphi$-function $\\varphi$ satisfies the $\\Delta_2$-condition, see e.g., \\cite{MU1,BAMUVI}. Finally, as last basic property of Orlicz spaces, we recall the following.\n\\begin{lemma}[\\cite{BAMA}] \\label{lemma1}\nThe space $C_c(\\R^n)$ is dense in $L^{\\varphi}(\\R^n)$ with respect to the modular topology, i.e., for every $f \\in L^{\\varphi}(\\R^n)$ and for every $\\ep>0$ there exists a constant $\\lambda>0$ and a function $g \\in C_c(\\R^n)$ such that $I^{\\varphi}[\\lambda(f-g)]<\\ep$.\n\\end{lemma}\n\n\n\\section{The sampling Kantorovich operators}\n\nIn this section we recall the definition of the operators with which we will work. We will denote by \n$t_{\\underline{k}}=\\left(t_{k_1},...,t_{k_n}\\right)$ a vector where each element $(t_{k_i})_{k_i \\in \\Z}$, $i=1,...,n$ is a sequence of real numbers with $-\\infty < t_{k_i}<t_{k_{i+1}}<+\\infty$, $\\lim_{k_i \\to \\pm \\infty}t_{k_i} = \\pm \t\\infty$, for every $i=1, ..., n$, and such that there exists $\\Delta$, $\\delta>0$ for which $\\delta \\leq \\Delta_{k_i}:= t_{k_{i+1}}-t_{k_{i}} \\leq  \\Delta$, for every $i=1, ... , n$.\nNote that, the elements of $(t_{k_i})_{k_i \\in \\Z}$ are not necessary equally spaced.  In what follows, we will identify with the symbol $\\Pi^n$ the sequence $(t_{\\underline{k}})_{\\kk \\in \\Z^n}$.\n\n  A function $\\chi: \\R^n \\to \\R$ will be called a kernel if it satisfies the following properties:\n\\begin{itemize}\n\t\\item[$(\\chi1)$] $\\chi \\in L^1(\\R^n)$ and is bounded in a neighborhood of $\\underline{0} \\in \\R^n$; \n\t\\item[$(\\chi 2)$] for every $\\underline{u} \\in \\R^n$, \\hskip0.2cm $\\displaystyle \\sum_{\\underline{k} \\in \\Z^n} \\chi(\\underline{u} - t_{\\underline{k}})\\ =\\ 1$;\n\t\\item[$(\\chi 3)$] for some $\\beta>0$,\n$$\nm_{\\beta, \\Pi^n }(\\chi)\\ =\\ \\sup_{\\underline{u} \\in \\R^n}\\sum_{\\underline{k} \\in \\Z^n}\\left|\\chi(\\underline{u}-t_{\\underline{k}})\\right|\\cdot \\left\\|\\underline{u}-t_{\\underline{k}}\\right\\|^{\\beta}_2\\ <\\ +\\infty,\n$$\nwhere $\\| \\cdot \\|_2$ denotes the usual Euclidean norm. \n\\end{itemize}\nWe now recall the definition of the linear multivariate sampling Kantorovich operators introduced in \\cite{COVI}. Define:\n\\begin{equation} \\label{KANTO}\n(S_w f)(\\underline{x})\\ :=\\ \\sum_{\\underline{k} \\in \\Z^n} \\chi(w\\underline{x}-t_{\\underline{k}})\\left[\\frac{w^n}{A_{\\underline{k}}} \\int_{R_{\\underline{k}}^w}f(\\underline{u})\\ d\\underline{u}\\right], \\hskip0.7cm (\\underline{x} \\in \\R^n),\n\\end{equation}\nwhere $f: \\R^n \\to \\R$ is a locally integrable function such that the above series is convergent for every $\\underline{x} \\in \\R^n$, where\n\\begin{displaymath}\nR_{\\underline{k}}^w\\ :=\\ \\left[\\frac{t_{k_1}}{w},\\frac{t_{k_1+1}}{w}\\right]\\times\\left[\\frac{t_{k_2}}{w},\\frac{t_{k_2+1}}{w}\\right]\\times...\\times\\left[\\frac{t_{k_n}}{w},\\frac{t_{k_n+1}}{w}\\right],\n\\end{displaymath}\n$w>0$ and $A_{\\underline{k}} = \\Delta_{k_1} \\cdot \\Delta_{k_2} \\cdot...\\cdot \\Delta_{k_n}$, $\\underline{k} \\in \\Z^n$. The operators in (\\ref{KANTO}) have been introduced in \\cite{BABUSTVI} in the univariate setting.\n\\begin{remark} \\label{remark1} \\rm\n(a) Under conditions $(\\chi 1)$ and $(\\chi 3)$, the following properties for the kernel $\\chi$ can be proved:\n$$\nm_{0, \\Pi^n}(\\chi)\\ :=\\ \\sup_{\\uu \\in \\R^n} \\sum_{\\kk \\in \\Z^n} \\left|\\chi(\\uu-\\tk)\\right|\\ <\\ +\\infty,\n$$\nand, for every $\\gamma >0$\n\\begin{equation} \\label{eee}\n\\lim_{w \\to +\\infty} \\sum_{\\left\\|w \\uu-\\tk\\right\\|_2 >\\gamma w}\\left|\\chi(w \\uu - \\tk)\\right|\\ =\\ 0,\n\\end{equation}\nuniformly with respect to $\\uu \\in \\R^n$, see \\cite{COVI}.\n\\vskip0.2cm\n\\noindent (b) By (a), we obtain that $S_w f$ with $f \\in L^{\\infty}(\\R^n)$ are well-defined. Indeed,\n\\begin{displaymath}\n\\left|(S_w f)(\\xx)\\right|\\ \\miu\\ m_{0, \\Pi^n}(\\chi)\\left\\|f\\right\\|_{\\infty}\\ < +\\infty,\n\\end{displaymath}\nfor every $\\xx \\in \\R^n$, i.e. $S_w : L^{\\infty}(\\R^n)\\rightarrow L^{\\infty}(\\R^n)$.\n\\end{remark}\n\\begin{remark} \\label{remark3} \\rm\nNote that, in the one-dimensional setting, choosing $t_{k}=k$, for every $k \\in \\Z$, condition $(\\chi 2)$ is equivalent to \n\\begin{displaymath}\n\\widehat{\\chi}(k) :=\\ \\left\\{\n\\begin{array}{l}\n0, \\hskip0.5cm k \\in \\Z\\setminus \\left\\{0\\right\\}, \\\\\n1, \\hskip0.5cm k=0,\n\\end{array}\n\\right.\n\\end{displaymath}\nwhere $\\widehat{\\chi}(v):=\\int_{\\R}\\chi(u)e^{-ivu}\\ du$, $v \\in \\R$, denotes the Fourier transform of $\\chi$; see \\cite{BUNE,BABUSTVI,COVI,COVI3}.\n\\end{remark}\n\n\n\n\n\\section{Convergence results}\n\nIn this section, we show the main approximation results for the multivariate sampling Kantorovich operators. In \\cite{COVI}, the following approximation theorem for our operators has been proved. \n\\begin{theorem} \\label{th1}\nLet $f \\in C^0(\\R^n)$. Then, for every $\\underline{x} \\in \\R^n$,\n\\begin{displaymath}\n\\lim_{w \\to +\\infty} (S_w f)(\\underline{x})\\ =\\ f(\\underline{x}).\n\\end{displaymath}\nIn particular, if $f \\in C(\\R)$, then\n\\begin{displaymath}\n\\lim_{w \\to +\\infty} \\left\\|S_w f -f\\right\\|_{\\infty}\\ =\\ 0.\n\\end{displaymath}\n\\end{theorem}\n\\begin{proof}\nHere we highlight the main points of the proof.\n\n\\noindent Let $f \\in C^0(\\R^n)$ and $\\xx \\in \\R^n$ be fixed. By the continuity of $f$ we have that for every fixed $\\ep>0$ there exists $\\gamma>0$ such that $|f(\\xx)-f(\\uu)|<\\ep$ for every $\\| \\xx -\\uu\\|_2 \\miu \\gamma$, $\\uu \\in \\R^n$. Then, by $(\\chi 2)$ we obtain:\n$$\n\\left|(S_w f)(\\xx)-f(\\xx)\\right|\\ \\miu\\ \\sum_{\\kk \\in \\Z^n} \\left|\\chi(w \\xx - \\tk)\\right| \\frac{w^n}{A_{\\kk}}\\int_{R^w_{\\kk}}\\left|f(\\uu)-f(\\xx)\\right|\\, d\\uu\\ \n$$\n\\vskip-0.5cm\n\\begin{eqnarray*}\n&=& (\\sum_{\\left\\|w \\xx-\\tk\\right\\|_2 \\miu w \\gamma\/2}+\\sum_{\\left\\|w \\xx-\\tk\\right\\|_2 > w \\gamma\/2})\\left|\\chi(w \\xx - \\tk)\\right|\\frac{w^n}{A_{\\kk}}\\int_{R^w_{\\kk}}\\left|f(\\uu)-f(\\xx)\\right|\\ d\\uu \\\\\n&=&\\ I_1+I_2.\n\\end{eqnarray*}\nFor $\\uu \\! \\in \\! R^w_{\\kk}$ and $\\left\\|w \\xx-\\tk\\right\\|_2\\! \\miu \\! w \\gamma\/2$ we have $\\|\\uu-\\xx\\|_2\\! \\miu \\! \\gamma$ for  $w\\! > \\!0$ sufficiently large, then by the continuity of $f$ we obtain $I_1 \\miu m_{0,\\Pi^n}(\\chi) \\ep$ (see Remark \\ref{remark1} (a)). Moreover, by the boundedness of $f$ and (\\ref{eee}) we obtain $I_2 \\miu 2\\|f\\|_{\\infty} \\ep$ for $w>0$ sufficiently large, then the first part of the theorem follows since $\\ep>0$ is arbitrary. The second part of the theorem follows similarly replacing $\\gamma>0$ with the parameter of the uniform continuity of $f$. \n\\end{proof}\n\n  In order to obtain a modular convergence result, the following norm-convergence theorem for the sampling Kantorovich operators (see \\cite{COVI}) can be formulated.\n\\begin{theorem} \\label{norm_conv} \nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in C_c(\\R^n)$ we have\n$$\n\\lim_{w \\to +\\infty} \\left\\|S_w f - f \\right\\|_{\\varphi}\\ =\\ 0.\n$$\n\\end{theorem}\nNow, we recall the following modular continuity property for $S_w$, useful to prove the modular convergence for the above operators in Orlicz spaces.\n\\begin{theorem} \\label{mod_cont} \nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in L^{\\varphi}(\\R^n)$ there holds\n$$\nI^{\\varphi}[\\lambda S_w f]\\ \\miu\\ \\frac{\\left\\|\\chi\\right\\|_1}{\\delta^n\\cdot m_{0,\\Pi^n}(\\chi)}I^{\\varphi}[\\lambda m_{0,\\Pi^n}(\\chi)f],\n$$\nfor some $\\lambda>0$. In particular, $S_w$ maps $L^{\\varphi}(\\R^n)$ in $L^{\\varphi}(\\R^n)$.\n\\end{theorem}\n\\noindent Now, the main result of this section follows (see \\cite{COVI}).\n\\begin{theorem} \\label{th2}\nLet $\\varphi$ be a convex $\\varphi$-function. For every $f \\in L^{\\varphi}(\\R^n)$, there exists $\\lambda>0$ such that\n$$\n\\lim_{w \\to +\\infty} I^{\\varphi}[\\lambda (S_w f-f)]\\ =\\ 0.\n$$\n\\end{theorem}\n\\begin{proof}\nLet $f \\in L^{\\varphi}(\\R^n)$ and $\\ep>0$ be fixed. By Lemma \\ref{lemma1}, there exists $\\overline{\\lambda}>0$ and $g \\in C_c(\\R^n)$ such that $I^{\\varphi}[\\overline{\\lambda} (f-g)]\\!<\\! \\ep$. Let now $\\lambda>0$ such that $3\\lambda(1 + m_{0,\\Pi^n}(\\chi)) \\miu \\overline{\\lambda}$. By the properties of $\\varphi$ and Theorem \\ref{mod_cont}, we have\n\\begin{eqnarray*}\n&& \\hskip-0.7cm I^{\\varphi}[\\lambda(S_w f-f)]\\ \\miu\\ I^{\\varphi}[3\\lambda(S_w f - S_w g)]+I^{\\varphi}[3\\lambda(S_w g - g)] + I^{\\varphi}[3\\lambda(f-g)]\\\\\n&\\miu&\\ \\frac{1}{m_{0,\\Pi^n}(\\chi)\\cdot \\delta^n}\\left\\|\\chi \\right\\|_1 I^{\\varphi}[\\overline{\\lambda} (f-g)] + I^{\\varphi}[3\\lambda(S_w g - g)] + I^{\\varphi}[\\overline{\\lambda} (f-g)] \\\\\n&<&\\ \\left(\\frac{1}{m_{0,\\Pi^n}(\\chi) \\cdot \\delta^n}\\left\\|\\chi \\right\\|_1+1\\right) \\ep + I^{\\varphi}[3\\lambda(S_w g - g)].\n\\end{eqnarray*}\nThe assertion follows from Theorem \\ref{norm_conv}.\n\\end{proof} \nThe setting of Orlicz spaces allows us to give a unitary approach for the reconstruction since we may obtain convergence results for particular cases of Orlicz spaces. For instance, choosing $\\varphi(u)=u^p$, $1 \\leq p < +\\infty$, we have that $L^{\\varphi}(\\R^n) = L^p(\\R^n)$ and $I^{\\varphi}[f]=\\|f\\|^p_{p}$, where $\\| \\cdot \\|_p$ is the usual $L^p$-norm. Then, from Theorem \\ref{mod_cont} and Theorem \\ref{th2} we obtain the following corollary.\n\\begin{corollary}\nFor every $f \\in L^p(\\R^n)$, $1 \\leq p < +\\infty$, \nthe following inequality holds:\n\\begin{displaymath}\n\\left\\|S_w f\\right\\|_p\\ \\miu\\ \\delta^{-n\/p}\\, [m_{0,\\Pi^n}(\\chi)]^{(p-1)\/p}\\, \\left\\|\\chi\\right\\|^{1\/p}_1\\, \\left\\|f\\right\\|_p.\n\\end{displaymath}\nMoreover, we have:\n$$\n\\lim_{w \\to +\\infty} \\|S_w f-f\\|_p\\ =\\ 0.\n$$\n\\end{corollary}\nThe corollary above, allows us to reconstruct $L^p$-signals (in $L^p$-sense), therefore signals\/images not necessarily continuous. Other examples of Orlicz spaces for which the above theory can be applied can be found e.g., in \\cite{MUORL,MU1,BAMUVI,BABUSTVI,COVI}. The theory of sampling Kantorovich operators in the general setting of Orlicz spaces allows us to obtain, by means of a unified treatment, several applications in many different contexts.\n\n\n\\section{Examples of special kernels} \\label{sec5}\n\nOne important fact in our theory is the choice of the kernels, which influence the order of approximation that can be achieved by our operators (see e.g. \\cite{COVI3} in one-dimensional setting).\n\n  For instance, one  can take into consideration  {\\em radial kernels}, i.e., functions for which the value depends on the Euclidean norm of the argument only. Example of such a kernel can be given, for example, by the Bochner-Riesz kernel, defined as follows $b^{\\alpha}(\\underline{x}):=2^{\\alpha}\\Gamma(\\alpha+1)\\|\\underline{x}\\|_2^{-(n\/2)+\\alpha}{\\cal B}_{(n\/2)+\\alpha}(\\|\\underline{x}\\|_2)$,\nfor $\\underline{x} \\in \\R^n$, where $\\alpha > (n-1)\/2$, ${\\cal B}_{\\lambda}$ is the Bessel function of order $\\lambda$ and $\\Gamma$ is the Euler function. For more  details about this matter,  see e.g. \\cite{BUFIST}. \n\n  To construct, in general, kernels satisfying all the assumptions $(\\chi_i)$, $i=1,2,3$ is not very easy.\n\n For this reason, here we show a procedure useful to construct examples using product of univariate kernels, see e.g. \\cite{BUFIST,COVI,COVI2}. In this case, we consider the case of uniform sampling scheme, i.e., $t_{\\underline{k}}=\\underline{k}$.\n\n Denote by $\\chi_1, ..., \\chi_n$, the univariate functions $\\chi_i : \\R \\to \\R$, $\\chi_i \\in L^1(\\R)$, satisfying\nthe following assumptions:\n\\begin{equation} \\label{uno-dim-}\nm_{\\beta,\\Pi^1}(\\chi_i)\\ :=\\ \\sup_{u \\in \\R}\\sum_{k \\in \\Z}\\left|\\chi_i(u-k)\\right| |u-k|^{\\beta}\\ <\\ +\\infty,\n\\end{equation}\nfor some $\\beta >0$, $\\chi_i$ is bounded in a neighborhood of the origin and \n\\begin{equation} \\label{sing}\n\\sum_{k \\in \\Z}\\chi_i(u-k) =1,\n\\end{equation}\nfor every $u \\in \\R$, for $i=1,...,n$. \nNow, setting $\\chi(\\underline{x}) := \\prod_{i=1}^n\\chi_i(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n$,\nit is easy to prove that $\\chi$ is a multivariate kernel for the operators $S_w$ satisfying all the assumptions of our theory, see e.g., \\cite{BUFIST,COVI}. \n\n  As a first example, consider the univariate Fej\\'{e}r's kernel defined by $F(x)\\ :=\\ \\frac{1}{2}\\mbox{sinc}^2\\left(\\frac{x}{2}\\right)$, $x \\in \\R$, where the sinc function is given by\n\\begin{displaymath}\n\\mbox{sinc}(x)\\ :=\\ \\left\\{\n\\begin{array}{l}\n\\disp \\frac{\\sin \\pi x}{\\pi x}, \\hskip1cm x \\in \\R\\setminus \\left\\{0\\right\\}, \\\\\n\\hskip0.5cm 1, \\hskip1.5cm x=0.\n\\end{array}\n\\right.\n\\end{displaymath}\nClearly, $F$ is bounded, belongs to $L^1(\\R)$ and satisfies the moment conditions $(\\ref{uno-dim-})$ for $\\beta = 1$, as shown in \\cite{BUNE,BABUSTVI,COVI}. \nFurthermore, taking into account that the Fourier transform of $F$ is given by (see \\cite{BUNE})\n\\begin{displaymath}\n\\widehat{F}(v) :=\\ \\left\\{\n\\begin{array}{l}\n1-|v\/\\pi|,\\ \\hskip0.5cm |v|\\miu \\pi, \\\\\n0,\\ \\hskip1.85cm |v|>\\pi,\n\\end{array}\n\\right.\n\\end{displaymath}\nwe obtain by Remark \\ref{remark3} that condition $(\\ref{sing})$ is fulfilled. Then, we can define $\\disp \\mathcal{F}_n(\\xx)= \\prod^n_{i=1}F(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n,$ the multivariate Fej\\'{e}r's kernel, satisfying the condition upon a multivariate kernel. The Fej\\'{e}r's kernel $F$ and the bivariate Fej\\'{e}r's kernel $F(x)\\cdot F(y)$ are plotted in Figure \\ref{fig1}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.22]{fejer}\n\\hskip0.8cm\n\\includegraphics[scale=0.22]{multidimFejer}\n\\caption{\\small{The Fej\\'{e}r's kernel $F$ (left) and the bivariate Fej\\'{e}r's kernel ${\\cal F}(x,y)$ (right).}} \\label{fig1}\n\\end{figure}\n\n  The Fej\\'{e}r's kernel $\\mathcal{F}_n$ is an example of kernel with unbounded support, then to evaluate our sampling Kantorovich series at any given $\\xx \\in \\R^n$, we need of an infinite number of mean values $w^n \\int_{R^w_{\\kk}}f(\\uu)\\ d\\uu$. However, if the function $f$ has compact support, this problem does not arise. In case of function having unbounded support the infinite sampling series must be truncated to a finite one, which leads to the so-called truncation error. In order to avoid the truncation error, one can take kernels $\\chi$ with bounded support. Remarkable examples of kernels with compact support, can be constructed using the well-known central B-spline of order $k \\in \\N$, defined by\n$$\nM_k(x) :=\\ \\frac{1}{(k-1)!}\\sum^k_{i=0}(-1)^i \\left(\\begin{array}{l} \\!\\! \nk\\\\\n\\hskip-0.1cm i\n\\end{array} \\!\\! \\right)\n\\left(\\frac{k}{2}+x-i\\right)^{k-1}_+,\n$$\nwhere the function $(x)_+ := \\max\\left\\{x,0\\right\\}$ denotes the positive part of $x \\in \\R$ (see \\cite{BABUSTVI,VIZA1,COVI}). The central B-spline $M_3$ and the bivariate B-spline kernel $M_3(x)\\cdot M_3(y)$ are plotted in Figure \\ref{fig2}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.22]{spline}\n\\hskip0.7cm\n\\includegraphics[scale=0.22]{bidimspline}\n\\caption{\\small{The B-spline $M_3$ (left) and the bivariate B-spline ${\\cal M}^2_3(x,y)$ (right).}} \\label{fig2}\n\\end{figure}\nWe have that, the Fourier transform of $M_k$ is given by\n$\\widehat{M_k}(v)\\ :=\\ \\mbox{sinc}^n\\left( \\frac{v}{2 \\pi} \\right)$, $v \\in \\R$, and then, if we consider the case of the uniform spaced sampling scheme, condition (\\ref{sing}) is satisfied by Remark \\ref{remark3}. Clearly, $M_k$ are bounded on $\\R$, with compact support $[-n\/2,n\/2]$, and hence $M_k \\in L^1(\\R)$, for all $k \\in \\N$. Moreover, it easy to deduce that condition $(\\ref{uno-dim-})$ is fulfilled for every $\\beta>0$, see \\cite{BABUSTVI}.\nHence $\\mathcal{M}^n_k(\\xx)\\ :=\\ \\prod^n_{i=1}M_k(x_i)$, $\\underline{x}=(x_1,...,x_n) \\in \\R^n$,\nis the multivariate B-spline kernel of order $k \\in \\N$.\n\n  Finally, other important examples of univariate kernels are given by the Jackson-type kernels, defined by\n$J_k(x)=c_k\\, \\mbox{sinc}^{2k}\\left(\\frac{x}{2k\\pi\\alpha}\\right)$, $x \\in  \\R$,\nwith $k \\in \\N$, $\\alpha \\geq 1$, where the normalization coefficients $c_k$ are given by\n$c_k\\ :=\\ \\left[ \\int_{\\R} \\mbox{sinc}^{2k}\\left(\\frac{u}{2 k \\pi \\alpha} \\right) \\, du \\right]^{-1}$.\nSince $J_k$ are band-limited to $[-1\/\\alpha,\\, 1\/\\alpha]$, i.e., their Fourier transform vanishes outside this interval, condition $(\\chi 2)$ is satisfied by Remark \\ref{remark3} and $(\\ref{uno-dim-})$ is satisfied since $J_k(x)=\\mathcal{O}(|x|^{-2k})$, as $x \\to \\pm \\infty$, $k \\in \\N$. In similar manner, by the previous procedure we can construct a multivariate version of Jackson kernels. For more details about Jackson-type kernels, and for others useful examples of kernels see e.g. \\cite{BUNE,BAMUVI,BABUSTVI0,BABUSTVI}.\n\n\n\\section{The sampling Kantorovich algorithm for image reconstruction}\n\nIn this section, we show applications of the multivariate sampling Kantorovich operators to image reconstruction. First of all, we recall that every bi-dimensional gray scale image $A$ is represented by a suitable matrix and can be modeled as a step function $I$, with compact support, belonging to $L^p(\\R^2)$, $1 \\leq p <+\\infty$. The definition of $I$ arise naturally as follows:\n$$\nI(x,y)\\ :=\\ \\sum^{m}_{i=1}\\sum^{m}_{j=1}a_{ij} \\cdot \\textbf{1}_{ij}(x,y), \\hskip0.3cm (x,y) \\in \\R^2,\n$$\nwhere ${\\bf 1}_{ij}(x,y)$, $i,j =1,2,...,m$, are the characteristics functions of the sets $(i-1,\\ i]\\times(j-1,\\ j]$ (i.e. ${\\bf 1}_{ij}(x,y)=1$, for $(x,y)\\ \\in\\ (i-1,\\ i]\\times(j-1,\\ j]$ and ${\\bf 1}_{ij}(x,y)=0$ otherwise). \n\n  The above function $I(x,y)$ is defined in such a way that, to every pixel $(i,j)$ it is associated the corresponding gray level $a_{ij}$. \n\n  We can now consider approximation of the original image by the bivariate sampling Kantorovich operators $(S_w I)_{w>0}$ based upon some kernel $\\chi$. \n\n  Then, in order to obtain a new image (matrix) that approximates in $L^p$-sense the original one, it is sufficient to sample $S_w I$ (for some $w>0$) with a fixed sampling rate. In particular, we can reconstruct the approximating images (matrices) taking into consideration different sampling rates and this is possible since we know the analytic expression of $S_w I$.\n\n If the sampling rate is chosen higher than the original sampling rate, one can get a new image that has a better resolution than the original one's. The above procedure has been implemented by using MATLAB and tools of matrix computation in order to obtain an optimized algorithm based on the multivariate sampling Kantorovich theory.\n\n  In the next sections, examples of thermographic images reconstructed by the sampling Kantorovich operators will be given to show the main applications of the theory to civil engineering. In particular, a real world case-study is analyzed in term of modal analysis to obtain a model useful to study the response of the building under seismic action.\n\n\n\\section{An application of thermography to civil engineering}\n\nIn the present application, thermographic images will be used to locate the resisting elements and to define their dimensions, and moreover to investigate the actual texture of the masonry wall, i.e., the arrangement of blocks (bricks and\/or stones) and mortar joints. \nIn general, thermographic images have a resolution too low to accurately analyze  the texture of the masonries,  therefore it has to be increased by means of suitable tools.\n\nIn order to obtain a consistent separation of the phases, that is a correct identification of the pixel which belong to the blocks and those who belong to mortar joints, the image is converted from gray-scale representation to black-and-white (binary) representation by means of an image texture algorithm, which employs techniques belonging to the field of digital image processing. \nThe image texture algorithm, described in details in \\cite{CACLGU}, leads to areas of white pixels identified as blocks and areas of black pixels identified as mortar joints.\nHowever, the direct application of the image texture algorithm to the thermographic images  (see Figure~\\ref{fig:original_ric}), can produce errors, as an incorrect separation between the bricks and the mortar (see Figure~\\ref{fig:original_BW}). \nTherefore, we can use the sampling Kantorovich operators to process the thermographic images. \nIn particular, here we used the operators $S_w$ based upon the bivariate Jackson-type kernel with $k=12$ (see Section \\ref{sec5}) and the parameter $w=40$ (see Figure~\\ref{fig:reconstructed_ric}). \nThe application of the image texture algorithm produces a consistent separation of the phases (see Figure~\\ref{fig:reconstructed_BW}).\n\n\\begin{figure}[!ht]\n\\centering\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{sez_im3_color} \\label{fig:original_ric}}\n\\hskip0.15cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{sez_im3_color_BW} \\label{fig:original_BW}}\n\\hskip0.15cm \n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ric_sez_im3_color_JACK_N=12_450pix_w=40} \\label{fig:reconstructed_ric}}\n\\hskip0.15cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ric_sez_im3_color_JACK_N=12_450pix_w=40_BW} \\label{fig:reconstructed_BW}} \n\\caption{{\\small (a) Original thermographic image ($75\\times 75$ pixel resolution) and (b) its texture; (c) Reconstructed thermographic image ($450\\times 450$ pixel resolution) and (d) its texture.}}\n\\label{fig:original}\n\\end{figure}\n\nIn order to perform structural analysis, the mechanical characteristics of an homogeneous material equivalent to the original heterogeneous material are sought. \nThe equivalence is in the sense that, when subjected to the same boundary conditions (b.c.), the overall responses in terms of mean values of stresses and deformations are the same.\nThe equivalent elastic properties are estimated by means of the ``test-window'' method \\cite{CLGU}.\n\n\n\\section{Case-study}\n\nThe image texture algorithm described previously  has been used to analyze a real-world case-study: a building  consisting of two levels and an attic, with a very simple architectural plan, a rectangle with sides 11 m and 11.4 m (see Figure~\\ref{fig:pianta_prospetto}). \nThe vertical structural elements consist of masonry walls. \nAt first level the masonry walls have thickness of 40 cm with blocks made of  stones, while at second level the masonry is made of squared tuff block and it has thickness of about 35 cm. \nBoth surfaces of the walls are plastered.\nThe slab can be assumed to be rigid in the horizontal plane and \nthe building is placed in a medium risk seismic area. \nAccording to a preliminary visual survey, the structure does not have strong asymmetries between principal directions and the distribution of the masses are quite uniform.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width = 0.75\\textwidth]{pianta_prospetto} \n\\caption{{\\small Case-study building: plant (left) and main facade (right).}}\n\\label{fig:pianta_prospetto}\n\\end{figure}\n\n\nThe response of the building is evaluated by means of three different models.\nIn the first model, only information that can be gathered by the visual survey have been used, and the characteristics of material has been assume according to Italian Building Code. \nIn the second model information concerning the actual geometry of vertical structural elements acquired by means of thermographic survey have been used; \nfor example, the survey showed that the openings at ground level in the main facade were once larger and have been subsequently partially filled with non-structural masonry, and thus the dimensions of all the masonry walls in the facade have to be reduced.\nEventually, in the third model information about the actual texture of masonries of ground level (chaotic masonry, Fig.~\\ref{fig:chaotic}) and of second level (periodic masonry, Fig.~\\ref{fig:periodic}) have been used. \nIn particular, the actual textures where established using the reconstructed thermographic images. \n\n\\begin{figure}[!ht]\n\\centering\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ir5463_dx_ric_BW} \\label{fig:chaotic} }\n\\hskip0.8cm\n\\subfigure[]{ \\includegraphics[width = 0.2\\textwidth]{ir5462_dx_ric_BW} \\label{fig:periodic}} \n\\caption{{\\small Texture of (a) ground level and (b) second level masonries.}}\n\\label{fig:masonries}\n\\end{figure}\n\nThe main characteristics of the models are reported in Table~\\ref{tab:char_models}.\nFor the mechanical characteristic the Young's modulus $E$ and the shear modulus $G$ are shown.\nIn the third model, the following mechanical characteristics (Young's modulus $E$ and Poisson's ratio $\\nu$) of the constituent phases have been used: for the stones of the ground level, $E = 25000\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$, for the bricks of the second level $E = 1700\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$, for the mortar of both levels  $E = 2500\\,\\mathrm{N\\,mm^{-2}}, \\nu = 0.2$.\n\n\\begin{table}[!ht]\n\\center\n{\\small\n\\begin{tabular}{c|c|cc|cc}\n\\hline\n & & \\multicolumn{2}{|c}{ground level} & \\multicolumn{2}{|c}{second level}\\\\\n & geometry & $E$ & $G$ & $E$ & $G$ \\\\\n & & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ & $\\mathrm{N\\,mm^{-2}}$ \\\\\n\\hline\n Model \\#1 & visual survey & 3346 & 1115 & 1620 & 540 \\\\\n Model \\#2 & thermogr. survey & 3346 & 1115 & 1620 & 540 \\\\\n Model \\#3 & thermogr. survey & 7050 & 2957 & 1996 & 833 \\\\\n \\hline\\end{tabular}\n}\n\\caption{{\\small Geometry and masonries' mechanical characteristics in the models.}}\n\\label{tab:char_models}\n\\end{table}\n\n\nThe behavior under seismic actions is estimated by means of modal analysis, using a commercial code based on the Finite Element Method.\nThe periods of the first three modes and the corresponding mass participating ratio  for the two principal direction of seismic action are reported in Tab.~\\ref{tab:periodi_masse}.\nFor all the models, the first mode is along the $y$ axis, the second is along the $x$ axis, and the third is mainly torsional. \nAnyway, the second mode is not a pure translational one since the asymmetry in the walls distribution produces also a torsional component. \n\n\\begin{table}[h!]\n\\center\n{\\small\n\\begin{tabular}{c|p{0.7cm}p{0.7cm}p{0.7cm}|p{0.7cm}p{0.7cm}p{0.7cm}|p{0.7cm}p{0.7cm}p{0.7cm}}\n\\hline\n& \\multicolumn{3}{|c|}{Periods}\n & \\multicolumn{6}{|c}{Mass partecipating ratio} \\\\\n & \\multicolumn{3}{|c|}{ }\n  & \\multicolumn{3}{|c|}{ $x$ direction} &    \\multicolumn{3}{|c}{$y$ direction} \\\\\n & 1st mode & 2nd mode & 3rd mode  & 1st mode & 2nd mode & 3rd mode  & 1st mode & 2nd mode & 3rd mode\\\\\n\\hline\n Model \\#1 & 0.22 & 0.15 & 0.13 & 0.00 & 0.64 & 0.13 & 0.73 & 0.00 & 0.00 \\\\\n Model \\#2 & 0.22 & 0.14 & 0.13 & 0.00 & 0.51 & 0.27 & 0.73 & 0.01 & 0.00 \\\\\n Model \\#3 & 0.18 & 0.13 & 0.11 & 0.00 & 0.46 & 0.26 & 0.68 & 0.01 & 0.00 \\\\\n \\hline\\end{tabular}\n}\n\\caption{{\\small Periods and mass participating ratio for two directions of seismic action of the first three modes .}}\n\\label{tab:periodi_masse}\n\\end{table}\n\nAs can be noted, the first two models have the same period for the fundamental mode in $y$ direction; nevertheless, in $x$ direction there is a slight difference due to the reduced dimensions of masonry walls width discovered by means of thermographic survey. In fact, the reduction of dimensions leads to a decrease of global stiffness while the total mass is almost the same.  \nThe periods of third model show a reduction of about 20\\% due to  the greater value of equivalent elastic moduli estimated by means of homogenized texture.\nAs already noted, for seismic action in $x$ direction the structure response is dominated by second mode, with also a significant contribution from the third mode (which is torsional); for seismic action in $y$ the response is dominated by first mode.\nIt is also worth noting that Model \\#3 shows a reduced mass participating ratio for the fundamental mode in each direction, and therefore a greater number of modes should be considered in the evaluation of seismic response in order to achieve a suitable accuracy. \n\n\n\n\n\n\n\\section{Concluding remarks}\n\nThe sampling Kantorovich operators and the corresponding MATLAB algorithm for image reconstruction are very useful to enhance the quality of thermographic images of portions of masonry walls. In particular, after the processing by sampling Kantorovich algorithm, the thermographic image has higher definition with respect to the original one and therefore, it was possible to estimate the mechanical characteristics of homogeneous materials equivalent to actual masonries, taking into account the texture (i.e., the arrangement of blocks and mortar joints). These materials were used to model the behavior of a case study under seismic action. This model has been compared with others constructed by well-know methods, using ``naked eyes'' survey and the mechanical parameters for materials taken from the Italian Building Code. Our method based on the processing of thermographic images by sampling Kantorovich operators enhances the quality of the model with respect to that based on visual survey only.\n\n\\noindent In particular the proposed approach allows to overcome some difficulties that arise when dealing with the vulnerability analysis of existing structures, which are: i) the knowledge of the actual geometry of the walls (in particular the identification of hidden doors and windows); ii) the identification of the actual texture of the masonry and the distribution of inclusions and mortar joints, and from this iii) the estimation of the elastic characteristics of the masonry.\nIt is noteworthy that, for item i) the engineer has usually limited knowledge, due to the lack of documentation, while for items ii) and iii) he usually use tables proposed in technical manuals and standards which however give large bounds in order to encompass the generality of the real masonries. Instead, the use of reconstruction techniques on thermographic images coupled with homogenization permits to reduce these latter uncertainty on the estimation of the mechanical characteristics of the masonry.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}