diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfkid" "b/data_all_eng_slimpj/shuffled/split2/finalzzfkid" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfkid" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction }\n{\\bf 1.1. Discrepancy.} Let $\\bx_0,\\bx_1,...$ be a sequence of points in $[0,1)^s$, $S \\subseteq [0,1)^s$,\n\\begin{equation}\\label{In1}\n\\Delta(S, (\\bx_{n})_{n=0}^{N-1} )= \\sum_{n=0}^{N-1} ( \\b1_{S}(\\bx_{n}) - \\lambda(S)),\n\\end{equation}\nwhere $\\d1_{S}(\\bx) =1, \\; {\\rm if} \\;\\bx \\in S$,\nand $ \\d1_{S}(\\bx) =0,$ if $\n\\bx \\notin S$. Here $\\lambda(S)$ denotes the $s$-dimensional Lebesgue-measure of $S$.\nWe define the star {\\it discrepancy} of an\n$N$-point set $(\\bx_{n})_{n=0}^{N-1}$ as\n\\begin{equation} \\label{In2}\n \\emph{D}^{*}((\\bx_{n})_{n=0}^{N-1}) =\n \\sup\\nolimits_{ 00 . $\nAccording to the well-known conjecture (see, e.g., [BeCh, p.283]), this estimate can be improved\n to\n$\n \\limsup\\nolimits_{N \\to \\infty } N (\\ln N)^{-s} \\emph{D}^{*}((\\bx_{n})_{n=0}^{N-1})>0\n$.\nSee [Bi] and [Le1] for the results on this conjecture.\n\n An $s$-dimensional sequence $((\\bx_{n})_{n \\geq 0})$ is of\n \\texttt{low discrepancy} (abbreviated\nl.d.s.) if $ \\emph{D}^{*}((\\bx_{n})_{n=0}^{N-1})=O(N^{-1}(\\ln\nN)^{s}) $ for $ N \\rightarrow \\infty $.\nFor examples of l.d.s. see, e.g., in [BeCh], [DiPi], [Ni]. \\\\ \\\\\n{\\bf 1.2. Digital Kronecker sequence.}\n\n\nFor an arbitrary prime power $b$, let $\\FF_b$ be the finite field of order $b$, $\\FF_b^{*} =\\FF_b\\setminus 0$, $\\ZZ_b = \\{0, 1, . . . , b-1 \\}$. Let $\\FF_b[z]$ be the\nset of all polynomials over $\\FF_b$,\nand let $\\FF_b((z^{-1}))$ be the field of formal Laurent series.\nEvery element $L$\nof $\\FF_b((z^{-1}))$ has a unique expansion into a formal Laurent series\n\\begin{equation} \\label{In10}\n L = \\sum_{k=w}^{\\infty} u_k z^{-k} \\quad \\with \\quad u_k \\in \\ZZ_b, \\;\\; \\and \\;\\; w\\in \\ZZ\n\t \\;\\; \\where \\;\\; u_w \\neq 0.\n\\end{equation}\nThe discrete exponential evaluation $\\nu$ of $L$ is defined by\n\\begin{equation} \\nonumber\n\\nu(L) := \u2212w, \\quad L\\neq 0, \\qquad \\nu(0) := \u2212\\infty.\n\\end{equation}\nFurthermore, we define the ``fractional part`` of $L$ by\n\\begin{equation} \\label{In14}\n \\{L\\} = \\sum_{k= \\max(1,w)}^{\\infty} u_k z^{-k} .\n\\end{equation}\nWe choose bijections $\\psi_r : \\ZZ_b \\to \\FF_b $ with $\\psi_r(0) = 0$, and for $i = 1, 2, ... , s$\nand $j = 1, 2, . . . $ we choose bijections $\\eta_{i,j} : \\FF_b \\to \\ZZ_b $.\nFor n = 0, 1, ..., let\n\\begin{equation} \\label{In16}\n n =\\sum_{r=0}^{\\infty} a_r(n) b^r\n\\end{equation}\nbe the digit expansion of $n$ in base $b$, where $a_r(n) \\in \\ZZ_b$ for $r \\geq 0$ and\n$a_r(n) = 0$ for all sufficiently large $r$.\n\n\n With every n = 0, 1, . . ., we associate the polynomial\n\\begin{equation} \\label{In18}\n n(z) =\\sum_{r=0}^{\\infty} \\psi_r(a_r(n)) z^r \\in \\FF_b[z]\n\\end{equation}\nand if $L \\in \\FF_b((z^{-1}))$ is as in (\\ref{In10}), then we define\n\\begin{equation} \\label{In20}\n \\eta^{(i)}(L) = \\sum_{k= \\max(1,w)}^{\\infty} \\eta_{i,k}(u_k) b^{-k} .\n\\end{equation}\n\n\n\n\n\nIn [Ni], Niederreiter introduced a non-Archimedean analogue of the classical Kronecker sequences.\nFor every $s$-tuple $\\bL = (L_1, . . . , L_s)$ of elements of $\\FF_b((z^{-1}))$, we define the sequence\n$S(\\bL) = (\\bl_n)_{n \\geq 0}$ by\n\\begin{equation} \\label{In22}\n\\bl_n = (l^{(1)}_n,..., l^{(s)}_n ), \\quad\nl^{(i)}_n = \\eta^{(i)}( n(z) L_i(z)),\n \\quad \\for \\quad \\; 1 \\leq i \\leq s, \\; n \\geq 0.\n\\end{equation}\nThis sequence is sometimes called a digital Kronecker sequence (see [LaPi, p.4]).\nIn analogy to classical Kronecker sequences, in [LaNi, Theorem 1], the following theorem has been\n proven \\\\ \\\\\n {\\bf Theorem A.} {\\it A digital\nKronecker sequence $S(L)$ is uniformly distributed in $[0,1)^s$ if and only if $1, L_1, . . . ,\nL_s$ are linearly independent over $\\ZZ_b[x]$.}\\\\\n\n\n\n\n\n\n By $\\mu_1$ we denote the normalized Haar-measure\non $\\FF_b((z^{-1}))$ and by $\\mu_s$ the $s$-fold product measure on $\\FF_b((z^{-1}))^s$.\nIn [La1], Larcher proved the following metrical upper bound on the star discrepancy\nof digital Kronecker sequences $ D_N(S(L)) = O(\nN^{-1}(\\log N)^s(\\log \\log N)^{2+\\epsilon})$.\nFor $\\mu_s$-almost all $L \\in \\FF_b((z^{-1}))^s$, $\\epsilon >0$.\n\nIn [LaPi, p.4], Larcher and Pillichshammer were able to give corresponding metrical\nlower bounds for the discrepancy of digital Kronecker\nsequences $ D_N(S(L)) \\geq\nc(b, s) N^{-1}\n(\\log N)^s \\log \\log N $\n for $\\mu_s$-almost all $L \\in \\FF_b((z^{-1}))^s$, for infinitely many $N \\geq 1$\nwith some $c(b, s) > 0$ not depending on N. \\\\ \\\\\n{\\bf 1.3. Bounded remainder set}.\\\\\n {\\bf Definition 1}. {\\it Let $\\bx_0,\\bx_1,...$ be a sequence of point in $[0,1)^s$.\n A subset $S$ of $ [0,1)^s$ is called a {\\sf bounded remainder set} for $(\\bx_n)_{n \\geq 0}$\nif the discrepancy function $\\Delta(S, (\\bx_n)_{n = 0}^{N-1})$ is bounded in $N$.}\\\\\n\n\n\n Let $\\alpha$ be an irrational number, let I be an interval in $[0,1)$ with the length $|I|$, let $\\{n\\alpha\\}$ be the fractional part of $n\\alpha$, $n=1,2,\u2026$.\n Hecke, Ostrowski and Kesten proved that $\\Delta(S, (\\{n\\alpha\\})_{n = 1}^N)$ is bounded\n if and only if $|I|=\\{k\\alpha\\}$ for some integer $k$ (see references in [GrLe]).\n\n The sets of bounded remainder for the classical $s$-dimensional Kronecker sequence were studied\nby Lev and Grepstad [GrLe]. The case of Halton's sequence was studied by Hellekalek [He].\nFor references to others investigations on bounded remainder set\n see [GrLe].\n\n\n\n\n\n\n\n\n\n\n\n\nLet $\\bgamma =(\\gamma_1,...,\\gamma_s)$,\n$\\gamma_i \\in (0, 1)$ with $b$-adic expansion $\\gamma_i= \\gamma_{i,1}b^{-1}+ \\gamma_{i,2}b^{-2}\\\\\n+...$, $i=1,...,s$.\n In this paper, we prove\\\\ \\\\\n{\\bf Theorem.} {\\it Let $(\\bl_n)_{n \\geq 0}$ be a uniformly distributed digital Kronecker sequence.\nThe set $[0,\\gamma_1) \\times ...\\times [0,\\gamma_s)$ is of bounded remainder with respect to\n $(\\bl_n)_{n \\geq 0}$ if and only if }\n\\begin{equation} \\label{Cond}\n \\max_{1 \\leq i \\leq s} \\max \\{ j \\geq 1 \\; | \\; \\gamma_{i,j} \\neq 0 \\} < \\infty.\n\\end{equation}\n\n\nIn [Le2], we proved similar results for digital sequences described in [DiPi, Sec. 8]. Note that according to\nLarcher's conjecture [La2, p.215], the assertion of the Theorem is true for all digital $(t,s)$-sequences in\nbase $b$.\n\n\n\n\n\\section{Notations.}\n\n\n\n\n\n\nA subinterval $E$ of $[0,1)^s$ of the form\n$$ E = \\prod_{i=1}^s [a_ib^{-d_i},(a_i+1)b^{-d_i}), $$\n with $a_i,d_i \\in \\ZZ, \\; d_i \\ge 0, \\; 0 \\le a_i < b^{d_i}$ for $1 \\le i \\le s$ is called an\n{\\it elementary interval in base $b \\geq 2$}.\\\\ \\\\\n{\\bf Definition 2}. {\\it Let $0 \\le t \\le m$ be integers. A {\\sf $(t,m,s)$-{\\sf net in base $b$}} is a point set\n$\\bx_0,...,\\bx_{b^m-1}$ in $ [0,1)^s $ such that $\\# \\{ n \\in [0,b^m -1] | x_n \\in E \\}=b^t$ for every elementary interval E in base $b$ with\n$\\vol(E)=b^{t-m}$.}\\\\ \\\\\n{\\bf Definition 3.} (\\cite[Definition 4.30]{DiPi}) { \\it For a given dimension $s \\geq 0$, an integer base $b \\geq 2$, and a\nfunction $\\bT : \\NN_0 \\to \\NN_0$ with $\\bT(m) \\leq m$ for all $m \\in \\NN_0$, a sequence $(\\bx_0,\\bx_1, . . .)$\nof points in $[0, 1)^s$ is called a $(\\bT, s)$-sequence in base $b$ if for all integers $m \\geq 1$\nand $k \\geq 0$, the point set consisting of the points $x_{kb^m}, . . . ,x_{kb^m+b^m-1}$ forms\na $(\\bT(m),m, s)$-net in base $b$. }\\\\\n\n A $(\\bT, s)$-sequence in base $b$ is called a strict $(\\bT, s)$-sequence in\nbase $b$ if for all functions $\\bU : \\NN_0 \\to \\NN_0$ with $\\bU(m) \\leq m$ for all $m \\in \\NN_0$ and with\n$\\bU(m) < \\bT(m)$ for at least one $m \\in \\NN_0$, it is not a $(\\bU, s)$-sequence in base $b$.\\\\ \\\\\n{\\bf Definition 4.} ([DiNi, Definition 1]) { \\it\nLet $m, s \\geq 1$ be integers. Let $C^{(1,m)},...,$ $C^{(s,m)}$ be $m \\times m$ matrices over $\\FF_b$.\nNow we construct $b^m$ points in $[0, 1)^s$.\n For $ n= 0, 1,...,b^m-1$, let $n =\\sum^{m-1}_{j=0} a_j(n) b^{j}$\nbe the $b$-adic expansion of $n$. For $r = 0,1,...$\nwe choose bijections $\\psi_r : \\ZZ_b \\to \\FF_b $ with $\\psi_r(0) = 0$, and for $i = 1, 2, . . . , s$\nand $j = 1, 2, . . . $ we choose bijections $\\eta_{i,j} : \\FF_b \\to \\ZZ_b $.\nWe map the vectors\n\\begin{equation} \\label{No1}\n\ty^{(i,m)}_{n}=(y^{(i,m)}_{n,1},...,y^{(i,m)}_{n,m}),\\quad y^{(i,m)}_{n,j}=\\sum_{r=0}^{m-1} \\psi_r(a_r(n)) c^{(i,m)}_{j,r}\\in \\FF_b\n\\end{equation}\nto the real numbers\n\\begin{equation} \\label{No2}\n x^{(i)}_n =\\sum_{j=1}^m \\eta_{i,j} (y^{(i,m)}_{n,j})\/b^j\n\\end{equation}\nto obtain the point\n\\begin{equation} \\nonumber\n \\bx_n= (x^{(1)}_n,...,x^{(s)}_n) \\in [0,1)^s.\n\\end{equation}\n \\\\\n\n\n\nThe point set $ \\{\\bx_0,...,\\bx_{b^m-1} \\}$ is called a {\\sf digital net} (over $\\FF_b$) (with {\\sf generating matrices} $(C^{(1,m)},...,C^{(s,m)}) $).\n\nFor $m = \\infty$, we obtain a sequence $\\bx_0, \\bx_1,...$ of points in $[0, 1)^s$ which is called a {\\sf digital sequence} $($over $\\FF_b)$ $($with {\\sf generating matrices} $(C^{(1,\\infty)},...,C^{(s,\\infty)}) )$.}\n\n\nWe abbreviate $C^{(i,m)}$ as $C^{(i)}$ for $m \\in \\NN$ and for $m=\\infty$. \\\\ \\\\\n {\\bf Lemma A} ([LaNi, ref. 1-8]). { \\it A digital Kronecker sequence in base $b$ can be expressed as some digital $(\\bT, s)$-sequence in base $b$.}\\\\ \\\\\n {\\bf Lemma B} (\\cite[Theorem 4.86]{DiPi}). {\\it Let $b$ be a prime power. A strict digital\n$(\\bT, s)$-sequence over $\\FF_b$ is\nuniformly distributed modulo one, if and only if $\\liminf_{m \\to \\infty} (m - \\bT(m))=\\infty$.}\\\\\n\n\nFor $m>n$, we put $\\sum_{j=m}^n c_j= 0$ and $\\prod_{j=m}^n c_j =1$.\nFor $x =\\sum_{j \\geq 1} x_{j} b^{-j}$,\nwhere $x_{i} \\in \\ZZ_b =\\{0,...,b-1\\}$, we define the truncation\n\\begin{equation} \\nonumber\n [x]_m =\\sum_{1 \\leq j \\leq m} x_{j}b^{-j} \\quad \\with \\quad m \\geq 1.\n\\end{equation}\nIf $\\bx = (x^{(1)}, . . . , x^{(s)}) \\in [0, 1)^s$, then the truncation $[\\bx]_m$ is defined coordinatewise, that is, $[\\bx]_m = ( [x^{(1)}]_m, . . . , [x^{(s)}]_m)$.\n\n\n\n For $x =\\sum_{j \\geq 1} x_{j} b^{-j}$ and $y =\\sum_{j \\geq 1} y_{j}b^{-j}$\nwhere $x_j ,y_j \\in \\ZZ_b$, we define the ($b$-adic) digital shifted point $v$ by\n$v = x \\oplus y := \\sum_{j \\geq 1} v_{j}b^{-j}$,\n where $v_j \\equiv x_j + y_j \\;(\\mod \\;b)$ and $v_j \\in \\ZZ_b$.\nFor $\\bx =(x^{(1)}, . . . , x^{(s)}) \\in [0, 1)^s$ and $\\by = (y^{(1)}, . . . , y^{(s)}) \\in [0, 1)^s$, we define the ($b$-adic) digital shifted point $\\bv$ by\n$ \\bv =\\bx \\oplus \\by =(x^{(1)} \\oplus y^{(1)}, . . . ,x^{(s)} \\oplus y^{(s)} )$.\n For $n_1,n_2 \\in [0,b^m)$, we define\n$n_1 \\oplus n_2 := (n_1 \/b^m\\oplus n_2)b^m)b^m$.\n\n\n\nFor $x =\\sum_{j \\geq 1} x_{j} b^{-j}$,\nwhere $x_{j} \\in \\ZZ_b$, $x_j=0 $ $(j=1,...,k)$ and $x_{k+1} \\neq 0$, we define the\nabsolute valuation $\\left\\|. \\right\\|_b $ of $x$ by $\\left\\|x \\right\\|_b =b^{-k-1}$.\nLet $\\left\\| n \\right\\|_b =b^k$ for $n \\in [b^k,b^{k+1})$.\\\\ \\\\\n{\\bf Definition 5.} {\\it A sequence $ (\\bx_{n})_{n \\geq 0} $\n\tin $[0,1)^s$ is {\\sf weakly admissible} in\nbase $b$ if}\n\\begin{equation} \\label{3}\n \\varkappa_m:= \\min_{0 \\leq k 0\\quad \\forall m \\geq 1\\; {\\rm where} \\;\\; \\left\\| \\bx \\right\\|_b := \\prod_{i=1}^s\n\t\\left\\|x^{(i)} \\right\\|_b .\n\\end{equation}\n\n\n\nLet $p$ be a prime, $b=p^{\\kappa}$,\n\\begin{equation} \\nonumber\n E(\\alpha) := exp (2\\pi i \\Tr(\\alpha)\/p), \\qquad \\alpha \\in \\FF_b,\n\\end{equation}\nwhere $\\Tr : \\FF_b \\to \\FF_p$ denotes the usual trace of an element of $\\FF_b$ in $\\FF_p$.\n\nLet\n\\begin{equation} \\label{Del1}\n \\delta(\\fT) = \\begin{cases}\n 1, & \\; {\\rm if} \\; \\fT \\;{\\rm is \\;true},\\\\\n 0, &{\\rm otherwise}.\n \\end{cases}\n\\end{equation}\\\\\t\n\n\t\n\t\nBy [LiNi, ref. 5.6 and ref. 5.8], we get\n\\begin{equation} \\label{Del2}\n \\frac{1}{q} \\sum_{\\beta \\in \\FF_b} E(\\alpha \\beta) = \\delta (\\alpha =0 ), \\quad \\where \\quad\n \\alpha \\in \\FF_b.\n\\end{equation}\n\n\n\n\n\n\n\n\n\\section{Proof}\n{\\bf Lemma 1.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a weakly admissible digital sequence in base $b$,\n $m \\geq 1$, $\\tau_m =[\\log_b (\\kappa_m)]+m $. Then we have for all integers $A \\geq 1$}\n\\begin{equation} \\nonumber\n | \\Delta([\\bs,\\bgamma),(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) -\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})| \\leq s,\\quad\n\t \\forall N\\in[1,b^m].\n\\end{equation}\\\\\n{\\bf Proof.} Let\n\\begin{equation} \\nonumber\n B=[\\bs,\\bgamma), \\quad B_i =\\prod_{1 \\leq j \\leq s, j\\neq i} [0, \\gamma^{(j)})\n\t\\times [0, [\\gamma^{(i)}]_{\\tau_m}) \\quad \\ad \\quad B_0 =\\cup_{i=1}^s(B \\setminus B_i).\n\\end{equation}\nIt is easy to see that $B= [\\bs,[\\bgamma]_{\\tau_m}) \\cup B_0$.\n By (\\ref{In1}), we get\n\\begin{equation} \\nonumber\n \\Delta([\\bs,\\bgamma),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})\n\\end{equation}\n\\begin{equation} \\nonumber\n = \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) +\\Delta(B_0,(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) .\n\\end{equation}\nHence\n\\begin{equation} \\nonumber\n |\\Delta([\\bs,\\bgamma),(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) -\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) |\n\\end{equation}\n\\begin{equation} \\label{Prof1}\n \\leq \\sum_{i=1}^s | \\Delta(B \\setminus B_i,(\\bx_{n})_{n=b^m A}^{b^m A+N-1})| .\n\\end{equation}\nSuppose that there exist $i \\in [1,s]$, $ k,n \\in [0, b^m),\\; k \\neq n$ and $A \\geq 1$\nsuch that $x_{n+ b^m A}, x_{k+ b^m A} \\in B \\setminus B_i$.\nTherefore\n\\begin{equation} \\nonumber\n x_{n+ b^m A,j}^{(i)} = x_{k+ b^m A,j}^{(i)} \\quad \\for \\quad j=1,...,\\tau_m.\n\\end{equation}\nFrom (\\ref{In16}), (\\ref{No1}) and \\eqref{No2}, we have\n\\begin{equation} \\nonumber\n y_{n+ b^m A,j}^{(i)} = y_{k+ b^m A,j}^{(i)} \\quad \\for \\quad j=1,...,\\tau_m,\n\\end{equation}\n\\begin{equation} \\nonumber\n y_{n+ b^m A,j}^{(i)}=y_{n,j}^{(i)}+y_{b^m A,j}^{(i)}, \\quad \\ad \\quad\n\t\t y_{k+ b^m A,j}^{(i)}=y_{k,j}^{(i)}+y_{b^m A,j}^{(i)} \\quad \\for \\; j=1,...,\\tau_m.\n\\end{equation}\nHence\n\\begin{equation} \\nonumber\n y_{n,j}^{(i)} = y_{k,j}^{(i)}, \\;\\; j=1,...,\\tau_m \\quad \\ad \\quad\n x_{n,j}^{(i)} = x_{k,j}^{(i)}, \\;\\; j=1,...,\\tau_m.\n\\end{equation}\nTherefore\n\\begin{equation} \\nonumber\n \\left\\| x_{n}^{(i)} \\ominus x_{k }^{(i)}\\right\\|_b < b^{-\\tau_m}\\leq \\kappa_m \\quad \\ad \\quad\n \\left\\| \\bx_{n} \\ominus \\bx_{k}\\right\\|_b \\geq \\varkappa_m.\n\\end{equation}\nBy (\\ref{3}) we have a contradiction. Thus\n\\begin{equation} \\nonumber\n\\card\\{n \\in [0,b^m) \\;|\\; \\bx_{n+ b^m A} \\in B \\setminus B_i \\} \\leq 1,\\; \\ad \\;\n | \\Delta(B \\setminus B_i,(\\bx_{n})_{n=b^m A}^{b^m A+N-1})| \\leq 1.\n\\end{equation}\nUsing (\\ref{Prof1}), we get the assertion of Lemma 1. \\qed \\\\\n\n\n\t\n\n Let $\\beta_1,...,\\beta_{\\kappa}$ be a $F_p$ basis of $\\FF_b$, and let $\\Tr$ be a standard trace function.\n\tLet\n\t\\begin{equation} \\label{Lemm1}\n \\omega(\\alpha) =\\sum_{j=1}^{\\kappa} p^{j-1} \\Tr(\\alpha \\beta_j), \\qquad b= p^{ \\kappa}.\n\\end{equation}\t\nWe use notations (\\ref{In16}), (\\ref{No1}) and (\\ref{No2}).\nLet $ n =\\sum_{r\\geq 0} a_r(n) b^{r}$\tbe the $b$-adic expansion of $n$, and let\n\\begin{equation} \\label{Lemm2}\n \\tilde{n} =\\sum_{r\\geq 0} \\omega(\\psi_r(a_r(n))) b^{r}.\n\\end{equation}\nTherefore\n\\begin{equation} \\label{Lemm26a}\n\\{ \\tilde{n}\\;| \\;0 \\leq n 0$, with\n$ k_j \\in \\ZZ_b$,\n\\begin{equation} \\label{Lemm7}\nv(k):= \\max \\{ j \\in [1, \\tau_m] \\; | \\; k_j \\neq 0\\}, \\quad v(0)=0.\n\\end{equation}\n\nSimilarly to [Ni, Theorem 3.10] (see also [DiPi, Lemma 14.8]), we consider the following\n Fourier series decomposition of the discrepancy function :\\\\ \\\\\n{\\bf Lemma 2.} {\\it Let $A \\geq 1$ be an integer, $N \\in [1,b^m]$, $\\gamma^{(s+1)}=N \/b^m $,\nand let $(\\bx_n)_{n \\geq 0}$ be a digital sequence in base $b$.\nThen}\n\\begin{equation} \\nonumber\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})\n\\end{equation}\n\\begin{equation} \\nonumber\n =\n \\sum_{n=0}^{b^m-1} \t\t\\sum_{(k^{(1)},...,k^{(s)}) \\in \\FF_b^{s\\tau_m}} \\; \\sum_{k^{(s+1)} \\in \\FF_b^{m}}\n\t E(\\mk \\cdot \\msu_{\\widetilde{n+b^m A}}) \\hat{\\d1}(\\mk)\n\t\t-b^m \\prod_{i=1}^{s+1} [\\gamma^{(i)}]_{\\tau_m} ,\n\\end{equation}\nwhere\n\\begin{equation} \\label{Lemm12}\n \\mk \\cdot \\msu_n = \\sum_{i=1}^{s} \\sum_{j=1}^{\\tau_m} k_{j}^{(i)}\nu^{(i)}_{n,j} + \\sum_{j=1}^m k_{j}^{(s+1)} u^{(s+1)}_{n,j},\n\\;\\; \\;\\; \\hat{\\d1}(\\mk) = \\prod_{i=0}^{s+1} \\hat{\\d1}^{(i)}(k^{(i)}),\n\\end{equation}\n $\\hat{\\d1}^{(i)}(0) =[\\gamma^{(1)}]_{\\tau_m} \\; (1 \\leq i \\leq s)$, $\\hat{\\d1}^{(s+1)}(0) =\\gamma^{(s+1)}\\;\\;$\n and\n\\begin{align} \\nonumber\n& \\hat{\\d1}^{(i)}(k) = b^{-v(k)} E\\Big(-\\sum_{j=1}^{v(k)-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)}) \\Big)\n\\Big( \\sum_{b=0}^{\\gamma_{v(k)}^{(i)} -1}\n\t E(-k_{v(k)} \\eta^{-1}_{i,v(k)}(b) )\\nonumber \\\\\n&+ E(-k_{v(k)}\\eta^{-1}_{i,v(k)}(\\gamma_{v(k)}^{(i)}))\n\t\t\\{b^{v(k)}[\\gamma^{(i)}]_{\\tau_m} \\}\\Big), \\qquad \\; i \\in [1,s], \\nonumber \\\\\n& \\hat{\\d1}^{(s+1)}(k) = b^{-v(k)}\n E\\Big( -\\sum_{j=1}^{{v(k)}-1} k_j\\psi_{j}(\\gamma_{j}^{(s+1)})\\Big) \\Big( \\sum_{b=0}^{\\gamma_{v(k)}^{(s+1)} -1}\n\t E(-k_{v(k)}\\psi_{v(k)}(b) ) \\nonumber \\\\\n&\n\t+ E(-k_{v(k)} \\psi_{v(k)}(\\gamma_{v(k)}^{(s+1)}))\n\t\t\\{b^{v(k)}\\gamma^{(s+1)} \\} \\Big). \\label{Lemm12a}\n\\end{align}\\\\\n{\\bf Proof.}\nLet $\\gamma = \\sum_{j = 1}^{\\dot{m}} \\gamma_{j} b^{-j} >0$, $w = \\sum_{j = 1}^{\\dot{m}} w_{j} b^{-j} $, with\n$ \\gamma_j,w_j \\in \\ZZ_b$.\nIt is easy to verify (see also [Ni, p. 37,38]) that\n\\begin{equation} \\nonumber\n \\d1_{[0,\\gamma)}(w) =\n\t\\sum_{r=1}^{\\dot{m}} \\sum_{b=0}^{\\gamma_r -1} \\prod_{j=1}^{r-1} \\delta(w_i=\\gamma_i)\n\t\\delta(w_r=b).\n\\end{equation}\t\nBy (\\ref{No2}) and (\\ref{Lemm3}), we have that\n\\begin{equation} \\nonumber\n x^{(i)}_{j,n} =b \\Longleftrightarrow y^{(i)}_{j,n} =\\eta^{-1}_{i,j}(b)\n\t\\Longleftrightarrow \tu^{(i)}_{j,\\tilde{n}} = \\eta^{-1}_{i,j}(b),\n\\end{equation}\t\nand\n\\begin{equation} \\nonumber\n \\d1_{[0, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n}) =\n\t\\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} \\prod_{j=1}^{r-1}\n\t\\delta(x^{(i)}_{j,n}=\\gamma_j^{(i)})\n\t \t\\delta(x^{(i)}_{r,n}=b)\n\\end{equation}\n\\begin{equation} \\nonumber\n =\n\t\\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} \\prod_{j=1}^{r-1}\n\t\\delta(y^{(i)}_{j,n}=\\eta_{i,j}^{-1}(\\gamma_j^{(i)}))\n\t \t\\delta(y^{(i)}_{r,n}=\\eta_{i,r}^{-1}(b))\n\\end{equation}\t\n\\begin{equation} \\nonumber\n =\n\t\\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} \\prod_{j=1}^{r-1}\n\t\\delta(u^{(i)}_{j,\\tilde{n}}=\\eta_{i,j}^{-1}(\\gamma_j^{(i)}))\n\t \t\\delta(u^{(i)}_{r,\\tilde{n}}=\\eta_{i,r}^{-1}(b)), \\quad i=1,...,s.\n\\end{equation}\t\nSimilarly, we derive\t\n\\begin{equation} \\label{Lemm12c}\n \\d1_{[0, \\gamma^{(s+1)})}( x^{(s+1)}_{n}) =\n\t\\sum_{r=1}^m \\sum_{b=0}^{\\gamma_r^{(s+1)} -1} \\prod_{j=1}^{r-1}\n\t\\delta(u^{(s+1)}_{j,\\tilde{n}}=\\psi_{j}^{-1}(\\gamma_j^{(s+1)}))\n\t \t\\delta(u^{(s+1)}_{r,\\tilde{n}}=\\psi_{r}^{-1}(b)) .\n\\end{equation}\t\t\nLet $ k \\cdot u^{(i)}_{\\tilde{n}} =\\sum_{j=1}^{\\tau_m} k_j u^{(i)}_{\\tilde{n},j}$.\n By (\\ref{Del2}), we have\n\\begin{align} \\nonumber\n & \\d1_{[0, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n}) =\n\t\\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r} \\sum_{k_1,...,k_r \\in \\FF_b} \\dot{\\d1}^{(i)}(k), \\quad\n\\where \\\\\n& \\dot{\\d1}^{(i)}(k)=\n\tE\\Big(\\sum_{j=1}^{r-1} k_j (u^{(i)}_{j,\\tilde{n}}-\\eta_{i,j}^{-1}(\\gamma_j^{(i)}))\n +\t \tk_r(u^{(i)}_{r_i,\\tilde{n}}-\\eta_{i,r}^{-1}(b)) \\Big) = E(k \\cdot u^{(i)}_{\\tilde{n}})\n \\tilde{\\d1}^{(i)}(k) \\nonumber \\\\\n& \\with \\quad \\tilde{\\d1}^{(i)}(k)= E\\Big(-\\sum_{j=1}^{r-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)})\n -\t \tk_r \\eta_{i,r}^{-1}(b) \\Big). \\label{Lemm12aa}\n\\end{align}\nHence\n\\begin{align} \\nonumber\n & \\d1_{[0, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n})= \\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r}\n \\sum_{k_1,...,k_{\\tau_m} \\in \\FF_b} \\delta( v(k) \\leq r)\n E(k \\cdot u^{(i)}_{\\tilde{n}}) \\tilde{\\d1}^{(i)}(k) \\\\\n &= \\sum_{k_1,...,k_{\\tau_m} \\in \\FF_b}\n\t\\sum_{r=1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r} \\delta( v(k) \\leq r)\n E(k \\cdot u^{(i)}_{\\tilde{n}}) \\tilde{\\d1}^{(i)}(k) \\nonumber \\\\\n& = \\sum_{k_1,...,k_{\\tau_m} \\in \\FF_b} E(k \\cdot u^{(i)}_{\\tilde{n}}) \\ddot{\\d1}^{(i)}(k), \\quad\n \\where \\quad\t\\ddot{\\d1}^{(i)}(k) =\\sum_{r= v(k)}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r}\n \\tilde{\\d1}^{(i)}(k) . \\nonumber\n\\end{align}\nApplying \\eqref{Lemm12a} and \\eqref{Lemm12aa}, we derive\n\\begin{align} \\nonumber\n& \\ddot{\\d1}^{(i)}(k) =\\sum_{r=v(k)}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r}\n E\\Big(-\\sum_{j=1}^{r-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)})\n -\t \tk_r \\eta_{i,r}^{-1}(b) \\Big) \\\\\n& = \\sum_{b=0}^{\\gamma_{ v(k)}^{(i)} -1} b^{-v(k)}\n E\\Big(-\\sum_{j=1}^{v(k)-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)})\n -\t \tk_{ v(k)} \\eta_{i,v(k)}^{-1}(b) \\Big) + \\nonumber \\\\\n& + E\\Big(-\\sum_{j=1}^{v(k)-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)}) \\Big)\n \\sum_{r= v(k)+1}^{\\tau_m} \\sum_{b=0}^{\\gamma_r^{(i)} -1} b^{-r}\n \\nonumber \\\\\n& = b^{-v(k)} E\\Big(-\\sum_{j=1}^{v(k)-1} k_j \\eta_{i,j}^{-1}(\\gamma_j^{(i)}) \\Big)\n \\Big( \\sum_{b=0}^{\\gamma_{ v(k)}^{(i)} -1}\n E \\big(-k_{ v(k)} \\eta_{i,v(k)}^{-1}(b) \\big) \\nonumber\\\\\n& + E\\big( -k_{ v(k)} ( \\eta_{i,v(k)}^{-1}(\\gamma_{ v(k)}^{(i)})) \\big)\n \\{ b^{v(k)} [\\gamma]_{\\tau_m}^{(i)} \\}\n\\Big) = \\hat{\\d1}^{(i)}(k) \\nonumber.\n\\end{align}\nHence\n\\begin{equation} \\nonumber\n \\d1_{[0, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n})=\\sum_{k_1,...,k_{\\tau_m} \\in \\FF_b} E(k \\cdot u^{(i)}_{\\tilde{n}}) \\hat{\\d1}^{(i)}(k).\n\\end{equation}\nSimilarly, we obtain from \\eqref{Lemm12a} and (\\ref{Lemm12c}) that\n\\begin{equation} \\nonumber\n \\d1_{[0, \\gamma^{(s+1)})}( x^{(s+1)}_{n})=\\sum_{k_1,...,k_{m} \\in \\FF_b} E(k \\cdot u^{(s+1)}_{\\tilde{n}}) \\hat{\\d1}^{(s+1)}(k).\n\\end{equation}\nUsing (\\ref{Lemm12}), we obtain\n\\begin{equation} \\label{Lemm20}\n \\prod_{i=1}^{s+1} \\d1_{[0, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n}) =\n\t\\sum_{(k^{(1)},...,k^{(s)}) \\in \\FF_b^{\\tau_m}} \\sum_{k^{(s+1)} \\in \\FF_b^{m}}\n\tE(\\mk \\cdot {\\bf u}_{\\tilde{n}}) \t\\hat{\\d1}(\\mk).\t\n\\end{equation}\nBearing in mind that $x^{(s+1)}_{n} =\\{ n\/b^m \\}$ and $\\gamma^{(s+1)}=N \/b^m $, we have\n\\begin{equation} \\nonumber\n \\bx_{n +b^m A} \\in [\\bs, [\\bgamma]_{\\tau_m}), n \\in [0 ,N) \\Longleftrightarrow\n\t (\\bx_{n +b^m A} , x^{(s+1)}_{n+b^m A}) \\in [\\bs, [\\bgamma]_{\\tau_m}) \\times [0,\\gamma^{(s+1)}).\n\\end{equation}\nFrom \\eqref{Lemm20} and \\eqref{In1}, we derive\n\\begin{equation} \\nonumber\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1}) =\n \\sum_{n=0}^{b^m-1} \\prod_{i=1}^{s+1} \\d1_{[\\bs, [\\gamma^{(i)}]_{\\tau_m})}( x^{(i)}_{n+b^m A})\n\t-b^m \\prod_{i=1}^{s+1} [\\gamma^{(i)}]_{\\tau_m}\n\\end{equation}\n\\begin{equation} \\nonumber\n =\n \\sum_{n=0}^{b^m-1}\t\\;\t\\sum_{(k^{(1)}...,k^{(s)}) \\in \\FF_b^{\\tau_m}} \\;\n \\sum_{k^{(s+1)} \\in \\FF_b^{m}}\n\t E(\\mk \\cdot \\msu_{\\widetilde{n+b^m A}}) \\prod_{i=1}^{s+1} \\hat{\\d1}^{(i)}(k^{(i)})\n\t\t-b^m \\prod_{i=1}^{s+1} [\\gamma^{(i)}]_{\\tau_m} .\n\\end{equation}\nHence Lemma 2 is proved. \\qed \\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet\n\\begin{equation} \\nonumber\n\\mk = (k^{(1)},...,k^{(s+1)}),\\;\n k^{(i)}=(k^{(i)}_{1},...,k^{(i)}_{\\tau_m}), i \\in [1,s], \\;\nk^{(s+1)}=(k^{(s+1)}_{1},...,k^{(s+1)}_{m}),\n\\end{equation}\n\\begin{equation} \\nonumber\n G_m =\\{ \\mk \\; | \\; \\;k^{(i)}_{j} \\in \\FF_b \\; \\with \\;\n j \\in [1, \\tau_m],\\; i \\in [1,s],\\;\\ad\\; j \\in [1, m] \\;\\for\\; i=s+1 \\},\n\\end{equation}\t\\\\\t\n$G_m^{*} = G_m \\setminus \\{ \\bs\\}$, and let\n\\begin{equation} \\label{Lemm25}\n D_m =\\{ \\mk \\in G_m \\; | \\; \\mk \\cdot \\msu_n =0 \\;\\; \\forall \\;\\;\n\t\t\t\t\t\tn \\in [0,b^m-1] \\}, \\quad \\;\\; D_m^{*} = D_m \\setminus \\{ \\bs\\}.\n\\end{equation}\t\t\t\t\t\t\nIt is easy to see that\n\\begin{equation} \\label{Lemm25a}\n \\mu \\mk \\in D_m^{*} \\quad \\fall \\quad \\mu \\in \\FF_b^{*}, \\; \\mk \\in D_m^{*}.\n\\end{equation}\t\\\\\n{\\bf Lemma 3.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a digital sequence in base $b$. Then}\n\\begin{equation} \\label{Lemm26}\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})\n =\n \t\t\\sum_{\\mk \\in G_m^{*}} \\hat{\\d1}(\\mk) \\sum_{n=0}^{b^m-1}\n\t E(\\mk \\cdot \\msu_n +\\mk \\cdot \\msu_{ \\widetilde{b^m A}}).\n\\end{equation} \\\\\n{\\bf Proof.} By (\\ref{Lemm12a}) we have $\\hat{\\d1}(\\bs) =\\prod_{i=1}^{s+1} [\\gamma^{(i)}]_{\\tau_m}$.\nApplying Lemma 2, we get\n\\begin{equation} \\nonumber\n \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})\n =\n \t\t\\sum_{\\mk \\in G_m^{*}} \\hat{\\d1}(\\mk) \\sum_{n=0}^{b^m-1}\n\t E(\\mk \\cdot \\msu_{ \\widetilde{n+b^m A}}).\n\\end{equation}\n Using (\\ref{Lemm2}), (\\ref{Lemm3}) and (\\ref{Lemm6}), we obtain\n\\begin{equation} \\nonumber\n \\widetilde{n+b^m A} =\\widetilde{n} + \\widetilde{b^m A} =\\widetilde{n} \\oplus \\widetilde{b^m A}\\quad \\ad \\quad\n\t\t\\msu_{ \\widetilde{n+b^m A}} =\\msu_{ \\widetilde{n}}\n\t\t+\\msu_{ \\widetilde{b^m A}}\n\\end{equation}\nNow from (\\ref{Lemm26a}), we get\n(\\ref{Lemm26}). Hence Lemma 3 is proved. \\qed \\\\ \\\\\n{\\bf Lemma 4.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a digital sequence in base $b$. Then }\n\\begin{equation} \\label{In195}\n \\sigma:= \\sum_{n=0}^{b^m-1} E( \\mk \\cdot \\msu_n ) =b^m \\delta(\\mk \\in D_m).\n\\end{equation} \\\\\n{\\bf Proof.} Using (\\ref{Lemm3}) and (\\ref{Lemm5}) and (\\ref{Lemm12}) , we have\n\\begin{equation} \\nonumber\n \\mk \\cdot \\msu_{\\tilde{n}} = \\sum_{i=1}^s \\sum_{j=1}^{\\tau_m} \\sum_{r=0}^{m-1}\n\tk_{j}^{(i)} \\psi_r(a_r(n)) c_{j,r}^{(i)} + \\sum_{j=1}^{m}\n\t k_{j}^{(s+1)} \\psi_{m-j}(a_{m-j}(n))\n\\end{equation}\n\\begin{equation} \\nonumber\n =\\sum_{r=0}^{m-1} \\psi_r(a_r(n)) \\Big(\\sum_{i=1}^s \\sum_{j=1}^{\\tau_m}\n\tk_{j}^{(i)} c_{j,r}^{(i)}\n\t + k_{m-r }^{(s+1)} \\Big) =\\sum_{r=0}^{m-1} f_r \\xi_r,\n\\end{equation}\nwhere\n\\begin{equation} \\label{Lem4-1}\nf_r=\\psi_r(a_r(n)) \\in \\FF_b \\quad \\ad \\quad\n\\xi_r=\\sum_{i=1}^s \\sum_{j=1}^{\\tau_m}\n\tk_{j}^{(i)} c_{j,r}^{(i)}\n\t + k_{m-r }^{(s+1)}.\n\\end{equation}\nBy (\\ref{Lemm26a}), (\\ref{In16}) and (\\ref{Del2}), we obtain\n\\begin{equation} \\nonumber\n \\sigma = \\sum_{\\tilde{n}=0}^{b^m-1} E( \\mk \\cdot \\msu_{\\tilde{n}}) =\\sum_{f_0,...,f_{m-1} \\in \\FF_b} E(\\sum_{r=0}^{m-1} f_r \\xi_r)=\n\tb^m \\prod_{r=0}^{m-1} \\delta (\\xi_r =0).\n\\end{equation}\nNow from (\\ref{Lemm25}), we get that $ \\mk \\in D_m$ and Lemma 4 follows. \\qed \\\\\n\n\nLet\n\\begin{equation} \\nonumber\n \\Lambda_m=\\{\\mk =(k^{(1)},...,k^{(s+1)}) \\in G_m \\; | \\; k^{(s+1)}=\\bs\\} ,\n\\end{equation}\n\\begin{equation} \\nonumber\n g_{\\bw} = \\{ A \\geq 1 \\; | \\; y^{(i)}_{b^m A,j}= w^{(i)}_j, \\;i \\in [1,s],\\;j \\in\n\t[1,\\tau_m] \\}, \\quad \\rho_{\\bw}= 0 \\;\\; \\for \\;\\; g_{\\bw} = \\emptyset,\n\\end{equation}\n\\begin{equation} \\label{End0}\n \\rho_{\\bw} = \\min_{A \\in g_{\\bw} } A \\;\\; \\for \\;\\; g_{\\bw} \\neq \\emptyset,\n\\quad \\;\\;\n M_m =\\{\\rho_{\\bw} \\; | \\; \\bw \\in \\Lambda_m \\}.\n\\end{equation}\nWe consider the following conditions :\n\\begin{equation} \\label{End}\n g_{\\bw} \\neq \\emptyset \\quad \\fall \\quad \\bw \\in \\Lambda_m\n\\end{equation}\nand\n\\begin{equation} \\label{End1}\n\\sigma_1:= \\frac{1}{\\card(R_m)} \\sum_{A \\in R_m}\n| \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})|^2\n =\\sum_{\\mk \\in D_m^{*}} b^{2m} |\\hat{\\d1} (\\mk )|^2\n\\end{equation}\nfor some finite set $R_m$.\\\\\nBearing in mind (\\ref{Lemm3}), we get\n\\begin{equation} \\label{End2}\n g_{\\bw} = \\{ A \\geq 1 \\; | \\; u^{(i)}_{\\widetilde{b^m A }}=w^{(i)}, \\; i \\in [1,s]\\},\n\t\\;\\; \\where \\;\\; w^{(i)}=(w^{(i)}_1,...,w^{(i)}_{\\tau_m}).\n\\end{equation}\\\\\n{\\bf Lemma 5.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a weakly admissible uniformly distributed digital\n $(\\bT,s)$-sequence in base $b$, satisfying to (\\ref{End}) for all $m \\geq m_0$ with some $m_0 \\geq 1$.\n Then (\\ref{End1}) is true for $R_m=M_m$}.\\\\ \\\\\n{\\bf Proof.}\nBy (\\ref{End0}) and (\\ref{Del2}), we obtain\n\\begin{equation} \\nonumber\n\\frac{1}{b^{s \\tau_m}} \t\\sum_{\\bw \\in \\Lambda_m} E(\\mk \\cdot \\bw) =\n\\frac{1}{b^{s \\tau_m}} \t\\sum_{w^{(i)}_j \\in \\FF_b, \\;i\\in[1,s],\\; j \\in [1,\\tau_m] }\n E\\Big(\\sum_{i=1}^s \\sum_{j=1}^{\\tau_m}\nk^{(i)}_j w^{(i)}_j \\Big)\n\\end{equation}\n\\begin{equation} \\label{In200}\n= \\prod_{i=1}^s\\prod_{j=1}^{\\tau_m} \\delta(k^{(i)}_j=0)\n= \\prod_{i=1}^s \\delta(k^{(i)}=0) ,\\quad \\where \\quad \\mk \\in G_m .\n\\end{equation}\nUsing (\\ref{Lemm26}), we derive\n\\begin{equation} \\label{In201}\n | \\Delta([\\bs,[\\bgamma]_{\\tau_m}),(\\bx_{n})_{n=b^m A}^{b^m A+N-1})|^2\n\\end{equation}\t\n\\begin{equation} \\nonumber\n =\t \\sum_{\\dot{\\mk}, \\ddot{\\mk} \\in G_m^{*}} \\hat{\\d1}(\\dot{\\mk} )\n\t\t\t\t \\overline{\\hat{\\d1}(\\ddot{\\mk} ) }\n\t\t\t\t\\sum_{\\dot{n},\\ddot{n}=0}^{b^{m}-1}\nE( \\dot{\\mk} \\cdot \\msu_{\\dot{n} } +\n \\dot{\\mk} \\cdot \\msu_{\\widetilde{b^m A}}\n -\\ddot{\\mk} \\cdot \\msu_{\\ddot{n} }\n\t\t\t\t\t\t-\\ddot{\\mk} \\cdot \\msu_{\\widetilde{b^m A}}).\n\\end{equation}\nIt is easy to see that if condition (\\ref{End}) is true, than $\\card( M_m) =b^{s \\tau_m}$ and\n $\\{ \\msu_{\\widetilde{b^m A}} \\; | \\; A \\in M_m \\} =\\Lambda_m$. \\\\\nApplying (\\ref{End0}), (\\ref{In200}), (\\ref{In201}) and (\\ref{End1}) with $R_m =M_m$, we have\n\\begin{equation} \\nonumber\n\\sigma_1=\n\t \\sum_{\\dot{\\mk}, \\ddot{\\mk} \\in G_m^{*}} \\hat{\\d1}(\\dot{\\mk} )\n\t\t\t\t \\overline{\\hat{\\d1}(\\ddot{\\mk} )}\n\t\t\t\t\\sum_{\\dot{n},\\ddot{n}=0}^{b^{m}-1}\n\t\t\t\tb^{-s\\tau_m} \\sum_{A \\in M_m}\nE( \\dot{\\mk} \\cdot \\msu_{\\dot{n}} -\n \\ddot{\\mk} \\cdot \\msu_{\\ddot{n}}\n\t\t\t\t\t\t+(\\dot{\\mk} -\\ddot{\\mk}) \\cdot \\msu_{\\widetilde{b^m A}} )\n\\end{equation}\n\\begin{equation} \\nonumber\n =\n\t \\sum_{\\dot{\\mk}, \\ddot{\\mk} \\in G_m^{*}} \\hat{\\d1}(\\dot{\\mk} )\n\t\t\t\t \\overline{\\hat{\\d1}(\\ddot{\\mk} )}\n\t\t\t\t\\sum_{\\dot{n},\\ddot{n}=0}^{b^{m}-1} \t\nE( \\dot{\\mk} \\cdot \\msu_{\\dot{n}} -\n \\ddot{\\mk} \\cdot \\msu_{\\ddot{n}} )\n\t\t\t\tb^{-s\\tau_m}\t\t\\sum_{\\bw \\in\\Lambda_m} E(\\dot{\\mk} -\\ddot{\\mk}) \\cdot \\bw )\n\\end{equation}\n\\begin{equation} \\nonumber\n =\n\t \\sum_{\\dot{\\mk}, \\ddot{\\mk} \\in G_m^{*}} \\hat{\\d1}(\\dot{\\mk} )\n\t\t\t\t \\overline{\\hat{\\d1}(\\ddot{\\mk} ) }\n\t\t\t\t\\sum_{\\dot{n},\\ddot{n}=0}^{b^{m}-1} \t\nE( \\dot{\\mk} \\cdot \\msu_{\\dot{n}} -\n \\ddot{\\mk} \\cdot \\msu_{\\ddot{n}} )\n\t\t\t\t\\prod_{i=1}^s \\delta( \\dot{k}^{(i)} = \\ddot{k}^{(i)}).\n\\end{equation}\nLet $\\dddot{n} =\\ddot{n} \\ominus \\dot{n}$.\nFrom (\\ref{In16}), we obtain $\\{ \\dddot{n}\\;| 0 \\leq \\ddot{n} 0$, $\\sa_{b+6} = \\sa_{b+7} =0$ such that}\n\\begin{equation} \\label{Lem2}\n \\Big| B_{k,c, \\psi}\\Big(\\sum_{j=1}^{b+7}\\frac{\\sa_j}{b^j}+\\frac{y}{b^{b+7}}\\Big) \\Big| \\geq b^{-b-7}, \\;\\;\\;\n\t\t\t\\forall \\;\t\t\tk \\in \\FF_b, c \\in \\ZZ_b, y\\in [0,1], \\psi \\in \\Psi\n\\end{equation}\nand\n\\begin{equation} \\label{Lem3}\n \\sum_{k \\in \\FF_b^{*}} |B_{k,c, \\psi}(x)|^2 \\geq b^{-2r} \\;\\; \\forall \\;\t\t c \\in \\ZZ_b,\t \\;\\; \\where\n \\;\\; \\langle x\t\\rangle \\geq b^{-r}, \\; r \\geq 1 .\n\\end{equation} \\\\\n{\\bf Proof.} Let\n\\begin{equation} \\nonumber\n \\dot{A}= \\{ \\theta_{k,c, \\psi}:=\\Re(A_{k,c, \\psi}) \\;|\\; k \\in \\FF_b,\\; c \\in \\ZZ_b, \\; \\psi \\in \\Psi\\}.\n\\end{equation}\nTaking into account that $\\card(\\Psi) =b!$, we get $\\card(\\dot{A}) \\leq b!b^2+2< b^{b+4}$.\\\\\nLet\n\\begin{equation} \\nonumber\n \\ddot{A}= \\{ \\ba =(\\sa_1,...,\\sa_{b+7}) \\in \\ZZ_b^{b+7} \\;|\\;\n\\sa_1^2+...+\\sa_{b+5}^2 >0, \\quad \\sa_{b+6} = \\sa_{b+7} =0 \\}\n\\end{equation}\nand let $z_{\\ba}=\\sa_1\/b+ \\cdots + \\sa_{b+7}\/b^{b+7} $.\nBy\t (\\ref{Lem2a}), we derive\n\\begin{equation} \\nonumber\n |B_{k,c,\\psi}(x)| =|A_{k,c,\\psi} +x| \\geq |\\Re(A_{k,c,\\psi}) +x|.\n\\end{equation}\nSuppose that (\\ref{Lem2}) is not true. Then for all $\\ba \\in \\ddot{A}$ there exist $k(\\ba),c(\\ba),\\psi(\\ba)$ and $y(\\ba)$\nsuch that\n\\begin{equation} \\nonumber\n b^{-b-7} > \\Big|B_{k(\\ba),c(\\ba),\\psi(\\ba)}\\Big(\\sum_{j=1}^{b+7}\\frac{\\sa_j}{b^j}+\\frac{y(\\ba)}{b^{b+7}}\\Big) \\Big| \\geq\n \\Big|\\theta_{k(\\ba),c(\\ba),\\psi(\\ba)} +z_{\\ba} +\\frac{y(\\ba)}{b^{b+7}} \\Big|.\n\\end{equation}\nHence $ |\\theta_{k(\\ba),c(\\ba),\\psi(\\ba)} +z_{\\ba} | < b^{-b-6} $.\nSuppose that $\\theta_{k(\\ba_1),c(\\ba_1),\\psi(\\ba_1)} = \\theta_{k(\\ba_2),c(\\ba_2),\\psi(\\ba_2)} $ for some $\\ba_1 , \\ba_2 \\in \\ddot{A} $, $\\ba_1 \\neq \\ba_2 $. Hence $|z_{\\ba_1} -z_{\\ba_2}| < b^{-b-5} $.\nBearing in mind that $|z_{\\ba_1} -z_{\\ba_2}| \\geq b^{-b-5} $ for all $\\ba_1 \\neq \\ba_2$, we\nget a contradiction. Therefore $\\theta_{k(\\ba_1),c(\\ba_1),\\psi(\\ba_1)} \\neq \\theta_{k(\\ba_2),c(\\ba_2),\\psi(\\ba_2)} $ for all $\\ba_1 , \\ba_2 \\in \\ddot{A} $, $\\ba_1 \\neq \\ba_2 $.\nThus $ \\card(\\dot{A}) \\geq \\card(\\ddot{A})$.\nHence\n\\begin{equation} \\nonumber\n b^{b+4} > \\card(\\dot{A}) \\geq \\card(\\ddot{A}) =b^{b+5}-1 > b^{b+4}.\n\\end{equation}\nWe have a contradiction. Therefore (\\ref{Lem2}) is true.\n\nNow we consider assertion (\\ref{Lem3}).\nIf $c=0$, then $|B_{k,c, \\psi}(x)|=|x|$ and (\\ref{Lem3}) follows. \\\\\n Now let $c \\in \\{ 1,...,b-1\\}$.\n By (\\ref{Lem2a}), we have\n\\begin{equation} \\label{Lem4}\n |B_{k,c, \\psi}(x)|^2 =|A_{k,c, \\psi}|^2 + x (A_{k,c, \\psi} + \\overline{A_{k,c, \\psi}}) +x^2.\n\\end{equation}\nUsing (\\ref{Del2}), we get\n\\begin{equation} \\nonumber\n \\sum_{k \\in \\FF_b^{*}} |A_{k,c, \\psi}|^2 =-c^2 + \\sum_{b_1,b_2=0}^{c-1} \t\n\t\\sum_{k \\in \\FF_b } E(k (\\psi(b_1)-\\psi(b_2))\n\\end{equation}\n\\begin{equation} \\nonumber\n =-c^2 +b\\sum_{b_1,b_2=0}^{c-1}\n\t\\delta(\\psi(b_1)=\\psi(b_2)) = -c^2 +bc.\n\\end{equation}\nTaking into account that $ \\psi(0) =0$, we obtain\n\\begin{equation} \\nonumber\n \\sum_{k \\in \\FF_b^{*}} A_{k,c, \\psi} =-c + \\sum_{b=0}^{c-1}\n\t\\sum_{k \\in \\FF_b } E(k \\psi(b) )=-c +b \\sum_{b=0}^{c-1} \\delta( \\psi(b) =0 ) \\geq b-c.\t\n\\end{equation}\nNow from (\\ref{Lem4}), we derive\n\\begin{equation} \\nonumber\n \\sum_{k \\in \\FF_b^{*}} |B_{k,c, \\psi}(x)|^2 \\geq c(b-c) +x^2 +2x(b-c) \\geq x^2\n\\end{equation}\nand (\\ref{Lem3}) follows. Thus Lemma 6 is proved. \\qed \\\\ \\\\\nApplying (\\ref{Lem2d}) - (\\ref{Lem3}), we have \\\\ \\\\\n{\\bf Corollary.} {\\it Let $\\sa_1,...,\\sa_{b+7}$ be integers chosen in Lemma 6 and let\n$\\gamma^{(s+1)}_{v(\\dot{k})+ j} =\\sa_j$, $j=1,...,b+7$, with some $\\dot{k} $.\nThen}\n\\begin{equation} \\label{Lem2ab}\n \t\t|\\hat{\\d1}^{(s+1)} (\\mu \\dot{k}) | \\geq b^{-v(\\dot{k})-b-7} \\;\\;\\;\n\t\t\t\\forall \\;\t\t\t\\mu \\in \\FF_b^{*}\n\\end{equation}\nand\n\\begin{equation} \\label{Lem3b}\n \\sum_{\\mu \\in \\FF_b^{*} } |\\hat{\\d1}^{(i)} (\\mu \\ddot{k}) |^2 \\geq b^{-2v(\\ddot{k})-2\\dot{r}} , \\;\\;\n\t\\where \\;\\; \\; \\langle b^{v(\\ddot{k})} \\gamma^{(i)}\t\\rangle \\geq b^{-\\dot{r}}.\n\\end{equation}\\\\\n{\\bf Lemma 7.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a digital sequence in base $b$ and\n let $\\rho \\in[2,m-2]$ be an integer.\n Then there exists $ \\mk \\in D_m^{*}$ such that $k^{(1)}=... =k^{(s-1)}=0$, $ k^{(s)}_{ v(k^{(s)})}=1$,\n$1 \\leq v(k^{(s)}) \\leq \\rho-1 $ and $v(k^{(s+1)}) \\leq m-\\rho+2$ }. \\\\ \\\\\n{\\bf Proof.} From (\\ref{Lemm2})-(\\ref{Lemm5}), (\\ref{Lemm25}) and (\\ref{Lem4-1}), we get that $ \\mk \\in D_m^{*}$ if and only if\n\\begin{equation} \\label{Lem61}\n \\sum_{i=1}^{s}\\sum_{j=1}^{\\tau_m}\n\tk_{j}^{(i)} c_{j,r}^{(i)}\n\t + k_{m-r }^{(s+1)} = 0, \\quad \\for \\; {\\rm all} \\quad r=0,1,...,m-1.\n\\end{equation}\nWe put $k^{(1)}=... =k^{(s-1)}=0, \\; k^{(s)}_{j} = 0$, for $j \\geq \\rho$ and $ k^{(s+1)}_{j} = 0$,\n for $j > m-\\rho +2$.\nHence (\\ref{Lem61}) is true if and only if\n\\begin{equation} \\label{Lem6a}\n k_{m-r }^{(s+1)} = -\\sum_{j=1}^{\\rho-1}\n\tk_{j}^{(s)} c_{j,r}^{(s)}\n\t\\;\\; \\for \\;\\; r=0,1,...,m -1, \\quad k_{m-r }^{(s+1)} = 0 \\; \\for \\; m-r > m-\\rho +2.\n\\end{equation}\nTherefore, in order to obtain the statement of the lemma, it is sufficient to show that there exists a nontrivial solution of the following system of linear equations\n\\begin{equation} \\label{Lem6}\n \\sum_{j=1}^{\\rho-1}\n\tk_{j}^{(s)} c_{j,r}^{(i)}\n\t + k_{m-r }^{(s+1)} \\delta(m-r \\leq m-\\rho+2)=0, \\quad r=0,...,m-1.\n\\end{equation}\n In this system, we have $m+1$ unknowns\n$k_{1}^{(s)},...,k_{\\rho-1}^{(s)}$, $k_{1}^{(s+1)},...,k_{m-\\rho+2}^{(s+1)}$\nand $m$ linear equations. Hence there exists a nontrivial solution of (\\ref{Lem6}).\n By (\\ref{Lem6}), we get that if $k^{(s)} =0$, then $k^{(s+1)} =0$. Hence $k^{(s)} \\neq 0$ and\n$1 \\leq v(k^{(s)}) \\leq \\rho-1 $.\nTaking into account that if $\\mk \\in D_m$ then $ \\mu \\mk \\in D_m$ for all $\\mu \\in \\FF_b^{*}$.\nTherefore there exists $ \\mk \\in D_m^{*}$ such that $ k^{(s)}_{ v(k^{(s)})}=1$ and\n$1 \\leq v(k^{(s)}) \\leq \\rho-1 $.\n Thus Lemma 7 is proved. \\qed \\\\ \\\\\n{\\bf Proposition.} {\\it Let $(\\bx_n)_{n \\geq 0}$ be a weakly admissible uniformly distributed\n digital $(\\bT,s)$-sequence in base $b$, satisfying to (\\ref{End}) for all $m \\geq m_0 \\geq 1$.\nThen $[0,\\gamma_1) \\times ...\\times [0,\\gamma_s)$ is of bounded remainder with respect to\n $(\\bx_n)_{n \\geq 0}$ if and only if (\\ref{Cond}) is true.}\\\\ \\\\\n{\\bf Proof.} The sufficient part of the Theorem and of the Proposition follows directly from the definition of $(\\bT, s)$\n sequence and Lemma B. We will consider only the necessary part of the Theorem and of the Proposition.\n\t\n\tSuppose that (\\ref{Cond}) does not true. Then\n\\begin{equation} \\nonumber\n \\max_{1 \\leq i \\leq s} \\card \\{ j \\geq 1 \\; | \\;\\gamma_{j}^{(i)} \\neq 0 \\} = \\infty.\n\\end{equation}\nLet, e.g.,\n\\begin{equation} \\nonumber\n \\card \\{ j \\geq 1 \\; | \\; \\gamma_{j}^{(s)} \\neq 0 \\} = \\infty.\n\\end{equation}\nLet\n\\begin{equation} \\label{Prop0}\n W = \\{ j \\geq 1 \\; | \\; \\gamma_{j}^{(s)} \\in \\{ 1,...,b-2\\} \\;\\oor\\;\n\\gamma_{j}^{(s)} =b-1, \\; \\and \\gamma_{j+1}^{(s)} =0 \\} .\n\\end{equation}\nBearing in mind that $ \\{ j \\geq 1 \\; | \\;\\gamma_{l}^{(s)} = b-1 \\;\\forall \\; l >j \\} =\\emptyset$,\n we obtain that $\\card (W)=\\infty$.\n\nSuppose that there exists $H>0$ such that $b^{2H}c_1 > 4H^2$, $ c_1 = \\gamma_0^2 b^{-4b-36} $,\n\\begin{equation} \\label{Prop1}\n |\\Delta([\\bs,\\bgamma),(\\bx_{n})_{n=M}^{M+N-1})| \\leq H -s\n\t\\quad \\fall \\quad \\;M\\geq 0, \\; N \\geq 1,\n\\end{equation}\nwith $[\\bs,\\bgamma) =[0,\\gamma_1)\\times \\cdots \\times [0,\\gamma_s)$,\n$ \\gamma_0 = \\gamma_1 \\gamma_2 \\cdots \\gamma_{s-1} $.\n\nLet $W=\\{\\dot{w}_j \\; | \\; \\dot{w}_i <\\dot{w}_j \\; \\for \\; i4H^2.\n\\end{equation}\nWe have a contradiction. Now let $\\card(H_1) =1$.\n\nBy (\\ref{Prop5}), we obtain that there exist $j,j_1 \\in [1,J]$ such that $j \\in H_1$, $j 4H^2 ,\n\\end{equation}\nwith $ c_1 = \\gamma_0^2 b^{-4b-36} $. We have a contradiction.\nBy \\eqref{In98}, the Proposition is proved. \\qed \\\\ \\\\\n{\\bf Completion of the proof of the Theorem.}\n By Lemma A,\n $S(\\bL^{(m)})$ is a uniformly distributed digital $(T,s)$-sequence in base $b$.\n\nBy Theorem A, we get that\n $1, L_1, . . . , L_s$ are\nlinearly independent over $\\FF_b[z]$. Hence $1, z^m L_1, ...,\nz^m L_s$ are\nlinearly independent over $\\FF_b[z]$. Let\n $\\bL^{(m)} = (z^m L_1, . . . ,z^m L_s)$, and let $S(\\bL^{(m)}) = (\\bl^{(m)}_n)_{n \\geq 0}$\n (see (\\ref{In22}))\nwith\n\\begin{equation} \\nonumber\n\\bl_n^{(m)} = (l^{(m,1)}_n,..., l^{(m,s)}_n ), \\quad\nl^{(m,i)}_n = \\eta^{(i)}( n(z)z^m L_i(z)),\n \\quad \\for \\quad \\; 1 \\leq i \\leq s, \\; n \\geq 0.\n\\end{equation}\n Using Theorem A, we obtain that $S(\\bL^{(m)})$ is a uniformly distributed sequence in $[0,1)^s$.\nTherefore, for all $\\bw \\in \\Lambda_m $ there exists an integer $A \\geq 1$ with\n\\begin{equation} \\nonumber\nl_{b^m A,j}^{(m,i)}=\\eta^{(i)}_j(w^{(i)}_j)\n \\quad \\for \\quad \\; 1 \\leq i \\leq s, \\; 1 \\leq j \\leq \\tau_m.\n\\end{equation}\nThus $S(\\bL^{(m)})$ satisfies the condition (\\ref{End}).\n\n\n\n\nBearing in mind that $1, L_1, . . . , L_s$ are\nlinearly independent over $\\FF_b[z]$, we get that $\\{ n(z) L_i\\} \\neq 0$ for all $n \\geq 1$. Hence $\\{ l^{(i)} (n)\\} \\neq 0$ for all $n \\geq 1$\n$(i=1,...,s)$. Therefore the sequence $S(\\bL)$ is weakly admissible.\n\n\n\n Applying the Proposition, we get the assertion of the Theorem. \\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and Motivation}\n\n\nNoncommutative geometry is a tool that f\\\/inds numerous applications in the description of a~wide range of physical systems. A celebrated example appearing in String Theory is in terms of the polarisation phenomenon discovered by Myers, in which $N$ D$p$-branes in the presence of transverse Ramond--Ramond f\\\/lux distribute themselves onto the surface of a higher-dimensional sphere~\\cite{Myers:1999ps}. The physics of the simplest case are captured by a $\\U(N)$ theory, with the solution involving fuzzy 2-spheres \\cite{Hoppe:1982,Hoppe:1988gk,Madore:1991bw}. These are related to families of Hermitian matrices obeying the $\\SU(2)$ algebra\n \\begin{gather}\\label{SU(2)condition}\n[ X^{i} , X^{j} ] = 2i \\epsilon^{ijk } X^k .\n\\end{gather}\nThe $X^i$ enter the physics as ground state solutions to the equations of motion via (\\ref{SU(2)condition}). Then their commutator action on the\nspace of all $N \\times N $ matrices organises the matrices into representations of $\\SU(2) \\simeq \\SO(3)$. An important aspect of the geometry\nof the fuzzy 2-sphere involves the construction of fuzzy (matrix) spherical harmonics in $\\SU(2)$ representations, which approach the space of\nall classical $S^2$ spherical harmonics in the limit of large matrices \\cite{Hoppe:1988gk}. This construction of fuzzy spherical harmonics allows\nthe analysis of f\\\/luctuations in a non-Abelian theory of D$p$-branes to be expressed at large $N$ in terms of an Abelian higher dimensional theory.\nThis describes a~D$(p+2)$ brane wrapping the sphere, with $N$ units of worldvolume magnetic f\\\/lux. At f\\\/inite $N$ the higher dimensional theory\nbecomes a noncommutative $\\U(1)$ with a UV cutof\\\/f \\cite{Iso:2001mg,Dasgupta:2002hx,Dasgupta:2002ru,Papageorgakis:2005xr}.\n\nIn this article we review a novel realisation of the fuzzy 2-sphere involving bifundamental matrices. The objects that crucially enter the construction are discrete versions of Killing spinors on the sphere \\cite{Nastase:2009ny,Nastase:2009zu}\\footnote{The work in \\cite{Nastase:2009ny} was carried out in collaboration with S.~Ramgoolam.}. The motivation is similar to the above and comes from the study of the model recently discovered by Aharony, Bergman, Jaf\\\/feris and Maldacena (ABJM) describing the dynamics of multiple parallel M2-branes on a $\\mathbb Z_k$ M-theory orbifold \\cite{Aharony:2008ug}, which followed the initial investigations of Bagger--Lambert and Gustavsson (BLG) \\cite{Bagger:2006sk,Bagger:2007jr,Bagger:2007vi,Gustavsson:2007vu}.\n\nThe ABJM theory is an $\\cN = 6$ superconformal Chern--Simons-matter theory with $\\SO(6)$ R-symmetry and gauge group $\\U(N) \\times \\U(\\bar N) $. The two Chern--Simons (CS) terms have equal but opposite levels $(k,-k)$ and the matter f\\\/ields transform in the bifundamental representation. One can use the inverse CS level $1\/k$ as a coupling constant to perform perturbative calculations. At $k=1$ the theory is strongly coupled and describes membranes in f\\\/lat space. For $k=1,2$ the supersymmetry and R-symmetry are nonperturbatively enhanced to $\\cN = 8$ and $\\SO(8)$ respectively \\cite{Aharony:2008ug,Gustavsson:2009pm,Kwon:2009ar}. It is then possible to use this action to investigate aspects of the $\\mathrm{AdS}_4\/\\mathrm{CFT}_3$ duality, with the role of the 't Hooft coupling played by $\\lambda = \\frac{N}{k}$. The action of the $\\mathbb Z_k$ orbifold on the $\\mathbb C^4$ space transverse to the M2's is such that taking $k\\to\\infty$ corresponds to shrinking the radius of the M-theory circle and entering a IIA string theory regime.\n\nOf particular interest are the ground-state solutions of the maximally supersymmetric massive deformation of ABJM found by Gomis, Rodr\\'iguez-G\\'omez,\nVan Raamsdonk and Verlinde (GRVV) \\cite{Gomis:2008vc}\\footnote{The mass-deformed theory was also presented in \\cite{Hosomichi:2008jb}.}. The theory\nstill has a $\\U(N) \\times \\U(\\bar N) $ gauge group and $\\mathcal N=6$ supersymmetry but conformal invariance is lost and the R-symmetry is broken down\nto $\\SU(2)\\times \\SU(2)\\times \\U(1)$. Its vacua are expected to describe a conf\\\/iguration of M2-branes blowing up into spherical M5-branes in the\npresence of transverse f\\\/lux through a generalisation of the Myers ef\\\/fect. At $k=1$ these solutions should have a dual description in terms of the\n$\\sfrac{1}{2}$-BPS M-theory geometries with f\\\/lux found in \\cite{Bena:2004jw,Lin:2004nb}.\n\nInterestingly, the matrix part of the above ground-state equation is given by the following simple relation, which we will refer to as the GRVV\nalgebra\\footnote{The same def\\\/ining matrix equation appears while looking for BPS funnel solutions in the undeformed ABJM theory and f\\\/irst appeared\nas such in \\cite{Terashima:2008sy}. Its relation to the M2--M5 system was also investigated in \\cite{Hanaki:2008cu}.}:\n\\begin{gather}\n G^\\a=G^\\a G^\\dagger_\\b G^\\b-G^\\b G^\\dagger_\\b G^\\a,\n\\label{equatio}\n\\end{gather}\nwhere $ G^\\alpha$ are $N\\times \\bar N$ and $ G_\\alpha^\\dagger$ are $\\bar N \\times N$ matrices respectively. Given that the Myers ef\\\/fect for\nthe M2-M5 system should employ a 3-dimensional surface, one might initially expect this to represent the def\\\/ining relation for a fuzzy 3-sphere.\nMoreover, the explicit irreducible solutions of (\\ref{equatio}) satisfy $G^\\a G^\\dagger_\\a=1$, which seems to suggest the desired fuzzy 3-sphere structure.\n\nHowever, we will see that the requisite $\\SO(4)$ R-symmetry, that would be needed for the existing fuzzy $S^3$ construction of Guralnik and Ramgoolam (GR) \\cite{Guralnik:2000pb,Ramgoolam:2001zx,Ramgoolam:2002wb}, is absent in this case. As was also shown in \\cite{Nastase:2009ny}, the GR fuzzy $S^3$ construction implies the following algebra\n\\begin{gather}\n\\epsilon^{mnpq}X^+_nX^-_pX^+_q = 2\\left(\\frac{(r+1)(r+3)+1}{r+2}\\right)X^+_m,\\nonumber\\\\\n\\epsilon^{mnpq}X^-_nX^+_pX^-_q = 2\\left(\\frac{(r+1)(r+3)+1}{r+2}\\right)X^-_m\\label{defrel} ,\n \\end{gather}\nwhich must be supplemented with the sphere condition\n\\begin{gather*}\n X_mX_m=X_m^+X_m^-+X_m^-X_m^+=\\frac{(r+1)(r+3)}{2}\\equiv N\n \\end{gather*}\n and the constraints\n\\begin{gather*}\n X^+_mX^+_n=X^-_mX^-_n=0.\n \\end{gather*}\nHere $r$ def\\\/ines a representation of $\\SO(4)\\simeq \\SU(2)\\times \\SU(2)$ by ${\\cal R}_r^+$ and ${\\cal R}_r^-$, with labels $(\\frac{r+1}{2}, \\frac{r-1}{2})$ and $(\\frac{r-1}{2}, \\frac{r+1}{2})$ respectively for the two groups, and the $X_m^\\pm$ are constructed from gamma mat\\-rices. Even\nthough the algebra (\\ref{defrel}) looks similar to the GRVV algebra~(\\ref{equatio}), they coincide only in the `fuzziest' case with $r=1$,\ni.e.\\ the BLG ${\\cal A}_4$-algebra, which in the Van Raamsdonk $\\SU(2)\\times \\SU(2)$ reformulation \\cite{VanRaamsdonk:2008ft} is\n\\begin{gather*}\nR^2X^m=-ik\\epsilon^{mnpq}X^nX^{\\dagger p}X^q \n\\end{gather*}\nThis fact suggests that equation~(\\ref{equatio}) does not describe a fuzzy $S^3$. Furthermore, the perturbative calculations that lead to the above equation are valid at large $k$, where the ABJM theory is describing IIA String Theory instead of M-theory and as a result a D2--D4 bound state in some nontrivial background.\n\nIn the following, we will review how solutions to equation~(\\ref{equatio}) actually correspond to a fuzzy 2-sphere, albeit in a realisation involving bifundamental instead of the usual adjoint matrices, by constructing the full spectrum of spherical harmonics. This is equivalent to the usual construction in terms of the $\\SU(2)$ algebra (\\ref{SU(2)condition}). In fact there is a one-to-one correspondence between the representations of the $\\SU(2)$ algebra $X_i$ and the representations in terms of bifundamental matrices. We will also show how the matrices $ G^\\alpha$, which are solutions of the GRVV algebra up to gauge transformations, correspond to fuzzy Killing spinors on the sphere, recovering the usual Killing spinors in the large $N$ limit.\n\nThe purpose of this article is to present the mathematical aspects of the above construction in a completely model-independent way and highlight some of its features simply starting from~(\\ref{equatio}). The reader who is interested in the full background and calculations in the context of the ABJM model is referred to \\cite{Nastase:2009ny,Nastase:2009zu}, where an analysis of small f\\\/luctuations around the ground-states at large $N$, $k$ showed that they can be organised in terms of a $\\U(1)$ theory on $\\mathbb R^{2,1}\\times S^2$, consistent with an interpretation as a D4-brane in Type IIA. The full 3-sphere expected from M-theory then appeared as the large $N$, $k=1$ limit of a fuzzy Hopf f\\\/ibration, $S^1\/\\mathbb Z_k\\hookrightarrow S_F^3\/\\mathbb Z_k\\stackrel{\\pi}{\\rightarrow} S_F^2$, in which the M-theory circle $S^1\/\\mathbb Z_k$ is f\\\/ibred over the noncommutative sphere base, $S^2_F$.\n\nWe also discuss how this bifundamental formulation af\\\/fects the twisting of the f\\\/ields\nwhen `deconstructing' a higher dimensional f\\\/ield theory. This is achieved by studying the f\\\/ield theory around a fuzzy sphere background, where the twisting is necessary in order to preserve supersymmetry. Even though the twisting is usually described in the context of compactifying the higher dimensional `deconstructed' theory, we show how this naturally arises from the bifundamental fuzzy sphere f\\\/ield theory point of view.\n\nThe rest of this paper is organised as follows. In Section~\\ref{constructing} we give the\nharmonic decomposition of the GRVV matrices and relate them to the fuzzy supersphere. In Section~\\ref{equivalence} we present a one-to-one map between the adjoint and bifundamental fuzzy sphere constructions, while in Section~\\ref{Hopffibration} we establish that connection in terms of the fuzzy Hopf f\\\/ibration and def\\\/ine the fuzzy version of Killing spinors on $S^2$. We then discuss the resulting `deconstruction' of higher dimensional\nf\\\/ield theories on the 2-sphere, specif\\\/ically the issue of twisting of the f\\\/ields in order to preserve supersymmetry.\nIn Section~\\ref{deconstruction} we review the process and discuss the dif\\\/ferences between the adjoint and bifundamental cases, while in Section~\\ref{supersymmetric} we brief\\\/ly discuss a particular application by summarising the results of \\cite{Nastase:2009ny,Nastase:2009zu}. We conclude with some closing remarks in Section~\\ref{conclusions}.\n\n\n\n\n\\section{Constructing the f\\\/luctuation expansion}\\label{constructing}\n\n\n{\\bf Notation.} In this section, we will denote by $k$, $l$, $m$, $n$ the matrix indices\/indices of states in a~vector space, while keeping $i,j=1,\\dots,3$ as vector indices on the fuzzy $S^2$. We will also use~$j$ for the $\\SU(2)$ spin and $Y_{lm}$ for $S^2$ spherical harmonics, following the standard notation. The distinction should be clear by the context.\n\n\n\\subsection{Ground-state matrices and symmetries}\n\nWe begin by writing the ground-state solutions to (\\ref{equatio}), found in \\cite{Gomis:2008vc} and given by\n\\begin{gather}\n\\big( G^1\\big)_{m,n } = \\sqrt { m- 1 } \\delta_{m,n},\\nonumber\\\\\n\\big( G^2\\big)_{m,n} = \\sqrt { ( N-m ) } \\delta_{ m+1 , n },\\nonumber\\\\\n\\big(G_1^{\\dagger} \\big)_{m,n} = \\sqrt { m-1} \\delta_{m,n},\\nonumber\\\\\n\\big( G_2^{\\dagger}\\big)_{m,n} = \\sqrt { (N-n ) } \\delta_{ n+1 , m } .\\label{BPSmatrices}\n\\end{gather}\nUsing the decomposition of the above complex into real coordinates\n\\begin{gather}\nG^1=X^1+iX^2 , \\qquad G^2=X^3+iX^4,\n\\label{complex}\n\\end{gather}\none easily sees that these satisfy\n\\begin{gather*}\n\\sum_{p=1}^4 X_pX^p\\equiv G^\\a G_\\a^\\dagger=N-1 ,\n\\end{gather*}\nwhich at f\\\/irst glance would seem to indicate a fuzzy $S^3$ structure. However, note that in the above $G^1=G_1^\\dagger$ for the ground-state solution. With the help of (\\ref{complex}) this results in $X_2=0$, which is instead indicative of a fuzzy $S^2$.\n\nAs usual in the case of fuzzy sphere constructions, the matrices $G^\\a$ will be used to construct both the symmetry operators (as bilinears in $G$, $G^\\dagger$, also acting on $G^\\a$ themselves) as well as fuzzy coordinates, used to expand in terms of spherical harmonics on the fuzzy sphere.\n\n\n\n\\subsubsection[$GG^\\dagger$ relations]{$\\boldsymbol{GG^\\dagger}$ relations}\\label{2.1.1}\n\n\n\nAs a f\\\/irst step towards uncovering the $S^2$ structure we calculate the $G \\Gd$ bilinears\n\\begin{gather*}\n \\big( G^1 G_1^{\\dagger} \\big)_{m,n} = (m-1) \\delta_{mn},\\nonumber\\\\\n \\big( G^2 \\Gd_2 \\big)_{mn} = ( N-m ) \\delta_{mn},\\nonumber\\\\\n \\big( G^{1} \\Gd_2 \\big)_{ mn} = \\sqrt { ( m-1) ( N-m+1) } \\delta_{ m , n+1 },\\nonumber\\\\\n \\big( G^2 \\Gd_1 \\big)_{mn} = \\sqrt { ( N-m ) m } \\delta_{m+1 , n },\\nonumber\\\\\n \\big( G^\\alpha G_\\alpha^{\\dagger} \\big)_{mn} = ( N-1 ) \\delta_{mn} \n\\end{gather*}\nDef\\\/ining $ J^{\\a}_{ \\b } = G^{\\a } \\Gd_{ \\b } $ we get the following commutation relation\n\\begin{gather*}\n[ J^{\\a}_{\\b} , J^{ \\mu }_{ \\nu } ] = \\delta^{\\mu }_{\\b } J^{\\a }_{\\nu}\n - \\delta^{\\a}_{\\nu} J^{\\mu }_{\\b } .\n\\end{gather*}\nThese are commutation relations of the generators of $\\U(2) $.\nThen the $ J_{i} = ( \\tilde \\s_i )^{\\a}_{\\b} J_{\\a}^{\\b} $\n are the generators of $\\SU(2)$ that result in the usual formulation\nof the fuzzy\\footnote{Note that more correctly, we should have written\n${J^\\a}_\\b=G^\\a G_\\b^\\dagger$ and\n\\[\nJ_i={(\\tilde\\s_i)^\\a}_\\b{J^\\b}_\\a={(\\s_i)_\\b}^\\a {J^\\b}_\\a ,\n\\]\nbut in the following we will stick to the notation $J^\\a_\\b$. The kind of matrix multiplication that one has will be made clear from the context.} $S^2$, in terms of the algebra\n\\begin{gather}\\label{normsu2}\n [ J_i , J_j ] = 2i \\epsilon_{ijk} J_k .\n\\end{gather}\nThe trace $J\\equiv J^\\a_\\a=N-1$ is a trivial $\\U(1)\\simeq\\U(2)\/\\SU(2)$ generator, commuting with everything else.\n\n\n\n\\subsubsection[$G^\\dagger G$ relations]{$\\boldsymbol{G^\\dagger G}$ relations}\\label{2.1.2}\n\nNext, we calculate the $ \\Gd G$ combinations\n\\begin{gather*}\n \\big( \\Gd_1 G^1 \\big)_{mn} = (m-1) \\delta_{ mn},\\nonumber\\\\\n \\big( \\Gd_2 G^2 \\big)_{mn} = (N-m+1) \\delta_{mn} - N \\delta_{m1} \\delta_{n1},\\nonumber\\\\\n \\big( \\Gd_1 G^2 \\big)_{mn} = \\sqrt { ( m-1 ) ( N-m ) } \\delta_{m+1 , n },\\nonumber\\\\\n \\big( \\Gd_2 G^1 \\big)_{mn} = \\sqrt { ( m-2)( N-m+1 ) } \\delta_{ m,n+1 },\\nonumber\\\\\n \\big( \\Gd_\\alpha G^\\alpha \\big)_{mn} = N \\delta_{mn } - N \\delta_{ m1 } \\delta_{n1 \n\\end{gather*}\nand def\\\/ine $ \\bar J_{\\a}^{\\b} = \\Gd_{\\a} G^{\\b}$. The commutation relations for the above then form another copy of~$\\U(2)$\n\\begin{gather*}\n [ \\bar J^{\\a}_{\\b} , \\bar J^{\\mu }_{\\nu } ] = \\delta_{\\b}^{\\mu } \\bar\n J^{\\a}_{\\nu } - \\delta_{\\nu }^{\\a} \\bar J^{\\mu }_{\\b }\n\\end{gather*}\nand similarly,\n$ \\bar{J}_{i} = ( \\tilde \\s_i )^{\\a}_{\\b} \\bar{J}_{\\a}^{\\b} $ once again satisfy the usual $\\SU(2)$ algebra\\footnote{Again, note that we should have written ${\\bar{J}_\\a\\,}^\\b=\\Gd_\\a G^\\b$ which emphasises that for $\\bar{J}$, the\nlower index is the f\\\/irst matrix index, and\n\\[\n\\bar{J}_i={(\\tilde{\\s}_i)^\\a}_\\b {\\bar{J}_\\a\\,}^\\b={(\\sigma_i)_\\b}^\\a{\\bar{J}_\\a\\,}^\\b ,\n\\]\nwhich emphasises that as matrices, the $\\bar{J}_i$ are def\\\/ined with the Pauli matrices, whereas $J_i$ was def\\\/ined with\ntheir transpose. However, we will again keep the notation $\\bar{J}^\\b_\\a$.}, for another fuzzy $S^2$\n\\[\n [ \\bar{J}_i , \\bar{J}_j ] = 2i \\epsilon_{ijk} \\bar{J}_k .\n\\]\n\nThe trace\n\\begin{gather}\n(\\bar{J})_{mn}=(\\bar{J}^\\a_\\a)_{mn}=N\\delta_{mn}-N\\delta_{m1}\\delta_{n1},\\label{barJ}\n\\end{gather}\nwhich is a $\\U(1)\\simeq\\U(2)\/\\SU(2)$ generator, commutes with the $\\SU(2)$ generators $\\bar{J}_i$, though as a~matrix does not commute with the generators $J^1_2$ and $J^2_1$ of the f\\\/irst set of $\\SU(2)$ generators.\n\nAt this point, it seems that we have two $\\SU(2)$'s, i.e.\\ $\\SO(4)\\simeq\\SU(2)\\times \\SU(2)$ as expected for a 3-sphere, even though we have not yet shown that these are proper space symmetries: We have only found that the $J, \\bar J $ satisfy a certain symmetry algebra. In fact, we will next see that these are not independent but rather combine into a single $\\SU(2)$.\n\n\\subsubsection[Symmetry acting on bifundamental $(N,\\bar{N})$ matrices]{Symmetry acting on bifundamental $\\boldsymbol{(N,\\bar{N})}$ matrices}\n\nAll the $(N,\\bar{N})$ bifundamental scalar matrices are of the type\n$ G$, $G G^{\\dagger} G$, $G G^{\\dagger} G G^{\\dagger} G$, $\\ldots $. The simplest such terms are the $G^\\a$ matrices themselves, the action of the symmetry generators on which we will next investigate.\n\nIt is easy to check that the matrices $G^\\a$ satisfy\n\\begin{gather*\nG^1 G^{\\dagger}_2 G^2 - G^2 G^{\\dagger}_2 G^1 = G^1 ,\\qquad\n G^2 G^{\\dagger}_1 G^1 - G^1 G^{\\dagger}_1 G^2 = G^2 .\n\\end{gather*}\nUsing the def\\\/initions of $J_i$ and $\\bar J_i$, we f\\\/ind\n\\begin{gather}\nJ_i G^{\\a } - G^{\\a} \\bar J_i = ( \\tilde \\sigma_i )^{\\a}_{\\b} G^{\\b} . \\label{offdiagtrans}\n\\end{gather}\nThe $G^1$, $G^2 $\ntransform like the $ (1,0) $ and $ (0,1)$ column\nvectors of the spin-$\\frac{1}{2}$ representation with\nthe $J$'s and $ \\bar J $'s matrices in the\n$ \\mathfrak u(N) \\times\\mathfrak u( \\bar N)$ Lie algebra.\n\nBy taking Hermitian conjugates in (\\ref{offdiagtrans}), we f\\\/ind that the antibifundamental f\\\/ields,\n$G^\\dagger_\\a$, transform as\n\\begin{gather}\\label{offdiagtransdagger}\nG^{\\dagger}_{\\a} J_i - \\bar J_i G^{\\dagger}_{\\a} =\n G^{\\dagger}_{\\b} ( \\tilde \\sigma_i )^{\\b}_{\\a } .\n\\end{gather}\n\nTherefore the $G^{\\a}$, $G^\\dagger_{\\a}$ form a representation when acted by both $J_i $ and $\\bar J_i $, but neither symmetry by itself gives a representation for $G^\\a$, $G^\\dagger_\\alpha$. This means that the geometry we will be constructing from bifundamental f\\\/luctuation modes has a single $\\SU(2)$ symmetry, as opposed to two. Equations~(\\ref{offdiagtrans}) and (\\ref{offdiagtransdagger}) imply relations giving transformations between $J_i$ and $\\bar J_i$, thus showing they represent the same symmetry\n\\[\n \\Gd_{\\g} J_i G^{\\g} = ( N+1 ) \\bar J_i ,\\qquad\n G^{\\g} \\bar J_i \\Gd_{\\g } = ( N-2) J_i .\n\\]\n\nWriting the action of the full $\\SU(2) \\times \\U(1) $ on the $G^\\a$, including the $\\U(1)$ trace $\\bar{J}$, we obtain\n\\begin{gather}\n J^{\\a}_{\\b} G^{\\g } - G^{\\g} \\bar J^{\\a}_{\\b}\n = \\delta^{\\g}_{\\b} G^{ \\a} - \\delta^{\\a}_{\\b} G^{\\g}, \\label{u2}\n\\end{gather}\nwhile taking Hermitian conjugates of (\\ref{u2}) we obtain the $\\U(2)$ transformation of $\\Gd_\\a$,\n\\[\n\\bar J^{\\a}_{\\b} \\Gd_{\\g} - \\Gd_{\\g} J^{\\a}_{\\b}\n= - \\delta^{\\a}_{\\g} \\Gd_{ \\b} + \\delta^{\\a}_{\\b} \\Gd_{\\g} .\n\\]\nThe consequence of the above equations is that $G^{ \\a } $ has charge\n$1$ under the $\\U(1)$ generator $ \\bar J $. Thus\na global $\\U(1)$ symmetry action on $G^{\\a}$ does not leave\nthe solution invariant, and we need to combine with the action of\n$ \\bar J$ from the gauge group to obtain an invariance.\n\n\nWe next turn to the construction of fuzzy spherical harmonics out of~$G^\\a$.\n\n\n\n\n\n\n\\subsection[Fuzzy $S^2$ harmonics from $\\U(N)\\times\\U(\\bar N)$ with bifundamentals]{Fuzzy $\\boldsymbol{S^2}$ harmonics from $\\boldsymbol{\\U(N)\\times\\U(\\bar N)}$ with bifundamentals}\\label{decomposition}\n\n\nAll bifundamental matrices of $\\U(N)\\times\\U(\\bar N)$, are maps between two dif\\\/ferent vector spaces. On the other hand, products of the bilinears $G\\Gd$ and $G^\\dagger G$ are adjoint matrices mapping back to the same vector space. Thus, the basis of `fuzzy spherical harmonics' on our fuzzy sphere will be constructed out of all possible combinations: $\\U(N)$ adjoints like $G\\Gd$, $G\\Gd G\\Gd$, $\\ldots$, $\\U(\\bar N)$ adjoints like $\\Gd G$, $\\Gd G\\Gd G$, $\\ldots$, and bifundamentals like $G$, $G\\Gd G$, $\\ldots$ and $\\Gd$, $\\Gd G \\Gd$, $\\ldots$.\n\n\\subsubsection{The adjoint of $\\U(N)$}\n\nMatrices like $ G \\Gd $ act on an $N$ dimensional vector space that we call $ {\\bf V}^+ $. Thus the space of linear maps from $ {\\bf V}^+ $ back\nto itself, $ End ( { \\bf V }_+ ) $, is the adjoint of the $\\U(N)$ factor in the $ \\U(N) \\times \\U( \\bar N ) $ gauge group and $G \\Gd$ are\nexamples of matrices belonging to it. The space $ {\\bf V}^+ $ forms an irreducible representation of $ \\SU(2) $ of spin $ j =\\frac{ N-1}{ 2 }$,\ndenoted by $V_N$\n\\begin{gather*\n \\bVp = V_N .\n\\end{gather*}\nThe set of all operators of the form $ G \\Gd$, $G \\Gd G \\Gd$, $\\ldots $ belong in ${\\rm End} ( { \\bf V }_+ ) $ and can be expanded in\na basis of `fuzzy spherical harmonics'\ndef\\\/ined using the $\\SU(2)$ structure. Through the $\\SU(2) $ generators $J_i$ we can form the fuzzy spherical harmonics as\n\\begin{gather*}\n Y^{0} = 1,\\qquad\n Y^1_{ i } = J_i,\\qquad\n Y^2_{ ((i_1 i_2 ))} = J_{ {((} i_1 } J_{ i_2 {))} },\\qquad\n Y^l_{ ((i_1 \\cdots i_l )) } = J_{ (( i_1 } \\cdots J_{ i_l )) } \n\\end{gather*}\nIn the above, the brackets $ (( i_1 \\cdots i_l )) $ denote traceless symmetrisation. The complete space of\n$N\\times N $ matrices can be expanded in the fuzzy\nspherical harmonics with $ 0 \\le l \\le 2j = N-1 $. One indeed checks that\n\\[\nN^2 = \\sum_{l=0}^{2j} ( 2l+1 ) .\n\\]\nThen, a general matrix in the adjoint of $\\U(N)$ can be expanded as\n\\[\nA=\\sum_{l=0}^{N-1}\\sum_{m=-l}^l a^{lm} Y_{lm}(J_i) ,\n\\]\nwhere\n\\[\n Y_{lm}(J_i) = \\sum_i f_{lm}^{((i_1\\cdots i_l))} J_{i_1}\\cdots J_{i_l} .\n\\]\nThe $ Y_{lm}(J_i)$ become the usual spherical harmonics in the `classical' limit, when $N\\to\\infty$ and the cut-of\\\/f in the angular momentum is removed.\n\nIn conclusion, all the matrices of $\\U(N)$ can be organised into irreps of $\\SU(2)$ constructed out of $J_i$, which form the fuzzy spherical harmonics $ Y_{lm}(J_i)$.\n\n\n\\subsubsection[The adjoint of $\\U ( \\bar N ) $]{The adjoint of $\\boldsymbol{\\U ( \\bar N )}$}\n\nIn a fashion similar to the $\\U(N)$ case, the matrices $ \\Gd G$, $\\Gd G \\Gd G$, $\\ldots $, are linear endomorphisms of $ {\\bf V}^-$. These matrices are in the adjoint of the $ \\U ( \\bar N ) $ factor of the $ \\U(N) \\times \\U ( \\bar N ) $ gauge group, and will be organised into irreps of the $\\SU(2)$ constructed out of $\\bar{J}_i$.\n\nHowever, we now have a new operator: We have already noticed in (\\ref{barJ}) that the $\\U(1)$ generator $\\bar{J}$ is nontrivial. We can express it as\n\\[\n\\bar J = \\Gd_{\\a} G^{\\a} = N - N \\bar E_{11} .\n\\]\nThis means that $ {\\rm End} ( {\\bf V}_- )$ contains in addition to the identity matrix, the matrix $ \\bar E_{11} $ which is invariant under $ \\SU(2) $. If we label the basis states in $ { \\bf V }^- $ as $ | e^-_{k} \\rangle $\n with $ k =1,\\dots , N $, then $ \\bar E_{11} = |e^{-}_1\\rangle \\langle e^{-}_1 | $. This in turn means that ${\\bf V}^- $ is a reducible representation\n\\begin{gather*\n {\\bf V}^- = V^-_{N-1 } \\oplus V^-_{1 } .\n\\end{gather*}\nThe f\\\/irst direct summand is the irrep of $\\SU(2) $ with dimension $N-1$ while the second is a one-dimensional irrep.\nIndeed, one checks that the $ \\bar J_i $'s annihilate the state $ | e^-_1 \\rangle $, which is necessary for the\nidentif\\\/ication with the one-dimensional irrep to make sense.\n\nAs a result, the space $ {\\rm End} ( {\\bf V}^- )$ decomposes as follows\n\\[\n {\\rm End} ( {\\bf V}^- ) = {\\rm End} ( V^-_{N-1 } ) \\oplus {\\rm End} ( V^-_1 )\n\\oplus {\\rm Hom} ( V^-_{N-1 } , V^-_1 ) \\oplus {\\rm Hom} ( V^-_{1} , V^-_{N-1} ) ,\n\\]\nthat is, the matrices split as $M_{\\mu\\nu}=(M_{ij},M_{11},M_{1i},M_{i1})$.\nThe f\\\/irst summand has a decomposition in terms of another set of fuzzy spherical harmonics\n\\[\n Y_{lm}(\\bar J_i) = \\sum_i f_{lm}^{((i_1\\cdots i_l))} \\bar J_{i_1}\\cdots \\bar J_{i_l} ,\n\\]\n for $ l $ going from $0$ to $N-2$, since\n\\[\n(N-1)^2 = \\sum_{l=0}^{N-2} ( 2l + 1 ) .\n\\]\nThis gives only matrices in the $(N-1)$ block, i.e.\\ the ${\\rm End} ( V^-_{N-1 } )$.\nThe second summand is just one matrix transforming in the trivial irrep, $\\bar E_{11}$.\nThe remaining two $N-1$ dimensional spaces of matrices cannot be expressed\nas products of $ \\bar J_i$. They are spanned by\n\\[\n\\bar E_{1k} = | e^-_1 \\rangle \\langle e^-_k | \\equiv g^{--}_{1k }, \\qquad\n\\bar E_{k1} = | e^-_k \\rangle \\langle e^-_1 | \\equiv g^{--}_{k 1} ,\n\\]\nwhich are like spherical harmonics for ${\\rm Hom} ( V^-_{N-1 } , V^-_1 ) \\oplus {\\rm Hom} ( V^-_{1} , V^-_{N-1} )$.\nThey transform in the $N-1$ dimensional irrep\nof $\\SU(2) $ under the adjoint action of $ \\bar J_i $ and are zero mode eigenfunctions of the $\\U(1)$ symmetry operator $\\bar J$.\n\nTherefore, one can expand a general matrix in the adjoint of $\\U ( \\bar N ) $ as\n\\begin{gather}\n\\bar{A} = \\bar a_{0} \\bar E_{11} + \\sum_{ l=0}^{N-2}\\sum_{m=-l}^l \\bar a_{l m }\nY_{lm}(\\bar J_i) + \\sum_{k=2 }^N b_{k } g^{--}_{1k } +\n\\sum_{k=2}^N \\bar b_k g^{--}_{k 1} \\label{ubarnexp}\n\\end{gather}\nand note that we could have replaced $\\bar{E}_{11}$ with the $\\U(1)$ generator\n$\\bar{J}$ by redef\\\/ining $\\bar{a}_0$ and $\\bar{a}_{00}$.\n\n{\\sloppy\nIn the large $N $ limit the $ Y_{lm}(\\bar J_i)$ become the ordinary spherical harmonics of $S^2$, just like~$Y_{lm}(J_i)$. There are order $N^2$ of these modes, which is appropriate as the fuzzy $S^2$ can roughly be thought of as a 2-dimensional space with each dimension discretised in $N$ units. The mode~$\\bar a_0$,~$b_k$ and~$\\bar b_k$ can be neglected at large $N$, as they have much less than~$N^2$ degrees of freedom.\n\n}\n\n\n\n\n\n\\subsubsection[$\\SU(2)$ harmonic decomposition of bifundamental matrices]{$\\boldsymbol{\\SU(2)}$ harmonic decomposition of bifundamental matrices}\\label{harmdec}\n\nAs in the case of the $\\U(\\bar{N})$ matrices, the bifundamental matrices of the form $ G$, $G \\Gd G$, $\\ldots $ giving physical f\\\/luctuating f\\\/ields, are not enough to completely f\\\/ill ${\\rm Hom} ( \\bV^- , \\bV^+ ) $. Given the decomposition $ \\bVm = V_{N-1}^- \\oplus V_{1 }^- $, we decompose ${\\rm Hom} ( \\bV^- , \\bV^+ ) $ as\n\\[\n{\\rm Hom} ( \\bV^- , \\bV^+ ) = {\\rm Hom} ( V_{N-1}^- , V^+_N ) \\oplus {\\rm Hom} ( V_1^- ,V^+_N ) ,\n\\]\ni.e.~the matrices $M_{\\mu\\nu}$ as $(M_{i\\nu},M_{1\\nu})$.\nThe f\\\/irst summand has dimension $ N ( N-1) $, while\nthe second has dimension $ N $ and forms an irreducible representation of $\\SU(2)$.\n\nSince the $ V_{N-1}^- $ and $V^+_N$ are irreps of $\\SU(2)$\nwe can label the states with the eigenvalue of~$\\bar J_3$,~$J_3 $\nrespectively. Given our normalisation of the $\\SU(2)$ generators\nin~(\\ref{normsu2}), the usual spin is $ \\frac{ J_3^{\\max} }{ 2 } $.\nThe matrices in ${\\rm Hom} ( V_{N-1}^- , V^+_N )$ are of the form $ | e^+_m \\rangle\\langle e^-_n | $, where $ m = \\frac{ -N+1}{ 2 } ,\\frac{ -N +3}{2 } ,\\dots ,\n\\frac{N-1}{2} $, $ n = \\frac{-N +2}{2} , \\frac{-N+4}{2} ,\\dots , \\frac{N-2}{2} $ denote the eigenvalues of $ \\frac{J_3}{2}$.\n These are spanned by matrices of the form $ G ( \\bar J_{i_1} ) ( \\bar J_{i_2} ) \\cdots ( \\bar J_{i_l} )$, i.e.\\ the matrix $G$ times matrices in ${\\rm End}(V_{N-1}^-)$.\n\nThe operators in ${\\rm Hom} ( V_{N-1}^- , V^+_N )$ transform\nin representations of spin $ l+\\frac{1}{2}$ for $ l = 0,\\dots , N-2 $.\nThe dimensions of these representations correctly add up to\n\\[\n\\sum_{ l=0 }^{ N-2} ( 2l + 2 ) = N ( N - 1 ) .\n\\]\nThis then gives the $\\SU(2) $ decomposition of ${\\rm Hom} ( V_{N-1}^- , V^+_N )$ as\n\\[\n{\\rm Hom} ( V_{N-1}^- , V^+_N ) = \\bigoplus_{l=0}^{N-2} V_{l+ 1\/2 } .\n\\]\n\nOn the other hand, matrices $ |e^+_{k}\\rangle \\langle e^-_1 | \\equiv \\hE_{k 1 }\\in {\\rm Hom} ( V_1^- , V^+_N ) $ cannot be written in terms\nof the $G$'s and $G^\\dagger$'s alone, because $G^{\\a} $ acting on $| e^{-}_1\\rangle $ gives zero. The index $k$ runs over the $N$ states\nin $ \\bV^+$. Here $\\hE_{k1}$ are eigenfunctions of the operator $\\bar { E}_{11} $ with unit charge,\n\\[\n\\hE_{k1}\\bar {E}_{11}=\\hE_{k1} .\n\\]\n\nCombining all of the above, the bifundamental f\\\/luctuations $ r^{ \\alpha } $\n can be expanded as follows\n\\[\nr^\\a= r^{ \\a }_{\\b} G^\\b +\\sum_{k=1}^Nt^\\a_k\\hE_{k1} ,\n\\]\nwith\n\\[\n r^{ \\a }_{\\b} = \\sum_{l=0}^{N-2}\\sum_{m=-l}^l(r^{lm})^\\a_\\b Y_{lm}(J_i) .\n\\]\nWe further decompose $ r^\\a_\\b $ into a trace and a traceless part\n and def\\\/ine\n\\begin{gather*}\ns^\\a_\\b = r^{ \\a }_{\\b} - \\frac{1}{2 } \\delta^{\\a}_{\\b } r^{\\g}_{\\g},\\qquad\nr = r^{\\g}_{\\g},\\qquad\nT^{ \\a } = t^{ \\alpha}_{ k } \\hE_{k 1} .\n\\end{gather*}\nThus the complete expansion of $r^\\a$ is given simply in terms of\n\\begin{gather}\nr^{ \\a } = r G^{ \\a } + s^{\\a}_{ \\b } G^{ \\b }+ T^{ \\alpha}. \\label{fluctuati}\n\\end{gather}\nWe could equivalently have written\n\\begin{gather*}\nr^\\a=\\sum_{l=0}^{N-2}\\sum_{m=-l}^l(r^{lm})^\\a_\\b G^\\b Y_{lm}(\\bar J_i)+\\sum_{k=1}^Nt^\\a_k\\hE_{k1}\n\\end{gather*}\nusing the spherical harmonics in $ \\bar J$ in (\\ref{ubarnexp}).\nIn the following, we will choose, without loss of generality, to work with (\\ref{fluctuati}).\n\nUntil now we have focused on matrices in ${\\rm Hom} ( \\bV^- , \\bV^+ ) $ but the case of ${\\rm Hom} ( \\bV^+ , \\bV^- ) $ is similar. The matrices $ \\Gd$, $\\Gd G \\Gd$, $\\ldots $ will also form a representation of $\\SU(2)$ given by $\\bar{J}\\sim \\Gd G $, times a $\\Gd$ matrix. Once again one needs to add an extra $ T^{\\dagger}_{ \\a } = ( t^{ \\alpha}_{ k } )^* \\hF_{1 k} $ f\\\/luctuation in order to express the matrices $\\hF_{ 1 k } \\equiv |e^-_{1}\\rangle \\langle e^+_k | \\in {\\rm Hom}(V_N^+,V_1^-)$. In fact, the result for the complete f\\\/luctuating f\\\/ield can be obtained by taking a Hermitian conjugate of (\\ref{fluctuati}), yielding\n\\begin{gather*}\nr^{ \\dagger }_{ \\alpha } = \\Gd_{ \\a } r + \\Gd_{ \\b } s^{\\b}_{\\a} + T^{\\dagger}_{\\alpha}.\n\\end{gather*}\n\n\n\n\\subsection{Fuzzy superalgebra}\n\n\n\nThe matrices $G^\\a$ and $J_i$ can be neatly packaged into supermatrices which form a representation of the orthosymplectic Lie\nsuperalgebra $\\text{OSp}(1|2)$. The supermatrix is nothing but the embedding of the $N\\times \\bar N$\nmatrices into $\\U(2N)$. The adjoint f\\\/ields live in the `even subspace', while the bifundamentals in the `odd subspace'. For a generic supermatrix\n\\[\nM = \\twobytwo{A}{B}{C}{D}\n\\]\nthe superadjoint operation is\n\\[\nM^\\ddagger = \\twobytwo{A^\\dagger}{C^\\dagger}{-B^\\dagger}{D^\\dagger} .\n\\]\nFor Hermitian supermatrices this is\n\\[\nX = \\twobytwo{A}{B}{-B^\\dagger}{D} ,\n\\]\nwith $A = A^\\dagger$ and $D = D^\\dagger$ \\cite{Hasebe:2004yp}. This gives the def\\\/inition of the supermatrices\n\\[\n{\\bf J}_i = \\twobytwo{J_i}{0}{0}{\\bar J_i}\\qquad\\textrm{and}\\qquad {\\bf J}_\\alpha = \\twobytwo{0}{\\sqrt N G_\\alpha}{-\\sqrt N G^\\dagger_\\alpha}{0} ,\n\\]\nwhere we raise and lower indices as $G_\\alpha = \\epsilon_{\\alpha\\beta} G^\\beta$,\nwith $\\epsilon = i \\ts_2 = -i\\s_2$. Then the SU(2) algebra together with the relation (\\ref{offdiagtrans})\nand the def\\\/inition of $J_i$, $\\bar J_i$ result in the following (anti)commutation relations\n\\begin{gather*}\n[{\\bf J}_i,{\\bf J}_j] = 2 i \\epsilon_{ijk} {\\bf J}_k, \\qquad\n[{\\bf J}_i,{\\bf J}_\\alpha] = {(\\ts_i)_\\alpha}_\\beta {\\bf J}^\\beta,\\nonumber\\\\\n\\{{\\bf J_\\alpha},{\\bf J_\\beta}\\} = - (\\tilde \\sigma_i)_{\\alpha\\beta} {\\bf J}_i= - (i\\tilde \\sigma_2 \\tilde \\sigma_i)_{\\alpha\\beta} {\\bf J}_i\n\\end{gather*}\nwhich is the def\\\/ining superalgebra $\\text{OSp}(1|2)$ for the fuzzy {\\it supersphere} of \\cite{Grosse}.\n\nIt is known that the only irreducible representations of $\\text{OSp}(1|2)$ split into the spin-$j$ plus the spin-$(j-\\frac{1}{2})$\nrepresentations of $\\SU(2)$, which correspond {\\em precisely} to the irreducible representation for the $J_i$ (spin $j$) and $\\bar J_i$\n(spin $j-1\/2$) that we are considering\nhere\\footnote{See for instance Appendix C of \\cite{Hasebe:2004yp}. The general spin-$j$ is the $J_i$ representation constructed from the GRVV matrices, while the general spin $j-\\frac{1}{2}$ is the $\\bar J_i$ representation constructed from the GRVV matrices.}.\n\nAs a result, the most general representations of the fuzzy superalgebra, including $G^\\a$ besides~$J_i$,~$\\bar J_i$, coincide with the most general representations of the two copies of $\\SU(2)$. This points to the fact that perhaps the representations in terms of $G^\\a$ are equivalent to the representations of $\\SU(2)$. Next we will see that this is indeed the case.\n\n\n\\section{Equivalence of fuzzy sphere constructions}\\label{equivalence}\n\nWe now prove that our new def\\\/inition of the fuzzy 2-sphere in terms of bifundamentals is equivalent to the usual\ndef\\\/inition in terms of adjoint representations of the $\\SU(2)$ algebra.\n\nThe ABJM bifundamental scalars are interpreted as Matrix Theory ($N\\times N$) versions of Euclidean coordinates. Accordingly,\nfor our fuzzy space solution in the large $N$-limit one writes $G^\\a\\rightarrow \\sqrt{N}g^\\a$, with $g^\\a$\nsome commuting classical objects, to be identif\\\/ied and better understood in due course.\nIn that limit, and similarly writing $J_i \\rightarrow N x_i$, $\\bar J_i \\rightarrow N \\bar x_i$, one has from Sections~\\ref{2.1.1} and~\\ref{2.1.2}\nthat the coordinates\n\\begin{gather}\nx_i = {(\\ts_i)^\\a}_\\b g^\\b g^*_\\a,\\qquad\n\\bar x_i = {(\\ts_i)^\\a}_\\b g^*_\\a g^\\b\\label{hopf}\n\\end{gather}\nare two versions of the same Euclidean coordinate on the 2-sphere, $x_i\\simeq \\bar x_i$.\n\nIn the above construction the 2-sphere coordinates $x_i, \\bar x_i$ are invariant under multiplication of the classical objects $g^\\a$ by a\n$\\U(1)$ phase, thus we can def\\\/ine objects $\\tilde{g}^\\a$ \\emph{modulo} such a phase, i.e.\\ $g^\\a=e^{i\\a(\\vec{x})}\\tilde g^\\a$. The GRVV matrices\n(\\ref{BPSmatrices}), that from now on we will denote by\n$\\tilde{G}^\\a$ instead of~$G^\\a$, are fuzzy versions of representatives of $\\tilde g^\\a$, chosen such that $\\tilde{g}^1=\\tilde{g}_1^\\dagger$\n(one could of course have chosen a dif\\\/ferent representative for $\\tilde{g}^\\a$ such that $\\tilde{g}^2=\\tilde{g}_2^\\dagger$ instead).\n\nIn terms of the $ g^\\alpha$, equation~(\\ref{hopf}) is the usual Hopf map from the 3-sphere $ g^\\a g^\\dagger_\\a=1$ onto the 2-sphere $x_ix_i=1$,\nas we will further discuss in the next section. In this picture, the phase is simply the coordinate on the $\\U(1)$ f\\\/ibre of the Hopf f\\\/ibration,\nwhile the $\\tilde g^\\alpha$'s are coordinates on the $S^2$ base. While $g^\\a$ are complex coordinates acted upon by $\\SU(2)$, the $\\tilde g^\\a$\nare real objects acted upon by the spinor representation of $\\SO(2)$, so they can be thought of as Lorentz spinors in two dimensions, i.e.\\\nspinors on the 2-sphere.\n\n\nThe fuzzy version of the full Hopf map, $J_i={(\\ts_i)^\\a}_\\b G^\\b G^\\dagger_\\a$, can be given either using $G^\\a= U \\tilde G^\\a $ or\n$G^\\alpha = \\tilde{\\hat G}^\\alpha\\hat{U}$. The $U$ and $\\hat{U}$ are unitary matrices that can themselves be expanded in terms of\nfuzzy spherical harmonics\n\\begin{gather*}\n U=\\sum_{lm} U_{lm} Y_{lm}(J_i) ,\n \\end{gather*} with $U U^\\dagger = \\hat U \\hat U^\\dagger =1$, implying that in the\nlarge-$N$ limit $(U,\\hat U)\\rightarrow e^{i\\a(\\vec{x})}$.\n\nThat means that by extracting a unitary matrix from the left or the right of $G^\\a$, i.e.\\ modulo a unitary matrix, the resulting algebra for $\\tilde{G}^\\a$\n\\begin{gather}\n-\\tilde{G}^\\a=\\tilde{G}^\\b \\tilde{G}^\\dagger_\\b \\tilde{G}^\\a-\\tilde{G}^\\a \\tilde{G}^\\dagger_\\b \\tilde{G}^\\b\n\\label{algebra}\n\\end{gather}\nshould then be exactly equivalent to the usual $\\SU(2)$ algebra that appears in the adjoint construction: Both should give the same description of the fuzzy 2-sphere. We would next like to prove this equivalence for all possible representations.\n\n\n\n\\subsection{Representations}\n\nWe f\\\/irst note that the irreducible representations of the algebra (\\ref{algebra}), given by the matri\\-ces~(\\ref{BPSmatrices}), indeed give the most general irreducible representations of SU(2). Def\\\/ining $J_{\\pm}=J_1\\pm iJ_2$, $\\bar J_{\\pm}=\\bar J_1\\pm i \\bar J_2$, we obtain from~(\\ref{BPSmatrices}) that\n\\begin{gather*}\n(J_+)_{m,m-1} = 2\\sqrt{(m-1)(N-m+1)}=2\\alpha_{\\frac{N-1}{2},m-\\frac{N+1}{2}},\\\\\n(J_-)_{n-1,n} = 2\\sqrt{(n-1)(N-n+1)}=2\\alpha_{\\frac{N-1}{2},n-\\frac{N+1}{2}},\\\\\n(J_3)_{mn} = 2\\left(m-\\frac{N+1}{2}\\right)\\delta_{mn}\n\\end{gather*}\nand\n\\begin{gather*}\n(\\bar J_+)_{m,m-1} = 2\\sqrt{(m-2)(N-m+1)}=2\\alpha_{\\frac{N-2}{2},m-\\frac{N+2}{2}},\\\\\n(\\bar J_-)_{n-1,n} = 2\\sqrt{(n-2)(N-n+1)}=2\\alpha_{\\frac{N-2}{2},n-\\frac{N+2}{2}},\\\\\n(\\bar J_3)_{mn} = 2\\left(m-\\frac{N+2}{2}\\right)\\delta_{mn}+N\\delta_{m1}\\delta_{n1} ,\n\\end{gather*}\nwhereas the general spin-$j$ representation of $\\SU(2)$ is\n\\begin{gather*}\n (J_+)_{m,m-1}=\\alpha_{j,m},\\qquad\n (J_-)_{n-1,n}=\\alpha_{j,n},\\qquad\n (J_3)_{mn}=m\\delta_{mn}\n\\end{gather*}\n(and the rest zero), where\n\\begin{gather*}\n\\alpha_{jm}\\equiv \\sqrt{(j+m)(j-m+1)}\n\\end{gather*}\nand $m\\in -j,\\dots ,+j$ takes $2j+1$ values. Thus the representation for $J_i$ is indeed the most general\n$N=2j+1$ dimensional representation, and since $(\\bar J_+)_{11}=(\\bar J_-)_{11}=(\\bar J_3)_{11}=0$, the representation\nfor $\\bar J_i$ is also the most general $(N-1)=2(j-\\frac{1}{2})+1$ dimensional representation.\n\nWe still have the $\\U(1)$ generators completing the $\\U(2)$ symmetry,\nwhich in the case of the irreducible GRVV matrices $\\tilde G^\\a$\nare diagonal and give the fuzzy sphere constraint $\\tilde G^\\a\\tilde G^\\dagger_\\a\\propto \\one$, $\\tilde G^\\dagger _\\a \\tilde G^\\a\\propto \\one$,\n\\begin{gather*}\n J={J^1}_1+{J^2}_2=(N-1)\\delta_{mn},\\qquad\n \\bar J ={\\bar J_1\\,}^1+{\\bar J_2\\,}^2=N\\delta_{mn}-N\\delta_{m1}\\delta_{n1} ,\n\\end{gather*}\nwhere again $(\\bar J)_{11}=0$, since $\\bar J_i$ is in the $N-1\\times N-1$ dimensional representation:\nThe element $E_{11}=\\delta_{m1}\\delta_{n1}$ is a special operator, so the f\\\/irst element of the vector space on which\nit acts is also special, i.e.\\ ${\\bf V}^-=V^-_{N-1}\\oplus V^-_1$.\n\nMoving to reducible representations of $\\SU(2)$, the Casimir operator $\\vec{J}^2=J_iJ_i$ giving the fuzzy sphere constraint is diagonal,\nwith blocks proportional to the identity. The analogous object that gives the fuzzy sphere constraint in our construction is the\noperator $J=G^\\a G^\\dagger_\\a$. Indeed, in the case of reducible matrices modulo unitary transformations, $\\tilde{G}^\\a$,\nwe f\\\/ind (in the same way as for $\\vec{J}^2=J_i J_i$ for the $\\SU(2)$ algebra)\n\\begin{gather}\nJ=\\text{diag}((N_1-1)\\one_{N_1 \\times N_1},(N_2-1)\\one_{N_2 \\times N_2},\\dots )\\label{j}\n\\end{gather}\nand similarly for $\\bar J=G^\\dagger_\\a G^\\a$\n\\begin{gather}\n\\bar J=\\text{diag}\\big(N_1\\big(1-E^{(1)}_{11}\\big)\\one_{N_1 \\times N_1}, N_2\\big(1-E^{(2)}_{11}\\big)\\one_{N_2 \\times N_2},\\dots \\big).\n\\label{jbar}\n\\end{gather}\n\n\\subsection[GRVV algebra $\\rightarrow \\SU(2)$ algebra]{GRVV algebra $\\boldsymbol{\\rightarrow \\SU(2)}$ algebra}\n\nFor this direction of the implementation one does not need to consider the particular representations of the algebra; the matrices $\\tilde{G}^\\a$ will be kept as arbitrary solutions. We def\\\/ine as before, but now for an arbitrary solution $G^\\a$,\n\\begin{gather}\nG^\\a G^\\dagger_\\b\\equiv {J^\\a}_\\b\\equiv\\frac{J_i {(\\ts_i)^\\a}_\\b +J\\delta^\\a_\\b }{2}.\\label{definiti}\n\\end{gather}\nUsing the GRVV algebra it is straightforward to verify that $G^\\a G^\\dagger_\\a\\equiv J$ commutes with $J_k$.\n\nMultiplying (\\ref{algebra}) from the right by ${(\\ts_k)^\\gamma}_\\a G^\\dagger_\\gamma$, one obtains\n\\[\n-J_k = G^\\b G^\\dagger _\\b J_k-{J^\\a}_\\b {J^\\b}_\\gamma {(\\ts_k)^\\gamma}_\\a .\n\\]\nUsing the def\\\/inition for the ${J^\\a}_\\b$ factors in (\\ref{definiti}) and the relation $[J,J_k]=0$, one\narrives at\n\\[\n-J_k=\\frac{i}{2}\\epsilon_{ijk} J_i J_j ,\n\\]\nwhich is just the usual SU(2) algebra.\n\nIt is also possible to def\\\/ine\n\\[\n G^\\dagger_\\a G^\\b\\equiv {\\bar J_\\a\\,}^\\b\\equiv \\frac{\\bar J_i{(\\ts_i)^\\b}_\\a+\\bar J\\delta^\\b_\\a}{2}\n\\]\n and similarly obtain $[\\bar J,\\bar J_k]=0$. By multiplying (\\ref{algebra}) from the left by ${(\\ts_k)^\\gamma}_\\a G^\\dagger_\\gamma$, we get in a~similar way\n\\[\n -\\bar J_k=\\frac{i}{2}\\epsilon_{ijk}\\bar J_i \\bar J_j .\n\\]\n\nThus the general SU(2) algebras for $J_i$ and $\\bar J_i$ indeed follow immediately from (\\ref{algebra}) without restricting to the irreducible GRVV matrices.\n\n\\subsection[$\\SU(2)$ algebra $\\rightarrow$ GRVV algebra]{$\\boldsymbol{\\SU(2)}$ algebra $\\boldsymbol{\\rightarrow}$ GRVV algebra}\n\nThis direction of the implementation is {\\it a priori} more problematic since, as we have already seen, the representations of $J_i$ and $\\bar J_i$ are not independent. For the irreducible case in particular, $V_N^+$ is replaced by the representation $V^-_{N-1}\\oplus V^-_1$, so we need to generalise this identif\\\/ication to reducible representations in order to prove our result. As we will obtain this relation at the end of this section and it should have been the starting point of the proof, we will close with some comments summarising the complete logic.\n\nWe will f\\\/irst try to understand\nthe classical limit. The Hopf f\\\/ibration (\\ref{hopf}) can be rewritten, together with the normalisation\ncondition, as\n\\[\n g^\\alpha g^*_\\beta = \\frac{1}{2} \\big[x_i {(\\ts_i)^\\alpha}_\\beta + \\delta^\\alpha_\\beta\\big] .\n\\]\n\nBy extracting a phase out of $g^\\a$, we should obtain the variables $\\tilde{g}^\\a$ on $S^2$ instead of $S^3$.\nIndeed, the above equations can be solved for $g^\\a$ by\n\\begin{gather}\ng^\\alpha = \\doublet{g^1}{g^2} = \\frac{e^{i \\phi}}{\\sqrt{2(1+x_3)}}{\\doublet{1+x_3}{x_1-ix_2}}=e^{i\\phi}\\tilde{g}^\\a,\n\\label{tildeg}\n\\end{gather}\nwhere $e^{i\\phi}$ is an arbitrary phase.\n\nIn the fuzzy case $G^\\a$ and $G^\\dagger_\\b$ do not commute, and there are two dif\\\/ferent kinds of equations corresponding to $J_i$ and $\\bar J_i$,\n\\begin{gather}\nG^\\alpha G^\\dagger_\\beta \\equiv \\frac{1}{2} \\big[J_i {(\\ts_i)^\\alpha}_\\beta + \\delta^\\alpha_\\beta J\\big],\\qquad\nG^\\dagger_\\beta G^\\alpha \\equiv \\frac{1}{2} \\big[\\bar J_i {(\\ts_i)^\\alpha}_\\beta + \\delta^\\alpha_\\beta \\bar J\\big] .\n\\label{reverse}\n\\end{gather}\nWe also impose that $[J,J_k]=0$, $[\\bar J,\\bar J_k]=0$, so that $J$ and $\\bar J$ are diagonal and proportional to the identity in the irreducible components of $J_i$.\n\nWe solve the f\\\/irst set of equations in (\\ref{reverse}) by writing $G^1G^\\dagger_1=\\frac{1}{2}(J+J_3)$, for which the\nmost general solution is $G_1 = T U$, with $T$ a Hermitian and $U$ a unitary matrix.\nSince $J+J_3$ is real and diagonal, by def\\\/ining\n\\begin{gather*}\nT = \\frac{1}{\\sqrt 2} (J+ J_3 )^{1\/2}\n\\end{gather*}\nwe obtain\n\\begin{gather}\\label{320}\nG^\\alpha = \\doublet{G^1}{G^2} = {\\doublet{J+J_3}{J_1-iJ_2}}\\frac{T^{-1}}{2}U_{N\\times N}\n=\\tilde{G}^\\alpha U_{N\\times N} .\n\\end{gather}\nThus $\\tilde{G}^\\alpha$ is also completely determined by $J_i$, $J$.\n\nSimilarly, the second set of equations in (\\ref{reverse}) can be solved by considering $G^\\dagger_1 G^1=\\frac{1}{2} (\\bar J+\\bar J_3)$, for which the most general solution is $G^1=\\hat{U}\\tilde{T}$, where as before\n\\[\n\\tilde{T} = \\frac{1}{\\sqrt 2}\\Big(\\bar J+ \\bar J_3\\Big)^{1\/2} ,\n\\]\nto obtain\n\\begin{gather}\\label{322}\nG^\\alpha = \\doublet{G^1}{G^2} = \\hat{U}_{\\bar N\\times\\bar N}\n\\frac{\\tilde{T}^{-1}}{2}{\\doublet{\\bar J+\\bar J_3}{\\bar J_1-i\\bar J_2}}=\\hat{U}\\tilde{\\hat G}^\\a .\n\\end{gather}\nThus $\\tilde{\\hat G}^\\a$ is completely determined by $\\bar J_i$, $\\bar J$.\n\nComparing the two formulae for $G^\\a$ we see that they are compatible if and only if\n\\begin{gather}\n\\hat{U}=TU\\tilde{T}^{-1}\\qquad\\text{and}\\qquad\n\\bar J_1-i\\bar J_2=\\tilde{T}^2U^{-1}T^{-1}(J_1-iJ_2)T^{-1}U ,\\label{equiva}\n\\end{gather}\nwhere $U$ is an arbitrary unitary matrix. These equations def\\\/ine an identif\\\/ication between the two representations of $\\SU(2)$, in terms of $J_i$ and $\\bar J_i$, needed in order to establish the equivalence with the GRVV matrices.\n\nWe now analyse the equivalence for specif\\\/ic representations.\nFor the irreducible representations of $\\SU(2)$, we def\\\/ine $\\bar J_i$ from $J_i$ as before ($V_N^+\\rightarrow V_{N-1}^-\\oplus V_1^-$) and $J=(N-1)\\one_{N \\times N}$, $\\bar J=N(1-E_{11})\\one_{N \\times N}$. For reducible representations of $\\SU(2)$, $J_i$ can be split such that $J_3$ is block-diagonal, with various irreps added on the diagonal. One must then take $J$ and $\\bar J$ of the form in (\\ref{j}) and (\\ref{jbar}). The condition (\\ref{equiva}) is solved by $U=1$ and $J_1$, $J_2$ block diagonal, with the blocks being the irreps of dimensions $N_1,N_2,N_3,\\dots $, and the $\\bar J_1$, $\\bar J_2$ being also block diagonal, but where each $N_k\\times N_k$ irrep block is replaced with the $(N_k-1)\\times (N_k-1)$ irrep block, plus an $E^{(k)}_{11}$, just as for the GRVV matrices.\n\nWe can hence summarise the proof {\\it a posteriori} in the following steps:\n\\begin{enumerate}\\itemsep=0pt\n\\item Start with $J_i$ ($i = 1,2,3$) in the reducible representation of $\\SU(2)$, i.e.\\ block diagonal with the blocks being irreps of dimensions $N_1, N_2, N_3, \\dots $.\n\\item Take $J = G^\\alpha G^\\dagger_\\alpha$ and $\\bar J = G^\\dagger_\\alpha G^\\alpha$ as in (\\ref{j}) and (\\ref{jbar}) since these are necessary conditions for the $G^\\alpha$ to satisfy the GRVV algebra. The condition $[J, J_k]=0$ is used here.\n\n\\item The $\\bar J_i$ are completely determined (up to conventions) from $J_i$, $J$ and $\\bar J$ by (\\ref{equiva}) and the condition $[\\bar J, \\bar J_k]=0$.\n\n\\item The $\\tilde G^\\alpha$ are then uniquely determined by (\\ref{320}), while the $\\tilde{\\hat G}^\\alpha$ by (\\ref{322}).\n\n\\item The $\\tilde G^\\alpha$ and $\\tilde{\\hat G}^\\alpha$ def\\\/ined as above indeed satisfy the GRVV algebra.\n\n\\end{enumerate}\n\n\\section{Fuzzy Hopf f\\\/ibration and fuzzy Killing spinors}\\label{Hopffibration}\n\nHaving established the equivalence between the adjoint (usual) and the bifundamental (in terms of $\\tilde{G}^\\a$) formulations of the fuzzy $S^2$ we turn towards ascribing an interpretation to the mat\\-ri\\-ces~$\\tilde{G}^\\a$ themselves.\n\n\\subsection{Hopf f\\\/ibration interpretation}\n\nOne such interpretation was alluded to already in (\\ref{complex}), where the fuzzy (matrix) coordinates $G^\\a$ were treated as complex spacetime coordinates. The irreducible GRVV matrices satisfy $ \\tilde G^1 \\tilde G^{\\dagger}_1 + \\tilde G^2 \\tilde G^{\\dagger}_2 = N-1 $ and $\\tilde G^1 = \\tilde G^{\\dagger}_1 $. The f\\\/irst relation suggests a fuzzy 3-sphere, but the second is an extra constraint which reduces the geometry to a 2d one. This is in agreement with the fuzzy $S^2$ equivalence that we already established in the previous section. The matrices~$\\tilde G^\\a$ are viewed as representatives when modding out the $\\U(N)$ symmetry, and the condition $\\tilde G^1=\\tilde G^\\dagger_1$ amounts to a choice of representative of the equivalence class.\n\nThe construction of the fuzzy $S^2$ in usual (Euclidean) coordinates\nwas obtained by\n\\begin{gather*}\n J_i = ( \\tilde \\sigma_i )^{\\a}_{\\b} G^{\\b} G^{\\dagger}_{\\a},\\\\\n x_i = \\frac{ J_i}{\\sqrt{ N^2-1} }\\ \\Rightarrow\n\\left\\{ \\begin{array}{l}\n\\displaystyle x_1 = \\frac{ J_1}{\\sqrt{ N^2-1} } = \\frac{ 1}{\\sqrt{ N^2-1} } \\big( G^1 G^{\\dagger}_2 +\n G^2 G^{\\dagger}_1\\big), \\vspace{1mm}\\\\\n \\displaystyle x_2 = \\frac{J_2}{\\sqrt{ N^2-1} } =\n\\frac{i}{\\sqrt{ N^2-1} } \\big( G^1 G^{\\dagger}_2 - G^2 G^{\\dagger}_1 \\big), \\vspace{1mm}\\\\\n\\displaystyle x_3 = \\frac{J_3}{\\sqrt{ N^2-1} } = \\frac{1}{\\sqrt{ N^2-1}} \\big( G^1 G_1^{\\dagger} - G^2 G_2^{\\dagger} \\big) ,\n\\end{array} \\right. \\\\\n\\frac{G^\\a}{\\sqrt{N}} \\rightarrow g^\\a\n\\end{gather*}\nand we already stated that the relation between $g^\\a$ and $x_i$ is the classical Hopf map $S^3\\stackrel{\\pi}{\\rightarrow} S^2$,~(\\ref{hopf}).\n\nIndeed, the description of the Hopf map in classical geometry\nis given as follows: One starts with Cartesian coordinates $X_1$, $X_2$, $X_3$, $X_4 $ on the unit $S^3$ with\n\\[\n X_1^2 + X_2^2 + X_3^2 + X_4^2 =1\n\\]\nand then goes to complex variables $Z^1 = X_1 + i X_2$, $Z^2 = X_3 + i X_4 $, satisfying $Z^\\alpha Z^*_\\alpha = 1$. The Hopf map def\\\/ines\nCartesian coordinates on the unit $S^2$ base of the f\\\/ibration by\n\\begin{gather}\\label{classHopf}\nx_i = ( \\tilde \\sigma_i )^{ \\a}_{\\b} Z^{ \\b} Z^*_{\\a} ,\n\\end{gather}\nwhich is invariant under an $S^1$ f\\\/ibre def\\\/ined by multiplication of $Z^\\a$ by a phase. The $x_i$ are Euclidean coordinates on an $S^2$ since\n\\[\n x_i x_i = ( \\tilde \\sigma_i )^{ \\a}_{\\b}\n( \\tilde \\sigma_i )^{ \\mu }_{\\nu } Z^{ \\b} Z^*_{\\a}\n Z^{ \\nu } Z^*_{\\mu } = 1\n\\]\nand this identif\\\/ies $Z^\\alpha \\equiv g^\\alpha$ from above.\n\nLet us now work in the opposite direction, starting from the classical limit and discretising the geometry by demoting the Hopf map (\\ref{classHopf}) from classical coordinates to f\\\/inite matrices. We need matrices for $Z^{\\a}$ which we call $ G^{\\alpha }$. The coordinates $x_i$ transform in the spin-$1$ representation of $\\SU(2)$. If we want to build them from bilinears of the form $G^{ \\dagger} G $ we need $G$, $\\Gd $ to transform in the spin-$\\frac{1}{2}$ representation. We also want a gauge symmetry to extend the $\\U(1)$ invariance of $Z^\\a$ (the $S^1$ f\\\/iber of the Hopf map), and for $N$-dimensional matrices $\\U(N)$ is the desired complex gauge invariance that plays that role.\n\nIn the usual fuzzy 2-sphere, the $x_i$ are operators mapping an irreducible $N$-dimensional $\\SU(2)$ representation $ V_N $ to itself. It is possible to do this in an $\\SU(2)$-covariant fashion because the tensor product of spin-$1$ with $V_N$ contains $V_N$. Since $G^{\\alpha } $ are spin-$\\frac{1}{2}$, and $ \\frac{ 1}{2 } \\otimes V_{N} = V_{N+1} \\oplus V_{N-1} $ does not contain $V_N$, we need to work with reducible representations in order to have $G^\\a$ map the representation back to itself. The simplest thing to do would be to consider the representation $ V_N \\oplus V_{N-1} $. The next simplest thing is to work with $ V_{N} \\oplus ( V_{N-1} \\oplus V_1 ) $ and this possibility is chosen by the GRVV matrices~\\cite{Gomis:2008vc} and allows a gauge group $\\U(N) \\times \\U(\\bar N)$ which has a $\\mathbb Z_2$ symmetry of exchange needed to preserve parity.\n\n{\\sloppy\nSo the unusual property of the GRVV matrices $\\tilde G$, the dif\\\/ference between $ \\bVp = V_N $ and $ \\bVm = V_{N-1} \\oplus V_1 $ follows from requiring a matrix realisation of the fuzzy $S^2$ base of the Hopf f\\\/ibration. These in turn lead to the $\\SU(2)$ decompositions of $ {\\rm End} ( \\bVp )$, ${\\rm End} ( \\bVm )$, ${\\rm Hom} ( \\bVp , \\bVm )$, ${\\rm Hom} ( \\bVm , \\bVp ) $, for the f\\\/luctuation matrices that we saw in Section~\\ref{decomposition}.\\footnote{The usual fuzzy $S^2$ has also been discussed in terms of the Hopf f\\\/ibration, where the realisation of the $\\SU(2)$ generators in terms of bilinears in Heisenberg algebra oscillators yields an inf\\\/inite dimensional space which admits various projections to f\\\/inite $N$ constructions \\cite{Balachandran:2005ew}. In that case the $x_i $ are not bilinears in f\\\/inite matrices.}\n\n}\n\n\nThe $x_i$, $G$, $\\Gd $ are operators in $ \\bVp \\oplus \\bVm$ which is isomorphic, as a vector space, to ${ \\bf V}_N \\otimes V_2 $. The endomorphisms of ${ \\bf V }_N$ correspond to the fuzzy sphere. The $N $ states of ${ \\bf V }_N $ generalise the notion of points on $S^2$ to noncommutative geometry. The 2-dimensional space $V_2$ is invariant under the $\\SU(2)$. It is acted on by $G$, $\\Gd $ which have charge $ +1$, $-1$ under the $\\U(1)$ (corresponding to $(J , \\bar J ) $) acting on the f\\\/ibre of the Hopf f\\\/ibration, so we also have two points on top of our fuzzy $S^2$.\n\nSince in this subsection we looked at a f\\\/ibration of $S^3$,\nwe need to emphasise that the f\\\/luctuation analysis does not have enough modes to describe the full space of functions on~$S^3$, even if we drop the requirement of $\\SO(4)$ covariance and allow for the possibility of an $\\SU(2) \\times \\U(1)$ description.\nAs we explained above, the only remnant of the circle in the matrix construction is the multiplicity associated with having states $|+\\rangle$, $|-\\rangle$ in $ \\bVp $ and $ \\bVm $. A classical description of the $S^3$ metric as a Hopf f\\\/ibration contains a coordinate $y$ transverse to the $S^2$. Instead, the matrix f\\\/luctuations of our solution are mapped to functions on $S^2$ and hence lead to a f\\\/ield theory on $S^2$.\n\n\n\n\n\\subsection{Killing spinor interpretation}\\label{killinginterp}\n\n\nWe will close this circle of arguments by interpreting the classical objects $\\tilde{g}^\\a$, obtained in the large-$N$ limit of $\\tilde{G}^\\a$, as Killing spinors\nand fuzzy Killing spinors on the 2-sphere respectively.\n\nWe have seen that in the classical limit the relation between $J_i$ and $G^\\a$ becomes the f\\\/irst Hopf map (\\ref{hopf}), and hence can be thought of as its {\\it fuzzy} version. However, the above Hopf relation is invariant under multiplication by an arbitrary phase corresponding to shifts on the~$S^1$ f\\\/ibre, so the objects~$\\tilde{g}^\\a$ obtained by extracting that phase in~(\\ref{tildeg}), i.e.\\\n\\begin{gather}\n \\tilde{g}^\\a=\\frac{1}{\\sqrt{2(1+x_3)}}{\\doublet{1+x_3}{x_1-ix_2}} ,\\label{doubletul}\n\\end{gather}\nare instead def\\\/ined on the classical $S^2$. In the Hopf f\\\/ibration, the index of $g^\\alpha$ is a spinor index of the global $\\SO(3)$ symmetry\nfor the 2-sphere. By extracting the $S^1$ phase one obtains a real (or rather, subject to a reality condition)\n$\\tilde{g}^\\a$ and the $\\a$ can be thought of as describing a (Majorana)\nspinor of the $\\SO(2)$ local Lorentz invariance on the 2-sphere. We will argue that the latter is related to a Killing spinor. Note that this\ntype of index identif\\\/ication easily extends to all even spheres.\n\nIn the fuzzy version of (\\ref{doubletul}), the $\\tilde{G}^\\a$ obtained from $G^\\alpha$ by extracting a unitary matrix, are real objects\ndef\\\/ined on the fuzzy $S^2$. They equal the GRVV matrices in the case of irreducible representations, or\n\\begin{gather*}\n\\tilde{G}={\\doublet{J+J_3}{J_1-iJ_2}}\\frac{T^{-1}}{2}\n\\end{gather*}\nin general.\n\nThe standard interpretation, inherited from the examples of the $\\SU(2)$ fuzzy 2-sphere and other spaces, is that the matrix indices give rise to the dependence on the sphere coordinates and the index $\\a$ is a \\emph{global} symmetry index. However, we have just seen that already in the classical picture one can identify the global symmetry spinor index with the \\emph{local} Lorentz spinor index. Therefore we argue that the correct interpretation of the classical limit for $\\tilde{G}^\\a$ is as a~spinor with both global \\emph{and} local Lorentz indices, i.e.\\ the Killing spinors on the sphere $\\eta^{\\a I}$. In the following we will use the index $\\a$ interchangeably for the two.\n\nIn order to facilitate the comparison with the Killing spinors, we express the classical limit of the $J_i$--$\\tilde{G}^\\a$ relation as\n\\begin{gather}\nx_i\\simeq\\bar x_i = {(\\sigma_i)_\\a}^\\b \\tilde{g}^\\dagger_\\b \\tilde{g}^\\a. \\label{classi}\n\\end{gather}\n\n\\subsubsection*{Killing spinors on $\\boldsymbol{S^n}$}\n\nWe now review some of the key facts about Killing spinors that we will need for our discussion. For more details, we refer the interested reader to e.g.~\\cite{vanNieu1983,Eastaugh1985,vanNieuwenhuizen:1984iz,Gunaydin:1984wc,Nastase:1999kf}.\n\nOn a general sphere $S^n$, one has Killing spinors satisfying\n\\begin{gather*}\nD_\\mu \\eta(x)=\\pm \\frac{i}{2}m\\gamma_\\mu \\eta(x).\n\\end{gather*}\nThere are two kinds of\nKilling spinors, $\\eta^+$ and $\\eta^-$, which in even dimensions are related by the chirality matrix, i.e.\\ $\\gamma_{n+1}$, through $\\eta^+=\\gamma_{n+1} \\eta^-$, as can be easily checked.\nThe Killing spinors on~$S^n$ satisfy orthogonality, completeness and a reality condition. The latter depends on the application, sometimes taken to be the \\emph{modified} Majorana condition, which mixes (or identif\\\/ies) the local Lorentz spinor index with the global symmetry spinor index of $S^n$.\nFor instance, on~$S^4$ the orthogonality and completeness are respectively\\footnote{The charge conjugation matrix in $n$ dimensions satisf\\\/ies in general\n\\[\nC^T=\\kappa C,\\qquad\n\\gamma_\\mu^T=\\lambda C \\gamma_\\mu C^{-1} ,\n\\]\nwhere $\\kappa=\\pm$, $\\lambda=\\pm$ and it is used to raise\/lower indices. The Majorana condition is then given by\n\\[\n\\bar \\eta =\\eta^T C .\n\\]},\n\\begin{gather*}\n\\bar \\eta^I \\eta^J=\\Omega^{IJ} \\qquad\\text{and}\\qquad\n\\eta^\\a_J\\bar \\eta^J_\\b=-\\delta_\\b^\\a ,\n\\end{gather*}\nwhere the index $I$ is an index in a spinorial representation of the $\\SO(n+1)_G$ invariance group of the sphere and the index $\\a$ is an index in a spinorial representation of the $\\SO(n)_L$ local Lorentz group on the sphere. The indices are then identif\\\/ied by the \\emph{modified} Majorana spinor condition as follows\\footnote{For more details on Majorana spinors and charge conjugation matrices see~\\cite{vanNieu1983,VanNieuwenhuizen:1981ae} and the Appendix of \\cite{Nastase:1999kf}.}\n\\[\n\\bar \\eta^I\\equiv \\big(\\eta^I\\big)^TC^{(n)}_{-}=-\\big(\\eta^J\\big)^\\dagger \\gamma_{n+1} \\Omega^{IJ} ,\n\\]\nwhere $\\Omega^{IJ} = i \\sigma_2 \\otimes \\one_{\\sfrac{n}{2}\\times \\sfrac{n}{2}}$ is the invariant tensor of $\\text{Sp}(\\sfrac{n}{2})$, satisfying $\\Omega^{IJ}\\Omega_{JK} = \\delta^I_K$.\n\nThe Euclidean coordinates of $S^n$ are bilinear in the Killing spinors\n\\begin{gather}\\label{bilinear}\nx_i=(\\Gamma_i)_{IJ}\\bar \\eta^I \\gamma_{n+1} \\eta^J ,\n\\end{gather}\nwhere $\\eta$ are of a single kind ($+$ or $-$), or equivalently $\\bar \\eta_+^I\\eta_-^J$. In the above the $\\Gamma$ are in $\\SO(n+1)_G$, while the $\\gamma$ in $\\SO(n)_L$.\n\nStarting from Killing spinors on $S^n$, one can construct all the higher spherical harmonics. As seen in equation~(\\ref{bilinear}), Euclidean coordinates on the sphere are spinor bilinears. In turn, symmetric traceless products of the $x_i$'s construct the scalar spherical harmonics $Y^k(x_i)$.\\footnote{These are the higher dimensional extensions of the usual spherical harmonics $Y^{lm}(x_i)$ for $S^2$.} One can also construct the set of spinorial spherical harmonics by acting with an appropriate operator on~$Y^k\\eta^I$\n\\begin{gather*}\n\\Xi^{k,+} = [(k+n-1+iD\\!\\!\\!\\!\/)Y^k]\\eta_+,\\\\\n\\Xi^{k,-} = [(k+n-1+iD\\!\\!\\!\\!\/)Y^k]\\eta_-=[(k+1+iD\\!\\!\\!\\!\/)Y^{k+1}]\\eta_+ .\n\\end{gather*}\nNote that in the above the derivatives act only on the scalar harmonics~$Y^k$.\n\nAny spinor on the sphere can be expanded in terms of spinorial spherical harmonics, $\\Psi=\\sum_k \\psi_k\\Xi^{k,\\pm}$. Consistency imposes that the $\\Xi^{k,\\pm}$ can only be commuting spinors. The Killing spinors are then themselves {\\em commuting} spinors, as they are used to construct the spinorial spherical harmonics.\n\nFor higher harmonics the construction extends in a similar way but the formulae are more complicated and, as we will not need them for our discussion, we will not present them here. The interested reader can consult e.g.~\\cite{Kim:1985ez}.\n\n\\subsubsection*{Killing spinors on $\\boldsymbol{S^2}$ and relation between spinors}\n\nFor the particular case of the $S^2$, $\\gamma_i=\\Gamma_i = \\sigma_i$ for both the $\\SO(2)_L$ and the $\\SO(3)_G$ Clif\\\/ford algebras. Then the two $C$-matrices can be chosen to be: $C_+=-\\sigma_1$, giving $\\kappa=\\lambda=+$, and $C_-=i\\sigma_2=\\epsilon$, giving $\\kappa=\\lambda=-$. Note that with these conventions one has $C_-\\gamma_3=i\\sigma_2\\sigma_3=-\\sigma_1=C_+$. In the following we will choose the Majorana condition to be def\\\/ined with respect to $C_-$.\n\n Equation~(\\ref{bilinear}) then gives for $n=2$\n\\begin{gather}\n\\bar \\eta^I =(\\eta^T)^I C_- \\ \\Rightarrow \\ x_i=(\\sigma_i)_{IJ}(\\eta^T)^I C_- \\gamma_3 \\eta^J. \\label{classiK}\n\\end{gather}\nThe orthonormality and completeness conditions for the Killing spinors on $S^2$ are\n\\begin{gather*}\n\\bar \\eta^I \\eta^J=\\epsilon^{IJ}\\qquad\\text{and}\\qquad\\eta^\\a_J \\bar \\eta^J_\\b =-\\delta^\\a_\\b ,\n\\end{gather*}\nwhile the modif\\\/ied Majorana condition is\n\\begin{gather*}\n(\\eta^J)^\\dagger=\\epsilon_{IJ}\\bar \\eta^I\\equiv \\epsilon_{IJ}(\\eta^I)^T C_- .\n\\end{gather*}\nSince $C_-=\\epsilon$, by making both indices explicit and by renaming the index $I$ as $\\dot\\a$ for later use, one also\nhas\n\\begin{gather}\\label{modifiedeta}\n(\\eta^{\\a\\dot\\a})^\\dagger=\\eta_{\\a\\dot\\a}\\equiv \\epsilon_{\\a\\b}\\epsilon_{\\dot\\a\\dot\\b}\\eta^{\\b\\dot\\b} .\n\\end{gather}\n\n\nFinally, the spinorial spherical harmonics on $S^2$ are\n\\begin{gather*}\n\\Xi^{\\pm}_{lm}=[(l+1+iD\\!\\!\\!\\!\/\\;)Y_{lm}]\\eta_{\\pm\n\\end{gather*}\nand thus the spherical harmonic expansion of an $S^2$-fermion is (writing explicitly the sphere fermionic index $\\a$)\n\\[\n\\psi^\\a=\\sum_{lm,\\pm} \\psi_{lm,\\pm}\\Xi^{\\pm,\\a}_{lm}=\\sum_{lm,\\pm} {[\\psi_{lm,\\pm}(l+1+iD\\!\\!\\!\\!\/\\;)Y_{lm}]^\\a\n}_\\b\\eta_{\\pm}^\\b .\n\\]\n\nTo construct explicitly the Killing spinor, we must f\\\/irst def\\\/ine a matrix $S$, that can be used to relate between the two dif\\\/ferent kinds\nof spinors on~$S^2$, spherical and Euclidean.\n\n\nOn the 2-sphere, one def\\\/ines the Killing vectors $K_i^a$ such that the adjoint action of the $\\SU(2)$ generators on the fuzzy sphere f\\\/ields becomes a derivation in the large-$N$ limit\\footnote{Precise expressions for the Killing vectors as well as a set of useful identities can be found in Appendix~A of~\\cite{Nastase:2009zu}.}\n\\begin{gather*}\n[J_i,\\cdot]\\to 2iK_i^a\\d_a=2i\\epsilon_{ijk}x_j\\d_k .\n\\end{gather*}\nOne can then explicitly check that $K_i^a$ produces a Lorentz transformation on the gamma matrices\\footnote{A Lorentz transformation on the spinors acts as\n${\\Lambda^\\mu}_\\nu \\gamma^\\nu = S\\gamma^\\mu S^{-1}$, with~$S$ unitary.}\n\\begin{gather*}\nK_i^a{(\\ts_i)^\\a}_\\b=-e^{am}{\\big(S \\sigma^m S^{-1}\\big)_\\b\\, }^\\a\\equiv -{\\big(S \\gamma^a S^{-1}\\big)_\\b\\,}^\\a,\n\\end{gather*}\nwhere $e^{am}$ is the vielbein on the sphere and $S$ is a unitary matrix def\\\/ining the transformation ($|a|=1$)\n\\begin{gather*\nS=a\\begin{pmatrix}\n -\\sin{\\frac{\\theta}{2}} \\,e^{i\\phi\/2}& \\displaystyle -i\\cos{\\frac{\\theta}{2}} \\,e^{i\\phi\/2}\\vspace{1mm}\\\\\n\\displaystyle \\cos{\\frac{\\theta}{2}} \\,e^{-i\\phi\/2}&\\displaystyle -i\\sin{\\frac{\\theta}{2}} \\,e^{-i\\phi\/2}\n\\end{pmatrix} .\n\\end{gather*}\nImposing the (symplectic) reality condition on $S$\n\\begin{gather}\\label{symplectic}\n\\epsilon_{\\a\\beta}{\\big(S^{-1}\\big)^\\b}_\\gamma \\epsilon^{\\gamma\\delta}={\\big(S^T\\big)_\\a}^\\delta=\n{S^\\delta}_\\a ,\n\\end{gather}\nwe f\\\/ix $a=\\sqrt{i}^*$ and obtain the relations\n\\begin{gather}\n{\\big(S \\sigma_i S^{-1}\\big)_\\a\\,}^\\b = {\\big(S\\sigma_i S^{-1}\\big)^\\b}_\\a,\\qquad\n{\\big(S\\gamma_3 S^{-1}\\big)^\\a}_\\b = -x_i{(\\ts_i)^\\a}_\\b,\\nonumber\\\\\n{\\big(S\\gamma_a S^{-1}\\big)^\\a}_\\b = -h_{ab }K_i^b{(\\ts_i)^\\a}_\\b.\\label{gamma3}\n\\end{gather}\n\nIf one has real spinors obeying\n\\begin{gather*}\n(\\chi_{\\a\\dot\\a})^\\dagger=\\chi^{\\a\\dot\\a}\\equiv \\epsilon^{\\a\\b}\\epsilon^{\\dot\\a\\dot\\b}\\chi_{\\b\\dot\\b},\n\\end{gather*}\nwhich was identif\\\/ied in (\\ref{modifiedeta}) as the {\\em modified} Majorana spinor condition,\nit follows from (\\ref{symplectic}) that rotation by the matrix $S$ preserves this relation, i.e.\\\n\\begin{gather}\n((\\chi_{\\dot\\a}S)_\\a)^\\dagger=\\big(S^{-1}\\chi^{\\dot\\a}\\big)^\\a\\equiv -\\epsilon^{\\dot\\a\\dot\\b}\\big(S^{-1}\\big)^{\\a\\b}\n\\chi_{\\b\\dot\\b}=\\epsilon^{\\dot\\a\\dot\\b}\\epsilon^{\\a\\b}(\\chi_{\\dot\\b}S)_\\b . \\label{rotreal}\n\\end{gather}\n\nWe can now def\\\/ine the explicit form of the Killing spinor\n\\begin{gather*\n\\eta^{I\\a}={\\big(S^{-1}\\big)^\\a}_\\b \\eta_0^{I\\b}= \\frac{1}{\\sqrt{2}} {\\big(S^{-1}\\big)^\\a}_\\b \\epsilon^{\\b I}=\\frac{1}{\\sqrt{2}}\n{S^I}_J \\epsilon^{\\a J},\n\\end{gather*}\nwhere in the last equality we used the (symplectic) reality condition (\\ref{symplectic}) on~$S$. From~(\\ref{rotreal}) it is clear that the $\\eta^{I\\a}$ obey the \\emph{modified} Majorana condition. It is then possible to use (\\ref{gamma3}) to prove that\n\\begin{gather*}\nx_i=(\\sigma_i)_{IJ}\\bar \\eta^I \\gamma_3 \\eta^J ,\n\\end{gather*}\nhence verifying that the $\\eta^{I\\a}$ are indeed Killing spinors. One can also explicitly check that\n\\begin{gather*}\nD_a\\big({\\big(S^{-1}\\big)^\\a}_\\b \\epsilon^{\\b I}\\big)=+\\frac{i}{2}{(\\gamma_a)^\\alpha}_\\beta {\\big(S^{-1}\\big)^\\beta}_\\gamma \\epsilon^{\\gamma I} ,\n\\end{gather*}\nwhich in turn means that\n\\begin{gather*}\n\\frac{1}{\\sqrt{2}} {\\big(S^{-1}\\big)^\\a}_\\b \\epsilon^{\\b I}=\\eta_+^{\\a I} .\n\\end{gather*}\n\n\n\\subsubsection*{Identif\\\/ication with Killing spinor}\n\n\nUsing (\\ref{modifiedeta}), we rewrite (\\ref{classiK}) as\n\\begin{gather}\\label{432}\nx_i={(\\sigma_i)^I}_J \\big(\\eta^I\\big)^\\dagger \\gamma_3\\eta^J\n={(\\ts_i)^I}_J\\big(\\sqrt{2}P_+\\eta^I\\big)^\\dagger\\big(\\sqrt{2}P_+\\eta^J\\big) ,\n\\end{gather}\nwhere $P_\\pm = \\frac{1}{2}(1\\pm\\gamma_3)$. Now comparing (\\ref{432}) with (\\ref{classi}) one is led to the following natural large-$N$ relation, $\\tilde{G}^\\a\\rightarrow \\sqrt{2N} P_+\\eta^I$, provided the spinor indices $\\a$ and $I$ get identif\\\/ied, i.e.\\\n\\begin{gather*}\n\\frac{\\tilde{G}^\\a}{\\sqrt{N}}\\equiv \\tilde{g}^\\a\\leftrightarrow \\tilde{g}^I\\equiv \\sqrt{2}P_+\\eta^I\n={(P_+)^\\a}_\\b {(S^{-1})^\\b}_\\g \\epsilon^{\\g I}\n={(P_+)^\\a}_\\b {S^I}_J \\epsilon^{\\b J}={S^I}_J{(P_-)^J}_K\n\\epsilon^{\\a K}\n\\end{gather*}\nThus, the Weyl projection can be thought of as `removing' either $\\a$ or $I$, since only one of the two spinor components is non-zero.\n\nIn order to further check this proposed identif\\\/ication at large-$N$ we now calculate\n\\begin{gather}\n\\d_a \\big(\\sqrt{2}P_+\\eta^I\\big)= -\\frac{i}{2} {\\big(S \\gamma_a S^{-1}\\big)^I}_J\n\\big(\\sqrt{2}P_+\\eta^J\\big)\n+\\tilde{T}_a \\big(\\sqrt{2}P_+\\eta^I\\big),\\label{killisp}\n\\end{gather}\nwhere $\\tilde{T}_\\theta=0$ and $\\tilde{T}_\\phi=\\frac{i}{2}\\cos\\theta$ and\n\\[\n(\\d_a S )S^{-1}=-\\frac{i}{2}S\\gamma_aS^{-1}+S T_a S^{-1}\n\\]\nby explicitly evaluation, with $T_\\theta=0$ and $T_\\phi=-\\frac{i}{2}\\cos\\theta \\gamma_3$.\n\nThis needs to be compared with the analogous result given in equation~(4.48) of \\cite{Nastase:2009ny} from the classical limit of the adjoint action of $J_i$ on $\\tilde{G}^\\a$, i.e.\\ from $[J_i,\\tilde{G}^\\a]$,\n\\begin{gather}\n\\d_a \\tilde{g}^\\a=\\frac{i}{2}\\hat h_{ab}K_i^b{(\\ts_i)^a}_\\beta \\tilde g^\\beta\n= -\\frac{i}{2}{\\big(S\\gamma_a S^{-1}\\big)^\\a}_\\b \\tilde{g}^\\b. \\label{classg}\n\\end{gather}\nIn \\cite{Nastase:2009ny} it was also verif\\\/ied that the above could reproduce the correct answer for $\\d_a x_i$, which can be rewritten as\n\\[\n\\d_a x_i=-\\frac{i}{2}\\tilde{g}^\\dagger_\\a\\big[{(\\ts_i)^\\a}_\\b {\\big(S\\gamma_a S^{-1}\\big)^\\b}_\\g-{\\big(S\\gamma_a S^{-1}\\big)^\\a}_\\b {(\\ts_i)^\\b}_\\g\\big]\n\\tilde{g}^\\g .\n\\]\n\nNote that even though there is a dif\\\/ference between (\\ref{killisp}) and (\\ref{classg}), given by the purely imaginary term $\\tilde{T}_a$\nthat is proportional to the identity, the two answers for $\\d_a x_i$ exactly agree, since in that case the extra contribution cancels.\nThis extra term is a ref\\\/lection of a double ambiguity: First, the extra index $\\a$ on $\\eta^I$ can be acted upon by matrices,\neven though it is Weyl-projected, in ef\\\/fect multiplying the Weyl-projected $\\eta^I$ by a complex number; if the complex number is a phase,\nit will not change any expressions where the extra index is contracted, thus we have an ambiguity against multiplication by a phase.\nSecond, $\\tilde{g}^\\a$ is just a representative of the reduction of $g^\\a$ by an arbitrary phase, so it is itself only def\\\/ined\nup to a phase. The net ef\\\/fect is that the identif\\\/ication of the objects in (\\ref{killisp}) and (\\ref{classg}) is only up to a phase.\nIndeed, locally, near $\\phi\\simeq 0$, one could write\n\\[\n\\tilde{g}^\\a e^{\\frac{i}{2}\\phi \\cos\\theta} \\ \\leftrightarrow \\ \\sqrt{2}P_+\\eta^I\n\\]\nbut it is not possible to get an explicit expression for the phase over the whole sphere.\n\n\n\\subsection{Generalisations}\n\n\n\nOn a general $S^{2n}$ some elements of the above analysis of fuzzy Killing spinors carry through.\nThat is because even though it is possible to write for every $S^{2n}$\n\\[\nx_A=\\bar\\eta^I (\\Gamma_A)_{IJ} \\gamma_{2n+1}\\eta^J ,\n\\]\nwhere $\\eta^I$ are the Killing spinors, one only has possible fuzzy versions of the quaternionic and octonionic Hopf maps to match it against,\ni.e.\\ for $2n=4,8$. We will next f\\\/ind and interpret the latter in terms of Killing spinors on the corresponding spheres.\n\n\\subsubsection*{$\\boldsymbol{S^4}$}\n\nThe second Hopf map, $S^7\\stackrel{\\pi}{\\rightarrow} S^4$, is related to the quaternionic algebra. Expressing the $S^7$ in terms of\ncomplex coordinates $g^\\a$, now with $\\a=1,\\dots ,4$, the sphere constraint becomes $g^\\a g^\\dagger_\\a=1$ ($g^\\a g^\\dagger_\\a=1\\Rightarrow x_A x_A=1$;\n$A = 1,\\dots ,5$). The map in this case is (see for instance\n\\cite{Wu:1988py})\n\\[\nx_A=g^\\b {(\\Gamma_A)^\\a}_\\b g^\\dagger_\\a,\n\\]\nwith ${(\\Gamma_A)^\\a}_\\b$ the $4\\times 4$ $\\SO(5)$ gamma matrices\\footnote{These are constructed as: $\\sigma_1$ and $\\sigma_3$ where $1$ is replaced by ${\\one}_{2\\times 2}$ and\n$\\sigma_2$ where $i$ is replaced by $i\\sigma_1, i\\sigma_2,i\\sigma_3$.}. Here we have identif\\\/ied the spinor index $I$ of $\\SO(5)$ with the Lorentz spinor index $\\a$ of $\\SO(4)$.\n\nInitially, the $g^\\a$'s are complex coordinates acted upon by $\\SU(4)$, but projecting down to the base of the Hopf f\\\/ibration we\nreplace $g^\\a$ in the above formula with real $\\tilde{g}^\\alpha$'s, instead acted upon by the spinorial representation of $\\SO(4)$, i.e.\\\nby spinors on the 4-sphere. This process is analogous to what we saw for the case of the 2-sphere. Once again, it is possible to\nidentify $\\tilde{g}^\\a$ with the Killing spinors, this time on $S^4$.\n\nThis suggest that one should also be able to write a spinorial version of the fuzzy 4-sphere\nfor some bifundamental matrices $\\tilde{G}^\\a$, satisfying\n\\[\nJ_A = \\tilde{G}^\\b {(\\Gamma_A)^\\a}_\\b \\tilde{G}^\\dagger_\\a, \\qquad\n\\bar J_A = \\tilde{G}^\\dagger_\\a {(\\Gamma_A)^\\a}_\\b \\tilde{G}^\\b ,\n\\]\nwhere $J_A$, $\\bar J_A$ generate an $\\SO(5)$\nspinor rotation on $\\tilde G^\\a$ by\n\\[\nJ_A\\tilde G^\\a- \\tilde G^\\a \\bar J_A={(\\Gamma_A)^\\a}_\\b\\tilde G^\\b .\n\\]\nThis in turn implies that the fuzzy sphere should be described by the same GRVV algebra as for the $S^2$ case\n\\[\n\\tilde G^\\a=\\tilde G^\\a\\tilde G^\\dagger_\\b\\tilde G^\\b -\\tilde G^\\b\\tilde G^\\dagger_\\b\\tilde G^\\a\n\\]\nbut now with $\\tilde G^\\a$ being 4 complex matrices that describe a fuzzy 4-sphere, which poses an interesting possibility that we\nwill however not further investigate here.\n\n\\subsubsection*{$\\boldsymbol{S^8}$}\nThe third Hopf map, $S^{15}\\stackrel{\\pi}{\\rightarrow} S^8$, is related to the octonionic algebra. The $S^{15}$ is expressed now\nby the real objects $g^T_\\a g^\\a=1$, $\\a=1,\\dots ,16$ that can be split into two groups ($1,\\dots ,8$ and $9,\\dots ,16$). The Hopf map is expressed by \\cite{Bernevig:2003yz} ($g^T_\\a g_\\a=1\\Rightarrow x_Ax_A=1$)\n\\[\nx_A=g^T_\\a(\\Gamma_A)^{\\a\\b}g_\\b ,\n\\]\nwhere $(\\Gamma_A)^{\\a\\b}$ are the $\\SO(9)$ gamma-matrices\\footnote{The gamma-matrices are constructed\nsimilarly to the $S^4$ case as follows: $\\Gamma_i=\\begin{pmatrix}0&\\lambda_i\\\\-\\lambda_i &0\\end{pmatrix}$, $\\Gamma_8=\n\\begin{pmatrix} 0&\\one_{8\\times 8}\\\\ \\one_{8\\times 8}&0\\end{pmatrix}$, $\\Gamma_9=\\begin{pmatrix}\\one_{8\\times 8}&0\\\\0&-\\one_{8\\times\n8}\\end{pmatrix}$, i.e.\\ from $\\sigma_2$ with $\\lambda_i$ replacing $i$, and from $\\sigma_1$ and $\\sigma_3$ with\n$1$ replaced by $\\one_{8\\times 8}$. The\n$\\lambda_i$ satisfy $\\{\\lambda_i,\\lambda_i\\}=-2\\delta_{ij}$ (similarly to the $i\\sigma_i$\nin the case of $S^4$) and are constructed from the structure constants of the algebra of the octonions~\\cite{Bernevig:2003yz}. An explicit\ninversion of the Hopf map is given by $g_\\a=[(1+x_9)\/2]^{1\/2}u_\\a$ for $\\a=1,\\dots ,8$ and $g_\\a=[2(1+x_9)]^{-1\/2}\n(x_8-x_i\\lambda_i)u_{\\a-8}$ for $\\a=9,\\dots,16$, with $u_\\a$ a real 8-component $\\SO(8)$ spinor satisfying $u^\\a u_\\a=1$\nthus parametrising the $S^7$ f\\\/ibre.}.\nSimilarly for the case of the $S^4$ above, even though $g^\\a$'s are initially 16-dimensional\nvariables acted by the spinor representation of $\\SO(9)$, one can project down to the base of the Hopf f\\\/ibration\nand replace the $g^\\a$'s with real 8-dimensional\nobjects on the 8-sphere $\\tilde{g}^\\a$. Then the $\\tilde{g}^\\a$'s are identif\\\/ied with the Killing spinors of $S^8$.\n\nThis once again suggests that one should be able to write a spinorial version of the fuzzy 8-sphere for some\nbifundamental matrices $\\tilde{G}^\\a$ satisfying\n\\[\nJ_A = \\tilde G_\\a(\\Gamma_A)^{\\a\\b}\\tilde G^T_\\b,\\qquad\n\\bar J_A = \\tilde G^T_\\a(\\Gamma_A)^{\\a\\b}\\tilde G_\\b ,\n\\]\nwhere $J_A$, $\\bar J_A$ generate an $\\SO(9)$\nspinor rotation on $\\tilde G^\\a$ by\n\\[\nJ_A\\tilde G_\\a - \\tilde G_\\a \\bar J_A={(\\Gamma_A)_\\a}^\\b\\tilde G_\\b\n\\]\nand implies the same GRVV algebra, but with the $\\tilde G^\\a$'s now being 16 dimensional real matrices that\ndescribe the fuzzy 8-sphere.\n\n\n\\section{Deconstruction vs.\\ twisted compactif\\\/ication}\\label{deconstruction}\n\nWe now describe certain changes which occur when `deconstructing' a supersymmetric f\\\/ield theory on\nthe bifundamental fuzzy $S^2$, in contrast to the usual $S^2$, and comparing with the compactif\\\/ied higher-dimensional theory.\n\nThe term `deconstruction' was f\\\/irst coined in \\cite{ArkaniHamed:2001ca} for a specif\\\/ic four-dimensional model but more generally extends to creating higher dimensional theories through f\\\/ield theories with matrix degrees of freedom of high rank. In our particular case, the fuzzy $S^2$ background arises as a~solution in a $d$-dimensional f\\\/ield theory and f\\\/luctuations around this background `deconstruct' a~$d+2$-dimensional f\\\/ield theory. We will focus on the case where the $d+2$-dimensional f\\\/ield theory compactif\\\/ied on $S^2$ is supersymmetric.\n\n\\subsection[Adjoint fuzzy $S^2$]{Adjoint fuzzy $\\boldsymbol{S^2}$}\n\nThis construction is familiar in the context of D-branes, though any f\\\/ield theory with a fuzzy~$S^2$ background will also do. For instance, the example we will follow is \\cite{Andrews:2006aw}, where an ${\\cal N}=1$ supersymmetric massive $\\SU(N)$ gauge theory around a fuzzy $S^2$ background solution, coming from the low energy theory on a stack of D3-branes in some nontrivial background, was identif\\\/ied with the Maldacena--N\\'u\\~nez theory of IIB 5-branes with twisted compactif\\\/ication on $S^2$ \\cite{Maldacena:2000mw}. This construction was known to give an ${\\cal N}=1$ massive theory after dimensional reduction that can be identif\\\/ied with the starting point, thus the D3-brane theory around the fuzzy sphere deconstructs the 5-brane theory.\n\nThe twisting of the 5-brane f\\\/ields can be understood both in the compactif\\\/ication as well as the deconstruction pictures. In compactif\\\/ication, and for the \\cite{Andrews:2006aw} model, it is known from \\cite{Bershadsky:1995qy} that in order to preserve supersymmetry on D-branes with curved worldvolumes one needs to twist the various D-brane f\\\/ields. Specif\\\/ically, that means embedding the $S^2$ spin connection, taking values in $\\SO(2)\\simeq\\U(1)$, into the R-symmetry. As a result, the maximal supersymmetry one can obtain after compactif\\\/ication is ${\\cal N}=1$ (corresponding to $\\U(1)_R$). On the other hand, in deconstruction, the need for twisting will instead appear by analysing the kinetic operators of the deconstructed f\\\/ields.\n\nThe brane intuition, though useful, is not necessary, and in the following we will understand the twisting as arising generally from requiring supersymmetry of the dimensionally reduced compactif\\\/ied theory. This will be matched by looking at the kinetic term diagonalisation of the deconstructed theory.\n\n\\subsubsection*{Compactif\\\/ication}\n\nOn a 2-sphere, scalar f\\\/ields are decomposed in the usual spherical harmonics $Y_{lm}(x_i)=Y_{lm}(\\theta,\\phi)$ and can thus give massless f\\\/ields after compactif\\\/ication (specif\\\/ically, the $l=0$ modes). However, that is no longer true for spinors and gauge f\\\/ields. In that case, the harmonic decomposition in terms of $Y_{lm}(x_i)$ must be redef\\\/ined in order to make explicit the Lorentz properties of spinors and vectors on the 2-sphere, i.e.\\ to make them eigenvectors of their corresponding operators.\n\nSpinors on the sphere are eigenvectors of the total angular momentum $J_i^2$. These are of two types: Eigenvectors $\\Omega$\nof the orbital angular momentum $L_i^2$ (Cartesian spherical spinors) and eigenvectors $\\Upsilon$ of the Dirac operator on the sphere\n$-i\\hat{\\nabla}_{S^2}=-i\\hat h^{ab} e^{m}_a\\sigma_m\\nabla_b$ (spherical basis spinors), whose square is\n$R^2(-i\\hat{\\nabla}_{S^2})^2=J_i^2+\\frac{1}{4}$. The two are related by a transformation with a sphere-dependent matrix $S$, already described in\nSection \\ref{killinginterp}. The former are decomposed in the spinorial spherical harmonics\n\\[\n\\Omega^{\\hat{\\a}}_{jlm}=\\sum_{\\mu=\\pm \\sfrac{1}{2}}C(l,\\sfrac{1}{2},j;m-\\mu, \\mu,m)Y_{l,m-\\mu}(\\theta,\\phi)\\chi_\\mu^{\\hat{\\a}} ,\n\\]\nwhere $j=q_{\\pm}=l\\pm \\frac{1}{2}$ and $\\hat{\\a}=1,2$, as\n\\[\n\\psi^{\\hat\\a}=\\sum_{lm}\\psi_{lm}^{(+)}\\Omega_{l+\\frac{1}{2},lm}^{\\hat\\a}+\\psi_{lm}^{(-)} \\Omega_{l-\\frac{1}{2},lm}^{\\hat\\a} .\n\\]\nBoth have a minimum mass of $\\frac{1}{2R}$, since the Dirac operator squares to $J_i^2+\\frac{1}{4}=j(j+1)+\\frac{1}{4}$. Similarly,\nthe vector f\\\/ields do not simply decompose in $Y_{lm}$'s, but rather in the vector spherical harmonics\n\\begin{gather*}\n \\frac{1}{R}{\\bf T}_{jm}=\\frac{1}{\\sqrt{j(j+1)}}\\big[\\sin\\theta \\d_\\theta Y_{jm} {\\bf \\hat\\phi}\n-\\csc\\theta \\d_\\phi Y_{jm}{\\bf\\hat\\theta}\\big],\\\\\n \\frac{1}{R}{\\bf S}_{jm}=\\frac{1}{\\sqrt{j(j+1)}}\\big[\\d_\\theta Y_{jm} {\\bf \\hat\\theta}\n+\\d_\\phi Y_{jm}{\\bf\\hat\\phi}\\big] ,\n\\end{gather*}\nwith $j\\geq 1$. It is more enlightening to show the decomposition of the f\\\/ield strength on the 2-sphere\n\\begin{gather*}\n\\frac{1}{R}\\csc\\theta F_{\\theta\\phi}=R^2\\sum_{lm}F_{lm}\\frac{1}{\\sqrt{l(l+1)}}\\Delta_{S^2}Y_{lm} ,\n\\end{gather*}\nwith $l=1,2,\\dots $. Thus again only massive and no massless modes are obtained after dimensional reduction \\cite{Andrews:2006aw}. Note that as we can see,\nthe expansion in spinorial or vector spherical harmonics corresponds to redef\\\/ining the expansion in terms of $Y_{lm}$ (rearranging its coef\\\/f\\\/icients).\n\nTherefore in the absence of twisting supersymmetry will be lost after dimensional reduction, since all $S^2$-fermions will be massive but some massless $S^2$-scalars will still remain. Twisting, however, allows for the presence of fermionic twisted-scalars (T-scalars), i.e.\\ fermions that are scalars of the twisted $\\SO(2)_T$ Lorentz invariance group (with charge $T$), which will stay massless. In this way the number of supersymmetries in the dimensionally reduced theory equals the number of fermionic T-scalars.\n\nOne chooses the twisted Lorentz invariance of the sphere as $Q_T=Q_{xy}+Q_A$, where $Q_{xy}$ is the charge under the original Lorentz invariance of the sphere $\\SO(2)_{xy}$, and $Q_A$ is the charge under the $\\U(1)$ subgroup of R-symmetry. This is necessary because one needs to identify the $\\U(1)$ spin connection (`gauge f\\\/ield of Lorentz invariance') with a corresponding connection in the R-symmetry subgroup, i.e.\\ a gauge f\\\/ield from the transverse manifold.\n\nAn example of an action for twisted f\\\/ields is provided by the\nresult of \\cite{Bershadsky:1995qy}, for a bosonic T-spinor $\\Xi$, fermionic T-scalars $\\Lambda$ and T-vectors $g_a$\n\\begin{gather}\n\\int\\! d^d x d^2\\sigma\\sqrt{h}\\!\\left[ -\\frac{i}{2}\\mu \\bar \\Lambda \\gamma^\\mu \\d_\\mu \\Lambda\\!-\\!\\frac{i}{2}\\mu\n\\bar g_a \\gamma^\\mu \\d_\\mu g^a\\!+\\!\\mu \\omega^{ab} \\bar G_{ab}\\Lambda\\!-\\!2\\d_\\mu \\Xi^\\dagger \\d^\\mu \\Xi\n\\!-\\!8\\Xi^\\dagger\\big({-}i\\hat \\nabla_{S^2}\\big)^2\\Xi\\right]\\! ,\\!\\!\\label{twistDorey}\n\\end{gather}\nwhere $\\mu$ is the mass parameter, $G_{ab}= \\d_{a} g_{b}- \\d_{b} g_{a}$ is the f\\\/ield strength of the fermionic T-vector, and as usual $\\omega^{ab}=\\frac{1}{\\sqrt{g}}\\epsilon^{ab}$ is the symplectic form on the sphere. We note that the kinetic terms in the f\\\/lat directions ($\\mu,\\nu$) are given by their bosonic or fermionic nature, while the type of kinetic terms in the sphere directions ($a,b$) are dictated by their T-spin and the number of derivatives on it are again dictated by their statistics (bosons have two derivatives, fermions only one).\n\nThese f\\\/ields are decomposed in spherical harmonics corresponding to their T-charge. Then e.g.\\ the fermionic T-scalar can have a massless ($l=0$) mode, which after dimensional reduction will still be a fermion and give ${\\cal N}=1$ supersymmetry.\n\n\\subsubsection*{Deconstruction}\n\nTo have a fuzzy sphere background of the usual type, we need in the worldvolume theory at least 3 scalar modes $\\phi_i$ to satisfy $[\\phi_i,\\phi_j]=2i\\epsilon_{ijk}\\phi_k$, but usually there are more. Then the need for e.g.\\ bosonic T-spinors is uncovered by diagonalising the kinetic term for all the scalar f\\\/luctuations around the fuzzy sphere background. For instance in \\cite{Andrews:2006aw}, there are 6 scalar modes forming 3 complex scalars $\\Phi_i$, with f\\\/luctuations $\\delta\\Phi_i=a_i+ib_i$ and kinetic term\n\\begin{gather*}\n\\int d^d x d^2 \\sigma \\sqrt{h} \\delta\\Phi_i^\\dagger\\big[\\big(1+J^2\\big)\\delta_{ij}-i\\epsilon_{ijk}J_k\\big]\\delta \\Phi_j .\n\\end{gather*}\nThe (complete set of) eigenvectors of this kinetic operator are given by the vector spherical harmonics $J_i Y_{lm}$ and the spinorial\nspherical harmonics $\\Omega^{\\hat{\\a}}_{jlm}$. This kinetic operator is then diagona\\-lised by def\\\/ining T-vectors $n_a$ coming from the\nvector spherical harmonics and T-spinors $\\xi^{\\hat{\\a}}$ coming from the spinor spherical harmonics.\nWhen completing this program, the deconstructed action is the same as the compactif\\\/ied one, e.g.\\ for \\cite{Andrews:2006aw} one again obtains\nthe twisted action~(\\ref{twistDorey}).\n\nAt f\\\/inite $N$, the matrices are expanded in the fuzzy spherical harmonics $Y_{lm}(J_i)$, becoming the $Y_{lm}(x_i)$ of classical $S^2$, but the above diagonalisation corresponds in the classical limit to re-organising the expansion (this includes a nontrivial action on the coef\\\/f\\\/icients of the expansion) to form the spinorial, vector, etc.\\ spherical harmonics.\n\nThus for the adjoint construction all the f\\\/ields on the classical $S^2$ appear as limits of functions expanded in the scalar fuzzy spherical harmonics, $Y_{lm}(J_i)$, and the various tensor structures of~$S^2$ f\\\/ields were made manifest by diagonalising the various kinetic operators.\n\n\n\n\\subsection[Bifundamental fuzzy $S^2$]{Bifundamental fuzzy $\\boldsymbol{S^2}$}\n\nThe case of the bifundamental fuzzy $S^2$ is richer. One wants to once again compare with the same compactif\\\/ication picture. However, the particulars of the deconstruction will be dif\\\/ferent.\n\n\\subsubsection*{Deconstruction}\n\nHere we need a fuzzy sphere background of GRVV type, hence at least 2 complex scalar modes~$R^\\a$ in the worldvolume theory giving the fuzzy sphere background in terms of $R^\\a=fG^\\a$, with~$G^\\a$ satisfying~(\\ref{equatio}). The f\\\/luctuation of this f\\\/ield will be called $r^\\a$.\n\nPerforming the deconstruction follows a set of steps similar to the adjoint fuzzy $S^2$, namely one wants to expand in the fuzzy spherical harmonics and in the classical limit reorganise the expansion (acting nontrivially on the coef\\\/f\\\/icients of the expansion) to construct the spinor, vector, etc.\\ spherical harmonics. However now there are some subtle points that one needs to take into account. We have two kinds of fuzzy spherical harmonics, $Y_{lm}(J_i)$ and $Y_{lm}(\\bar J_i)$, both giving the same $Y_{lm}(x_i)$ in the classical limit. Adjoint f\\\/ields, e.g.\\ the gauge f\\\/ields, will be decomposed in terms of one or the other according to their respective gauge groups. On the other hand for bifundamental f\\\/ields one must f\\\/irst `extract' a bifundamental GRVV matrix, $\\tilde G^\\a$~or~$\\tilde G^\\dagger_a$, before one is left with adjoints that can be decomposed in the same way. We detailed this procedure for~$r^\\a$ in Section~\\ref{harmdec}. The expansion in $Y_{lm}(x_i)$ must be then reorganised as in the usual fuzzy~$S^2$ in order to diagonalise the kinetic operator, thus producing the spinor, vector, etc.\\ spherical harmonics.\n\nThe most important dif\\\/ference is that $\\tilde G^\\a$ has a spinor index on $S^2$; in particular we saw in Section~\\ref{killinginterp} that in the classical limit $\\tilde g^\\a$ is identif\\\/ied with a Killing spinor. That means that the operation of `extracting' $\\tilde G^a$ corresponds to automatically twisting the f\\\/ields! Let us make this concrete by considering a specif\\\/ic example.\n\n In the mass-deformed ABJM theory, one has besides the $R^\\a$ f\\\/ield a doublet of scalar f\\\/ields~$Q^{\\dot\\a}$ with f\\\/luctuation $q^{\\dot\\a}$, where $\\dot\\a$ is an $\\SU(2)$ index transverse to the sphere. Thus the $q^{\\dot\\a}$ start of\\\/f life as scalars. However, due to their bifundamental nature, one must f\\\/irst `extract' $\\tilde G^\\a\\rightarrow \\sqrt{N}\\tilde g^\\a$, by writing $q^{\\dot\\a}=Q_\\a^{\\dot\\a}\\tilde G^\\a$. In order to diagonalise the kinetic operator, we perform an S-transformation and construct\n\\begin{gather}\n \\Xi^\\alpha_{\\dot \\alpha} = i(P_+ S^{-1} Q_{\\dot \\alpha})^\\alpha + \\big(P_- S^{-1} Q_{\\dot \\alpha}\\big)^\\alpha,\\label{Qredef}\n\\end{gather}\n after which the kinetic term becomes the twisted action\n\\begin{gather}\n\\label{finaltransverse}\n N^2\\int d^3 x d^2\\sigma \\sqrt {\\hat h}\\left[ \\frac{1}{2} \\bar \\Xi^{\\dot \\alpha} (- i2 \\mu\\hat\\nabla_{S^2})^2 \\Xi_{\\dot \\alpha} -\\frac{1}{2} \\d_\\mu \\bar \\Xi^{\\dot \\alpha}\\d^\\mu \\Xi_{\\dot \\alpha} - 3 \\mu^2\\bar \\Xi^{\\dot \\alpha} \\Xi_{\\dot \\alpha} \\right] .\n\\end{gather}\n\nMore generally, the functions on the sphere are actually sections of the appropriate bundle: Either ordinary functions, sections of the spinor or the line bundle. Specif\\\/ically, anything without an $\\alpha$ index is a T-scalar, one $\\a$ index implies a T-spinor and two $\\a$ indices a T-scalar plus a T-vector in a $( {\\bf 1}\\oplus {\\bf 3})$ decomposition. That is, the $\\U(1)_T$ invariance is identif\\\/ied with the $\\SO(2)_L\\simeq \\U(1)_L$ Lorentz invariance of the sphere, described by the index~$\\a$.\n\nIn addition to this, an interesting new alternative to the above construction also arises. We can choose to keep $\\tilde G^\\a$ in the spherical harmonic expansion (by considering it as part of the spherical harmonic in the classical limit). The derivative of the spherical harmonic expansion then includes the derivative of $\\tilde g^\\a$ given in (\\ref{classg}) and one obtains a fuzzy version of the classical derivative operator\n\\begin{gather*}\nq^\\dagger_{\\dot{\\beta}}J_i-\\bar J_i q^\\dagger_{\\dot{\\b}} \\ \\rightarrow \\\n2i K_i^a \\d_a q^\\dagger_{\\dot{\\b}}+q^\\dagger_{\\dot\\b}x_i .\n\\end{gather*}\nThis operator acts on all bifundamental f\\\/ields, including the ABJM fermions $\\psi^{\\dagger\\a}$.\nIn this new kind of expansion, we recover the usual Lorentz covariant kinetic term. For instance for the scalar f\\\/ields $q^{\\dot\\a}$ of ABJM\nwe obtain (after a rescaling of the f\\\/ields)\n\\[\n\\frac{1}{g_{YM}^2}\\int d^3 x d^2 \\s \\sqrt { h }\\big[ {-}\\d^A q_{\\dot\\a}^\\dagger \\d_A q^{\\dot\\a}\\big] ,\n\\]\nwhere $A=\\mu, a$ is a total (worldvolume + fuzzy sphere) index. The price one pays for this simplicity (compared to~(\\ref{finaltransverse})) is however that the classical $N\\rightarrow \\infty$ limit of the supersymmetry transformation is very\nsubtle, since a naive application will relate f\\\/ields with dif\\\/ferent f\\\/inite $N$ gauge structures (bifundamentals with adjoints),\nnaively implying a gauge-dependent supersymmetry parameter.\n\nBut at least formally, by keeping $\\tilde G^\\a$ inside the spherical harmonic expansion, we obtain an un-twisted, fully supersymmetric version of the action on the whole worldvolume plus the fuzzy sphere.\n\n\\section[Supersymmetric D4-brane action on fuzzy $S^2$ from ABJM]{Supersymmetric D4-brane action on fuzzy $\\boldsymbol{S^2}$ from ABJM}\\label{supersymmetric}\n\nAs a concrete application of the whole discussion thus far, we present the f\\\/inal results for the Lagrangian obtained by studying f\\\/luctuations around the fuzzy $S^2$ ground-state of the mass-deformed ABJM model.\n\nThe f\\\/luctuating f\\\/ields are the $r^\\a$ scalars forming the fuzzy sphere background, transverse scalars $q^{\\dot\\a}$,\ngauge f\\\/ields $A_\\mu$ and $\\hat A_\\mu$, fermions $\\psi_\\a$ and $\\chi_{\\dot\\a}$.\nThe spherical harmonic expansion on the fuzzy sphere is for each of the above\n\\begin{gather*}\nr^\\a = r\\tilde{G}^\\a+{s^\\a}_\\b \\tilde{G}^\\b=\\big[(r)_{lm}\\delta^\\a_\\b+({s^\\a}_\\b)_{lm}\\big]Y_{lm}(J_i)\\tilde{G}^\\b,\n\\nonumber\\\\\nq^{\\dot\\a} = Q^{\\dot\\a}_\\a\\tilde{G}^\\a=(Q^{\\dot\\a}_\\a)_{lm}Y_{lm}(J_i)\\tilde{g}^\\a,\\nonumber\\\\\n\\psi_\\a = \\tilde{\\psi}\\tilde{G}_\\a+{U_\\a}^\\b \\tilde{G}_\\b=\\big[(\\tilde{\\psi})_{lm}\\delta_\\a^\\b+({U_\\a}^\\b)_{lm}\\big]\nY_{lm}(J_i)\\tilde{G}_\\b,\\nonumber\\\\\n\\chi_{\\dot\\a} = \\chi_{\\dot\\a \\a}\\tilde{G}^\\a=(\\chi_{\\dot\\a \\a})_{lm}Y_{lm}(J_i)\\tilde{G}^\\a,\\nonumber\\\\\nA_\\mu = A_\\mu^{lm}Y_{lm}(J_i),\\qquad\n\\hat A_\\mu = \\hat A_\\mu^{lm}Y_{lm}(\\bar J_i) \n\\end{gather*}\nbecoming in the classical limit\n\\begin{gather*}\nr^\\a = r\\tilde{g}^\\a+{s^\\a}_\\b \\tilde{g}^\\b=\\big[(r)_{lm}\\delta^\\a_\\b+({s^\\a}_\\b)_{lm}\\big]Y_{lm}(x_i)\\tilde{g}^\\b,\n \\\\\n q^{\\dot\\a} = Q^{\\dot\\a}_\\a\\tilde{g}^\\a=(Q^{\\dot\\a}_\\a)_{lm}Y_{lm}(x_i)\\tilde{g}^\\a, \\\\\n \\psi_\\a = \\tilde{\\psi}\\tilde{g}_\\a+{U_\\a}^\\b \\tilde{g}_\\b=\\big[(\\tilde{\\psi})_{lm}\\delta_\\a^\\b+({U_\\a}^\\b)_{lm}\\big]\nY_{lm}(x_i)\\tilde{g}_\\b,\\\\\n \\chi_{\\dot\\a} = \\chi_{\\dot\\a \\a}\\tilde{g}^\\a=(\\chi_{\\dot\\a \\a})_{lm}Y_{lm}(x_i)\\tilde{g}^\\a,\\\\\n A_\\mu = A_\\mu^{lm}Y_{lm}(x_i),\\qquad\n \\hat A_\\mu = \\hat A_\\mu^{lm}Y_{lm}(x_i) \n\\end{gather*}\nThese can be further redef\\\/ined as\n\\begin{gather*}\n {s^\\a}_\\b {(\\ts_i)^\\b}_\\a=K_i^aA_a+x_i\\phi,\\qquad\n \\Upsilon_{\\dot\\a}^\\a=(P_-S^{-1}\\chi_{\\dot\\a})^\\a ,\n\\end{gather*}\nwith $A_a$ becoming the sphere component of the gauge f\\\/ield and $\\Phi=2r+\\phi$ becoming a scalar, while $2r-\\phi$ is `eaten' by the gauge f\\\/ield\nin a Higgs mechanism that takes us from nonpropagating CS gauge f\\\/ield to propagating YM f\\\/ield in 3d \\cite{Mukhi:2008ux}. The f\\\/inal supersymmetric version of the\naction is then\n\\begin{gather*}\nS_{\\rm phys} = \\frac{1}{g_{\\rm YM}^2}\\int d^3 x d^2 \\s \\sqrt { h }\n \\Bigg[{-}\\frac{1}{4} F_{AB} F^{AB}-\\frac{1}{2} \\partial_A\n\\Phi \\partial^A\\Phi - \\frac{\\mu^2}{2} \\Phi^2 -\\d^A q_{\\dot\\a}^\\dagger \\d_A q^{\\dot\\a}\n+ \\frac{\\mu}{2}\\; \\omega^{ab} F_{ab}\\Phi \\\\\n\\phantom{S_{\\rm phys} =}{}\n+\\left(\\frac{1}{2}\\bar \\Upsilon^{\\dot \\alpha} \\tilde D_5 \\Upsilon_{\\dot\\alpha} +\n\\frac{i}{2} \\mu \\bar \\Upsilon^{\\dot\\alpha} \\Upsilon_{\\dot{\\a}}+{\\rm h.c.}\\right)\n-(\\psi S) \\tilde D_5 \\big(S^{-1}\\psi^\\dagger\\big) +\\frac{i}{2}\\mu (\\psi S) \\big(S^{-1}\\psi^\\dagger\\big) \\Bigg] .\n\\end{gather*}\n\nThe twisting of the f\\\/ields that have a $\\tilde G^\\a$ in their spherical harmonic expansion is done as follows: First, we twist\nby expressing $q^{\\dot\\a}$ as $Q^{\\dot\\a}_\\a$ and $\\psi_\\a$ as $\\tilde{\\psi}$, ${U_\\a}^\\b$. We then redef\\\/ine the twisted f\\\/ields in order to\ndiagonalise their kinetic operator by further writing $Q_\\a^{\\dot\\a}$ according to (\\ref{Qredef}) and\n\\begin{alignat*}{3}\n& {U_\\a}^\\b = \\frac{1}{2}U_i{(\\ts_i)_\\a}^\\b ,\\qquad &&\n{\\bar U_\\a\\,}^\\b=\\frac{1}{2} U_i {(\\ts_i)_\\a}^\\b, & \\\\\n& U_i = K_i^a g_a+\\hat\\psi x_i ,\\qquad &&\n\\bar U_i=K_i^a\\bar g_a+\\bar{\\hat\\psi} x_i .\n\\end{alignat*}\n\nThe f\\\/inal twisted action is\n\\begin{gather*}\nS_{\\rm phys} = \\frac{1}{g_{\\rm YM}^2}\\int d^3 x d^2 \\s \\sqrt { h }\n \\Bigg[-\\frac{1}{4} F_{AB} F^{AB}-\\frac{1}{2} \\partial_A\n\\Phi \\partial^A\\Phi - \\frac{\\mu^2}{2} \\Phi^2\n+ \\frac{\\mu}{2} \\omega^{ab} F_{ab}\\Phi\\\\\n\\phantom{S_{\\rm phys} =}{}\n +\\left(\\frac{1}{2}\\bar \\Upsilon^{\\dot \\alpha} D_5 \\Upsilon_{\\dot\\alpha} + \\frac{i}{2} \\mu \\bar\n\\Upsilon^{\\dot\\alpha} \\Upsilon_{\\dot{\\a}}+{\\rm h.c.}\\right)\n+\\frac{1}{4} \\bar \\Xi^{\\dot \\alpha} \\left(- \\frac{2i}{\\mu} \\nabla_{S^2}\\right)^2 \\Xi_{\\dot \\alpha} -\n\\d_\\mu \\bar \\Xi^{\\dot \\alpha}\\d^\\mu \\Xi_{\\dot \\alpha} \\\\\n\\phantom{S_{\\rm phys} =}{}\n- \\frac{3}{2} \\mu^2\\bar \\Xi^{\\dot \\alpha} \\Xi_{\\dot \\alpha}\n +\\frac{1}{4}\\bar \\Lambda \\slashed{\\d}\\Lambda +\\frac{1}{4} \\bar g_a \\slashed{\\d} g^a +\n\\frac{i}{4}\\omega^{ab}\\bar G_{ab}\\Lambda + \\frac{i}{2}\\mu \\bar \\Lambda \\Lambda\n\\Bigg] .\n\\end{gather*}\n\n\\section{Conclusions}\\label{conclusions}\n\nIn this paper we reviewed our fuzzy $S^2$ construction in terms of bifundamental matrices, originally obtained in the context of the ABJM model in~\\cite{Nastase:2009ny,Nastase:2009zu}, focusing on its model-independent mathematical aspects. We found that this is completely equivalent to the usual adjoint $\\SU(2)$ construction, but that it involves fuzzy versions of Killing spinors on the 2-sphere, which we def\\\/ined. We described the qualitative dif\\\/ferences that appear when using the bifundamental $S^2$ to `deconstruct' higher dimensional f\\\/ield theories. The expansion of the f\\\/ields involving fuzzy Killing spinors result in an automatic twisting of the former on the sphere. Alternatively, including the Killing spinors in the fuzzy spherical harmonic expansion provides a new approach to the construction of f\\\/ields on~$S^2$.\n\nWe expect that the generality of the construction will lead to it f\\\/inding a place in numerous applications both in the context of\nphysical systems involving bifundamental matter, e.g.\\ quiver gauge theories as in~\\cite{Maldacena:2009mw}, as well as noncommutative\ngeometry. We hope to further report on both of these aspects in the future.\n\n\\subsection*{Acknowledgements}\n\nIt is a pleasure to thank Sanjaye Ramgoolam for many comments, discussions and collaboration in~\\cite{Nastase:2009ny}. CP is supported by the STFC grant ST\/G000395\/1.\n\n\n\\pdfbookmark[1]{References}{ref}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOn your walk home, a runner whisks past you. Her feet flying over the concrete and leaves, they make a blur of a small but unmistakable check mark. This remarkably simple logo, dubbed the swoosh, perfectly embodies motion and speed, attributes of the winged Goddess of victory in Greek mythology, Nike.\n\nFashion is all about identity. From luxury splurges to mass retail sneakers, logos have been considered a key status symbol. Over time however, as buying habits change, the status symbols evolve. Since the rise of the No Logo movement~\\cite{nologo}, some brands have embraced minimalism. Louis Vuitton made news in 2013 when it pulled back on the use of its iconic LVs on purses.\nGood branding is more than a logo. It is storytelling; a visual story woven into every piece. Here's a test: if you cover up the logo on a product, can you still tell the brand?\n\nUniqueness is a vital factor for a successful clothing business. Shoppers not only want to be fashionable, but also want to express themselves. In 1992 Christian Louboutin decided to create a signature style that hints at sensuality and power simultaneously, and they painted the soles of their shoes red! There are a million ways that designers make memorable brand expressions. Sometimes they bring life to a logo, other times they use patterns to make a brand recognizable. Some make eccentric products in shape and geometry and others make name for themselves by unique color combinations, folds and cuts. Figure~\\ref{fig:introduction_different_brand_strategy} shows examples drawn from the wide spectrum of visual expressions fashion brands adopt. While some use colorful graphics or repeated logo prints, others design unique patterns or mainly invest on logos.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{sample_brands_flat.pdf}\n\\caption{Visual brands spectrum. Brands use a wide range of visual expressions. Experts can identify brands even in the absence of logo. While some use colorful graphics, others adopt unique patterns or choose to rely on logo.}\n\\label{fig:introduction_different_brand_strategy}\n\\end{figure}\n\nWith the recent success of computer vision and the rise of online commerce, there is a huge excitement to turn computers into visual experts. The ever-changing landscape of fashion industry has provided a unique opportunity to leverage computational algorithms on large data to achieve the knowledge and expertise unattainable for any individual fashion expert. Previous researchers have worked on clothing parsing~\\cite{yamaguchi:paperdoll,gong:lookinto}, outfit compatibility and recommendation~\\cite{veit:learningvisual,yamaguchi:mix}, style and trend recognition~\\cite{kiapour:hipsterwars,alhalah:fashionforward}, attribute recognition~\\cite{chen:deepdomain,chen:describing,liu:fashionlandmark} and retrieval~\\cite{kiapour:wtb,liu:deepfashion}. In order to interpret deep visual representations, studies have discovered neurons that can predict semantic attributes shared among categories~\\cite{ozeki:understandingconvolutional,escorcia:ontherelationship} and grand-mother-cell like features~\\cite{agrawal:analyzingtheperformance} and probed the neuron activations to discover concepts~\\cite{vittayakorn:automaticattribute,bau:networkdissection,zhou:objectdetectors}. Another body of research rely on attention paradigm to find parts of the image that are most responsible for the classification~\\cite{ruth:interpretable,selvaraju:gradcam,zeiler:visualizing,springenberg:striving}. Our work builds upon the top-down attention mechanism of Zhang et al.~\\cite{zhang:topdown} to uncover what computer vision models learn in order to distinguish fine-grained fashion brands across a wide variety of products. Specifically, we aim to answer the following questions:\n\\begin{itemize}\n\\item How can we quantify visual brand representations?\n\\item How do deep networks distinguish between very similar products?\n\\item What are the key visual expressions that brands adopt?\n\\item Which visual representations are shared or unique across brands?\n\\item How well does the learned representations align with human perception?\n\\end{itemize}\n\n\n\\section{Methodology}\n\\noindent\n\\textbf{Data}. We collect a new large dataset of $3,828,735$ clothing product images from $1219$ brands taken from a global online marketplace reported in Table~\\ref{tab:data_stats}. The dataset contains diverse images from stock quality photos taken professionally with white background to photos of used products photographed by amateurs in challenging viewpoints and lighting. We grouped the products to fall into five broad categories:\\emph{Bags, Footwear, Bottom wear, Outerwear} and \\emph{Tops}.\n\n\\noindent\n\\textbf{Classification Network}.\nIn deep learning, fine-tuning a convolutional network, pretrained on large data, is considered as a simple transfer learning to provide good initialization~\\cite{transferlearning}. We fine-tune the ResNet-50 model on ImageNet~\\cite{he:deepresidual,krizhevsky:imagenet} for classification among the $1219$ brands in our dataset and achieve $47.1\\%$ top-1 accuracy. Next we use an attention mechanism to generate brand-specific attention maps on the convolution layers. In our experiments, we study \\emph{res5b} maps due to its manageable size and proximity to the final classification layer. Our method can be applied to any convolution layer in deep networks.\n\n\\begin{table}\n\\centering\n\\caption{Fashion brands dataset collected from online e-commerce sites.}\n\\begin{tabular}{llccc}\n\\hline\n\\textbf{Category} & \\textbf{Subcategories} & \\textbf{\\#Brands} & \\textbf{\\#Train} & \\textbf{\\#Test}\\\\\n\\hline\nBags & Handbags, Purses & $132$ & $206,232$ & $18,427$ \\\\ \\hline\n \nFootwear & Shoes, Heels, Boots & $235$ & $368,846$ & $37,825$ \\\\ \\hline\n \nBottom wear & Pants, Jeans, Skirts & $218$ & $431,568$ & $44,408$ \\\\ \\hline\n \nOuter wear & Jackets, Coats & $238$ & $442,950$ & $42,962$ \\\\ \\hline\n\nTops & Tops, Blouses, Dresses & $556$ & $926,033$ & $89,762$ \\\\ \\hline\n \nAll & {} & $1219$ & $3,480,575$ & $348,160$ \\\\\n \\hline\n\\end{tabular}\n\\label{tab:data_stats}\n\\end{table}\n\n\n\\noindent\n\\textbf{Top-Down Excitation Maps}.\nOur goal is to interpret the deep model's predictions in order to explain the visual characteristics of fashion brands. Using the Excitation Backprop method~\\cite{zhang:topdown}, we generate marginal probabilities on intermediate layers for the brand predicted with the maximum posterior probability, hence the name top-down. We assume the response in convolutional layers is positively correlated with their confidence of detection. This probabilistic framework produces well-normalized excitation maps efficiently via a single backward pass down to the target layer. We define two measures to encode the excitations:\n\n\\smallskip\n\\noindent{\\textbf{Strength}}. Strength is calculated by computing the maximum over the excitation maps of a convolution layer. For every input image $x$, we compute excitation map $M_{k}(x)$ of every internal convolution unit $k$. \nWe denote the excitation strength of convolutional layer by $S(x) = \\max_{s\\in{h_{k}}{{w_{k}}}}{E_{s}{(x)}}$, where $E_s(x) = \\sum_{k=1\\dots K}{M_{k}{(x)}}$\nand $K$ is the total number of individual convolution units.\n\n\n\n\n\n\\smallskip\n\\noindent{\\textbf{Extent}}. Extent is a measure to encode the spatial support of high activations in excitation maps. Specifically, we first calculate the excitation map at every location $s$ across all units. Next we compute the ratio of locations where their excitation exceeds the mean value of all the excitations, represented by $T$. We define excitation extent of input image $x$ by $E(x) = \\frac{1}{{h_{k}}{w_{k}}}\\sum_{s\\in{h_{k}}{{w_{k}}}}{\\textbf{1}\\left[E_s{(x)} > T\\right]}$.\n\n\n\\smallskip\n\\noindent\n\\textbf{Discriminability}.\nWe aim to find units\/neurons that often get high excitement values corresponding to a given brand. For every convolutional unit, we calculate the maximum value over the entire excitation map for every image $I$ and compute two distributions for positive $P^{+}$ and negative $P^{-}$ images associated to a brand $b$. For every unit $k$, we compute the symmetric KL divergence~\\cite{vittayakorn:automaticattribute}: $D_k(b|I) = KL(P^+ || P^-) + KL(P^- || P^+)$. The units that maximize the distance between the class conditional probabilities are deemed to have higher discriminability.\n\n\n\\section{Experiments}\n\n\n\n\n\\subsection{Brand Representations}\nHow do clothing brands make their products stand out among others? What do fashion designers do to appeal to shoppers? In order to answer these questions we begin by exploring two ends of the spectrum of visual branding: brands which make themselves stand out through a localized mark, sign or logo, e.g. Chanel bags or Polo Ralph Lauren Shirts, and brands that convey their message via a spread design using colors and patterns, think colorful Vera Bradley or woven leather Bottega Veneta bags. In the following, we conduct our experiments on the bags category as it depicts a wide range of brand visualization strategies and receives the best classification score among the categories in our dataset.\n\n\\smallskip\n\\noindent\\textbf{Strength.}\nFigure~\\ref{fig:experiments_top_brands_strength_extent} depicts brands that obtain the highest excitation strengths. For each brand we compute median of the predicted strength across all samples in the test set. We find that brands such as Fjallraven, Jansport and Coach, design their bags with a unique logo or mark their goods with their brand name.\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{top_strength_extent_samples_nocaption.pdf}\n\\caption{Left: Brands with high Strength. Examples of bags from brands with high excitation strength are shown. All brands show concentrated logos or printed brand names. Right: Brands with high Extent. Examples of bags with high excitation extent are shown. Some brands print a large graphic on their products while other have a repeated pattern or logo. Composition of image can affect the extent signal as shown in the examples in the last column with large, repeated or close-up logos.}\n\\label{fig:experiments_top_brands_strength_extent}\n\\end{figure}\n\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Extent.}\nWhat are the brands that are not as invested in logos and instead are interested to convey their message via unique patterns? Figure~\\ref{fig:experiments_top_brands_strength_extent} depicts brands with the highest excitation extent values. For each brand, we find the extent decile to which it belongs to by computing the median of all extent values in the test set. Brands such as For U Designs, print large graphics of animals, nature or galaxy on their bags. Louis Vuitton makes their products remarkably recognizable via a unique checkered pattern or the famous repeated LV monogram. Vera Bradley is filled with colorful floral and paisley patterns and MCM repeats it's logo across a large region of the product. We also see how composition of images in a brand can contributes to large extent levels. The illustrated examples of Supreme brand are photographed in close-up and show the brand name in large size which leads to expanded excitations.\n\n\n\n\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Extent vs. Strength.}\nNext we explore the space of Extent and Strength jointly. We ask, which brands have high extent but no single strong excitation value in their maps or vice versa? Are there examples that have both high or low extent and strengths? In order to answer these interesting questions, we plot the samples of top and bottom brands with samples falling in the highest and lowest extents in Figure~\\ref{fig:extent_vs_strength_bags}. We find that brands such as Burberry, Gucci or For U Designs are concentrated in the higher half of the spectrum, while logo-heavy brands such as Tommy Hilfiger, Tony Burch and Herschel Supply Co. bags are spread along the strength axis with low extent values. Interestingly, we observe that the model picks up signals, however weak, in the straps of Tommy Hilfiger totes, striped in iconic colors of Tommy Hilfiger. Comparing Tory Burch bags along the spectrum, the logos are hard to capture in the examples falling on the lower side of the strength axis while images of the same brand with high strength show fully visible logos. We also probe the middle region and observe an interesting phenomena. Louis Vuitton and Burberry images that fall in between, show a mix of logo monograms and brand names instead of just patterns.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{extent_vs_strength_8_rd.pdf}\n\\caption{Extent vs Strength. Depicts the transition of brands across the spectrum. We show samples that fall across the spectrum of extent and strength. Samples with high extent show a repeated texture or a large pattern. Items with high strength show a localized mark, logo or brand name. Hard examples to recognize such as Tommy Hilfiger bags that only show a specific type of stripes require specialized neurons to detect them.}\n\\label{fig:extent_vs_strength_bags}\n\\end{figure}\n\n\\subsection{Versatility of Convolutional Units}\nNext, we go one step deeper and rank the convolutional units\/neurons of the layer for each brand based on symmetric KL divergence score. We observe some neurons detect complex entangled concepts while others are more interpretable and specialized towards disentangled visual features. Figure~\\ref{fig:experiments_specialist_generalist} left, shows top detection examples of such neurons for two sample brands. Some Adidas neurons detect the logos while others are specialized to detect vertical or horizontal stripes. For Burberry, we find units that detect diagonal or straight patterns, while another unit is more sensitive to the horse rider in the Burberry knight logo.\n\nWe further investigate ``specialist\" vs. ``generalist\" units. We compute the number of brands activated for each unit. Specialists units are activated for only one or few brands. Figure~\\ref{fig:experiments_specialist_generalist} right, shows examples of specialist units and the brands they activate. Unit $253$ is an expert only in detecting the Harley Davidson logo, which is unique and can happen in many locations over the object and requires its own specialized unit. Meanwhile, units $1631$ and $770$ detect floral and natural patterns that are more general and shared among brands such as Vera Bradley and Mary Frances. Unit $1250$ is specialized to detect hobo-shaped bags with a large crescent-shaped bottom and a shoulder strap that represents multiple brands. By analyzing the space of specialist units we can discover unique visual expressions that sets a brand apart. Generalist units point us to features shared by several brands.\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\caption{Left: Top activated neurons for Adidas and Burberry brands. Three top neurons specialized in recognizing in (a) Adidas and (b) Burberry. First column show a specialized neuron for recognizing the logo associated with a brand name, second column is specialized for three stripes, last column recognizes the smaller scale logo. First column in (b) recognizes vertical and horizontal stripes, second shows examples for neuron specialized in diagonal patterns and lastly is the logo detector for Louis Vuitton. Right: Specialist vs generalist units.}\n\\includegraphics[width=1.0\\textwidth]{specialist_generalist_rd.pdf}\n\\label{fig:experiments_specialist_generalist}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\textwidth]{human_exp.pdf}\n\\caption{Left: Individual brands ranked in the logo visibility spectrum. Brands are sorted based on their fraction of samples labeled by humans as (i) Logo, (ii) No Logo or (iii) Repeated Logo. Right: Pearson correlation of Strength and Extent of excitations with brand visibility variations.}\n\\label{fig:human_exp}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Human Experiment}\nWe conduct a human study asking 5 subjects on Amazon Mechanical Turk to label each product image in the bags category according to the visibility of the logo into one of three groups: (i) Logo (ii) Repeated Logo, when a pattern of repeated logos or monogram and (iii) No Logo. $46\\%$ contain a visible logo and $51\\%$ contain no logo. This is particularly interesting as a recent study shows that one third of the handbags purchased in the U.S. did not have a visible logo~\\cite{cnbc:nologo}. The classifier correctly predicts $65.01\\%$, $68.67\\%$ and $54.46\\%$ of the brands in groups (i), (ii) and (iii) respectively. This is significant, given that group (iii) constitutes the majority of the dataset and confirms that deep classifiers learn unique visual characteristics of all three groups.\n\n\n\n\\smallskip\n\\noindent\n\\textbf{Logo Visibility in Brands.}\nWe further study individual brands by ranking them based on the ratio of samples that fall into each of the three logo visibility groups. Figure~\\ref{fig:human_exp}, shows that brands such as Fjallraven and The North Face are logo-based. On the other hand, Lucky Brand does not depend on logo. Instead, they claim to give your look ``the added flare\" by embellishments such as fringe and embroidered detailing. Fendi and MCM opt in to repeat their logo in their design. In fact, the ``Shopper\" totes from MCM are reversible with their logo printed on both inside and outside! \n\n\\smallskip\n\\noindent\n\\textbf{Correlation with Strength and Extent}\nFinally we compute the Pearson correlation between the predicted strength and extent of the excitation maps and report the results in Figure~\\ref{fig:human_exp}. We find that excitation strength has strong correlation with samples depicting a logo while extent has a negative correlation with logo-oriented products. Products with repeated logo produce scattered signals with low strength and high extent. For an item with no logo, the network needs to aggregate signals from various spatial locations and hence it is positively correlated with extent and negatively correlated with strength.\n\n\\noindent\n\\textbf{Conclusion}.\nIn this work, we quantify the deep representations to analyze and interpret visual characteristics of fashion brands. We find units that are specialized to detect specific brands as well as versatile units that detect shared concepts. A human experiment confirms the proposed measures are aligned with human perception.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe operation cabling for classical braids studied in~\\cite{CW}.\n For virtual pure braid group $VP_n$ this operation gives new generators for $VP_n$ (see \\cite{BW}). It was proved that for $n\\geq 3$, the group $VP_n$ is generated by the $n$-strand virtual braids obtained by taking $(k,l)$-cabling on the standard generators $\\lambda_{1,2}$ and $\\lambda_{2,1}$ of $VP_2$ together with adding trivial strands $n-k-l$ to the end for $1\\leq k\\leq n-1$ and $2\\leq k+l\\leq n$, where a $(k,l)$-cabling on a $2$-strand virtual braid means to take $k$-cabling on the first strand and $l$-cabling on the second strand.\n\n\nDifferent from the classical situation~\\cite{CW} that the $n$-strand braids cabled from the standard generator $A_{1,2}$ for $P_2$ generates a free group of rank $n-1$, the subgroup of $VP_n$ generated by $n$-strand virtual braids cabled from $\\lambda_{1,2}$ and $\\lambda_{2,1}$, which is denoted by $T_{n-1}$, is no longer free for $n\\geq3$.\nFor the first nontrivial case that $n=3$, a presentation of $T_2$ has been explored with producing a decomposition theorem for $VP_3$ using cabled generators~\\cite{BMVW}.\n\n\nIn the present article we continue to study $VP_n$ in cabled generators, which we started in \\cite{BW}. We find some sufficient condition under which a simplicial group $G_*$ is contractible. In particular, we prove that the simplicial group $VAP_* = \\{VP_i\\}_{i=1,2,\\ldots}$ is contractible.\nAlso, we prove the lifting theorem for the virtual pure braid groups. From this theorem follows that if we know the structure of $VP_4$, $T_3$ or $P_4$, then using degeneracy maps we can find the structure of $VP_n$, $T_n$ or $P_n$ for all bigger $n$. On the other side we prove that if we know a presentation of $VP_n$, $n \\geq 4$, then conjugated it by elements $\\rho_n$, $\\rho_n \\rho_{n-1}$, $\\ldots$, $\\rho_n \\rho_{n-1} \\ldots \\rho_1 \\in VB_{n+1}$ we can find the presentation of $VP_{n+1}$.\n\nThe article is organized as follows. In Section \\ref{virt}, we give a review on braid groups and virtual braid groups. The simplicial structure on virtual pure braid groups will be discussed in Section~\\ref{simp}. Is Section \\ref{lift} we prove the lifting theorem.\nIn Section ~\\ref{s41}, we discuss the cabling operation on classical pure braid group $P_n$ as subgroup of$VP_n$. We know two types of decompositions of $P_n$ as semi-direct products (see, for example, \\cite{B1}). In Section \\ref{s41} we construct new decomposition of this type in terms of the cabled generators.\nIn the last Section \\ref{fin} we formulate some questions for further research.\n\n\n\\subsection{Acknowledgements}\nThis article was written when the first author visited College of Mathematics and information Science Hebei Normal University. He thanks the administration for good working conditions.\n\n\n\\section{Braid and virtual braid groups} \\label{virt}\n\n\\subsection{Braid group} The braid group $B_n$ on $n$ strings is generated by $\\sigma_1,\\, \\sigma_2, \\, \\ldots , \\, \\sigma_{n-1}$ and is defined by relations\n\\begin{align*}\n& \\sigma_i \\sigma_{i+1} \\sigma_i = \\sigma_{i+1} \\sigma_i \\sigma_{i+1},~~~i = 1, 2, \\ldots, n-2, \\\\\n& \\sigma_i \\sigma_{j} = \\sigma_j \\sigma_{i},~~|i-j|>1.\n\\end{align*}\n\nLet $S_n$, $n \\geq 1$ be the symmetric group which is generated by $\\rho_1, \\, \\rho_2, \\, \\ldots , \\, \\rho_{n-1}$ and is defined by relations\n\\begin{align*}\n& \\rho_i^2 = 1,~~~i = 1, 2, \\ldots, n-1,\\\\\n& \\rho_i \\rho_{i+1} \\rho_i = \\rho_{i+1} \\rho_i \\rho_{i+1},~~~i = 1, 2, \\ldots, n-2,\\\\\n& \\rho_i \\rho_{j} = \\rho_j \\rho_{i},~~|i-j|>1.\n\\end{align*}\n\nThere is a homomorphism $B_n \\to S_n$, which sends $\\sigma_i$ to $\\rho_i$. Its kernel is the pure braid group $P_n$. This group is generated by elements $A_{i,j}$, $1 \\leq i < j \\leq n$, where\n$$\nA_{i,i+1} = \\sigma_i^2,\n$$\n$$\nA_{i,j} = \\sigma_{j-1} \\sigma_{j-2} \\ldots \\sigma_{i+1} \\sigma_i^2 \\sigma_{i+1}^{-1} \\ldots \\sigma_{j-2}^{-1} \\sigma_{j-1}^{-1},~~~i+1 < j \\leq n,\n$$\nand is defined by relations (where $\\varepsilon = \\pm 1$):\n\\begin{align*}\n& A_{ik}^{-\\varepsilon} A_{kj} A_{ik}^{\\varepsilon} = (A_{ij} A_{kj})^{\\varepsilon} A_{kj} (A_{ij} A_{kj})^{-\\varepsilon},\\\\\n& A_{km}^{-\\varepsilon} A_{kj} A_{km}^{\\varepsilon} = (A_{kj} A_{mj})^{\\varepsilon} A_{kj} (A_{kj} A_{mj})^{-\\varepsilon},~~m < j, \\\\\n& A_{im}^{-\\varepsilon} A_{kj} A_{im}^{\\varepsilon} = [A_{ij}^{-\\varepsilon}, A_{mj}^{-\\varepsilon}]^{\\varepsilon} A_{kj} [A_{ij}^{-\\varepsilon}, A_{mj}^{-\\varepsilon}]^{-\\varepsilon}, ~~i < k < m,\\\\\n& A_{im}^{-\\varepsilon} A_{kj} A_{im}^{\\varepsilon} = A_{kj}, ~~k < i, m < j~\\mbox{or}~ m < k,\n\\end{align*}\nHere and further $[a,b] = a^{-1} b^{-1} a b$ is the commutator of $a$ and $b$.\n\nThere is an epimorphism of $P_n$ to $P_{n-1}$ what is removing of the $n$-th string. Its kernel $U_n = \\langle A_{1n}, A_{2n}, \\ldots, A_{n-1,n} \\rangle$ is a free group of rank $n-1$ and $P_n = U_n \\leftthreetimes P_{n-1}$ is a semi-direct product of $U_n$ and $P_{n-1}$. Hence,\n$$\nP_n = U_n \\leftthreetimes (U_{n-1} \\leftthreetimes (\\ldots \\leftthreetimes (U_3 \\leftthreetimes U_2)) \\ldots ),\n$$\nis a semi-direct product of free groups and $U_2 = \\langle A_{12}\\rangle$ is the infinite cyclic group.\n\n\n\n\n\\subsection{Virtual braid group} \\label{virt1}\n\nThe virtual braid group $VB_n$ is generated by elements\n$$\n\\sigma_1,\\, \\sigma_2, \\, \\ldots , \\, \\sigma_{n-1}, \\, \\rho_1, \\, \\rho_2, \\, \\ldots , \\, \\rho_{n-1},\n$$\nwhere $\\sigma_1,\\, \\sigma_2, \\, \\ldots , \\, \\sigma_{n-1}$ generate the classical braid group $B_n$ and\nthe elements $\\rho_1$, $\\rho_2$, $\\ldots $, $\\rho_{n-1}$ generate the symmetric group\n$S_n$. Hence, $VB_n$ is defined by relations of $B_n$, relations of $S_n$\nand mixed relation:\n$$\n\\sigma_i \\rho_j = \\rho_j \\sigma_i,~~~|i-j| > 1,\n$$\n$$\n\\rho_i \\rho_{i+1} \\sigma_i = \\sigma_{i+1} \\rho_i \\rho_{i+1}~~~i = 1, 2, \\ldots, n-2.\n$$\n\nAs for the classical braid groups there exists the canonical\nepimorphism of $VB_n$ onto the symmetric group $VB_n\\to S_n$ with the\nkernel called the {\\it virtual pure braid group} $VP_n$. So we have a\nshort exact sequence\n\\begin{equation*}\n1 \\to VP_n \\to VB_n \\to S_n \\to 1.\n\\end{equation*}\nDefine the following elements in $VP_n$:\n$$\n\\lambda_{i,i+1} = \\rho_i \\, \\sigma_i^{-1},~~~\n\\lambda_{i+1,i} = \\rho_i \\, \\lambda_{i,i+1} \\, \\rho_i = \\sigma_i^{-1} \\, \\rho_i,\n~~~i=1, 2, \\ldots, n-1,\n$$\n$$\n\\lambda_{ij} = \\rho_{j-1} \\, \\rho_{j-2} \\ldots \\rho_{i+1} \\, \\lambda_{i,i+1} \\, \\rho_{i+1}\n\\ldots \\rho_{j-2} \\, \\rho_{j-1},\n$$\n$$\n\\lambda_{ji} = \\rho_{j-1} \\, \\rho_{j-2} \\ldots \\rho_{i+1} \\, \\lambda_{i+1,i} \\, \\rho_{i+1}\n\\ldots \\rho_{j-2} \\, \\rho_{j-1}, ~~~1 \\leq i < j-1 \\leq n-1.\n$$\nIt is shown in \\cite{B} that the group $VP_n$, $n\\geq 2$ admits a\npresentation with the generators $\\lambda_{ij},\\ 1\\leq i\\neq j\\leq n,$\nand the following relations:\n\\begin{align}\n& \\lambda_{ij}\\lambda_{kl}=\\lambda_{kl}\\lambda_{ij} \\label{rel},\\\\\n&\n\\lambda_{ki}\\lambda_{kj}\\lambda_{ij}=\\lambda_{ij}\\lambda_{kj}\\lambda_{ki}\n\\label{relation},\n\\end{align}\nwhere distinct letters stand for distinct indices.\n\nLike the classical pure braid groups, groups $VP_n$ admit a\nsemi-direct product decompositions \\cite{B}: for $n\\geq 2,$ the\n$n$-th virtual pure braid group can be decomposed as\n\\begin{equation}\nVP_n=V_{n-1}^*\\rtimes VP_{n-1},~~n \\geq 2,\n\\label{eq:s_d_dec}\n\\end{equation}\nwhere $V_{n-1}^*$ is a subgroup of $VP_{n}$, $V_1^* = F_2$, $VP_1$ is supposed\nto be the trivial group.\n\n\n\n\\section{Simplicial groups} \\label{simp}\n\n\n\\subsection{Simplicial sets and simplicial groups} Recall the definition of simplicial groups (see \\cite[p.~300]{MP} or \\cite{BCWW}). A sequence of sets $X_* = \\{ X_n \\}_{n \\geq 0}$ is called a\n{\\it simplicial set} if there are face maps:\n$$\nd_i : X_n \\longrightarrow X_{n-1} ~\\mbox{for}~0 \\leq i \\leq n\n$$\nand degeneracy maps\n$$\ns_i : X_n \\longrightarrow X_{n+1} ~\\mbox{for}~0 \\leq i \\leq n,\n$$\nthat are satisfy the following simplicial identities:\n\\begin{enumerate}\n\\item $d_i d_j = d_{j-1} d_i$ if $i < j$,\n\\item $s_i s_j = s_{j+1} s_i$ if $i \\leq j$,\n\\item $d_i s_j = s_{j-1} d_i$ if $i < j$,\n\\item $d_j s_j = id = d_{j+1} s_j$,\n\\item $d_i s_j = s_{j} d_{i-1}$ if $i > j+1$.\n\\end{enumerate}\nHere $X_n$ can be geometrically viewed as the set of $n$-simplices including all possible degenerate simplices.\n\nA {\\it simplicial group} is a simplicial set $X_*$ such that each $X_n$ is a group and all face and degeneracy operations are group homomorphism.\n Let $G_*$ be a simplicial group. The \\textit{Moore cycles} $\\mathrm{Z}_n(G_*)\\leq G_n$ is defined by\n$$\n\\mathrm{Z}_n(G_*)=\\bigcap_{i=0}^n\\mathrm{Ker}(d_i\\colon G_n\\to G_{n-1})\n$$\nand the \\textit{Moore boundaries} $\\mathcal{B}_n(G_*)\\leq G_n$ is defined by\n$$\n\\mathcal{B}_n(G_*)=d_0\\left(\\bigcap_{i=1}^{n+1}\\mathrm{Ker}(d_i\\colon G_{n+1}\\to G_n)\\right).\n$$\nSimplicial identities guarantees that $\\mathcal{B}_n(G_*)$ is a (normal) subgroup of $\\mathrm{Z}_n(G_*)$. The \\textit{Moore homotopy group} $\\pi_n(G_*)$ is defined by\n$$\n\\pi_n(G_*)=\\mathrm{Z}_n(G_*)\/\\mathcal{B}_n(G_*).\n$$\nIt is a classical result due to J. C. Moore ~\\cite{Moore} that $\\pi_n(G_*)$ is isomorphic to the $n$-th homotopy group of the geometric realization of $G_*$.\n\n\n\n\n\n\n\\subsection{Simplicial group on virtual pure braid groups}\nBy using the same ideas in the work~\\cite{BCWW,CW} on the classical braids, in \\cite{BW} was introduced a simplcial group\n$$\n\\VAP_* :\\ \\ \\ \\ldots\\ \\begin{matrix}\\longrightarrow\\\\[-3.5mm] \\ldots\\\\[-2.5mm]\\longrightarrow\\\\[-3.5mm]\n\\longleftarrow\\\\[-3.5mm]\\ldots\\\\[-2.5mm]\\longleftarrow \\end{matrix}\\ VP_4 \\ \\begin{matrix}\\longrightarrow\\\\[-3.5mm]\\longrightarrow\\\\[-3.5mm]\\longrightarrow\\\\[-3.5mm]\\longrightarrow\\\\[-3.5mm]\\longleftarrow\\\\[-3.5mm]\n\\longleftarrow\\\\[-3.5mm]\\longleftarrow\n\\end{matrix}\\ VP_3\\ \\begin{matrix}\\longrightarrow\\\\[-3.5mm] \\longrightarrow\\\\[-3.5mm]\\longrightarrow\\\\[-3.5mm]\n\\longleftarrow\\\\[-3.5mm]\\longleftarrow \\end{matrix}\\ VP_2\\ \\begin{matrix} \\longrightarrow\\\\[-3.5mm]\\longrightarrow\\\\[-3.5mm]\n\\longleftarrow \\end{matrix}\\ VP_1$$\non pure virtual braid groups with $\\VAP_n=VP_{n+1}$, the face homomorphism\n$$\nd_i : \\VAP_n=VP_{n+1} \\longrightarrow \\VAP_{n-1}=VP_n\n$$\ngiven by deleting $(i+1)$th strand for $0\\leq i\\leq n$, and the degeneracy homomorphism\n$$\ns_i : \\VAP_n=VP_{n+1} \\longrightarrow \\VAP_{n+1}=VP_{n+2}\n$$\ngiven by doubling the $(i+1)$th strand for $0\\leq i\\leq n$.\n\nLet $\\iota_n\\colon VP_n\\to VP_{n+1}$ be the inclusion. Geometrically $\\iota_n$ is the group homomorphism by adding a trivial strand on the end. From geometric information, we have the following formulae:\n\\begin{equation}\\label{formula1}\ns_j\\iota_n=\\iota_{n+1}s_j\\colon VP_n\\longrightarrow VP_{n+1} \\textrm{ for } 0\\leq j\\leq n-1,\n\\end{equation}\n\\begin{equation}\\label{formula2}\nd_j\\iota_n=\\left\\{\n\\begin{array}{lcl}\n\\iota_{n-1}d_j&\\textrm{ if }& j l,\n\\end{array}\n\\right.\n$$\n$$\ns_{i-1} (\\lambda_{l,k}) = \\left\\{\n\\begin{array}{lr}\n\\lambda_{l+1,k+1} & for ~i < k,\\\\\n\\lambda_{l+1,k+1} \\lambda_{l+1,k} & for ~i = k, \\\\\n\\lambda_{l+1,k} & for ~k < i < l,\\\\\n& \\\\\n\\lambda_{l,k} \\, \\lambda_{l+1,k} & for ~i = l, \\\\\n& \\\\\n\\lambda_{l,k} & for ~i > l.\n\\end{array}\n\\right.\n$$\n\n\\end{prop}\n\n\n\n\\subsection{Lifting defining relations of $VP_{3}$ to $VP_4$}\n\nIn the group $VP_3$ we have 6 relations:\n $$\n\\lambda_{12} \\lambda_{13} \\lambda_{23} = \\lambda_{23} \\lambda_{13} \\lambda_{12},~~~\n\\lambda_{21} \\lambda_{23} \\lambda_{13} = \\lambda_{13} \\lambda_{23} \\lambda_{21},~~~\n\\lambda_{13} \\lambda_{12} \\lambda_{32} = \\lambda_{32} \\lambda_{12} \\lambda_{13},\n$$\n$$\n\\lambda_{31} \\lambda_{32} \\lambda_{12} = \\lambda_{12} \\lambda_{32} \\lambda_{31},~~~\n\\lambda_{23} \\lambda_{21} \\lambda_{31} = \\lambda_{31} \\lambda_{21} \\lambda_{23},~~~\n\\lambda_{32} \\lambda_{31} \\lambda_{21} = \\lambda_{21} \\lambda_{31} \\lambda_{32}.\n$$\nActing on these relations by degeneracy map $s_2$ we get 6 relations in $VP_4$. Let us analise these relations.\n\n\n1) The image of the first relation has the form\n$$\n\\lambda_{12} \\cdot \\lambda_{14} (\\lambda_{13} \\cdot \\lambda_{24}) \\lambda_{23} = \\lambda_{24} (\\lambda_{23} \\cdot \\lambda_{14}) \\lambda_{13} \\cdot \\lambda_{12}.\n$$\nUsing the commutativity relation\n$$\n\\lambda_{13} \\lambda_{24} = \\lambda_{24} \\lambda_{13},~~~\\lambda_{23} \\lambda_{14} = \\lambda_{14} \\lambda_{23},\n$$\nwe get\n$$\n\\lambda_{12} \\lambda_{14} \\lambda_{24} \\cdot \\lambda_{13} \\lambda_{23} = \\lambda_{24} \\lambda_{14} (\\lambda_{23} \\lambda_{13} \\lambda_{12}).\n$$\nUsing the following relation of $VP_3$:\n$$\n\\lambda_{23} \\lambda_{13} \\lambda_{12} = \\lambda_{12} \\lambda_{13} \\lambda_{23},\n$$\nwe get\n$$\n\\lambda_{12} \\lambda_{14} \\lambda_{24} = \\lambda_{24} \\lambda_{14} \\lambda_{12}.\n$$\nthat is the long relation in $VP_4$.\n\n2) The image of the second relation has the form\n$$\n \\lambda_{21} \\cdot \\lambda_{24} (\\lambda_{23} \\cdot \\lambda_{14}) \\lambda_{13} = \\lambda_{14} (\\lambda_{13} \\cdot \\lambda_{24}) \\lambda_{23} \\cdot \\lambda_{21}.\n$$\nUsing the the commutativity relations\n$$\n\\lambda_{23} \\lambda_{14} = \\lambda_{14} \\lambda_{23},~~~\\lambda_{13} \\lambda_{24} = \\lambda_{24} \\lambda_{13},\n$$\nwe get\n$$\n \\lambda_{21} \\lambda_{24} \\lambda_{14} \\lambda_{23} \\lambda_{13} = \\lambda_{14} \\lambda_{24} (\\lambda_{13} \\lambda_{23} \\lambda_{21}).\n$$\nFrom the relation of $VP_3$:\n$$\n \\lambda_{13} \\lambda_{23} \\lambda_{21} = \\lambda_{24} \\lambda_{14} \\lambda_{12},\n$$\nwe get\n$$\n \\lambda_{21} \\lambda_{24} \\lambda_{14} = \\lambda_{14} \\lambda_{24} \\lambda_{21},\n$$\ni.e. the long relation in $VP_4$.\n\n3) The image of the third relation has the form\n$$\n \\lambda_{14} (\\lambda_{13} \\cdot \\lambda_{12} \\cdot \\lambda_{32}) \\lambda_{42} = \\lambda_{32} \\lambda_{42} \\cdot \\lambda_{12} \\cdot \\lambda_{14} \\lambda_{13}.\n$$\nUsing the following relation from $VP_3$:\n$$\n \\lambda_{13} \\lambda_{12} \\lambda_{32} = \\lambda_{32} \\lambda_{12} \\lambda_{13},\n$$\nwe get\n$$\n(\\lambda_{14} \\lambda_{32}) \\lambda_{12} (\\lambda_{13} \\lambda_{42}) = \\lambda_{32} \\lambda_{42} \\lambda_{12} \\lambda_{14} \\lambda_{13}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{14} \\lambda_{32} = \\lambda_{32} \\lambda_{14},~~~\\lambda_{13} \\lambda_{42} = \\lambda_{42} \\lambda_{13},\n$$\nwe have\n$$\n\\lambda_{32} \\lambda_{14} \\lambda_{12} \\lambda_{42} \\lambda_{13} = \\lambda_{32} \\lambda_{42} \\lambda_{12} \\lambda_{14} \\lambda_{13}.\n$$\nAfter cancellation\nwe get\n$$\n \\lambda_{14} \\lambda_{12} \\lambda_{42} = \\lambda_{42} \\lambda_{12} \\lambda_{14},\n$$\ni.e. the long relation in $VP_4$.\n\n4) The image of the forth relation has the form\n$$\n \\lambda_{31} (\\lambda_{41} \\cdot \\lambda_{32}) \\lambda_{42} \\cdot \\lambda_{12} = \\lambda_{12} \\cdot \\lambda_{32} (\\lambda_{42} \\cdot \\lambda_{31}) \\lambda_{41}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{41} \\lambda_{32} = \\lambda_{32} \\lambda_{41},~~~\\lambda_{42} \\cdot \\lambda_{31} = \\lambda_{31} \\cdot \\lambda_{42},\n$$\nwe get\n$$\n \\lambda_{31} \\lambda_{32} \\lambda_{41} \\lambda_{42} \\lambda_{12} = (\\lambda_{12} \\lambda_{32} \\lambda_{31}) \\lambda_{42} \\lambda_{41}.\n$$\nUsing the following relation from $VP_3$:\n$$\n \\lambda_{12} \\lambda_{32} \\lambda_{31} = \\lambda_{31} \\lambda_{32} \\lambda_{12},\n$$\nafter cancelations we get\n$$\n \\lambda_{41} \\lambda_{42} \\lambda_{12} = \\lambda_{12} \\lambda_{42} \\lambda_{41},\n$$\ni.e. the long relation in $VP_4$.\n\n\n5) The image of the firth relation has the form\n$$\n \\lambda_{24} (\\lambda_{23} \\cdot \\lambda_{21} \\cdot \\lambda_{31}) \\lambda_{41} = \\lambda_{31} \\lambda_{41} \\cdot \\lambda_{21} \\cdot \\lambda_{24} \\lambda_{23}.\n$$\nUsing the following relation from $VP_3$:\n$$\n \\lambda_{23} \\lambda_{21} \\lambda_{31} = \\lambda_{31} \\lambda_{21} \\lambda_{23},\n$$\nand the commutativity relations\n$$\n\\lambda_{24} \\lambda_{13} = \\lambda_{13} \\lambda_{24},~~~\\lambda_{23} \\lambda_{41} = \\lambda_{41} \\lambda_{23},\n$$\nwe get\n$$\n \\lambda_{24} \\lambda_{21} \\lambda_{41} = \\lambda_{41} \\lambda_{21} \\lambda_{24},\n$$\ni.e. the long relation in $VP_4$.\n\n6) The image of the sixth relation has the form\n$$\n \\lambda_{32} (\\lambda_{42} \\cdot \\lambda_{31}) \\lambda_{41} \\cdot \\lambda_{21} = \\lambda_{21} \\cdot \\lambda_{31} (\\lambda_{41} \\cdot \\lambda_{32}) \\lambda_{42}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{42} \\lambda_{31} = \\lambda_{31} \\lambda_{42},~~~\\lambda_{41} \\lambda_{32} = \\lambda_{32} \\lambda_{41},\n$$\nwe get\n$$\n \\lambda_{32} \\lambda_{31} \\lambda_{42} \\lambda_{41} \\lambda_{21} = (\\lambda_{21} \\lambda_{31} \\lambda_{32}) \\lambda_{41} \\lambda_{42}.\n$$\nUsing the\nfollowing relation from $VP_3$:\n$$\n \\lambda_{21} \\lambda_{31} \\lambda_{32} = \\lambda_{32} \\lambda_{31} \\lambda_{21},\n$$\nwe get\n$$\n \\lambda_{42} \\lambda_{41} \\lambda_{21} = \\lambda_{21} \\lambda_{41} \\lambda_{42},\n$$\ni.e. the long relation in $VP_4$. Hence, we proved\n\n\\begin{lem} \\label{l1}\nFrom relations $\\mathcal{R}^V(3)$, relations $s_2(\\mathcal{R}^V(3))$ and the commutativity relations in $\\mathcal{R}^V(4)$ follows the next set of relations in $\\mathcal{R}^V(4)$:\n$$\n\\lambda_{12} \\lambda_{14} \\lambda_{24} = \\lambda_{24} \\lambda_{14} \\lambda_{12},~~~\n\\lambda_{21} \\lambda_{24} \\lambda_{14} = \\lambda_{14} \\lambda_{24} \\lambda_{21},~~~\n\\lambda_{14} \\lambda_{12} \\lambda_{42} = \\lambda_{42} \\lambda_{12} \\lambda_{14},\n$$\n$$\n\\lambda_{41} \\lambda_{42} \\lambda_{12} = \\lambda_{12} \\lambda_{42} \\lambda_{41},~~~\n\\lambda_{24} \\lambda_{21} \\lambda_{41} = \\lambda_{41} \\lambda_{21} \\lambda_{24},~~~\n\\lambda_{42} \\lambda_{41} \\lambda_{21} = \\lambda_{21} \\lambda_{41} \\lambda_{42},\n$$\ni.e. the set of relations where the indexes of the generators lie in the set $\\{ 1, 2, 4 \\}$.\n\\end{lem}\n\n\\medskip\n\nTake the set $s_1(\\mathcal{R}^V(3))$.\n\n1) The image of the first relation has the form\n$$\n \\lambda_{13} (\\lambda_{12} \\cdot \\lambda_{14} \\cdot \\lambda_{24}) \\lambda_{34} = \\lambda_{24} \\lambda_{34} \\cdot \\lambda_{14} \\cdot \\lambda_{13} \\lambda_{12}.\n$$\nUsing the the following relation from Lemma \\ref{l1}:\n$$\n \\lambda_{12} \\lambda_{14} \\lambda_{24} = \\lambda_{24} \\lambda_{14} \\lambda_{12},\n$$\nwe get\n$$\n(\\lambda_{13} \\lambda_{24}) \\lambda_{14} (\\lambda_{12} \\lambda_{34}) = \\lambda_{24} \\lambda_{34} \\lambda_{14} \\lambda_{13} \\lambda_{12}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{13} \\lambda_{24} = \\lambda_{24} \\lambda_{13},~~~\\lambda_{12} \\lambda_{34} = \\lambda_{34} \\lambda_{2}\n$$\nwe have\n$$\n\\lambda_{13} \\lambda_{14} \\lambda_{34} = \\lambda_{34} \\lambda_{14} \\lambda_{13},\n$$\ni.e. the long relation in $VP_4$.\n\n2) The image of the second relation has the form\n$$\n \\lambda_{21} (\\lambda_{31} \\cdot \\lambda_{24}) \\lambda_{34} \\cdot \\lambda_{14} = \\lambda_{14} \\cdot \\lambda_{24} (\\lambda_{34} \\cdot \\lambda_{21}) \\lambda_{31}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{31} \\lambda_{24} = \\lambda_{24} \\lambda_{31},~~~\\lambda_{34} \\lambda_{21} = \\lambda_{21} \\lambda_{34}\n$$\nwe have\n$$\n \\lambda_{21} \\lambda_{24} \\lambda_{31} \\lambda_{34} \\lambda_{14} = (\\lambda_{14} \\lambda_{24} \\lambda_{21}) \\lambda_{34} \\lambda_{31}.\n$$\nUsing the second relation from Lemma \\ref{l1}:\n$$\n\\lambda_{14} \\lambda_{24} \\lambda_{21} = \\lambda_{21} \\lambda_{24} \\lambda_{14},\n$$\nwe get\n$$\n \\lambda_{31} \\lambda_{34} \\lambda_{14} = \\lambda_{14} \\lambda_{34} \\lambda_{31},\n$$\ni.e. the long relation in $VP_4$.\n\n\n3) The image of the third relation has the form\n$$\n \\lambda_{14} \\cdot \\lambda_{13} (\\lambda_{12} \\cdot \\lambda_{43}) \\lambda_{42} = \\lambda_{43} (\\lambda_{42} \\cdot \\lambda_{13}) \\lambda_{12} \\cdot \\lambda_{14}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{12} \\lambda_{43} = \\lambda_{43} \\lambda_{12},~~~\\lambda_{42} \\lambda_{13} = \\lambda_{13} \\lambda_{42}\n$$\nwe have\n$$\n \\lambda_{14} \\lambda_{13} \\lambda_{43} \\lambda_{12} \\lambda_{42} = \\lambda_{43} \\lambda_{13} (\\lambda_{42} \\lambda_{12} \\lambda_{14}).\n$$\nUsing the third relation from Lemma \\ref{l1}:\n$$\n\\lambda_{42} \\lambda_{12} \\lambda_{14} = \\lambda_{14} \\lambda_{12} \\lambda_{42},\n$$\nwe get\n$$\n \\lambda_{14} \\lambda_{13} \\lambda_{43} = \\lambda_{43} \\lambda_{13} \\lambda_{14},\n$$\ni.e. the long relation in $VP_4$.\n\n4) The image of the forth relation has the form\n$$\n \\lambda_{41} \\cdot \\lambda_{43} (\\lambda_{42} \\cdot \\lambda_{13}) \\lambda_{12} = \\lambda_{13} (\\lambda_{12} \\cdot \\lambda_{43}) \\lambda_{42} \\cdot \\lambda_{41}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{42} \\lambda_{13} = \\lambda_{13} \\lambda_{42},~~~\\lambda_{12} \\lambda_{43} = \\lambda_{43} \\lambda_{12}\n$$\nwe have\n$$\n \\lambda_{41} \\lambda_{43} \\lambda_{13} \\lambda_{42} \\lambda_{12} = \\lambda_{13} \\lambda_{43} (\\lambda_{12} \\lambda_{42} \\lambda_{41}).\n$$\nUsing the forth relation from Lemma \\ref{l1}:\n$$\n\\lambda_{12} \\lambda_{42} \\lambda_{41} = \\lambda_{41} \\lambda_{42} \\lambda_{12},\n$$\nwe get\n$$\n \\lambda_{41} \\lambda_{43} \\lambda_{13} = \\lambda_{13} \\lambda_{43} \\lambda_{41},\n$$\ni.e. the long relation in $VP_4$.\n\n5) The image of the firth relation has the form\n$$\n \\lambda_{24} (\\lambda_{34} \\cdot \\lambda_{21}) \\lambda_{31} \\cdot \\lambda_{41} = \\lambda_{41} \\cdot \\lambda_{21} (\\lambda_{31} \\cdot \\lambda_{24}) \\lambda_{34}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{34} \\lambda_{21} = \\lambda_{21} \\lambda_{34},~~~\\lambda_{31} \\lambda_{24} = \\lambda_{24} \\lambda_{31}\n$$\nwe have\n$$\n \\lambda_{24} \\lambda_{21} \\lambda_{34} \\lambda_{31} \\lambda_{41} = (\\lambda_{41} \\lambda_{21} \\lambda_{24}) \\lambda_{31} \\lambda_{34}.\n$$\nUsing the firth relation from Lemma \\ref{l1}:\n$$\n \\lambda_{41} \\lambda_{21} \\lambda_{24} = \\lambda_{24} \\lambda_{21} \\lambda_{41},\n$$\nwe get\n$$\n \\lambda_{34} \\lambda_{31} \\lambda_{41} = \\lambda_{41} \\lambda_{31} \\lambda_{34},\n$$\ni.e. the long relation in $VP_4$.\n\n6) The image of the sixth relation has the form\n$$\n \\lambda_{43} (\\lambda_{42} \\cdot \\lambda_{41} \\cdot \\lambda_{21}) \\lambda_{31} = \\lambda_{21} \\lambda_{31} \\cdot \\lambda_{41} \\cdot \\lambda_{43} \\lambda_{42}.\n$$\nusing the\nsixth relation from Lemma \\ref{l1}:\n$$\n \\lambda_{42} \\lambda_{41} \\lambda_{21} = \\lambda_{21} \\lambda_{41} \\lambda_{42},\n$$\nwe get\n$$\n(\\lambda_{43} \\lambda_{21}) \\lambda_{41} (\\lambda_{42} \\lambda_{31}) = \\lambda_{21} \\lambda_{31} \\lambda_{41} \\lambda_{43} \\lambda_{42}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{43} \\lambda_{21} = \\lambda_{21} \\lambda_{43},~~~\\lambda_{42} \\lambda_{31} = \\lambda_{31} \\lambda_{42},\n$$\nwe get\n$$\n \\lambda_{43} \\lambda_{41} \\lambda_{31} = \\lambda_{31} \\lambda_{41} \\lambda_{43},\n$$\ni.e. the long relation in $VP_4$. Hence, we proved\n\n\\begin{lem} \\label{l2}\nFrom relations $\\mathcal{R}^V(3)$, relations $s_1(\\mathcal{R}^V(3))$, $s_2(\\mathcal{R}^V(3))$, and commutativity relations in $\\mathcal{R}^V(4)$ follows the next set of relations in $\\mathcal{R}^V(4)$:\n$$\n\\lambda_{13} \\lambda_{14} \\lambda_{34} = \\lambda_{34} \\lambda_{14} \\lambda_{13},~~~\n\\lambda_{31} \\lambda_{34} \\lambda_{14} = \\lambda_{14} \\lambda_{34} \\lambda_{31},~~~\n\\lambda_{14} \\lambda_{13} \\lambda_{43} = \\lambda_{43} \\lambda_{13} \\lambda_{14},\n$$\n$$\n\\lambda_{41} \\lambda_{43} \\lambda_{13} = \\lambda_{13} \\lambda_{43} \\lambda_{41},~~~\n\\lambda_{34} \\lambda_{31} \\lambda_{41} = \\lambda_{41} \\lambda_{31} \\lambda_{34},~~~\n\\lambda_{43} \\lambda_{41} \\lambda_{31} = \\lambda_{31} \\lambda_{41} \\lambda_{43},\n$$\ni.e. the set of relations where the indexes of the generators lie in the set $\\{ 1, 3, 4 \\}$.\n\\end{lem}\n\n\\medskip\n\nTake the set of relations $s_0(\\mathcal{R}^V(3))$.\n\n\n\n1) The image of the first relation has the form\n$$\n\\lambda_{13} (\\lambda_{23} \\cdot \\lambda_{14}) \\lambda_{24} \\cdot \\lambda_{34} = \\lambda_{34} \\cdot \\lambda_{14} (\\lambda_{24} \\cdot \\lambda_{13}) \\lambda_{23}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{23} \\lambda_{14} = \\lambda_{14} \\lambda_{23},~~~\\lambda_{24} \\lambda_{13} = \\lambda_{13} \\lambda_{24},\n$$\n we rewrite it in the form\n$$\n\\lambda_{13} \\lambda_{14} \\lambda_{23} \\lambda_{24} \\lambda_{34} = (\\lambda_{34} \\lambda_{14} \\lambda_{13}) \\lambda_{24} \\lambda_{23}.\n$$\nUsing the first relation from Lemma \\ref{l2}:\n$$\n\\lambda_{34} \\lambda_{14} \\lambda_{13} = \\lambda_{13} \\lambda_{14} \\lambda_{34},\n$$\nwe get\n the following\n long relation in $VP_4$:\n$$\n \\lambda_{23} \\lambda_{24} \\lambda_{34} = \\lambda_{34} \\lambda_{24} \\lambda_{23}.\n$$\n\n\n\n2) The image of the sixth relation has the form\n$$\n \\lambda_{32} (\\lambda_{31} \\cdot \\lambda_{34} \\cdot \\lambda_{14}) \\lambda_{24} = \\lambda_{14} \\lambda_{24} \\cdot \\lambda_{34} \\cdot \\lambda_{32} \\lambda_{31}.\n$$\nUsing the second relation from Lemma \\ref{l2}:\n$$\n\\lambda_{31} \\lambda_{34} \\lambda_{14} = \\lambda_{14} \\lambda_{34} \\lambda_{31},\n$$\nand the commutativity relations\n$$\n\\lambda_{32} \\lambda_{14} = \\lambda_{14} \\lambda_{32},~~~\\lambda_{31} \\lambda_{24} = \\lambda_{24} \\lambda_{31}\n$$\nwe have\n$$\n \\lambda_{32} \\lambda_{34} \\lambda_{24} = \\lambda_{24} \\lambda_{34} \\lambda_{32},\n$$\ni.e. the long relation in $VP_4$.\n\n\n3) The image of the third relation has the form\n$$\n \\lambda_{14} (\\lambda_{24} \\cdot \\lambda_{13}) \\lambda_{23} \\cdot \\lambda_{43} = \\lambda_{43} \\cdot \\lambda_{13} (\\lambda_{23} \\cdot \\lambda_{14}) \\lambda_{24}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{24} \\lambda_{13} = \\lambda_{13} \\lambda_{24},~~~\\lambda_{23} \\lambda_{14} = \\lambda_{14} \\lambda_{23},\n$$\nwe have\n$$\n \\lambda_{14} \\lambda_{13} \\lambda_{24} \\lambda_{23} \\lambda_{43} = (\\lambda_{43} \\lambda_{13} \\lambda_{14}) \\lambda_{23} \\lambda_{24}.\n$$\nUsing the third relation from Lemma \\ref{l2}:\n$$\n\\lambda_{43} \\lambda_{13} \\lambda_{14} = \\lambda_{14} \\lambda_{13} \\lambda_{43},\n$$\nwe get\n$$\n \\lambda_{24} \\lambda_{23} \\lambda_{43} = \\lambda_{43} \\lambda_{23} \\lambda_{24},\n$$\ni.e. the long relation in $VP_4$.\n\n4) The image of the forth relation has the form\n$$\n \\lambda_{42} (\\lambda_{41} \\cdot \\lambda_{43} \\cdot \\lambda_{13}) \\lambda_{23} = \\lambda_{13} \\lambda_{23} \\cdot \\lambda_{43} \\cdot \\lambda_{42} \\lambda_{41}.\n$$\nUsing the forth relation from Lemma \\ref{l2}:\n$$\n\\lambda_{41} \\lambda_{43} \\lambda_{13} = \\lambda_{13} \\lambda_{43} \\lambda_{41},\n$$\nwe get\n$$\n (\\lambda_{42} \\lambda_{13}) \\lambda_{43} (\\lambda_{41} \\lambda_{23}) = \\lambda_{23} \\lambda_{43} \\lambda_{42} \\lambda_{41}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{42} \\lambda_{13} = \\lambda_{13} \\lambda_{42},~~~\\lambda_{41} \\lambda_{23} = \\lambda_{23} \\lambda_{41}\n$$\nwe get\n$$\n \\lambda_{42} \\lambda_{43} \\lambda_{23} = \\lambda_{23} \\lambda_{43} \\lambda_{42},\n$$\ni.e. the long relation in $VP_4$.\n\n5) The image of the firth relation has the form\n$$\n \\lambda_{34} \\cdot \\lambda_{32} (\\lambda_{31} \\cdot \\lambda_{42}) \\lambda_{41} = \\lambda_{42} (\\lambda_{41} \\cdot \\lambda_{32}) \\lambda_{31} \\cdot \\lambda_{34}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{31} \\lambda_{42} = \\lambda_{42} \\lambda_{31},~~~\\lambda_{41} \\lambda_{32} = \\lambda_{32} \\lambda_{41},\n$$\nwe get\n$$\n \\lambda_{34} \\lambda_{32} \\lambda_{42} \\lambda_{31} \\lambda_{41} = \\lambda_{42} \\lambda_{32} (\\lambda_{41} \\lambda_{31} \\lambda_{34}).\n$$\nUsing the firth relation from Lemma \\ref{l2}:\n$$\n\\lambda_{41} \\lambda_{31} \\lambda_{34} = \\lambda_{34} \\lambda_{31} \\lambda_{41},\n$$\nwe get\n$$\n \\lambda_{34} \\lambda_{32} \\lambda_{42} = \\lambda_{42} \\lambda_{32} \\lambda_{34},\n$$\ni.e. the long relation in $VP_4$.\n\n6) The image of the sixth relation has the form\n$$\n \\lambda_{43} \\cdot \\lambda_{42} (\\lambda_{41} \\cdot \\lambda_{32}) \\lambda_{31} = \\lambda_{32} (\\lambda_{31} \\cdot \\lambda_{42}) \\lambda_{41} \\cdot \\lambda_{43}.\n$$\nUsing the commutativity relations\n$$\n\\lambda_{41} \\lambda_{32} = \\lambda_{32} \\lambda_{41},~~~\\lambda_{31} \\lambda_{42} = \\lambda_{42} \\lambda_{31},\n$$\nwe get\n$$\n \\lambda_{43} \\lambda_{42} \\lambda_{32} \\lambda_{41} \\lambda_{31} = \\lambda_{32} \\lambda_{42} (\\lambda_{31} \\lambda_{41} \\lambda_{43}).\n$$\nUsing the\nsixth relation from Lemma \\ref{l2}:\n$$\n\\lambda_{31} \\lambda_{41} \\lambda_{43} = \\lambda_{43} \\lambda_{41} \\lambda_{31},\n$$\nwe get\n$$\n \\lambda_{43} \\lambda_{42} \\lambda_{32} = \\lambda_{32} \\lambda_{42} \\lambda_{43},\n$$\ni.e. the long relation in $VP_4$. Hence, we proved\n\n\\begin{lem}\nFrom relations $\\mathcal{R}^V(3)$, relations $s_i(\\mathcal{R}^V(3))$, $i = 0, 1, 2$, and commutativity relations in $\\mathcal{R}^V(4)$ follows the next set relations in $\\mathcal{R}^V(4)$:\n$$\n\\lambda_{23} \\lambda_{24} \\lambda_{34} = \\lambda_{34} \\lambda_{24} \\lambda_{23},~~~\n\\lambda_{32} \\lambda_{34} \\lambda_{24} = \\lambda_{24} \\lambda_{34} \\lambda_{32},~~~\n\\lambda_{24} \\lambda_{23} \\lambda_{43} = \\lambda_{43} \\lambda_{23} \\lambda_{24},\n$$\n$$\n\\lambda_{42} \\lambda_{43} \\lambda_{23} = \\lambda_{23} \\lambda_{43} \\lambda_{42},~~~\n\\lambda_{34} \\lambda_{32} \\lambda_{42} = \\lambda_{42} \\lambda_{32} \\lambda_{34},~~~\n\\lambda_{43} \\lambda_{42} \\lambda_{32} = \\lambda_{32} \\lambda_{42} \\lambda_{43},\n$$\ni.e. the set of relations where the indexes of the generators lie in the set $\\{ 2, 3, 4 \\}$.\n\\end{lem}\n\n\n\n\n\n\n\n\\subsection{Lifting the commutativity relations from $\\mathcal{R}^V(4)$ into $\\mathcal{R}^V(5)$}\n\nWe have to show that $\\mathcal{R}^V(5) = \\langle \\mathcal{R}^V(4), s_i(\\mathcal{R}^V(4)), i = 0, 1, 2, 3 \\rangle$. At first consider the commutativity relations\n$$\n[\\lambda_{i4}^*, \\lambda_{kl}^*], ~~1 \\leq i \\leq 3,~~ 1 \\leq k < l \\leq 3,\n$$\nin $\\mathcal{R}^V(4)$. We divide them on the four groups:\n\n1-st group: $[\\lambda_{34}, \\lambda_{12}] = [\\lambda_{24}, \\lambda_{13}] = [\\lambda_{14}, \\lambda_{23}] = 1$;\n\n2-nd group: $[\\lambda_{34}, \\lambda_{21}] = [\\lambda_{24}, \\lambda_{31}] = [\\lambda_{14}, \\lambda_{32}] = 1$;\n\n3-d group: $[\\lambda_{43}, \\lambda_{21}] = [\\lambda_{42}, \\lambda_{31}] = [\\lambda_{41}, \\lambda_{32}] = 1$;\n\n4-th group: $[\\lambda_{43}, \\lambda_{12}] = [\\lambda_{42}, \\lambda_{13}] = [\\lambda_{41}, \\lambda_{23}] = 1$.\n\n\nTake the third relation from the 1-st group and acting on it by $s_i$, $i = 0, 1, 2, 3$, we get the following relations:\n$$\n[\\lambda_{15} \\lambda_{25}, \\lambda_{34}] = [\\lambda_{15}, \\lambda_{24} \\lambda_{34}] = [\\lambda_{15}, \\lambda_{24} \\lambda_{23}] = [\\lambda_{15} \\lambda_{14}, \\lambda_{23}] = 1.\n$$\nUsing the commutativity relation\n$$\n\\lambda_{14} \\lambda_{23} = \\lambda_{23} \\lambda_{14},\n$$\nwhich hold in $VP_4$, from the last relation we have\n\\begin{equation} \\label{cc1}\n[\\lambda_{15}, \\lambda_{23}] = 1.\n\\end{equation}\nWith considering (\\ref{cc1}) we get\n$$\n[\\lambda_{15}, \\lambda_{24}] = 1.\n$$\nThen from the second relation follows relation $[\\lambda_{15}, \\lambda_{34}] = 1$ and from the first relation follows $[\\lambda_{25}, \\lambda_{34}] = 1$.\nHence, we have proven\n\n\\begin{lem} \\label{cl1}\nFrom the lifting $s_i$, $i = 0, 1, 2, 3$, of the relation $[\\lambda_{14}, \\lambda_{23}] = 1$ and the commutativity relations in $\\mathcal{R}^V(4)$ follows the commutativity relations\n$$\n[\\lambda_{15}, \\lambda_{23}] = [\\lambda_{15}, \\lambda_{24}] =[\\lambda_{15}, \\lambda_{34}] = [\\lambda_{25}, \\lambda_{34}] = 1,\n$$\nfrom $\\mathcal{R}^V(5)$.\n\\end{lem}\n\n\nTake the second relation in the 1-st group and acting on it by $s_i$, $i = 0, 1, 2, 3$, we get the following relations:\n$$\n[\\lambda_{35}, \\lambda_{14} \\lambda_{24}] = [\\lambda_{25} \\lambda_{35}, \\lambda_{14}] = [\\lambda_{25}, \\lambda_{14} \\lambda_{13}] = [\\lambda_{25} \\lambda_{24}, \\lambda_{13}] = 1.\n$$\nUsing the commutativity relation\n$$\n\\lambda_{24} \\lambda_{13} = \\lambda_{13} \\lambda_{24},\n$$\nwhich hold in $VP_4$, from the last relation we have\n\\begin{equation} \\label{c1}\n[\\lambda_{25}, \\lambda_{14}] = 1.\n\\end{equation}\nThen from the third relation follows $[\\lambda_{25}, \\lambda_{13}] = 1$. From the second relation follows $[\\lambda_{35}, \\lambda_{14}] = 1$\n and from the first relation follows $[\\lambda_{35}, \\lambda_{24}] = 1$.\nHence, we have proven\n\n\\begin{lem} \\label{cll2}\nFrom the lifting $s_i$, $i = 0, 1, 2, 3$, of the relation $[\\lambda_{24}, \\lambda_{13}] = 1$ and the commutativity relations in $\\mathcal{R}^V(4)$ follows the commutativity relations\n$$\n[\\lambda_{25}, \\lambda_{14}] = [\\lambda_{25}, \\lambda_{13}] =[\\lambda_{35}, \\lambda_{14}] = [\\lambda_{35}, \\lambda_{24}] = 1,\n$$\nfrom $\\mathcal{R}^V(5)$.\n\\end{lem}\n\n\n\nTake the first relation in the 1-st group and acting on it by $s_i$, $i = 0, 1, 2, 3$, we get the following relations:\n$$\n[\\lambda_{45}, \\lambda_{13} \\lambda_{23}] = [\\lambda_{45}, \\lambda_{13} \\lambda_{12}] = [\\lambda_{35} \\lambda_{45}, \\lambda_{12}] = [ \\lambda_{35} \\lambda_{34}, \\lambda_{12}] = 1.\n$$\nUsing the commutativity relation $[\\lambda_{34}, \\lambda_{12}] = 1$, rewrite the last relation in the form\n$$\n[\\lambda_{35}, \\lambda_{12}] = 1.\n$$\n Then from the third relation follows\n$[\\lambda_{45}, \\lambda_{12}] = 1.$\nThen from the second relation follows\n$[\\lambda_{45}, \\lambda_{13}] = 1,$\nand hence from the first relation we get\n$[\\lambda_{45}, \\lambda_{23}] = 1.$\nHence, we have proven\n\n\\begin{lem} \\label{cl2}\nFrom the lifting $s_i$, $i = 0, 1, 2, 3$, of the relation $[\\lambda_{34}, \\lambda_{12}] = 1$, the commutativity relations in $\\mathcal{R}^V(4)$ and relations from Lemma \\ref{cll2} follow the commutativity relations\n$$\n[\\lambda_{35}, \\lambda_{12}] = [\\lambda_{45}, \\lambda_{12}] =[\\lambda_{45}, \\lambda_{13}] = [\\lambda_{45}, \\lambda_{23}] = 1,\n$$\nfrom $VP_5$.\n\\end{lem}\n\n\\bigskip\n\n\nTake the 2-nd group of commutativity relations:\n$$\n[\\lambda_{34}, \\lambda_{21}] = [\\lambda_{24}, \\lambda_{31}] = [\\lambda_{14}, \\lambda_{32}] = 1.\n$$\nActing on the third relation by $s_i$, $i = 0, 1, 2, 3$, we get\n$$\n[\\lambda_{15} \\lambda_{25}, \\lambda_{43}] = [\\lambda_{15}, \\lambda_{43} \\lambda_{42}] = [\\lambda_{15}, \\lambda_{32} \\lambda_{42}] = [\\lambda_{15} \\lambda_{14}, \\lambda_{32}] = 1.\n$$\nUsing the commutativity relations in $VP_4$, rewrite the last relation in the form\n\\begin{equation} \\label{com22}\n[\\lambda_{15}, \\lambda_{32}] = 1.\n\\end{equation}\nFrom (\\ref{com22}), the third relation gives $[\\lambda_{15}, \\lambda_{42}] = 1$. Then from the second relation follows that $[\\lambda_{15}, \\lambda_{43}] = 1$ and from the first relation follows that $[\\lambda_{25}, \\lambda_{43}] = 1$. We proved\n\n\\begin{lem}\nLifting $s_i$, $i = 0, 1, 2, 3$, the commutativity relation $[\\lambda_{14}, \\lambda_{32}] = 1$ and the commutativity relations of $VP_4$ give relations\n$$\n[\\lambda_{15}, \\lambda_{32}] = [\\lambda_{15}, \\lambda_{42}] = [\\lambda_{15}, \\lambda_{43}] = [\\lambda_{25}, \\lambda_{43}] = 1.\n$$\n\\end{lem}\n\n\nActing on the second relation by $s_i$, $i = 0, 1, 2, 3$, we get\n$$\n[\\lambda_{35}, \\lambda_{42} \\lambda_{41}] = [\\lambda_{25} \\lambda_{35}, \\lambda_{41}] = [\\lambda_{25}, \\lambda_{31} \\lambda_{41}] = [\\lambda_{25} \\lambda_{24}, \\lambda_{31}] = 1.\n$$\nUsing the relation $[\\lambda_{24}, \\lambda_{31}] = 1$ in $VP_4$ from the last relation follows that\n$$\n[\\lambda_{25}, \\lambda_{31}] = 1.\n$$\nHence from the third relation follows that $[\\lambda_{25}, \\lambda_{41}] = 1$, from the second relation follows that $[\\lambda_{35}, \\lambda_{41}] = $ and from the first relation follows that $[\\lambda_{35}, \\lambda_{42}] = 1$. We proved\n\n\\begin{lem}\nLifting $s_i$, $i = 0, 1, 2, 3$, of the commutativity relation $[\\lambda_{24}, \\lambda_{31}] = 1$ and the commutativity relations of $VP_4$ give relations\n$$\n[\\lambda_{25}, \\lambda_{31}] = [\\lambda_{25}, \\lambda_{41}] = [\\lambda_{35}, \\lambda_{41}] = [\\lambda_{35}, \\lambda_{42}] = 1.\n$$\n\\end{lem}\n\n\\medskip\n\nActing on the first relation by $s_i$, $i = 0, 1, 2, 3$, we get\n$$\n[\\lambda_{45}, \\lambda_{32} \\lambda_{31}] = [\\lambda_{45}, \\lambda_{21} \\lambda_{31}] = [\\lambda_{35} \\lambda_{45}, \\lambda_{21}] = [\\lambda_{35} \\lambda_{34}, \\lambda_{21}] = 1.\n$$\nUsing the commutativity relations in $VP_4$ from the last relation follows that $[\\lambda_{35}, \\lambda_{21}] = 1$. Then from the third relation we have $[\\lambda_{45}, \\lambda_{21}] = 1$, from the second relation: $[\\lambda_{45}, \\lambda_{31}] = 1$ and from the first relation: $[\\lambda_{45}, \\lambda_{32}] = 1$.\n\n\n\\begin{lem}\nLifting $s_i$, $i = 0, 1, 2, 3$, of the commutativity relation $[\\lambda_{34}, \\lambda_{21}] = 1$ and the commutativity relations of $VP_4$ give relations\n$$\n[\\lambda_{35}, \\lambda_{21}] = [\\lambda_{45}, \\lambda_{21}] = [\\lambda_{45}, \\lambda_{31}] = [\\lambda_{45}, \\lambda_{32}] = 1.\n$$\n\\end{lem}\n\n\nIn $VP_5$ we have 24 commutativity relations of the form $[\\lambda_{i5}, \\lambda_{kl}^*] = 1$, $\\lambda_{kl}^* \\in \\{ \\lambda_{kl}, \\lambda_{lk}\\}$, where $1 \\leq i < 5$, $1 \\leq k < l \\leq 4$:\n$$\n[\\lambda_{45}, \\lambda_{12}^*] = [\\lambda_{45}, \\lambda_{13}^*] = [\\lambda_{45}, \\lambda_{23}^*] =\n[\\lambda_{35}, \\lambda_{12}^*] = [\\lambda_{35}, \\lambda_{14}^*] = [\\lambda_{35}, \\lambda_{24}^*] = 1,\n$$\n$$\n[\\lambda_{25}, \\lambda_{13}^*] = [\\lambda_{25}, \\lambda_{14}^*] = [\\lambda_{25}, \\lambda_{34}^*] =\n[\\lambda_{15}, \\lambda_{23}^*] = [\\lambda_{15}, \\lambda_{24}^*] = [\\lambda_{15}, \\lambda_{34}^*] = 1.\n$$\nThese relations follow from the 1-st and from the 2-nd groups of commutativity relations in $\\mathcal{R}^V(4)$. The other commutativity relations from $\\mathcal{R}^V(5)\\setminus \\mathcal{R}^V(4)$ follow by the same way from the 3-d and from the 4-th groups of relations.\n\n\n\\subsection{Lifting the commutativity relations from $\\mathcal{R}^V(n)$ to $\\mathcal{R}^V(n+1)$, $n \\geq 5$}\n\nWe have to show that $\\mathcal{R}^V(n+1) = \\langle \\mathcal{R}^V(n), s_i(\\mathcal{R}^V(n)), i = 0, 1, \\ldots, n-1 \\rangle$. At first consider the commutativity relations\n$$\n[\\lambda_{mn}^*, \\lambda_{kl}^*], ~~1 \\leq m < n,~~ 1 \\leq k < l < n,\n$$\nin $VP_n$, which are not commutativity relations in $VP_{n-1}$. We divide them on the four groups:\n\n1-st group: $[\\lambda_{mn}, \\lambda_{kl}] = 1$;\n\n2-nd group: $[\\lambda_{mn}, \\lambda_{lk}] = 1$;\n\n3-d group: $[\\lambda_{nm}, \\lambda_{lk}] = 1$;\n\n4-th group: $[\\lambda_{nm}, \\lambda_{kl}] = 1$.\n\nConsider the relations from the 1-st group and divide them on some subgroups.\n\n1) Suppose that $m < k < l < n.$\n\nActing on the relation $[\\lambda_{mn}, \\lambda_{kl}] = 1$ by $s_{n-1}$ and using Proposition \\ref{p3.1}, we get the relation\n$$\n[\\lambda_{m,n+1} \\lambda_{mn}, \\lambda_{kl}] = 1.\n$$\nSince $[\\lambda_{mn}, \\lambda_{kl}] = 1$ and this relation is a relation in $VP_n$, we have relation in $VP_{n+1}$:\n\\begin{equation} \\label{r10}\n[\\lambda_{m,n+1}, \\lambda_{kl}] = 1.\n\\end{equation}\n\nLet $i$ be such that $m < k < l < i < n.$ Acting by $s_{i-1}$ on the relation $[\\lambda_{mn}, \\lambda_{kl}] = 1$ we get\n$$\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{kl}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\nLet $i = l$, then\n$$\ns_{l-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k,l+1} \\lambda_{kl}] = 1.\n$$\nUsing the commutativity relations in $VP_n$ and relation (\\ref{r10}), we have\n\\begin{equation} \\label{r11}\n[\\lambda_{m,n+1}, \\lambda_{k,l+1}] = 1,\n\\end{equation}\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ satisfies the inequality $m < k < i < l < n.$ Acting by $s_{i-1}$ we get\n$$\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k,l+1}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\nLet $i=k$. Acting by $s_{k-1}$ we get\n$$\ns_{k-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k,l+1} \\lambda_{k+1,l+1}] = 1.\n$$\nUsing the relation (\\ref{r11}), we have\n\\begin{equation} \\label{r12}\n[\\lambda_{m,n+1}, \\lambda_{k+1,l+1}] = 1,\n\\end{equation}\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ satisfies the inequality $m < i < k < l < n.$ Acting by $s_{i-1}$ we get\n$$\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k+1,l+1}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\n\nLet $i=m$. Acting by $s_{m-1}$ we get\n$$\ns_{m-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1} \\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1.\n$$\nUsing the relation (\\ref{r12}), we have\n$$\n[\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1,\n$$\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ be satisfied the inequality $i < m < k < l < n.$ Acting by $s_{i-1}$ we get\n$$\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\n\\medskip\n\n2) Suppose that $k < m < l < n.$\n\nActing on the relation $[\\lambda_{mn}, \\lambda_{kl}] = 1$ by $s_{n-1}$, we get the relation\n$$\n[\\lambda_{m,n+1}\\lambda_{mn}, \\lambda_{kl}] = 1.\n$$\nSince $[\\lambda_{mn}, \\lambda_{kl}] = 1$ that follows from the relations in $VP_n$, we have relation:\n$$\n[\\lambda_{m,n+1}, \\lambda_{kl}] = 1.\n$$\n\n\nLet $i$ be such that $k < m < l < i < n.$ Acting by $s_{i-1}$ on the relation $[\\lambda_{mn}, \\lambda_{kl}] = 1$ we get\n\\begin{equation} \\label{r15}\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{kl}] = 1\n\\end{equation}\nthat is a relation in $VP_{n+1}$.\n\n\nLet $i = l$, then\n$$\ns_{l-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k,l+1} \\lambda_{kl}] = 1\n$$\nUsing the commutativity relations in $VP_n$ and relation (\\ref{r15}), we have\n$$\n[\\lambda_{m,n+1}, \\lambda_{k,l+1}] = 1,\n$$\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ be satisfied the inequality $k < m < i < l < n.$ Acting by $s_{i-1}$ we get\n\\begin{equation} \\label{r16}\ns_i ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{k,l+1}] = 1\n\\end{equation}\nthat is a relation in $VP_{n+1}$.\n\n\nLet $i=m$. Acting by $s_{m-1}$ we get\n\\begin{equation} \\label{r17}\ns_m ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1} \\lambda_{m+1,n+1}, \\lambda_{k,l+1}] = 1.\n\\end{equation}\nUsing the relation (\\ref{r16}), we have\n$$\n[\\lambda_{m+1,n+1}, \\lambda_{k,l+1}] = 1,\n$$\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ be satisfied the inequality $k < i < m < l < n.$ Acting by $s_{i-1}$ we get\n\\begin{equation} \\label{r18}\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k,l+1}] = 1\n\\end{equation}\nthat is a relation in $VP_{n+1}$.\n\nLet $i=k$. Acting by $s_{k-1}$ we get\n\\begin{equation} \\label{r19}\ns_{k-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k,l+1} \\lambda_{k+1,l+1}] = 1.\n\\end{equation}\nUsing the relation (\\ref{r18}), we have\n$$\n[\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1,\n$$\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ be satisfied the inequality $i < k < m < l < n.$ Acting by $s_{i-1}$ we get\n\\begin{equation} \\label{r20}\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1\n\\end{equation}\nthat is a relation in $VP_{n+1}$.\n\n\n\n\\medskip\n\n3) Suppose that $k < l < m < n.$\n\nActing on $[\\lambda_{mn}, \\lambda_{kl}] = 1,$ by $s_{n-1}$, we get the relation\n$$\ns_{n-1}([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}\\lambda_{mn}, \\lambda_{kl}] = 1.\n$$\nUsing commutativity relations in $VP_n$ and the commutativity relations in $VP_{n+1}$, which were proved in 2), from our relation follows\n\\begin{equation} \\label{r21}\n[\\lambda_{m,n+1}, \\lambda_{kl}] = 1.\n\\end{equation}\n\n\nLet $i$ be such that $k < l < m < i < n.$ Acting by $s_{i-1}$ on the relation $[\\lambda_{mn}, \\lambda_{kl}] = 1$ we get\n$$\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1}, \\lambda_{kl}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\n\nLet $i = m$, then\n$$\ns_{m-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m,n+1} \\lambda_{m+1,n+1}, \\lambda_{kl}] = 1.\n$$\nUsing relation (\\ref{r21}), we have\n\\begin{equation} \\label{r22}\n[\\lambda_{m+1,n+1}, \\lambda_{k,l}] = 1,\n\\end{equation}\ni.e. a commutativity relation in $VP_{n+1}$.\n\nLet $i$ satisfies the inequality $k < l < i < m < n.$ Acting by $s_{i-1}$ we get\n$$\ns_i ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,l+1}, \\lambda_{kl}] = 1\n$$\nthat is a relation in $VP_{n+1}$.\n\n\nLet $i=l$. Acting by $s_{l-1}$ we get\n$$\ns_{l-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k,l+1} \\lambda_{kl}] = 1.\n$$\nUsing (\\ref{r22}) we get\n\\begin{equation} \\label{r23}\n[\\lambda_{m+1,n+1}, \\lambda_{k,l+1}] = 1.\n\\end{equation}\n\n\nLet $i$ satisfies the inequality $k < i < l < m < n.$ Acting by $s_{i-1}$ we get\n\\begin{equation} \\label{r24}\ns_{i-1} ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k,l+1}] = 1\n\\end{equation}\nthat is a relation in $VP_{n+1}$.\n\nLet $i=k$. Acting by $s_{k-1}$ we get\n$$\ns_k ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k,l+1} \\lambda_{k+1,l+1}] = 1.\n$$\nUsing the relation (\\ref{r23}), we have\n$$\n[\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1.\n$$\n\nLet $i$ be satisfied the inequality $i < k < l < m < n.$ Acting by $s_{i-1}$ we get\n$$\ns_i ([\\lambda_{mn}, \\lambda_{kl}]) = [\\lambda_{m+1,n+1}, \\lambda_{k+1,l+1}] = 1.\n$$\n\n\\medskip\n\nWe considered only the 1-st group of relations. The proof for the other groups is similar.\n\n\\subsection{Lifting the long relations from $\\mathcal{R}^V(n)$ to $\\mathcal{R}^V(n+1)$, $n \\geq 4$}\n\nDenote by $R_{ijk}$ the following set of long relations\n$$\n\\lambda_{ij} \\lambda_{ik} \\lambda_{jk} = \\lambda_{jk} \\lambda_{ik} \\lambda_{ij},~~~\n\\lambda_{ji} \\lambda_{jk} \\lambda_{ik} = \\lambda_{ik} \\lambda_{jk} \\lambda_{ji},\n$$\n$$\n\\lambda_{ik} \\lambda_{ij} \\lambda_{kj} = \\lambda_{kj} \\lambda_{ij} \\lambda_{ik},~~~\n\\lambda_{ki} \\lambda_{kj} \\lambda_{ij} = \\lambda_{ij} \\lambda_{kj} \\lambda_{ki},\n$$\n$$\n\\lambda_{jk} \\lambda_{ji} \\lambda_{ki} = \\lambda_{ki} \\lambda_{ji} \\lambda_{jk},~~~\n\\lambda_{kj} \\lambda_{ki} \\lambda_{ji} = \\lambda_{ji} \\lambda_{ki} \\lambda_{kj},\n$$\ni.e. relations which contains the generators with indexes from the set $\\{ i, j, k \\}$.\n\n\nWe have to prove that relations $R_{i,j,n+1}$ follows from relations of $\\mathcal{R}^V(n)$, $s_l(\\mathcal{R}^V(n))$, $l = 0, 1, \\ldots, n-1$ and commutativity relations of $\\mathcal{R}^V(n+1)$.\n\n\\begin{thm}\nThe long relations $R_{i,j,n+1}$ in $\\mathcal{R}^V(n+1)$ follow from the relations of $\\mathcal{R}^V(n)$, $s_l(\\mathcal{R}^V(n))$, $l = 0, 1, \\ldots, n-1$ and commutativity relations of $\\mathcal{R}^V(n+1)$.\n\\end{thm}\n\nTo prove this theorem we start with the following\n\n\\begin{lem} \\label{lem1}\nLet $n \\geq 4$ and for the set of integer numbers $\\{ i, j, n+1 \\}$, $1 \\leq i < j \\leq n+1$ one of the following conditions holds\n\n1) $i \\geq 3$;\n\n2) $j - i \\geq 3$;\n\n3) $n+1 - j \\geq 3$.\n\nThen there is an integer $k$, $1 \\leq k \\leq n$, such that the relations $R_{i,j,n+1} \\subseteq \\mathcal{R}^V(n+1)$ follows from the relations $s_{k-1}(\\mathcal{R}^V(n))$.\n\\end{lem}\n\n\\begin{proof}\n1) Suppose that the condition 1) holds. Put $k=1$ and consider the relations $R_{i-1,j-1,n}$ in $\\mathcal{R}^V(n)$. It is not difficult to see that $s_0(R_{i-1,j-1,n}) = R_{i,j,n+1}$.\n\n2) Suppose that the condition 2) holds. Put $k=i+1$ and consider the relations $R_{i,j-1,n}$ in $\\mathcal{R}^V(n)$. It is not difficult to see that $s_{i}(R_{i,j-1,n}) = R_{i,j,n+1}$.\n\n3) Suppose that the condition 3) holds. Put $k=j+1$ and consider the relations $R_{i,j,n}$ in $\\mathcal{R}^V(n)$. It is not difficult to see that $s_{j}(R_{i,j,n}) = R_{i,j,n+1}$.\n\\end{proof}\n\nNow suppose that $i = 2$ and for the set $\\{ i, j, n+1 \\}$ none of the conditions of the lemma is satisfied. Take the set of relations $R_{1,j-1,n}$ and find $s_0(R_{1,j-1,n})$. The first relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{1,j-1} \\lambda_{1n} \\lambda_{j-1,n} = \\lambda_{j-1,n} \\lambda_{1n} \\lambda_{1,j-1}.\n$$\nActing by $s_0$ we get the relation\n$$\n(\\lambda_{1,j} \\lambda_{2,j}) (\\lambda_{1,n+1} \\lambda_{2,n+1}) \\lambda_{j,n+1} = \\lambda_{j,n+1} (\\lambda_{1,n+1} \\lambda_{2,n+1}) (\\lambda_{1,j} \\lambda_{2,j}).\n$$\nSince $\\lambda_{2,j} \\lambda_{1,n+1} = \\lambda_{1,n+1} \\lambda_{2,j}$ and $\\lambda_{2,n+1} \\lambda_{1,j} = \\lambda_{1,j} \\lambda_{2,n+1}$, rewrite the last relation in the form\n\\begin{equation} \\label{r30}\n\\lambda_{1,j} \\lambda_{1,n+1} \\lambda_{2,j} \\lambda_{2,n+1} \\lambda_{j,n+1} = (\\lambda_{j,n+1} \\lambda_{1,n+1} \\lambda_{1,j}) \\lambda_{2,n+1} \\lambda_{2,j}.\n\\end{equation}\nTake the set $\\{ 1, j, n+1 \\}$. Since $n \\geq 4$, then for this set condition 2) or condition 3) of Lemma \\ref{lem1} holds. Then the set of relation $R_{1,j,n+1}$ comes from relations of $VP_n$. In particular, the relation\n$$\n\\lambda_{j,n+1} \\lambda_{1,n+1} \\lambda_{1,j} = \\lambda_{1,j} \\lambda_{1,n+1} \\lambda_{j,n+1}\n$$\nholds. Using this relation, rewrite (\\ref{r30}):\n$$\n\\lambda_{1,j} \\lambda_{1,n+1} \\lambda_{2,j} \\lambda_{2,n+1} \\lambda_{j,n+1} = (\\lambda_{1,j} \\lambda_{1,n+1} \\lambda_{j,n+1}) \\lambda_{2,n+1} \\lambda_{2,j}.\n$$\nAfter cancelations we have\n$$\n \\lambda_{2,j} \\lambda_{2,n+1} \\lambda_{j,n+1} = \\lambda_{j,n+1} \\lambda_{2,n+1} \\lambda_{2,j}.\n$$\nIt is the first relation from $R_{2,j,n+1}$.\n\n\nThe second relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{j-1,1} \\lambda_{j-1,n} \\lambda_{1,n} = \\lambda_{1,n} \\lambda_{j-1,n} \\lambda_{j-1,1}.\n$$\nActing by $s_0$ we get the relation\n$$\n(\\lambda_{j2} \\lambda_{j1}) \\lambda_{j,n+1} (\\lambda_{1,n+1} \\lambda_{2,n+1}) = (\\lambda_{1,n+1} \\lambda_{2,n+1}) \\lambda_{j,n+1} (\\lambda_{j2} \\lambda_{j1}).\n$$\nAs we seen before the set of relation $R_{1,j,n+1}$ holds in $VP_{n+1}$. Using the relation\n$$\n\\lambda_{j1} \\lambda_{j,n+1} \\lambda_{1,n+1} = \\lambda_{1,n+1} \\lambda_{j,n+1} \\lambda_{j1}\n$$\nrewrite our relation in the form:\n$$\n\\lambda_{j2} (\\lambda_{1,n+1} \\lambda_{j,n+1} \\lambda_{j1}) \\lambda_{2,n+1} = \\lambda_{1,n+1} \\lambda_{2,n+1} \\lambda_{j,n+1}\\lambda_{j2} \\lambda_{j1}.\n$$\nUsing the commutativity relations $\\lambda_{j2} \\lambda_{1,n+1} = \\lambda_{1,n+1} \\lambda_{j2}$ and $\\lambda_{j1} \\lambda_{2,n+1} = \\lambda_{2,n+1} \\lambda_{j1}$ we have\n$$\n(\\lambda_{1,n+1} \\lambda_{j2}) \\lambda_{j,n+1} (\\lambda_{2,n+1} \\lambda_{j1}) = \\lambda_{1,n+1} \\lambda_{2,n+1} \\lambda_{j,n+1}\\lambda_{j2} \\lambda_{j1}.\n$$\nAfter cancelations we get\n$$\n \\lambda_{j2} \\lambda_{j,n+1} \\lambda_{2,n+1} = \\lambda_{2,n+1} \\lambda_{j,n+1} \\lambda_{j2}.\n$$\nIt is the second relation from $R_{1,j,n+1}$.\n\n\nThe third relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{1n} \\lambda_{1,j-1} \\lambda_{n,j-1} = \\lambda_{n,j-1} \\lambda_{1,j-1} \\lambda_{1n}.\n$$\nActing by $s_0$ we get the relation\n$$\n(\\lambda_{1,n+1} \\lambda_{2,n+1}) (\\lambda_{1j} \\lambda_{2j}) \\lambda_{n+1,j} = \\lambda_{n+1,j} (\\lambda_{1j} \\lambda_{2j}) (\\lambda_{1,n+1} \\lambda_{2,n+1}).\n$$\nSince $\\lambda_{2,n+1} \\lambda_{1j} = \\lambda_{1j} \\lambda_{2,n+1}$ and $\\lambda_{2j} \\lambda_{1,n+1} = \\lambda_{1,n+1} \\lambda_{2j}$, rewrite the last relation in the form\n\\begin{equation} \\label{r31}\n\\lambda_{1,n+1} \\lambda_{1j} \\lambda_{2,n+1} \\lambda_{2j} \\lambda_{n+1,j} = (\\lambda_{n+1,j} \\lambda_{1j} \\lambda_{1,n+1}) \\lambda_{2j} \\lambda_{2,n+1}.\n\\end{equation}\nAs we seen, the set of relation $R_{1,j,n+1}$ comes from relations of $VP_n$. In particular, the relation\n$$\n\\lambda_{n+1,j} \\lambda_{1j} \\lambda_{1,n+1} = \\lambda_{1,n+1} \\lambda_{1j} \\lambda_{n+1,j}\n$$\nholds. Using this relation, rewrite (\\ref{r31}):\n$$\n\\lambda_{1,n+1} \\lambda_{1j} \\lambda_{2,n+1} \\lambda_{2j} \\lambda_{n+1,j} = (\\lambda_{1,n+1} \\lambda_{1j} \\lambda_{n+1,j}) \\lambda_{2j} \\lambda_{2,n+1}.\n$$\nAfter cancelations we have\n$$\n \\lambda_{2,n+1} \\lambda_{2j} \\lambda_{n+1,j} = \\lambda_{n+1,j} \\lambda_{2j} \\lambda_{2,n+1}.\n$$\nIt is the third relation from $R_{1,j,n+1}$.\n\n\nThe forth relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{n1} \\lambda_{n,j-1} \\lambda_{1,j-1} = \\lambda_{1,j-1} \\lambda_{n,j-1} \\lambda_{n1}.\n$$\nActing by $s_0$ we get the relation\n$$\n(\\lambda_{n+1,2} \\lambda_{n+1,1}) \\lambda_{n+1,j} (\\lambda_{1j} \\lambda_{2j}) = (\\lambda_{1j} \\lambda_{2j}) \\lambda_{n+1,j} (\\lambda_{n+1,2} \\lambda_{n+1,1}).\n$$\nAs we seen before the set of relation $R_{1,j,n+1}$ holds in $VP_{n+1}$. Using the relation\n$$\n\\lambda_{n+1,1} \\lambda_{n+1,j} \\lambda_{1j} = \\lambda_{1j} \\lambda_{n+1,j} \\lambda_{n+1,1}\n$$\nrewrite our relation in the form:\n$$\n\\lambda_{n+1,2} (\\lambda_{1j} \\lambda_{n+1,j} \\lambda_{n+1,1}) \\lambda_{2j} = \\lambda_{1j} \\lambda_{2j} \\lambda_{n+1,j} \\lambda_{n+1,2} \\lambda_{n+1,1}.\n$$\nUsing the commutativity relations $\\lambda_{n+1,2} \\lambda_{1j} = \\lambda_{1j} \\lambda_{n+1,2}$ and $\\lambda_{n+1,1} \\lambda_{2j} = \\lambda_{2j} \\lambda_{n+1,1}$ we have\n$$\n(\\lambda_{1j} \\lambda_{n+1,2}) \\lambda_{n+1,j} (\\lambda_{2j} \\lambda_{n+1,1}) = \\lambda_{1j} \\lambda_{2j} \\lambda_{n+1,j} \\lambda_{n+1,2} \\lambda_{n+1,1}.\n$$\nAfter cancelations we get\n$$\n \\lambda_{n+1,2} \\lambda_{n+1,j} \\lambda_{2j} = \\lambda_{2j} \\lambda_{n+1,j} \\lambda_{n+1,2}.\n$$\nIt is the forth relation from $R_{1,j,n+1}$.\n\n\n\n\nThe firth relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{j-1,n} \\lambda_{j-1,1} \\lambda_{n1} = \\lambda_{n1} \\lambda_{j-1,1} \\lambda_{j-1,n}.\n$$\nActing by $s_0$ we get the relation\n$$\n\\lambda_{j,n+1} (\\lambda_{j2} \\lambda_{j1}) (\\lambda_{n+1,2} \\lambda_{n+1,1}) = (\\lambda_{n+1,2} \\lambda_{n+1,1}) (\\lambda_{j2} \\lambda_{j1}) \\lambda_{j,n+1}.\n$$\nSince $\\lambda_{j1} \\lambda_{n+1,2} = \\lambda_{n+1,2} \\lambda_{j1}$ and $\\lambda_{n+1,1} \\lambda_{j2} = \\lambda_{j2} \\lambda_{n+1,1}$, rewrite the last relation in the form\n\\begin{equation} \\label{r32}\n\\lambda_{j,n+1} \\lambda_{j2} (\\lambda_{n+1,2} \\lambda_{j1}) \\lambda_{n+1,1} = \\lambda_{n+1,2} (\\lambda_{j2} \\lambda_{n+1,1}) \\lambda_{j1} \\lambda_{j,n+1}.\n\\end{equation}\nAs we note before the set of relation $R_{1,j,n+1}$ comes from relations of $VP_n$ and in particular, the relation\n$$\n\\lambda_{n+1,1} \\lambda_{j1} \\lambda_{j,n+1} = \\lambda_{j,n+1} \\lambda_{j1} \\lambda_{n+1,1}\n$$\nholds. Using this relation, rewrite (\\ref{r32}):\n$$\n\\lambda_{j,n+1} \\lambda_{j2} (\\lambda_{j,n+1} \\lambda_{j1} \\lambda_{n+1,1} = \\lambda_{n+1,2} \\lambda_{j2} \\lambda_{j,n+1} \\lambda_{j1} \\lambda_{n+1,1}.\n$$\nAfter cancelations we have\n$$\n\\lambda_{j,n+1} \\lambda_{j2} \\lambda_{j,n+1} = \\lambda_{j,n+1} \\lambda_{j2} \\lambda_{j,n+1}.\n$$\nIt is the firth relation from $R_{1,j,n+1}$.\n\n\nThe sixth relation in $R_{1,j-1,n}$ has the form\n$$\n\\lambda_{n,j-1} \\lambda_{n1} \\lambda_{j-1,1} = \\lambda_{j-1,1} \\lambda_{n1} \\lambda_{n,j-1}.\n$$\nActing by $s_0$ we get the relation\n$$\n\\lambda_{n+1,j} (\\lambda_{n+1,2} \\lambda_{n+1,1}) (\\lambda_{j2} \\lambda_{j1}) = (\\lambda_{j2} \\lambda_{j1}) (\\lambda_{n+1,2} \\lambda_{n+1,1}) \\lambda_{n+1,j}.\n$$\nUsing the commutativity relations $\\lambda_{n+1,1} \\lambda_{j2} = \\lambda_{j2} \\lambda_{n+1,1}$ and $\\lambda_{j1} \\lambda_{n+1,2} = \\lambda_{n+1,2} \\lambda_{j1}$ we have\n$$\n\\lambda_{n+1,j} \\lambda_{n+1,2} \\lambda_{j2} \\lambda_{n+1,1} \\lambda_{j1} = \\lambda_{j2} \\lambda_{n+1,2} (\\lambda_{j1} \\lambda_{n+1,1} \\lambda_{n+1,j}).\n$$\nUsing the relation\n$$\n\\lambda_{j1} \\lambda_{n+1,1} \\lambda_{n+1,j} = \\lambda_{n+1,j} \\lambda_{n+1,1} \\lambda_{j1},\n$$\n rewrite our relation in the form:\n$$\n\\lambda_{n+1,j} \\lambda_{n+1,2} \\lambda_{j2} \\lambda_{n+1,1} \\lambda_{j1} = \\lambda_{j2} \\lambda_{n+1,2} (\\lambda_{n+1,j} \\lambda_{n+1,1} \\lambda_{j1}).\n$$\nAfter cancelations we get\n$$\n\\lambda_{n+1,j} \\lambda_{n+1,2} \\lambda_{j2} = \\lambda_{j2} \\lambda_{n+1,2} \\lambda_{n+1,j}.\n$$\nIt is the sixth relation from $R_{2,j,n+1}$.\n\nHence, we have proven\n\n\\begin{lem}\nLet $n \\geq 4$. Acting on the relations $R_{1,j-1,n}$ of $VP_n$ by $s_0$ and using the relations, which we got in Lemma \\ref{lem1}, we get relations $R_{2,j,n+1}$ in $VP_{n+1}$.\n\\end{lem}\n\nNext, suppose that $i = 1$ in the set $\\{ i, j, n+1 \\}$. Since $n \\geq 4$ and we can not use Lemma \\ref{lem1} for the relations $R_{i,j,n+1}$, we see that it is possible only in the case $j = 3$, $n+1 = 5$. Hence we have to prove that the relations $R_{1,3,5}$ follow from relations $s_k(\\mathcal{R}^V(4))$ for some $k$.\n\nConsider relations $R_{1,2,4}$ in $VP_4$ and acting on them by $s_1$. The first relation in $R_{1,2,4}$ has the form\n$$\n\\lambda_{12} \\lambda_{14} \\lambda_{24} = \\lambda_{24} \\lambda_{14} \\lambda_{12}.\n$$\nActing on it by $s_1$ we get\n$$\n(\\lambda_{13} \\lambda_{12}) \\lambda_{15} (\\lambda_{25} \\lambda_{35}) = (\\lambda_{25} \\lambda_{35}) \\lambda_{15} (\\lambda_{13} \\lambda_{12}).\n$$\nNote that relations $R_{1,2,5}$ satisfy condition 3) in Lemma \\ref{lem1}. Using the first relation from this set:\n$$\n\\lambda_{12} \\lambda_{15} \\lambda_{25} = \\lambda_{25} \\lambda_{15} \\lambda_{12},\n$$\nwe get\n$$\n\\lambda_{13} (\\lambda_{25} \\lambda_{15} \\lambda_{12}) \\lambda_{35} = \\lambda_{25} \\lambda_{35} \\lambda_{15} \\lambda_{13} \\lambda_{12}.\n$$\nUsing the commutativity relations $\\lambda_{13} \\lambda_{25} = \\lambda_{25} \\lambda_{13}$ and $\\lambda_{12} \\lambda_{35} = \\lambda_{35} \\lambda_{12}$, we have\n$$\n(\\lambda_{25} \\lambda_{13}) \\lambda_{15} (\\lambda_{35} \\lambda_{12}) = \\lambda_{25} \\lambda_{35} \\lambda_{15} \\lambda_{13} \\lambda_{12}.\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{13} \\lambda_{15} \\lambda_{35} = \\lambda_{35} \\lambda_{15} \\lambda_{13}.\n$$\nThis is the first relation from $R_{1,3,5}$.\n\nThe second relation in $R_{1,2,4}$ has the form\n$$\n\\lambda_{21} \\lambda_{24} \\lambda_{14} = \\lambda_{14} \\lambda_{24} \\lambda_{21}.\n$$\nActing on it by $s_1$ we get\n$$\n(\\lambda_{21} \\lambda_{31}) (\\lambda_{25} \\lambda_{35}) \\lambda_{15} = \\lambda_{15} (\\lambda_{25} \\lambda_{35}) (\\lambda_{21} \\lambda_{31}).\n$$\nUsing the commutativity relation $\\lambda_{31} \\lambda_{25} = \\lambda_{25} \\lambda_{31}$ and $\\lambda_{35} \\lambda_{21} = \\lambda_{21} \\lambda_{35}$, we have\n$$\n\\lambda_{21} (\\lambda_{25} \\lambda_{31}) \\lambda_{35} \\lambda_{15} = \\lambda_{15} \\lambda_{25} (\\lambda_{21} \\lambda_{35}) \\lambda_{31}.\n$$\nBy Lemma \\ref{lem1} we have relation\n$$\n\\lambda_{15} \\lambda_{25} \\lambda_{21} = \\lambda_{21} \\lambda_{25} \\lambda_{15}.\n$$\nUsing it we get\n$$\n\\lambda_{21} \\lambda_{25} \\lambda_{31} \\lambda_{35} \\lambda_{15} = (\\lambda_{21} \\lambda_{25} \\lambda_{15}) \\lambda_{35} \\lambda_{31}.\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{31} \\lambda_{35} \\lambda_{15} = \\lambda_{15} \\lambda_{35} \\lambda_{31}.\n$$\nThis is the second relation from $R_{1,3,5}$.\n\nUsing the third relation in the set $R_{1,2,4}$:\n$$\n\\lambda_{14} \\lambda_{12} \\lambda_{42} = \\lambda_{42} \\lambda_{12} \\lambda_{14}\n$$\nand acting by $s_1$ we get\n$$\n\\lambda_{15} (\\lambda_{13} \\lambda_{12}) (\\lambda_{53} \\lambda_{52}) = (\\lambda_{53} \\lambda_{52}) (\\lambda_{13} \\lambda_{12}) \\lambda_{15}.\n$$\nUsing the commutativity relation $\\lambda_{12} \\lambda_{53} = \\lambda_{53} \\lambda_{12}$ and $\\lambda_{52} \\lambda_{13} = \\lambda_{13} \\lambda_{52}$, we have\n$$\n\\lambda_{15} \\lambda_{13} (\\lambda_{53} \\lambda_{12}) \\lambda_{52} = \\lambda_{53} (\\lambda_{13} \\lambda_{52}) \\lambda_{12} \\lambda_{15}.\n$$\nUsing the relation\n$$\n\\lambda_{52} \\lambda_{12} \\lambda_{15} = \\lambda_{15} \\lambda_{12} \\lambda_{52},\n$$\nwhich we have by Lemma \\ref{lem1} we get\n$$\n\\lambda_{15} \\lambda_{13} \\lambda_{53} \\lambda_{12} \\lambda_{52} = \\lambda_{53} \\lambda_{13} (\\lambda_{15} \\lambda_{12} \\lambda_{52}).\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{15} \\lambda_{13} \\lambda_{53} = \\lambda_{53} \\lambda_{13} \\lambda_{15}.\n$$\nThis is the third relation in $R_{1,3,5}$.\n\nThe forth relation in $R_{1,2,4}$ has the form\n$$\n\\lambda_{41} \\lambda_{42} \\lambda_{12} = \\lambda_{12} \\lambda_{42} \\lambda_{41}.\n$$\nActing on it by $s_1$ we get\n$$\n\\lambda_{51} (\\lambda_{53} \\lambda_{52}) (\\lambda_{13} \\lambda_{12}) = (\\lambda_{13} \\lambda_{12}) (\\lambda_{53} \\lambda_{52}) \\lambda_{51}.\n$$\nUsing the commutativity relation $\\lambda_{52} \\lambda_{13} = \\lambda_{13} \\lambda_{52}$ and $\\lambda_{12} \\lambda_{53} = \\lambda_{53} \\lambda_{12}$, we have\n$$\n\\lambda_{51} \\lambda_{53} (\\lambda_{13} \\lambda_{52}) \\lambda_{12} = \\lambda_{13} (\\lambda_{53} \\lambda_{12}) \\lambda_{52} \\lambda_{51}.\n$$\nBy Lemma \\ref{lem1} we have relation\n$$\n\\lambda_{12} \\lambda_{52} \\lambda_{51} = \\lambda_{51} \\lambda_{52} \\lambda_{12}.\n$$\nUsing it we get\n$$\n\\lambda_{51} \\lambda_{53} \\lambda_{13} \\lambda_{52} \\lambda_{12} = \\lambda_{13} \\lambda_{53} (\\lambda_{51} \\lambda_{52} \\lambda_{12}).\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{51} \\lambda_{53} \\lambda_{13} = \\lambda_{13} \\lambda_{53} \\lambda_{51}.\n$$\nThis is the forth relation in $R_{1,3,5}$.\n\n\nUsing the firth relation in the set $R_{1,2,4}$:\n$$\n\\lambda_{24} \\lambda_{21} \\lambda_{41} = \\lambda_{41} \\lambda_{21} \\lambda_{24}\n$$\nand acting by $s_1$ we get\n$$\n(\\lambda_{25} \\lambda_{35}) (\\lambda_{21} \\lambda_{31}) \\lambda_{51} = \\lambda_{51} (\\lambda_{21} \\lambda_{31}) (\\lambda_{25} \\lambda_{35}).\n$$\nUsing the commutativity relation $\\lambda_{35} \\lambda_{21} = \\lambda_{21} \\lambda_{35}$ and $\\lambda_{31} \\lambda_{25} = \\lambda_{25} \\lambda_{31}$, we have\n$$\n\\lambda_{25} (\\lambda_{21} \\lambda_{35}) \\lambda_{31} \\lambda_{51} = \\lambda_{51} \\lambda_{21} (\\lambda_{25} \\lambda_{31}) \\lambda_{35}.\n$$\nUsing the relation\n$$\n\\lambda_{51} \\lambda_{21} \\lambda_{25} = \\lambda_{25} \\lambda_{21} \\lambda_{51},\n$$\nwhich we have by Lemma \\ref{lem1} we get\n$$\n\\lambda_{25} \\lambda_{21} \\lambda_{35} \\lambda_{31} \\lambda_{51} = (\\lambda_{25} \\lambda_{21} \\lambda_{51}) \\lambda_{31} \\lambda_{35}.\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{35} \\lambda_{31} \\lambda_{51} = \\lambda_{51} \\lambda_{31} \\lambda_{35}.\n$$\nThis is the firth relation from $R_{1,3,5}$.\n\nThe sixth relation in $R_{1,2,4}$ has the form\n$$\n\\lambda_{42} \\lambda_{41} \\lambda_{21} = \\lambda_{21} \\lambda_{41} \\lambda_{42}.\n$$\nActing on it by $s_1$ we get\n$$\n(\\lambda_{53} \\lambda_{52}) \\lambda_{51} (\\lambda_{21} \\lambda_{31}) = (\\lambda_{21} \\lambda_{31}) \\lambda_{51} (\\lambda_{53} \\lambda_{52}).\n$$\nBy Lemma \\ref{lem1} we have relation\n$$\n\\lambda_{52} \\lambda_{51} \\lambda_{21} = \\lambda_{21} \\lambda_{51} \\lambda_{52},\n$$\nfrom which\n$$\n\\lambda_{53} (\\lambda_{21} \\lambda_{51} \\lambda_{52}) \\lambda_{31} = \\lambda_{21} \\lambda_{31} \\lambda_{51} \\lambda_{53} \\lambda_{52}.\n$$\nUsing the commutativity relation $\\lambda_{53} \\lambda_{21} = \\lambda_{21} \\lambda_{53}$ and $\\lambda_{52} \\lambda_{31} = \\lambda_{31} \\lambda_{52}$, we have\n$$\n(\\lambda_{21} \\lambda_{53}) \\lambda_{51} (\\lambda_{31} \\lambda_{52}) = \\lambda_{21} \\lambda_{31} \\lambda_{51} \\lambda_{53} \\lambda_{52}.\n$$\nAfter cancelation we arrive to the relation\n$$\n\\lambda_{53} \\lambda_{51} \\lambda_{31} = \\lambda_{31} \\lambda_{51} \\lambda_{53}.\n$$\nThis is the sixth relation from $R_{1,3,5}$.\n\n\n\\subsection{Simplicial group $T_*$} \\label{T}\n\nThe simplicial group $T_*$ was defined in the paper \\cite{BW}. In the same paper was proved that $T_3$ is generated by elements\n$$\na_{31},~~a_{22},~~a_{13},~~b_{31},~~b_{22},~~b_{13}\n$$\nand is defined by relations\n$$\n[a_{31}, a_{22}]^{c_{11}^k c_{21}^m} = [a_{31}, a_{13}]^{c_{11}^k c_{21}^m} = [a_{22}, a_{13}]^{c_{11}^k c_{21}^m} = 1,\n$$\n$$\n[b_{31}, b_{22}]^{c_{11}^k c_{21}^m} = [b_{31}, b_{13}]^{c_{11}^k c_{21}^m} = [b_{22}, b_{13}]^{c_{11}^k c_{21}^m} = 1,\n$$\n that can be written in the form\n$$\n[a_{31}, a_{22}^{c_{22}^{m} c_{31}^{-m}}] = [a_{31}, a_{13}^{c_{13}^{k} c_{22}^{m-k} c_{31}^{-m}}] = [a_{22}^{c_{22}^m c_{31}^{-m}}, a_{13}^{c_{13}^k c_{22}^{m-k} c_{31}^{-m}}] = 1,\n$$\n$$\n[b_{31}, b_{22}^{c_{22}^{m} c_{31}^{-m}}] = [b_{31}, b_{13}^{c_{13}^{k} c_{22}^{m-k} c_{31}^{-m}}] = [b_{22}^{c_{22}^m c_{31}^{-m}}, b_{13}^{c_{13}^k c_{22}^{m-k} c_{31}^{-m}}] = 1.\n$$\nwhere $k, m \\in \\mathbb{Z}$.\n\n\nIn the general case we will prove\n\n\\begin{thm}\\label{T_n}\nThe group $T_n$, $n \\geq 2$ is generated by elements\n$$\na_{i,n+1-i},~~b_{i,n+1-i},~~i = 1, 2, \\ldots n,\n$$\nand is defined by relations\n$$\n[a_{i,n+1-i}, a_{j,n+1-j}]^{c_{11}^{k_1} c_{21}^{k_2} \\ldots c_{n-1,1}^{k_{n-1}} },\n$$\n$$\n[b_{i,n+1-i}, b_{j,n+1-j}]^{c_{11}^{k_1} c_{21}^{k_2} \\ldots c_{n-1,1}^{k_{n-1}} },\n$$\nwhere $1 \\leq i \\not= j \\leq n$, $k_l \\in \\mathbb{Z}$.\n\\end{thm}\n\n\\section{$VP_n$ as a subgroup of $VB_{n+1}$} \\label{conj}\n\nIn the previous section we shown how it is possible to construct $VP_n$ from $VP_{n-1}$ using operation cabling. In this section we will show how it is possible to construct $VP_{n+1}$, using the action of the symmetric group $S_{n+1} = \\langle \\rho_1, \\rho_2, \\ldots, \\rho_{n_1} \\rangle$, which is a subgroup of the virtual braid group $VB_{n+1} = VP_{n+1} \\leftthreetimes S_{n+1}$. Recall that $S_{n+1}$ acts on the generators of $VP_{n+1}$ by the rule\n$$\n\\rho_k \\lambda_{ij} \\rho_k = \\lambda_{\\rho_k(i),\\rho_k(j)},~~~k = 1, 2, \\ldots, n-1.\n$$\n\nThe symmetric group $S_{n+1}$ ia s disjoint union of cosets by $S_n$:\n$$\nS_{n+1} = S_{n} e \\sqcup S_{n} \\rho_n \\sqcup S_{n} \\rho_n \\rho_{n-1} \\sqcup \\ldots \\sqcup S_{n} \\rho_n \\rho_{n-1} \\ldots \\rho_{1}.\n$$\n\nWe will denote $\\mathcal{X}_k$ the set of generators of $VP_k$, $k \\geq 2$, i.e.\n$$\n\\mathcal{X}_k = \\{ \\lambda_{ij} ~|~1 \\leq i \\not= j \\leq k \\};\n$$\n$\\mathcal{R}_k$ will denote the set of defining relations of $VP_k$. In particular, $\\mathcal{LR}_k$ will denote the set of long relations and $\\mathcal{CR}_k$ the set of commutativity relations. It is evident that\n$$\n\\mathcal{R}_k = \\mathcal{LR}_k \\cup \\mathcal{CR}_k.\n$$\nSince, $VP_3$ does not contain commutativity relations, then $\\mathcal{R}_3 = \\mathcal{LR}_3.$\n\nLet $k > 2$ and $1 \\leq i < j < l \\leq k$ be three distinct integer numbers. Denote by $\\mathcal{R}^{ijl}_k$ the following set of long defining relations from $\\mathcal{R}_k$:\n$$\n\\lambda_{ij} \\lambda_{il} \\lambda_{jl} = \\lambda_{jl} \\lambda_{il} \\lambda_{ij},~~~\\lambda_{ji} \\lambda_{jl} \\lambda_{il} = \\lambda_{il} \\lambda_{jl} \\lambda_{ji},\n$$\n$$\n\\lambda_{il} \\lambda_{ij} \\lambda_{lj} = \\lambda_{lj} \\lambda_{ij} \\lambda_{il},~~~\\lambda_{li} \\lambda_{lj} \\lambda_{ij} = \\lambda_{ij} \\lambda_{lj} \\lambda_{li},\n$$\n$$\n\\lambda_{jl} \\lambda_{ji} \\lambda_{li} = \\lambda_{li} \\lambda_{ji} \\lambda_{jl},~~~\\lambda_{lj} \\lambda_{li} \\lambda_{ji} = \\lambda_{ji} \\lambda_{li} \\lambda_{lj}.\n$$\nThen\n$$\n\\mathcal{LR}_k = \\bigsqcup_{1 \\leq i m.\n$$\nDenote\n$$\n\\mathcal{R}^{i,j,l,m}_k = \\{ \\lambda_{ij}^* \\lambda_{lm}^* = \\lambda_{lm}^* \\lambda_{ij}^* \\}\n$$\nthe set of four commutativity relation with fixed indexes, then\n$$\n\\mathcal{CR}_k = \\bigsqcup_{ i < j,~ l < m,~ j > m} \\mathcal{R}^{i,j,l,m}_k\n$$\nis the full set of the commutativity relations in $VP_k$\n\nTake the set of generators of $VP_3$:\n$$\n\\mathcal{X}_3 = \\{ \\lambda_{12}, \\lambda_{21}, \\lambda_{13}, \\lambda_{23}, \\lambda_{31}, \\lambda_{32} \\}\n$$\nand acting on it by coset representatives of $S_4$ by $S_3$ we get\n$$\n\\mathcal{X}_3^{\\rho_3} = \\{ \\lambda_{12}, \\lambda_{21}, \\lambda_{14}, \\lambda_{24}, \\lambda_{41}, \\lambda_{42} \\},\n$$\n$$\n\\mathcal{X}_3^{\\rho_3 \\rho_2} = \\{ \\lambda_{13}, \\lambda_{31}, \\lambda_{14}, \\lambda_{34}, \\lambda_{41}, \\lambda_{43} \\},\n$$\n$$\n\\mathcal{X}_3^{\\rho_3 \\rho_2 \\rho_1} = \\{ \\lambda_{23}, \\lambda_{32}, \\lambda_{24}, \\lambda_{34}, \\lambda_{42}, \\lambda_{43} \\}.\n$$\nWe see that\n$$\n\\mathcal{X}_4 = \\mathcal{X}_3 \\cup \\mathcal{X}_3^{\\rho_3} \\cup \\mathcal{X}_3^{\\rho_3 \\rho_2}.\n$$\nIn the general case we have the similar result\n\n\\begin{prop}\nFor $n \\geq 3$ the following equality holds\n$$\n\\mathcal{X}_{n+1} = \\mathcal{X}_n \\cup \\mathcal{X}_n^{\\rho_n} \\cup \\mathcal{X}_n^{\\rho_n \\rho_{n-1}}.\n$$\n\\end{prop}\n\n\\begin{proof}\nAny generator in $\\mathcal{X}_{n+1} \\setminus \\mathcal{X}_{n}$ has the form $\\lambda_{i,n+1}^*$ for some $i$, $1\\leq i \\leq n$. Take the generator $\\lambda_{1n}^* \\in \\mathcal{X}_{n}$ and acting on it by conjugation of $\\rho_n$:\n$$\n\\left( \\lambda_{1n}^* \\right)^{\\rho_n} = \\lambda_{1,n+1}^*,~~~\\left( \\lambda_{2n}^* \\right)^{\\rho_n} = \\lambda_{2,n+1}^*, \\ldots, \\left( \\lambda_{n-1,n}^* \\right)^{\\rho_n} = \\lambda_{n-1,n+1}^*.\n$$\nTo find the last generator $\\lambda_{n,n+1}^*$, take the generator $\\lambda_{n-1,n}^*$ and acting of conjugation by $\\rho_n \\rho_{n-1}$ we get\n$$\n\\left( \\lambda_{n-1,n}^* \\right)^{\\rho_n \\rho_{n-1}} = \\left( \\lambda_{n-1,n+1}^* \\right)^{\\rho_{n-1}} = \\lambda_{n,n+1}^*.\n$$\n\n\\end{proof}\n\nTo find the set of defining relations in $\\mathcal{R}_4$, take the defining relations of $\\mathcal{R}_3 = \\mathcal{R}^{123}$ and acting by coset representatives we get\n$$\n\\mathcal{R}_3^{\\rho_3} = \\mathcal{R}_4^{124},~~~\\mathcal{R}_3^{\\rho_3 \\rho_2} = \\mathcal{R}_4^{134},~~~\\mathcal{R}_3^{\\rho_3 \\rho_2 \\rho_1} = \\mathcal{R}_4^{234}.\n$$\nSince\n$$\n\\mathcal{LR}_4 = \\mathcal{R}_4^{123} \\sqcup \\mathcal{R}_4^{124} \\sqcup \\mathcal{R}_4^{134} \\sqcup \\mathcal{R}_4^{234} ~~\\mbox{and}~~\\mathcal{R}_4^{123} = \\mathcal{R}_3^{123} = \\mathcal{R}_3,\n$$\nwe get\n$$\n\\mathcal{LR}_4 = \\mathcal{R}_3 \\sqcup \\mathcal{R}_3^{\\rho_3} \\sqcup \\mathcal{R}_3^{\\rho_3 \\rho_2} \\sqcup \\mathcal{R}_3^{\\rho_3 \\rho_2 \\rho_1}.\n$$\nIn $VP_3$ we don't have commutativity relations hence, we have\n\n\\begin{prop}\n$$\n\\mathcal{R}_4 = \\mathcal{R}_3 \\sqcup \\mathcal{R}_3^{\\rho_3} \\sqcup \\mathcal{R}_3^{\\rho_3 \\rho_2} \\sqcup \\mathcal{R}_3^{\\rho_3 \\rho_2 \\rho_1} \\sqcup \\mathcal{CR}_4.\n$$\n\\end{prop}\n\nIn the general case we can prove\n\n\\begin{thm}\nFor $n \\geq 4$ we have\n$$\n\\mathcal{R}_{n+1} = \\mathcal{R}_n \\sqcup \\mathcal{R}_n^{\\rho_n} \\sqcup \\mathcal{R}_n^{\\rho_n \\rho_{n-1}} \\sqcup \\ldots \\sqcup \\mathcal{R}_n^{\\rho_n \\rho_{n-1} \\ldots \\rho_1}.\n$$\n\\end{thm}\n\n\\begin{proof}\nConsider the set of long relations $\\mathcal{R}_{n+1}^{i,j,n+1}$ which does not lie in $\\mathcal{R}_{n}$. If $j\\not= n$, then the relations $\\mathcal{R}_{n}^{i,j,n}$ lie in $\\mathcal{R}_{n}$ and acting by $\\rho_n$ we get\n$$\n\\left( \\mathcal{R}_{n}^{i,j,n} \\right)^{\\rho_n} = \\mathcal{R}_{n+1}^{i,j,n+1}.\n$$\nIf $j=n$, but $i\\not= n-1$, then\n$$\n\\left( \\mathcal{R}_{n}^{i,n-1,n}\\right)^{\\rho_n \\rho_{n-1}} = \\left( \\mathcal{R}_{n+1}^{i,n-1,n+1} \\right)^{\\rho_{n-1}} = \\mathcal{R}_{n+1}^{i,n,n+1}.\n$$\nIf $j=n$, $i = n-1$, then\n$$\n\\left( \\mathcal{R}_{n}^{n-2,n-1,n} \\right)^{\\rho_n \\rho_{n-1} \\rho_{n-2}} = \\left( \\mathcal{R}_{n+1}^{n-2,n-1,n+1} \\right)^{\\rho_{n-1} \\rho_{n-2}} = \\left( \\mathcal{R}_{n+1}^{n-2,n,n+1} \\right)^{\\rho_{n-2}} = \\mathcal{R}_{n+1}^{n-1,n,n+1}.\n$$\n\nConsider a set of commutativity relations\n$$\n\\mathcal{R}^{i,n+1,l,m}_{n+1} \\in \\mathcal{R}_{n+1} \\setminus \\mathcal{R}_{n}.\n$$\nWe will assume that $i < l < m$. Proofs for other cases is similar.\n\nIf $m \\not= n$, then\n$$\n\\left( \\mathcal{R}^{i,n,l,m}_{n} \\right)^{\\rho_n} = \\mathcal{R}^{i,n+1,l,m}_{n+1}.\n$$\nIf $m= n$, but $l \\not= n-1$, then\n$$\n\\left( \\mathcal{R}^{i,n,l,n-1}_{n} \\right)^{\\rho_n \\rho_{n-1}} = \\left( \\mathcal{R}^{i,n+1,l,n-1}_{n+1} \\right)^{\\rho_{n-1}} = \\mathcal{R}^{i,n+1,l,n}_{n+1}.\n$$\nIf $m= n$, $l = n-1$, but $i \\not= n-2$, then\n$$\n\\left( \\mathcal{R}^{i,n,n-2,n-1}_{n} \\right)^{\\rho_n \\rho_{n-1} \\rho_{n-2}} = \\left( \\mathcal{R}^{i,n+1,n-2,n-1}_{n+1} \\right)^{\\rho_{n-1} \\rho_{n-2}} = \\left( \\mathcal{R}^{i,n+1,n-2,n}_{n} \\right)^{ \\rho_{n-2}} = \\mathcal{R}^{i,n+1,n-1,n}_{n+1}.\n$$\nIf $m= n$, $l = n-1$ and $i = n-2$, then\n$$\n\\left( \\mathcal{R}^{n-3,n,n-2,n-1}_{n} \\right)^{\\rho_n \\rho_{n-1} \\rho_{n-2} \\rho_{n-3}} = \\left( \\mathcal{R}^{n-3,n+1,n-2,n-1}_{n+1} \\right)^{\\rho_{n-1} \\rho_{n-2} \\rho_{n-3}} =\n$$\n$$\n= \\left( \\mathcal{R}^{n-3,n+1,n-2,n}_{n} \\right)^{ \\rho_{n-2} \\rho_{n-3}} = \\left( \\mathcal{R}^{n-3,n+1,n-1,n}_{n+1}\\right)^{\\rho_{n-3}} = \\mathcal{R}^{n-2,n+1,n-1,n}_{n+1}.\n$$\n\\end{proof}\n\n\n\n\\medskip\n\n\n\\section{Cabling of the Artin pure braid group} \\label{s41}\n\n\n\nIn the paper \\cite{CW} was defined a cabling on the the set of pure braid groups $\\{P_n\\}_{n=2,3,\\ldots}$. It was proven that in fact that all generators of $P_n$ come from the unique generator $A_{12}$ of $U_2$, using cabling.\nIn this section we find a set of defining relation of $P_4$ in these generators.\n\n\nIn the previous section we define elements $c_{ij} = b_{ij} a_{ij}$. Put\n$$\nT_k^c = \\langle c_{ij}~|~i+j = k+1 \\rangle,~~k = 1, 2, \\ldots, n-1.\n$$\nAny group $T_k^c$ for $k>1$ is getting from $T_{k-1}^c$ using cabling, i.e.\n$$\nT_k^c = \\langle s_0(T_{k-1}^c), s_1(T_{k-1}^c), \\ldots, s_{k-2}(T_{k-1}^c) \\rangle.\n$$\nThen $P_n = \\langle T_1^c, T_2^c, \\ldots, T_{n-1}^c \\rangle$.\n\n\n\nIn the paper \\cite{BW} was found set of defining relations of $P_4$ in the cabled generators $c_{ij}$, more precisely was proven\n\n\\begin{prop} The group $P_4$ is generated by elements\n$$\nc_{11},~~ c_{21},~~ c_{12},~~ c_{31},~~c_{22},~~ c_{13}\n$$\nand is defined by relations (where $\\varepsilon = \\pm 1$):\n$$\nc_{21}^{c_{11}^{\\varepsilon}} = c_{21},~~~c_{12}^{c_{11}^{\\varepsilon}} = c_{12}^{c_{21}^{-\\varepsilon}},~~~c_{31}^{c_{11}^{\\varepsilon}} = c_{31},~~~c_{22}^{c_{11}^{\\varepsilon}} = c_{22},~~~c_{13}^{c_{11}^{\\varepsilon}} = c_{13}^{c_{22}^{-\\varepsilon}},\n$$\n$$\nc_{31}^{c_{21}^{\\varepsilon}} = c_{31},~~~c_{22}^{c_{21}^{\\varepsilon}} = c_{22}^{c_{31}^{-\\varepsilon}},~~~c_{13}^{c_{21}^{\\varepsilon}} = c_{13}^{c_{22}^{\\varepsilon} c_{31}^{-\\varepsilon}},\n$$\n$$\nc_{31}^{c_{12}^{\\varepsilon}} = c_{31},~~~c_{13}^{c_{12}^{\\varepsilon}} = c_{13}^{c_{31}^{-\\varepsilon}}.\n$$\n$$\nc_{22}^{c_{12}^{-1}} = [c_{31}, c_{13}^{-1}] \\, [c_{13}^{-1}, c_{22}] \\, c_{22} \\, [c_{21}^2, c_{12}^{-1}] = c_{13}^{c_{31}} c_{13}^{-c_{22}} c_{22} [c_{21}^2, c_{12}^{-1}],\n$$\n$$\nc_{22}^{c_{12}} = [c_{12}, c_{21}^{-2}] \\, c_{22} \\, [c_{22}^{-3}, c_{13}] \\, [c_{13}, c_{31}^{-1}] = [c_{12}, c_{21}^{-2}] \\, c_{13}^{-c_{22}^{-2}} \\, c_{22} \\, c_{13}^{c_{31}^{-1}} .\n$$\n\n\\end{prop}\n\n\\medskip\n\n Define the following subgroups of $P_4$:\n$$\nV_1 = \\langle c_{11}, c_{12}, c_{13} \\rangle,~~~V_2 = \\langle c_{21}, c_{22} \\rangle,~~~V_3 = \\langle c_{31} \\rangle.\n$$\nThen\n\n\\begin{thm}\n$$\nP_4 = V_1 \\leftthreetimes (V_2 \\leftthreetimes V_3).\n$$\n\\end{thm}\n\n\\begin{proof}\nAt first prove that $\\langle V_2, V_3 \\rangle = V_2 \\leftthreetimes V_3$. Indeed, this group is defined by relations.\n$$\n[c_{31}, c_{21}] = 1,~~c_{22}^{c_{21}} = c_{22}^{c_{31}^{-1}}.\n$$\nSince the first relation we can write in the form\n$$\nc_{21}^{c_{31}} = c_{21},\n$$\nwe have the need decomposition.\n\nFrom the defining relations of $P_4$ find the following formulas of conjugation by $c_{31}$:\n$$\nc_{11}^{c_{31}} = c_{11},~~~c_{12}^{c_{31}} = c_{12 },~~~c_{13}^{c_{31}} = c_{13}^{c_{12}^{-1}}.\n$$\nHence\n$$\nP_4 = \\langle V_1, V_2 \\rangle \\leftthreetimes V_3.\n$$\n\nFind the formulas of conjugations by $c_{21}$:\n$$\nc_{11}^{c_{21}} = c_{11},~~~c_{12}^{c_{21}} = c_{12}^{c_{11}^{-1}},~~~c_{13}^{c_{21}} = c_{13}^{c_{12} c_{11}^{-1}}.\n$$\nAlso we have two formulas of conjugation by $c_{22}$:\n$$\nc_{11}^{c_{22}} = c_{11},~~~c_{13}^{c_{22}} = c_{13}^{c_{11}^{-1}}.\n$$\nTo finish the proof we need to find a formula for the conjugation $c_{12}^{c_{22}}$ and $c_{12}^{c_{22}^{-1}}$.\n\nIn the proof of the previous theorem we have found relation:\n$$\nc_{21} c_{22}^{-1} c_{13} c_{12}^{-1} = c_{21}^{-1} c_{12}^{-1} c_{21}^{2} c_{22}^{-1} (c_{22}^{-1} c_{13} c_{22}).\n$$\nMultiply both sides on $c_{21}^{-1}$ to the left and using relation\n$$\nc_{22}^{-1} c_{13} c_{22} = c_{11} c_{13} c_{11}^{-1},\n$$\nwe get\n$$\nc_{22}^{-1} c_{13} c_{12}^{-1} = (c_{21}^{-2} c_{12}^{-1} c_{21}^{2}) (c_{11} c_{13} c_{11}^{-1})^{c_{22}} c_{22}^{-1}.\n$$\nUsing the conjugation formulas:\n$$\nc_{21}^{-2} c_{12}^{-1} c_{21}^{2} = c_{11}^{2} c_{12}^{-1} c_{11}^{-2},~~~(c_{11} c_{13} c_{11}^{-1})^{c_{22}} = c_{11}^2 c_{13} c_{11}^{-2},\n$$\nwe get\n$$\n(c_{13} c_{12}^{-1})^{c_{22}} = c_{11}^{2} c_{12}^{-1} c_{13} c_{11}^{-2}.\n$$\nUsing the conjugation formula:\n$$\nc_{13}^{c_{22}} = c_{13}^{c_{11}^{-1}}\n$$\nwe have\n$$\nc_{13} c_{11}^{-1} c_{12}^{-c_{22}} = c_{11} c_{12}^{-1} c_{13} c_{11}^{-2}.\n$$\nFrom this relation we get the need formula:\n$$\nc_{12}^{c_{22}} = c_{11}^2 c_{13}^{-1} c_{12} c_{11}^{-1} c_{13} c_{11}^{-1}.\n$$\n\nConjugating both sides by $c_{22}^{-1}$ we find\n$$\nc_{12}^{c_{22}^{-1}} = c_{11}^{-1} c_{13} c_{11}^{-1} c_{12} c_{13}^{-1} c_{11}^{2}.\n$$\n\\end{proof}\n\nIn this theorem we used full set of defining relations for $P_4$. Let us consider the group $P_3$. It has the following presentation\n$$\nP_3 = \\langle c_{11}, c_{21}, c_{12}~|~c_{11}^{c_{21}} = c_{11},~~c_{12}^{c_{21}} = c_{12}^{c_{11}^{-1}} \\rangle.\n$$\nUsing degeneracy maps $s_0, s_1, s_2$, we construct the following subgroups of $P_4$:\n$$\ns_0(P_3) = \\langle c_{21}, c_{31}, c_{22}~|~c_{21}^{c_{31}} = c_{21},~~c_{22}^{c_{31}} = c_{22}^{c_{21}^{-1}} \\rangle,\n$$\n$$\ns_1(P_3) = \\langle c_{12}, c_{31}, c_{13}~|~c_{12}^{c_{31}} = c_{12},~~c_{13}^{c_{31}} = c_{13}^{c_{12}^{-1}} \\rangle,\n$$\n$$\ns_2(P_3) = \\langle c_{11}, c_{22}, c_{13}~|~c_{11}^{c_{22}} = c_{11},~~c_{13}^{c_{22}} = c_{13}^{c_{11}^{-1}} \\rangle.\n$$\n\n\nFrom the list of relations in $P_3$, $s_i(P_3)$, $i = 0, 1, 2$, we see that it is not the full list of relations for $P_4$. To have a full list we can add the relations\n$$\nc_{11}^{c_{31}} = c_{11},~~c_{13}^{c_{21}} = c_{13}^{c_{12} c_{11}^{-1}},~~c_{12}^{c_{22}} = c_{11}^2 \\, c_{13}^{-1} \\, c_{12} \\, c_{11}^{-1} \\, c_{13} \\, c_{11}^{-1}.\n$$\n\n\nBut us follows from Theorem \\ref{lift}, for $n \\geq 5$ the full list of relations for $P_n$ comes from relations of $P_{n-1}$, $s_i(P_{n-1})$, $i = 0, 1, \\ldots, n-2$. Using induction by $n$ we can find relations of $P_n$. We get the following relations:\n\n-- conjugations by $c_{n-1,1}$\n\n$$\nc_{n-k,k}^{c_{n-1,1}} = c_{n-k,k}^{c_{n-k,k-1}^{-1}},~~~k = 2, 3, \\ldots, n-1;~~~c_{ij}^{c_{n-1,1}} = c_{ij}~~~\\mbox{if}~i+j < n;\n$$\n\n-- conjugations by $c_{n-2,2}$\n\n$$\nc_{n-k,k}^{c_{n-2,2}} = c_{n-k,k}^{c_{n-k,k-2}^{-1}},~~~k = 2, 3, \\ldots, n-1;~~~c_{ij}^{c_{n-2,2}} = c_{11}^2 \\, c_{13}^{-1} \\, c_{ij} \\, c_{11}^{-1} \\, c_{13} \\, c_{11}^{-1},~~~i+j < n;\n$$\n$$\nc_{lm}^{c_{n-2,}} = c_{lm}~~~\\mbox{in all other cases};\n$$\n\n\nIn the general case we prove\n\n\\begin{thm}\nFor $n \\geq 3$ the pure braid group $P_n$ is the semi-direct product of free groups:\n$$\nP_n = V_1 \\leftthreetimes (V_{2} \\leftthreetimes ( \\ldots ( V_{n-2} \\leftthreetimes V_{n-1})\\ldots)),\n$$\nwhere\n$$\nV_{n-1} = \\langle c_{n-1,1} \\rangle,\n$$\n$$\nV_{n-2} = \\langle c_{c_{n-2,1}, n-2,2} \\rangle,\n$$\n$$\n......................................\n$$\n$$\nV_{1} = \\langle c_{11}, c_{12}, \\ldots, c_{1,n-1} \\rangle.\n$$\n\\end{thm}\n\n\\begin{proof}\nThe theorem is true for $n=4$.\nWe prove that $P_n = V_1 \\leftthreetimes P_{n-1}$ for $n > 4$. By the lifting theorem the set of defining relations for $P_n$ come from the set of defining relations for $P_{n-1}$ by degeneracy maps. Using this fact let us prove that $V_1$ is normal in $P_n$.\n\\end{proof}\n\n\n\n\\section{Directions for further research} \\label{fin}\n\nWe know some generalizations oh the Artin braid group $B_n$, for example, welded braid group, singular braid groups and others (see \\cite{B}). In these groups it is possible to define pure subgroups. It is interesting to study presentations of these subgroups in cabled generators, define analogs of simplicial group $T_*$ and find its homotopy type.\n\nFor example, the welded braid group $WB_n$ contains the group of basis conjugating automorphisms $Cb_n$.\n\n\\begin{question}\nThe group of basic conjugating automorphisms $Cb_2$ is generated by two automorphisms $\\varepsilon_{21}$ and $\\varepsilon_{12}$ which generate a free group of rank 2. Using operation cabling find a presentation of $Cb_n$ in the cable generators.\n\\end{question}\n\n\\begin{question}\nLet $\\varphi : VP_n \\to Cb_n$ be a homomorphism which sends $\\lambda_{ij}$ to $\\varepsilon_{ij}$. Is it true that $T_{n-1}$ is isomorphic to its image $\\varphi(T_{n-1})$?\n\\end{question}\n\nWe know Artin and Gassner representations of $P_n$ (see \\cite[Chapter 3]{Bir}).\n\n\\begin{question}\nFind analogs of Artin and Gassner representations of $P_n$, using decomposition from Section \\ref{s41}. Are they equivalent to the classical representations?\n\\end{question}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Q-balls in the false vacuum}\n\nLet us consider a complex scalar field $\\varphi$ with a potential $U(\\varphi)$ \nthat has a local minimum at $\\varphi=0$ and is invariant under a U(1) symmetry\n$\\varphi \\rightarrow \\exp(i\\theta) \\varphi$. We also require that $U(\\varphi)$ admit\nQ-balls, that is \\cite{coleman1} \n\n\\begin{equation}\nU(\\varphi) \\left\/ |\\varphi|^2 \\right. = {\\rm min},\n\\ \\ {\\rm for} \\ \n\\varphi=\\varphi_0 \\neq 0. \n\\label{condmin}\n\\end{equation}\nThen a Q-ball solution \\cite{coleman1} of the form\n$\\varphi(x,t)=e^{i\\omega t} \\bar{\\varphi}(x)$, where $\\bar{\\varphi(x)}$ is real and\nindependent of time, minimizes energy for a fixed charge\n\n\n\\begin{equation}\nQ= \\frac{1}{2i} \\int \\varphi(x,t)^* \\stackrel{\\leftrightarrow}{\\partial}_t \n\\varphi(x,t) \\, d^3x \\ = \\omega \\int \\bar{\\varphi}^2(x) \\ d^3x.\n\\label{Q}\n\\end{equation}\n\nThe energy (mass) of the Q-ball can be written as \n\n\\begin{equation}\nE(Q) = \\frac{Q^2}{ 2 \\int \\bar{\\varphi}^2 d^3x} \\ + \\ \nT \\ + \\ V, \n\\label{E} \n\\end{equation}\nwhere $T= \\int \\frac{1}{2} (\\nabla \\bar{\\varphi})^2 \\ d^3x $\nis the gradient energy, and $V=\\int U(\\bar{\\varphi}) d^3x $ is the potential\nenergy of the Q-ball. \n\nA large Q-ball is a spherical object, inside which the field is close to \n$\\varphi=\\varphi_0$ defined in equation (\\ref{condmin}), while $\\varphi=0$ outside. \nWe now assume that $U(\\varphi_0)<0$, so that $V<0$ for a large enough Q-ball. \n\nLet us examine the energy of a fixed-charge field configuration as a\nfunction of its size using a one-parameter family of the test functions \nobtained from the Q-ball solution $\\bar{\\varphi}$ by expanding (contracting) it\nby a factor $\\alpha$: \n \n\\begin{equation}\n\\varphi_\\alpha(x) = \\bar{\\varphi}(\\alpha x)\n\\label{alpha}\n\\end{equation}\nThe corresponding energy is \n\n\\begin{equation}\nE_\\alpha (Q) = \\frac{1}{\\alpha^3} \\, \n\\frac{Q^2}{ 2 \\int \\bar{\\varphi}^2 d^3x} \\ + \\ \n\\alpha \\, T + \\alpha^3 \\, V, \n\\label{Ealpha} \n\\end{equation}\nwhere $T>0$, but $V<0$. By definition, $E_\\alpha$ has a minimum at\n$\\alpha=1$. Therefore, $(d E_\\alpha\/d\\alpha)|_{\\alpha=1} =0$. However, \nin general the equation $(d E_\\alpha\/d\\alpha)=0$ can have two roots, the \none corresponding to a minimum ($\\alpha_1=1$), and the \none that corresponds to a maximum ($\\alpha_2 > 1$). \n\nThe $\\alpha=1$ solution, $\\bar{\\varphi}(x)$, is the Q-ball and is a \n(local) minimum of energy. The $\\alpha=\\alpha_2$ solution is unstable. \nIt represents a kind of a Q-bounce, a critical bubble of the true vacuum, \nsuch that $\\varphi_\\alpha(x)$ will expand indefinitely for $\\alpha>\\alpha_2$,\nor will contract to a Q-ball for $\\alpha<\\alpha_2$.\n\n\\begin{figure}\n\\setlength{\\epsfxsize}{3.3in}\n\\centerline{\\epsfbox{figure_pt1.eps}}\n\\caption{\nThe energy of a fixed-charge solution in the false vacuum as a function of\nits size. \n}\n\\label{fig1}\n\\end{figure}\n\n\n\nIf $Q$ is small, $E_\\alpha(Q)$ in equation (\\ref{Ealpha}) has both a\nminimum (Q-ball) and a maximum (Q-bounce) as shown in Fig. 1. \nHowever, as the charge $Q$ approaches \na critical value $Q_c$, the minimum and the maximum come closer together \nand coincide for $Q=Q_c$. At this point, there is only one non-trivial\nfield configuration. It is unstable. \nAfter a critical $Q_c$-ball is formed, it will expand filling the space\nwith the true vacuum, much like a critical bubble in the event of\ntunneling. The value of the critical charge is given by \n\n\\begin{equation}\n\\left. \\frac{d E_\\alpha}{d\\alpha} \\right |_{\\alpha=1} = \n\\left. \\frac{d^2 E_\\alpha}{d\\alpha^2} \\right |_{\\alpha=1} = 0,\n\\label{dE2da}\n\\end{equation}\nand is defined implicitly by the relation \n\n\\begin{equation}\nQ_c=\n\\left [ -\\int U[\\bar{\\varphi}(x)] d^3x \n\\right ]\n\\left [ \\int \\bar{\\varphi}^2(x) d^3x \n\\right ].\n\\label{Qc_imp}\n\\end{equation}\nEquation (\\ref{Qc_imp}) is exact but implicit since the solution\n$\\bar{\\varphi}(x)$ is itself a function of $Q_c$. Through a virial theorem\n\\cite{ak_qb}, it also implies that, for a critical charge Q-ball, the\nidentity holds: $2T=-9V$. \n\nTo obtain an explicit formula for the critical charge $Q_c$, we will use\nthe thin-wall approximation \\cite{coleman1}, which is valid in the limit of\nlarge $Q$. (In what follows we will consider cases with $Q_c \\gg 1$ only.) \nIn the thin-wall limit, \na Q-ball is approximated by a spherical bubble of radius $R$ with \n$\\varphi(r) =\\varphi_0$ inside, for $r \\equiv \\sqrt{x_i x^i} R$. The thin-wall approximation is good if \n$|U(\\varphi_0)| \\ll (S_1)^{4\/3}$, where $S_1 = \\int_0^{\\varphi_0}\\sqrt{2 \n\\hat{U}_\\omega(\\varphi)} d \\varphi$, $\\hat{U}_\\omega(\\varphi)= U(\\varphi)-(\\omega^2\/2) \n\\varphi^2$, and $\\omega = Q\/(\\frac{4}{3} \\pi R^3 \\varphi_0^2)$. A large charge\nrequires a small value of $\\omega$ \\cite{ak_qb}, so that $U(\\varphi)\\approx\n\\hat{U}_\\omega (\\varphi)$ for large $Q$ in the false vacuum\\footnote{\nThis is not the case in the true vacuum, where $\\omega \\rightarrow\n\\omega_0>0$ for $Q \\rightarrow \\infty$ \\cite{coleman1,ak_qb}. In the false\nvacuum $\\omega_0=0$. Therefore, for large $Q$ one can approximate $S_1$ by\nits value at $\\omega=0$.}. \n\nIn the thin-wall limit, the Q-ball energy is \n\n\\begin{equation}\nE(Q) = - |U(\\varphi_0)| \\left (\\frac{4}{3} \\pi R^3 \\right ) + 4\\pi R^2 S_1 +\n\\frac{Q^2}{2 \\varphi_0^2 \\, (4\/3) \\pi R^3}.\n\\label{ER}\n\\end{equation}\nThe charge $Q=Q_c$ when $dE\/dR = 0$ and $d^2E\/dR^2 = 0$ simultaneously. \nFrom equation (\\ref{ER}), \n\n\\begin{equation}\nQ_c= \\frac{100 \\pi \\sqrt{10}}{81} \\, \\frac{\\varphi_0 S_1^3}{U_0^{5\/2}},\n\\label{Qc}\n\\end{equation}\nwhere $U_0 \\equiv -U(\\varphi_0) > 0$. \n\nOne can ask how large should the critical charge $Q_c$ be to destabilize\na false vacuum which is otherwise stable on the cosmological time\nscale. We will consider two cases: (a)~a metastable false vacuum at zero\ntemperature with a lifetime that exceeds the age of the Universe, and \n(b)~a finite-temperature case: a false vacuum at $T>0$, whose decay\nwidth is small in comparison to the rate of expansion of the Universe. \nAt zero temperature, the decay probability of the false vacuum per unit \nfour-volume is \n\n\\begin{equation}\n\\Gamma\/{\\sf V} = A \\, e^{-B}, \n\\label{gamma}\n\\end{equation}\nwhere $B$ is the Euclidean action of the bounce \\cite{tunn}. \n\nSuppose the Universe rests in the false vacuum whose lifetime exceeds \n$t_0\\sim 10^{10}$ years. This implies that $\\Gamma\/{\\sf V} \\ll 1$ for \n${\\sf V} \\sim t_0^4$. Taking the pre-exponential factor to be of order\n$A\\sim (100 \\, {\\rm GeV})^4$, one obtains the constraint \n$B \\stackrel{>}{_{\\scriptstyle \\sim}} 400$ for a false vacuum to \nbe considered stable. \n\nOn the other hand, in the thin-wall approximation, the action of the bounce\nassociated with tunneling into a global minimum at $\\varphi=\\varphi_0$ is\n\\cite{tunn} \n\n\\begin{equation}\nB= \\frac{27 \\pi^2}{2} \\, \\frac{S_1^4}{U_0^{3}}. \n\\label{B}\n\\end{equation}\nFrom equations (\\ref{Qc}) and (\\ref{B}), \n\n\\begin{equation}\nQ_c= \\frac{200}{729} \\sqrt{\\frac{5}{\\pi}} \\, \\left ( \\frac{2 B^3}{3} \n\\right )^{1\/4} \n\\frac{\\varphi_0 }{U_0^{1\/4}}\n\\label{QcB}\n\\end{equation}\n\nFor $B\\sim 400$, $Q_c \\sim 28 (\\varphi_0\/U_0^{1\/4})$ is sufficient to\ndestabilize the vacuum. Similarly, for the first-order phase transition at\nfinite temperature \\cite{linde}, \n\n\\begin{equation}\n\\Gamma\/{\\sf V} = A(T) \\, e^{-S_3\/T}, \n\\label{gammaT}\n\\end{equation}\nwhere $S_3$ is the tree-dimensional bounce action, for which the thin-wall\napproximation yields \\cite{linde} $S_3=16 \\pi S_1^3\/(3 U_0^2(T))$. The\ncritical charge can once again be re-expressed in terms of the bounce\naction and the (now temperature-dependent) depth of the true vacuum:\n\n\\begin{equation}\nQ_c=\\frac{25}{54} \\sqrt{\\frac{5}{2}} \\, \\frac{\\varphi_0 S_3}{U_0^{1\/2}(T)}\n\\label{QcT}\n\\end{equation}\nTunneling rate is negligible in comparison to the expansion rate of the\nUniverse at the electroweak scale temperatures if $S_3\/T \n\\stackrel{>}{_{\\scriptstyle \\sim}} 200$. A Q-ball with a charge $Q> Q_c \n\\sim 146 \\varphi_0 T\/U_0^{1\/2}(T)$ will facilitate an otherwise impossible \nphase transition in such a system. \n\nIn the next section, we discuss the conditions under which Q-balls of\ncritical size can be produced in thermal equilibrium via the accretion of\ncharge. The result is a rapid phase transition that proceeds via\nnucleation of charged critical bubbles of the true vacuum. \n\n\n\\section{Synthesis of critical Q-balls}\n\nSolitosynthesis \\cite{foga,gk} of large Q-balls in (approximate) thermal\nequilibrium is possible as long as the chain of requisite reactions leading\nfrom small to large solitons is not hampered by freeze-out. \nFirst, we require that the equilibrium number density of the $\\varphi$\nparticles is maintained via some microscopic processes, \n\n\\begin{equation}\n\\varphi \\bar{\\varphi} \\longleftrightarrow {\\rm light \\ particles}, \n\\label{reaction}\n\\end{equation}\nwhich are sufficiently fast in comparison to the expansion rate of the\nUniverse. \n\nSecond, the following chain of reactions is essential for the\nsolitosynthesis: \n\n\\begin{eqnarray}\n\\varphi + \\varphi & \\longleftrightarrow & \\Phi(2) \\nonumber \\\\ \n{\\rm a \\ few} \\ \\varphi'{\\rm s} & \\longleftrightarrow & \\Phi({\\rm few})\n\\label{firststep} \\\\\n\\nonumber \\\\\n& \\cdot \\ \\cdot \\ \\cdot & \\nonumber \\\\\n\\Phi(Q) + \\varphi & \\longleftrightarrow & \\Phi(Q+1) \\label{nextstep} \\\\\n\\nonumber \\\\\n& \\cdot \\ \\cdot \\ \\cdot , & \\nonumber \n\\end{eqnarray}\nwhere $\\Phi(Q)$ denotes a soliton with the charge $Q$. \n\nIt is crucial that, unlike some other non-topological solitons, \nclassically stable Q-balls exist for very small (integer) \ncharges $Q \\ge 1$ \\cite{ak_qb}. If the minimal charge of a soliton \n$Q_{min} \\gg 1$, then the chain of solitosynthesis reactions would have to \nstart from a process $\\varphi+\\varphi+...+\\varphi \\leftrightarrow \\Phi(Q_{min})$ \nthat is increasingly rare for large $Q_{min}$ and cannot be in\nequilibrium. Synthesis of non-topological solitons with $Q_{min} \\gg 1$ \nis stymied by the lack of the first step reaction that could produce the\nseeds for subsequent accretion. \nIn the case of Q-balls, there is no minimal charge\\footnote{\nA naive application of the thin-wall formuli to small Q-balls, \noutside the thin-wall limit, could lead one to an erroneous conclusion\nthat Q-balls must have a large charge to be stable. This apparent \nconstraint is merely an artifact of the thin-wall approximation\n\\cite{ak_qb}.}, except the quantization condition requires that $Q$ be an\ninteger; hence $Q \\ge 1$. The first of these reactions (\\ref{firststep}) \nmay also be supplemented by the pair production of small solitons\nvia charge fluctuations \\cite{gkm}. A freeze-out temperature of the\nreactions (\\ref{reaction}) -- (\\ref{nextstep}), $T_f$, is an important \nmodel-dependent parameter. \n\nIn chemical equilibrium, reactions (\\ref{firststep}) -- (\\ref{nextstep})\nenforce a relation between the chemical potentials of $\\varphi $ \nand $\\Phi (Q)$: $\\mu (Q) = Q \\mu(\\varphi)$. This allows one to express the\n$Q$-ball number density $n_{_Q}$ in terms of the $\\varphi$-number density, \n$n_\\varphi $ (see, {\\it e.\\,g.}, Ref. \\cite{foga}): \n\n\\begin{equation}\nn_{_Q} = \\frac{g_{_Q}}{g_\\varphi^Q} n_\\varphi^Q \\, \\left ( \\frac{E(Q)}{m_\\varphi} \n\\right )^{3\/2} \\, \\left ( \\frac{2\\pi}{m_\\varphi T} \\right )^{3(Q-1)\/2} \\, \n\\exp \\left ( \\frac{B_{_Q}}{T} \\right ),\n\\label{nQ}\n\\end{equation}\nwhere $B_{_Q}= Q m_\\varphi - E(Q)$, $g_{_Q}$ is the internal partition\nfunction of the soliton, and $g_\\varphi$ is the number of degrees of freedom\nassociated with the $\\varphi$ field. A soliton has $B_{_Q}>0$. Charge\nconservation implies: \n\n\\begin{equation}\nn_\\varphi = \\eta_\\varphi n_\\gamma - \\sum_Q Q n_{_Q}, \n\\label{Qconst}\n\\end{equation}\nwhere $\\eta_\\varphi $ is the Q-charge asymmetry. Here we have assumed that\nthere is no light fermions or vector bosons carrying the same $U_{_Q}(1)$\ncharge. In the presence of light charged particles, the charge asymmetry\n$\\eta$ can be accommodated by the light particle densities, allowing\n$n_\\varphi$ to drop to its $\\mu \\ll m_\\varphi$ equilibrium value suppressed by\n$\\exp (-m_\\varphi\/T)$, which is too low for the solitosynthesis to take\nplace\\footnote{The author thanks M. Shaposhnikov for illuminating \ndiscussions of this and other issues related to solitosynthesis.}.\n\nEquations (\\ref{nQ}) and (\\ref{Qconst}), usually solved numerically, \ndescribe an explosive growth of solitons below some critical temperature\n$T_s$ \\cite{foga,gk}. This exponentially fast growth is hampered \neventually, when the population of the $\\varphi$ particles is depleted in\naccordance with the charge conservation (\\ref{Qconst}). One can show that\nthe growing population of Q-balls is dominated by those with a large\ncharge. \n\nIf a single soliton with a critical charge $Q_c$ is produced per Hubble\nvolume, a phase transition takes place. In fact, at $T=T_s$\ncritical solitons are copiously produced. We will obtain a crude\nanalytical estimate of $T_s$, the temperature at which $n_{Q_c}$ starts \ngrowing. \n\nBased on the results of the preceding section, we take $Q_c \\gg 1 $. \nIn the limit of large $Q$, $g_{_{Q+1}}\/g_{_Q} \\approx 1$, $E(Q+1)\/E(Q) \n\\approx 1$, and one can write the Saha equation in the form \n\n\\begin{equation}\n\\frac{d}{dQ} (\\ln n_{_Q}) = \\frac{n_{_{Q+1}}}{n_{_Q}} -1 = \n\\left ( \\frac{n_\\varphi}{g_\\varphi} \\right )\n\\left ( \\frac{2\\pi}{m_\\varphi T} \\right )^{3\/2} \\exp\n\\left ( \\frac{b_{_Q}}{T} \\right) -1, \n\\label{saha_diff}\n\\end{equation}\nwhere $b_{_Q} = m_\\varphi -[E(Q+1)-E(Q)] \\approx m_\\varphi - dE(Q)\/dQ$. \n\nDifferential equation (\\ref{saha_diff}) describes a growing (with $Q$) \nequilibrium population of large solitons when its right-hand side is\npositive. We wish to determine \nthe temperature $T_s$, at which the growth of critical Q-balls begins. \nFor the initial growth of $n_{_{Q_c}}$, we neglect the back reaction \non the $\\varphi$ population and take $n_\\varphi = \\eta_\\varphi n_\\gamma \\sim \n\\eta_\\varphi T^3$ in accordance with equation (\\ref{Qconst}). The right-hand\nside of equation (\\ref{saha_diff}) is positive for temperatures below $T_s$\ndefined by \n\n\\begin{equation}\nT_s = \\frac{b_{_Q}}{|\\ln \\eta_\\varphi |+(3\/2) \\ln (m_\\varphi\/T_s)-\\ln g_\\varphi}. \n\\label{Ts1}\n\\end{equation}\n\nA copious production of critical solitons is possible if \n\n\\begin{equation}\nT_s > T_f.\n\\label{cond}\n\\end{equation}\nAs was explained earlier, another condition for solitosynthesis is that all\nthe particles that carry the $U_{_Q}(1)$ charge, including fermions and gauge\nbosons, must have masses that are large in comparison to $T_s$.\nFor scalars this condition is satisfied automatically. \n\n\\section{Phase transitions in supersymmetric models}\n\nA particularly interesting example is \nbaryon (B) and lepton (L) balls in the MSSM \\cite{ak_mssm}. We will\ncontinue to use $\\varphi$ as a generic name for the squarks and sleptons. \nA $B$- or $L$- ball can grow via absorption of the SU(2)-doublet sleptons\n$\\tilde{L}_{_L}$ and squarks $\\tilde{Q}_{_L}$, as well as the corresponding singlets $\\tilde{q}_{_R}$ and\n$\\tilde{l}_{_R}$. Their equilibrium population is maintained through several\nreactions,\nwhose total cross-section $\\sigma$, and hence the freeze-out temperature \n$T_f \\approx m_\\varphi\/\\ln (m_\\varphi m_{_P}\\sigma)$ depend on the parameters of\nthe model. \n\nA typical cross-section of the processes (\\ref{nextstep}) is\ncharacterized by a relatively large Q-ball size, $\\sigma \\sim 4\\pi\nR_{_Q}^2 $. These reactions, therefore, are not expected to \nfreeze out before the annihilation reactions (\\ref{reaction}). Therefore,\nthe relevant freeze-out temperature is determined by the total annihilation\ncross-section $\\sigma$ of the squarks (sleptons). \n\nAs an example, let us consider a local lepton number breaking ($\\, \\backslash \\! \\! \\! \\! L \\ $) minimum \nthat develops along the $H_1 \\tilde{L}_{_L} \\tilde{l}_{_R} \\neq 0 $ direction \\cite{clm} due to\nthe corresponding (supersymmetric) tri-linear coupling \nproportional to the $\\mu$-term\\footnote{Although in the preceding section \n$\\mu$ was used to denote a chemical potential, we trust that no confusion\nwill arise as we adopt the common notation for the $\\mu$-term in the MSSM.}.\nIt arises from the superpotential $W=\\lambda H_2 L_{_L} l_{_R} + \\mu H_1\nH_2+...$ The analysis of \nthe MSSM scalar potential shows that both $H_1$ and $H_2$ have large VEV's\nin this minimum, $\\langle H_{1,2} \\rangle \\sim \\mu\/\\lambda = \\tilde{v}$,\nwhere $\\lambda \\mu $ is the corresponding tri-linear coupling and $\\lambda\n$ is the associated Yukawa coupling. Both $\\tilde{Q}_{_L}$ and $\\tilde{q}_{_R}$ have vanishing VEV\nin the $\\, \\backslash \\! \\! \\! \\! L \\ $ minimum. In the standard vacuum, $H_1^2 + H_2^2 =\nv^2$, where $v \\approx 250$ GeV. \n\nWe consider the case of a metastable $\\, \\backslash \\! \\! \\! \\! L \\ $ vacuum decaying into a deeper \nstandard (lepton number conserving) true vacuum (Fig. \\ref{fig2}). Both\nvacua conserve the baryon number since $\\langle \\tilde{Q}_{_L} \\rangle =\\langle \\tilde{q}_{_R}\n\\rangle = 0$, and because the $SU(2)\\times U(1)$ gauge group is broken, so \nthe sphaleron transitions are suppressed. For simplicity, we will assume\nthat there is no other false vacuum with energy density below that of the\n$\\, \\backslash \\! \\! \\! \\! L \\ $ minimum. \n\nIf the Yukawa coupling $\\lambda$ is small and $\\mu$ is of the order the \nelectroweak scale, then $\\tilde{v}\/v \\gg 1$ and the $\\, \\backslash \\! \\! \\! \\! L \\ $ vacuum is\n``very far'' from the true vacuum. The tunneling rate is then\nsuppressed by the huge size of the barrier and is negligible on the\ncosmological time scale (the probability of tunneling goes, roughly, as\n$\\exp(-const\/\\lambda^2)$ \\cite{ehnt}). \n\nWe will show that for $\\tilde{v}\/v \\gg 1$, a rapid transition from the \nfalse $\\, \\backslash \\! \\! \\! \\! L \\ $ vacuum to the standard true vacuum can be precipitated by \nthe solitosynthesis of baryonic balls. \n\n\\begin{figure}\n\\setlength{\\epsfxsize}{3.3in}\n\\centerline{\\epsfbox{figure_pt2.eps}}\n\\caption{\nThe energy of a large baryonic ball is minimized when {$\\varphi(x)=\\varphi_0$}, \n{$U(\\varphi_0)<0$}, in its interior. Although both the true ({$\\tilde{L}_{_L}=0$}) and the \nfalse ({$\\tilde{L}_{_L}=\\tilde{v}$}) vacua conserve the baryon number, {$B$}-balls\nprecipitate a phase transition into a negative-energy attractive domain\nof the true vacuum (shaded region). \n}\n\\label{fig2}\n\\end{figure}\n\nIt is easy to see that there is always a value of charge (baryon number, in\nour example) $Q_1$, such that for $Q>Q_1$ the energy (\\ref{E}) of a B-ball\nin the false vacuum is minimal when the potential energy in its \ninterior is negative ($V<0$). Since the origin (Fig. \\ref{fig2}) is a\nglobal minimum, there is a domain with a negative energy density in the\nfield space around the origin. This domain is shown as a shaded region in \nFig. 2. By assumption, $U(\\varphi) \\ge 0$ everywhere\noutside this region. A nucleation of a critical B-ball with $\\varphi(0)=\\varphi_0,\n\\ U(\\varphi_0)<0$, will, therefore,\nconvert the false vacuum into a phase that will eventually relax to the \ntrue vacuum at the origin. From this example one can see, in particular, \nthat Q-balls associated with some $U_{_Q}(1)$ symmetry can mediate phase \ntransitions between the two vacua, both of which preserve $U_{_Q}(1)$. \n\nAs follows from the discussion in the last section, solitosynthesis of\nbaryonic balls can take place only in the absence of light baryons. \nFor a large enough Higgs VEV in the false vacuum, the the masses of \nquarks will be approximately equal to those of squarks. The mass\nsplittings are due to the soft supersymmetry breaking terms \nindependent of the Higgs VEV. Since the Higgs VEV in the $\\, \\backslash \\! \\! \\! \\! L \\ $ vacuum\nassociated with a small Yukawa coupling $\\lambda$ is large ($\\sim\n\\mu\/\\lambda$), it can be much greater than the SUSY breaking terms\nresponsible for the mass difference. As a result, the quark masses are of\norder the squark masses in the false vacuum, where the typical scale $m_\\varphi\n\\sim \\mu\/\\lambda$ can be as large as $10^3...10^6$ GeV. \n\nWe will examine condition (\\ref{cond}) for $m_\\varphi \\sim 1$ TeV in the false \n$\\, \\backslash \\! \\! \\! \\! L \\ $ vacuum (while $m_\\varphi \\sim 10^2$ GeV in the true vacuum). \nWe take the baryon (lepton) asymmetry $\\eta_\\varphi \\sim 10^{-10}$. \nProduction of critical solitons can take place at the temperature \n\n\\begin{equation}\nT_s \n\\approx b_{_Q}\/\\ln(1\/\\eta_\\varphi) \\approx \\frac{1}{23} \n\\left ( m_\\varphi - \\left. \\frac{d}{dQ} E(Q) \\right |_{Q=Q_c} \\right )\n\\end{equation}\nif \n\n\\begin{equation}\nT_s > \\ T_f \\approx m_\\varphi\/\\ln(m_\\varphi m_{_P} \\sigma) \\approx m_\\varphi\/40\n\\label{cond_mssm}\n\\end{equation}\nOne can easily check that the charge accretion is many orders of\nmagnitude faster than the Hubble time scale at these temperatures. \nThe characteristic time scale associated with the growth of a Q-ball \nwith the charge $Q\\sim Q_c$ can be estimated as \n\n\\begin{equation}\n\\tau_{requir}^{-1} \\sim \\frac{dQ}{dt} \\sim n_\\varphi \\sigma v_\\varphi \\sim \n\\eta_\\varphi n_\\gamma (4 \\pi R_{Q_c}^2) \\sqrt{T_s\/m_\\varphi} \\sim \n10^{-7} GeV\n\\label{time1}\n\\end{equation}\nwhile the Hubble scale at these temperatures is $\\tau_{avail}^{-1} \n\\sim T_s^2\/m_{_P} \\sim 10^{-15}$ GeV. Clearly, $\\tau_{requir} \\ll \n\\tau_{avail}$, hence the Q-balls have plenty of time to grow when the \ncondition (\\ref{cond_mssm}) is satisfied. \n\nWe note in passing that although $dE\/dQ$ is a constant for large Q-balls \nin the true vacuum \\cite{coleman1}, this is generally \nnot the case for Q-balls in the false vacuum. For instance, in the limit\n$U_0 \\rightarrow 0$, $E(Q) \\propto Q^{4\/5}$ \\cite{spector}. \n\nIn summary, the first-order phase transition of a new kind can result \nfrom the solitosynthesis of Q-balls (in particular, B- and L-balls in the\nMSSM), which can destabilize a metastable vacuum even when the tunneling\nrate is negligible. At a certain temperature \n(\\ref{Ts1}), Q-balls build up rapidly by absorbing charged particles\nfrom the outside until the critical size is reached, at which point they\nexpand and fill the Universe with the ``true vacuum'' phase. This\nmechanism has important implications for models with low-energy\nsupersymmetry, in which the presence of baryonic and leptonic Q-balls is\ngeneric. \n\nThe author thanks M. Shaposhnikov for many stimulating discussions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}