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+{"text":"\\section{Supersymmetric Lagrangian and Susy breaking}\n\nThe most general superfield action for a homogeneous scalar supermultiplet \ninteracting with the scalar factor in the supersymmetric FRW model\n\\cite{ORT1,TRM} has the form\n\\begin{eqnarray} \n\\rm S &=& \\rm S_{FRW} + S_{\\rm mat}, \\nonumber \\\\\nS_{FRW}&=& \\int 6\\left[- \\frac{1}{2\\kappa^2}\\frac{{I\\!\\!R}}{{I\\!\\!N}} \n{\\cal D}_{\\bar \\eta} {I\\!\\!R} {\\cal D}_\\eta {I\\!\\!R}\n+\\frac{\\sqrt{k}}{2\\kappa^2} {I\\!\\!R}^2 \\right]\nd\\eta d {\\bar \\eta} dt\\, , \\nonumber\\\\            \n S_{ mat} &=&  \\int \\left[ \\frac{1}{2} \\frac{{I\\!\\!R}^3}{{I\\!\\!N}} \n{\\cal D}_{\\bar \\eta} {\\Phi} {\\cal D}_\\eta {\\Phi}  \n- 2 {I\\!\\!R}^3 g({\\Phi} )\\right] d\\eta d {\\bar \\eta} dt\\, ,  \n\\label{action1}\n\\end{eqnarray}\nwith $k=0,1$ stands for flat and closed space, and \n$\\kappa^2 =8\\pi G_N$, where $G_N$ is Newton's constant of gravity,\n $(\\hbar =c=1)$. The units for the constants and fields in this\nwork are the following: $[\\kappa^2]=\\ell^2$, $[{I\\!\\!N}]=\\ell^0 , \n[{I\\!\\!R}]=\\ell^{1}, [{\\Phi} ]=\\ell^{-1}$  and\nsuperpotential $[g({\\Phi} )]=\\ell^{-3}$, where $\\ell $ correspond to units of \nlenght\n\nFor the one-dimensional gravity superfield ${I\\!\\!N} (t,\\eta ,\\bar\\eta)$ \n$({I\\!\\!N}^\\dagger ={I\\!\\!N})$ we have the following series expansion\n\n\\begin{equation}\n{I\\!\\!N}(t,\\eta ,\\bar\\eta )=N(t)+i\\eta\\bar\\psi^\\prime (t)+i\\bar\\eta\\psi^\\prime\n(t)+\\eta\\bar\\eta {\\cal V}^\\prime (t) \\, ,\n\\label{expansion1}\n\\end{equation}\nwhere $N(t)$ is the lapse function and also we have introduced the \nreparametrization $\\psi^\\prime (t)= N^{\\frac{1}{2}} (t)\\psi (t)$ \nand ${\\cal V}^\\prime (t) = {I\\!\\!N}(t) {\\cal V}(t) +\\bar \\psi (t)\\psi (t)$.\n\nThe Taylor series expansion for the superfield ${I\\!\\!R} (t,\\eta ,\\bar\\eta)$\nwith the scalar factor $R(t)$ has the similar form\n\n\\begin{equation}\n{I\\!\\!R} (t,\\eta ,\\bar\\eta)= R(t)+i\\eta\\bar\\lambda^\\prime (t)\n+i\\bar\\eta\\lambda^\\prime (t) +\\eta\\bar\\eta {\\cal B}^\\prime (t),\n\\label{expansion2}\n\\end{equation}\nwhere $\\lambda^\\prime (t) =\\kappa N^{\\frac{1}{2}} (t)\\lambda (t)$ and\n${\\cal B}^\\prime (t)=\\kappa N(t){\\cal B}(t)+\\frac{\\kappa}{2} \n(\\bar\\psi \\lambda -\\psi \\bar\\lambda)$.  \n\nThe component of the (scalar) matter superfields $\\Phi (t,\\eta ,\\bar\\eta)$\nmay be written as $(\\Phi^+ =\\Phi )$\n\n\\begin{equation}\n\\Phi =\\varphi (t)+i\\eta\\bar\\chi^\\prime (t)+i\\bar\\eta \\chi^\\prime (t)+F^\\prime\n(t)\\eta \\bar \\eta\n\\label{matter}\n\\end{equation}\nwhere $\\chi^\\prime (t) =N^{\\frac{1}{2}} (t)\\chi (t)$ and $F^\\prime (t)=NF+\n\\frac{1}{2} (\\bar \\psi \\chi -\\psi \\bar \\chi )$, \nbesides,\n${\\cal D}_\\eta = \\frac{\\partial}{\\partial\\eta}+i \\bar\\eta \n\\frac{\\partial}{\\partial t}$ and ${\\cal D}_{\\bar\\eta}=-\n\\frac{\\partial}{\\partial\\bar\\eta}-i\\eta \\frac{\\partial}{\\partial t}$ are the\nsupercovariant derivatives of the conformal supersymmetry $n=2$, which has\ndimension $[{\\cal D}_\\eta ]=[{\\cal D}_{\\bar\\eta}] =\\ell^{-\\frac{1}{2}}$. \n\nAfter the integration over Grassmann  complex coordinate $\\eta$ and \n$\\bar\\eta$, and making the following redefinition of the ``fermion\"  fields \n(Grassmann variables) \n$\\lambda (t) \\to \\frac{1}{3} R^{-\\frac{1}{2}}(t)\\lambda(t)$ and \n$\\chi (t) \\to R^{-3\/2} (t) \\chi (t)$, we find the Lagrangian in which \nthe fields ${\\cal B}(t)$ and $F(t)$\nare auxiliary and they can be eliminated with the help of their equations \nof motion.\n\nFinally, the Lagrangian in terms of components of the superfields \n${I\\!\\!R}$, ${I\\!\\!N}$, $\\Phi$ takes the form\n\n\\begin{eqnarray}\nL &=&-\\frac{3}{\\kappa^2} \\frac{R({\\cal D} R)^2}{N} +\\frac{2}{3} i\\bar\\lambda\n{\\cal D}\\lambda +\\frac{\\sqrt{k}}{\\kappa} R^{\\frac{1}{2}} (\\bar\\psi \\lambda -\n\\psi\\bar\\lambda ) \\nonumber \\\\\n&+& \\frac{1}{3} NR^{-1} \\sqrt{k} \\bar\\lambda\\lambda +\\frac{3k}{\\kappa^2} NR+\n\\frac{R^3}{2}\\frac{({\\cal D}\\varphi)^2}{N}-i\\bar\\chi {\\cal D}\\chi \\nonumber \\\\\n&-& \\frac{3}{2} \\sqrt{k}NR^{-1}\\bar\\chi\\chi -\\kappa^2 N g(\\varphi )\\bar\\lambda\n\\lambda -6\\sqrt{k} N g(\\varphi)R^2  \\label{lagra}  \\\\\n&-&NR^3 V(\\varphi)+\\frac{3}{2}\\kappa^2 N g(\\varphi)\\bar\\chi\\chi \n+\\frac{i\\kappa}{2} {\\cal D}\\varphi (\\bar\\lambda \\chi +\\lambda\\bar\\chi ) \n\\nonumber \\\\\n&-& 2 N \\frac{\\partial^2 g(\\varphi)}{\\partial\\varphi^2} \\bar\\chi\\chi \n-\\kappa N \\frac{\\partial g(\\varphi)}{\\partial\\varphi}(\\bar\\lambda\\chi \n- \\lambda\\bar\\chi ) + \n\\frac{\\kappa^2}{4}R^{-3\/2}(\\psi\\bar\\lambda -\\bar\\psi\\lambda )\\bar\\chi\\chi \n\\nonumber \\\\\n&-& \\kappa R^{3\/2} (\\bar\\psi\\lambda - \\psi\\bar\\lambda) g(\\varphi)+R^{3\/2} \n\\frac{\\partial g(\\varphi)}{\\partial \\varphi} (\\bar\\psi \\chi -\\psi \\bar \\chi)\\,\n ,  \\nonumber\n\\end{eqnarray} \nwhere ${\\cal D} R=\\dot R-\\frac{i\\kappa}{6} R^{-\\frac{1}{2}} \n(\\psi\\bar\\lambda +\\bar\\psi\\lambda )$ and \n${\\cal D}\\varphi =\\dot \\varphi -\\frac{i}{2} R^{-\\frac{3}{2}}\n(\\bar\\psi\\chi +\\psi\\bar\\chi)$ are the supercovariant derivatives, and \n${\\cal D}\\lambda =\\dot\\lambda -\\frac{i}{2} {\\cal V}\\lambda$, \n${\\cal D}\\chi =\\dot\\chi -\\frac{i}{2}{\\cal V}\\chi$ are the ${\\cal U}$ (1) \ncovariant derivatives.\n\nThe potential for the homogeneous scalar fields \n\\begin{equation}\nV(\\varphi ) =2 \\left( \\frac{\\partial g(\\varphi)}{\\partial\\varphi}\\right)^2\n- 3\\kappa^2 g^2(\\varphi),\n\\label{potential}\n\\end{equation}\nconsists of two terms, one of them is the potential for the scalar field in \nthe case of global supersymmetry. The potential (\\ref{potential}) \n is not positive semi-definite in contrast with the standard \nsupersymmetric quantum mechanics.\nUnlike the standard supersymmetric quantum mechanics, our model,  \n describing the minisuperspace approach to\nsupergravity coupled to matter, allows the  supersymmetry breaking when the \nvacuum energy is equal to zero $V(\\varphi)=0$.  \n\nIn order to give some implications of spontaneous supersymmetry breaking we \ndisplay the potential $V(\\varphi)$ (\\ref{potential}) in terms of the \nauxiliary fields $F(t)$ and ${\\cal B}(t)$, \n\n\\begin{equation}\nV(\\varphi) = \\frac{1}{2} F^2 - \\frac{3}{\\kappa^2 R^2} {\\cal B}^2\\, ,\n\\label{po-aux} \n\\end{equation}\nwhere the bosonic $F$ and ${\\cal B}$  are\n\\begin{equation}\nF=2 \\frac{\\partial g(\\varphi)}{\\partial\\varphi } \\, , \\,\\qquad \n{\\cal B}= -\\kappa^2 R\\, g(\\varphi) \\, .\n\\end{equation}\n\nThe selection rules for the occurrence of spontaneous supersymmetry breaking \nare\n\n\\begin{eqnarray}\n&{\\rm i)}&\\qquad \\frac{\\partial V(\\varphi)}{\\partial\\varphi} \n= 4\\frac{\\partial g}{\\partial\\varphi} \n\\left[ \\frac{\\partial^2 g}{\\partial\\varphi^2} \n-\\frac{3}{2} \\kappa^2 g \\right]=0, \n\\,\\, {\\rm at} \\,\\, \\varphi = \\varphi_0 \\label{uno}\\\\\n&{\\rm ii)}&\\qquad V (\\varphi_0 )= 0 \\Rightarrow\n \\left[ \\left( \\frac{\\partial g}{\\partial\\varphi} \\right)^2 -\\frac{3}{2} \n\\kappa^2 g^2 \\right] =0 , \\label{dos}\\\\\n&{\\rm iii)}&\\qquad F= 2\\frac{\\partial g(\\varphi)}{\\partial\\varphi} \\not =\n0 \\, , \\, {\\rm at} \\, \\varphi = \\varphi_0, \\label{tres} \n\\end{eqnarray}\n\nThe first condition implies the existence of a minimum, in the scalar field;\nthe second condition  is the absence of the cosmological constant, the third \ncondition  is for the  breaking of supersymmetry.\nThe measure of this breakdown is the term $-\\kappa^2 g(\\varphi )\nN\\bar\\lambda\\lambda$ in the lagrangian (\\ref{lagra}). Furthermore, we can \nidentify  \n\\begin{equation}\nm_{3\/2} = \\kappa^2 g(\\varphi_0)= \\frac{g(\\varphi_0)}{M^2_{pl}}\n\\label{mass}\n\\end{equation}\nas the gravitino mass in the effective supergravity theory, and \n$M_{pl}=\\frac{1}{\\kappa} = \\frac{1}{\\sqrt{ 8\\pi G_N}}= 2.4 \\times 10^{18}Gev$\nis the reduced Planck mass.\n \nThe factor $R$ in the kinetic terms of the  scalar factor \n$-\\frac{3}{\\kappa^2N} R (\\dot R)^2$ plays the role of a ``metric'' tensor in \nthe Lagrangian, the kinetic energy for the scalar factor is negative \ndefinite, this is due to the fact, that the particle-like fluctuations do not \ncorrespond to the scalar factor, and the kinetic energy of the scalar \nfields $\\frac{R^3}{2N} \\dot\\varphi^2$ \nis positive.\n\\section{The corresponding supersymmetric quantum mechanics}\n\nThe hamiltonian can be calculated in the usual way. We have the classical \ncanonical Hamiltonian\n\n\\begin{equation}\nH_{can} =NH+\\frac{1}{2}\\bar\\psi \\, S-\\frac{1}{2}\\psi \\, \\bar S+ \n\\frac{1}{2} {\\cal V} \\, {\\cal F} ,\n\\label{hamiltonian}\n\\end{equation} \nwhere $H$ is the Hamiltonian of the system, $S$, $\\bar S$  are supercharges and\n${\\cal F}$ is the  ${\\cal U}(1)$ rotation generator. The form of the canonical\nHamiltonian (\\ref{hamiltonian}) explains the fact, that $N, \\psi, \\bar\\psi$ \nand ${\\cal V}$ are  Lagrange multipliers which enforce  only the first class \nconstraints $H=0, S=0, \\bar S=0$ and ${\\cal F}=0$, which express the \ninvariance of the conformal\n$n=2$ supersymmetric transformations. As usual with the Grassmann variables,\nwe have the second-class constraints which can be eliminated by the Dirac \nprocedure, as a result only the following non-zero \nDirac brackets $\\left\\{\\,\\, ,\\,\\,\\right\\}$ remain. For Grassmann  variables \n$\\lambda ,\\bar\\lambda ,\\chi$\nand $\\bar\\chi$ \n\n\\begin{equation}\n\\left\\{\\lambda ,\\bar\\lambda \\right\\}_*=+\\frac{3}{2}i \\, , \\, \n\\left\\{\\chi ,\\bar\\chi \\right\\}_* =-i \\, .\n\\label{bracket1}\n\\end{equation}\n\nThe canonical Poisson brackets for the $R, \\pi_R$ and $\\varphi, \\pi_\\varphi$,\n have the following form\n\\begin{equation}\n\\left\\{R, \\pi_R\\right\\}_{Pb} =-1 \\, , \\, \\left\\{\\varphi ,\\pi_\\varphi \n\\right\\}_{Pb} =-1 \\, .\n\\label{bracket2}\n\\end{equation}\n\nIn a quantum theory the brackets (\\ref{bracket1}) and (\\ref{bracket2}) must be \nreplaced by anticommutator\n\n\\begin{equation}\n\\left\\{\\lambda , \\bar\\lambda \\right\\}=-\\frac{3}{2} \\, , \\, \n\\left \\{\\chi ,\\bar\\chi \\right\\} =1 ,\n\\label{bra}\n\\end{equation}\nand can be considered as generators of the Clifford algebra,  as well as the \ncommutators\n\\begin{equation}\n\\left[R,\\pi_R\\right] =-i \\, , \\,\\left [\\varphi , \\pi_\\varphi \\right] =-i\\, .\n\\end{equation}\n\nThe quantization procedure takes  into account the dependence of the \nLagrangian on the metric ``R\", but at the quantum level we must consider the\nnature of the Grassmann variables $\\lambda ,\\bar\\lambda ,\\chi$ and $\\bar\\chi$,\nand with respect to these ones we perform the antisymmetrizations, then\n we can write the bilinear combination in the form of the commutators e.g. \n$\\bar\\chi \\chi \\to \\frac{1}{2}\\left [\\bar\\chi ,\\chi \\right]$ and this leads \nto the \nfollowing quantum \nHamiltonian $H$\n\\begin{eqnarray}\nH &=& -\\frac{\\kappa^2}{12} R^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \n\\pi_R -\n\\frac{3k R}{\\kappa^2} -\\frac{1}{6} \\frac{\\sqrt{k}}{R} \\left[\\bar\\lambda , \n\\lambda \\right]\n+ \\frac{\\pi^2_\\varphi}{2R^3}  \\nonumber \\\\\n&&- \\frac{i\\kappa}{4R^3}\\pi_\\varphi \\left(\\left[\\bar\\lambda ,\\chi \\right]\n+\\left[\\lambda ,\\bar\\chi \\right] \\right)\n-\\frac{\\kappa^2}{16R^3} \\left[\\bar\\lambda ,\\lambda \\right] \n\\left[\\bar\\chi ,\\chi \\right]+ \\frac{3\\sqrt{k}}{4 R} \\left[\\bar\\chi , \\chi \n\\right] \\nonumber \\\\\n&&+ \\frac{\\kappa^2}{2} g(\\varphi) \\left[\\bar\\lambda ,\\lambda \\right]\n+6\\sqrt{k}\\,  g(\\varphi) R^2 + R^3 V(\\varphi) +\\frac{3}{4} \\kappa^2 \ng(\\varphi) \\left[\\bar\\chi ,\\chi \\right]  \\nonumber \\\\\n&& + \\frac{\\partial^2 g(\\varphi)}{\\partial \\varphi^2} \\left[\\bar\\chi ,\\chi \n\\right]+ \\frac{1}{2}\\kappa \\frac{\\partial g(\\varphi)}{\\partial\\varphi} \n\\left( \\left[ \\bar\\lambda ,\\chi \\right]-\\left[\\lambda ,\\bar\\chi \\right]\n\\right) \\, ,\n\\label{hamil}\n\\end{eqnarray}\nwhere $\\pi_R = i \\frac{\\partial}{\\partial R}$ and\n$\\pi_\\varphi = i \\frac{\\partial}{\\partial \\varphi}$. \nThe supercharges $S$, $S^\\dagger$ and the fermion number operator ${\\cal F}$ \nhave the following structures\n\n\\begin{equation}\nS=A\\lambda +B\\chi \\, , \\qquad \\, S^\\dagger =A^\\dagger \\lambda^\\dagger \n+B^\\dagger  \\chi^\\dagger \\, ,\n\\end{equation}\n where  \n\\begin{eqnarray}\nA&=&\\frac{i}{3}\\kappa R^{-\\frac{1}{2}} \\pi_R - \\frac{2\\sqrt k}{\\kappa}\nR^{\\frac{1}{2}} + 2\\kappa R^{\\frac{3}{2}} g(\\varphi) + \\frac{\\kappa}{4}\nR^{-\\frac{3}{2}} \\left[ \\bar \\chi, \\chi \\right]\\, , \\nonumber \\\\\nB&=& i R^{-\\frac{3}{2}} \\pi_\\varphi + 2 R^{\\frac{3}{2}} \n\\frac{\\partial g(\\varphi)}{\\partial \\varphi} \\, ,\n\\end{eqnarray}\nwith $A^\\dagger$ and $B^\\dagger$ are hermitian to A and B respectively, \nand \n\\begin{equation}\n{\\cal F}=-\\frac{1}{3}\\left[\\bar\\lambda ,\\lambda \\right]\n+\\frac{1}{2} \\left[\\bar\\chi ,\\chi \\right] .\n\\end{equation}\n\nAn ambiguity exist in the factor ordering of these operators, such \nambiguities always arise, when the operator expression contains the product of\nnon-commuting operators $R$ and $\\pi_R$ as in our case. It is then \nnecessary to find some criteria to know which factor ordering should be \nselected. We propose the following: to integrate with measure \n$R^{\\frac{1}{2}}dR$ in \nthe inner product of two states \\cite{AFF}. In this measure the \nconjugate momentum  $\\pi_R$ is non-hermitan with \n$\\pi^\\dagger_R=R^{-\\frac{1}{2}} \\pi_R R^{\\frac{1}{2}}$,\nhowever, the combination $(R^{-\\frac{1}{2}}\\pi_R)^\\dagger =\\pi^\\dagger_R \nR^{-\\frac{1}{2}} = R^{-\\frac{1}{2}} \\pi_R$ is a hermitian one, and  \n$(R^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \\pi_R)^\\dagger =\nR^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \\pi_R$ is \nhermitian too, with this the parameter $p=1\/2$ is fixed.\n\nThe anticommutator value $\\{\\lambda , \\bar\\lambda \\}=-\\frac{3}{2}$ of the\nsuperpartners $\\lambda$ and $\\bar\\lambda$ of the scalar factor $R$ is negative,\nunlike the anticommutation relation for $\\chi$ and $\\bar\\chi$ in (\\ref{bra}),\n which is positive. They can be redefined and one becomes the following \nconjugate operation for the operators $\\lambda$ and $\\chi$\n\n\\begin{equation}\n\\bar\\lambda = \\xi^{-1} \\lambda^\\dagger \\xi =-\\lambda^\\dagger \\, , \\, \\qquad\n\\bar\\chi = \\xi^{-1}\n\\chi^\\dagger \\xi = \\chi^\\dagger ,\n\\end{equation}\nwhere $\\{\\lambda ,\\lambda^\\dagger \\} =\\frac{3}{2}$, and the operator $\\xi$ \npossesses the following properties\n\n\\begin{equation}\n\\lambda^\\dagger \\xi =-\\xi\\lambda^\\dagger \\, , \\,\\qquad \n\\chi^\\dagger \\xi=\\xi \\chi^\\dagger \\,\\qquad {\\rm and} \\,\\qquad\n\\xi^\\dagger = \\xi \\, .\n\\end{equation}\n\nSo, for the supercharge operator $\\bar S$ we have  the following equation\n\n\\begin{equation}\n\\bar S = \\xi^{-1} S^\\dagger \\xi \\, .\n\\end{equation}\n\nFor the quantum generators $H, S, \\bar S$ and ${\\cal F}$ we obtain the \nfollowing superalgebra \n\n\\begin{eqnarray}\n\\left\\{ S, \\bar S\\right\\} &=&2H \\, ,\\quad \\, S^2=\\bar S^2 =0 \\, , \\qquad \\, \n\\left[S,H\\right]=0, \\nonumber\\\\\n\\left[\\bar S,H\\right]&=&0 \\, , \\quad \\left [{\\cal F}, S\\right]=-S \\, ,\\quad \n\\,\\left [{\\cal F}, \\bar S \\right ]=\\bar S \\,  . \n\\label{superalgebra}\n\\end{eqnarray}\nWe can see, that the anticommutator of supercharges $S$ and their conjugate\n$\\bar S$ under our conjugate operation has the form\n\n\\begin{equation}\n\\overline{ \\left \\{ S,\\bar S \\right\\} }= \\xi^{-1} \n\\left\\{ S,\\bar S \\right \\}^\\dagger \n\\xi = \\left\\{S,\\bar S\\right\\} , \n\\end{equation}\nand the  Hamiltonian operator is self-conjugate under the operation \n$\\bar H =\\xi^{-1} H^\\dagger \\xi$. In the case of standard supersymmetric \nquantum mechanics\nwe would have $\\bar\\lambda =\\lambda^\\dagger$ and $\\bar S=S^\\dagger$, and \nthe Hamiltonian would be positive. In our case, the algebra \n(\\ref{superalgebra}) \ndoes not define positive-definite Hamiltonian in a full agreement with the \ncircumstance that the  potential $V(\\varphi)$ (\\ref{potential}) of the scalar \nfield is not positive semi-definite in general, in contrast with the \nstandard supersymmetric quantum mechanics.\n\nWe can choose the following matrix representation for the operators \n$\\lambda ,\\bar\\lambda , \\chi , \\bar\\chi$ and $\\xi$ in the form of the \ntensorial product of matrices $2\\times 2$\n\\begin{eqnarray}\n\\lambda &=&\\sqrt{\\frac{3}{2}}\\sigma_{(-)}\\otimes 1 \\, \n,\\qquad \\,\\bar\\lambda =-\\sqrt{\\frac{3}{2}}\\sigma_{(+)}\n\\otimes 1 , \\nonumber\\\\\n\\chi &=& \\sigma_3 \\otimes \\sigma_{(-)} \\, ,\\qquad \\, \\bar\\chi =\\sigma_3 \\otimes\n\\sigma_{(+)} \\, , \\qquad \\,\n\\xi = \\sigma_3 \\otimes 1 \\, ,\n\\end{eqnarray}\nwhere $\\sigma_{\\pm}=\\frac{\\sigma_1 \\pm i \\sigma_2}{2}$,  $\\sigma_1$,\n$\\sigma_2$ and $\\sigma_3$ are the Pauli matrices.\n\n\\section{The wave function for the zero energy state}\n\n\nIn the quantum theory the first-class constraints $H=0, S=0, \\bar S=0$ and \n${\\cal F}=0$ associated with the invariant action (\\ref{action1}) under the\nlocal $n=2$  conformal supersymmetry become conditions on the wave function \n$\\Psi$. So that any\nphysical state must obey the following quantum constrains\n\n\\begin{equation}\n\\hat H \\, \\Psi =0, \\hat S \\, \\Psi =0, {\\hat{\\bar{S}}}\\, \\Psi = 0 \\,\\, \n{\\rm and} \\,\\,\n\\hat{\\cal F} \\, \\Psi =0 ,\n\\label{physical}\n\\end{equation}\nwhere the first equation in (\\ref{physical}) is the Wheeler-DeWitt equation \nfor the minisuperspace model. The eigenstates of the Hamiltonian (\\ref{hamil}) \nhave four-components \n\n\\begin{equation}\n\\Psi (R,\\varphi ) =\n\\left[ \\begin{array}{c}\n \\psi_1(R,\\varphi ) \\\\\n \\psi_2(R,\\varphi ) \\\\\n \\psi_3(R,\\varphi ) \\\\\n \\psi_4(R,\\varphi )  \n\\end{array}\n\\right] \\, .\n\\end{equation}\n\nWe have rewritten the equations $S\\, \\Psi=0$ and $\\bar S \\, \\Psi=0$ in the \nfollowing form \n\n\\begin{eqnarray}\n\\left(\\lambda\\bar S - \\bar\\lambda S\\right) |\\Psi>& =& \\left[- \\left( \n\\frac{A-A^\\dagger}{2}\\right) \\left\\{ \\bar \\lambda , \\lambda \\right\\}\n-\\left( \\frac{A+A^\\dagger}{2} \\right) \\left[\\bar\\lambda ,\\lambda \\right]\n\\right. \\nonumber \\\\\n&&- \\left. \\left(\\frac{B-B^\\dagger}{2}\\right)\n(\\bar\\lambda\\chi +\\lambda\\bar\\chi )+\\left( \\frac{B+B^\\dagger}{2} \\right) \n(\\lambda\\bar\\chi -\\bar\\lambda\\chi )\\right] |\\Psi> =0, \\\\ \n\\left( \\bar\\chi S -\\chi\\bar S\\right)|\\Psi> &=&\\left[ \\left(\n\\frac{A-A^\\dagger}{2}\\right)(\\bar\\chi\\lambda +\\chi\\bar\\lambda )\n+\\left(\\frac{A+A^+}{2}\\right)(\\bar\\chi\\lambda -\\chi\\bar\\lambda) \\right. \n\\nonumber \\\\\n&& -\\left. \\left(\\frac{B-B^\\dagger}{2}\\right)\\left \\{\\bar \\chi , \\chi \\right\\}\n+\\left(\\frac{B+B^\\dagger}{2}\\right) \\left[\\bar\\chi ,\\chi\\right] \n\\right]|\\Psi> = 0 \\, .\n\\end{eqnarray}\n\nUsing a matrix representation for $\\lambda, \\bar \\lambda, \\chi$ and\n$\\bar \\chi$, we found that only $\\psi_4$ have the right behaviour when\n $R\\to \\infty$ because $\\psi_4 \\to 0$, \nand due that the others components $\\psi_1 , \\psi_2$ and $\\psi_3$ for the wave \nfunction at $R\\to \\infty$ are infinite, we consider these components as no\nphysical. Thus, there is a normalizable component $\\psi_4$ for H such that \n$S \\Psi = \\bar S \\Psi =0$, and this eigenstate corresponds to the\nground state with eigenvalue $E=0$. The wave function $\\Psi$ has  also\nthe non-normalizable components  $\\psi_1, \\psi_2$ and $\\psi_3$  for $E=0$,\nbut for them the eigenvalue of $H$ is non-zero, for this case all components\nare normalizable\n\nSo, the partial differencial equations for $\\psi_4 (R,\\varphi)$ have \nthe following forms\n\n\\begin{eqnarray}\n\\left[-R^{-\\frac{1}{2}} \\frac{\\partial}{\\partial R} - 6 g(\\varphi)\nR^{3\/2}+\n6\\sqrt{k} M^2_{pl} R^{\\frac{1}{2}}+\\frac{3}{4} R^{-3\/2} \\right] \\psi_4 &=&0,\n\\label{factor}\\\\\n\\left[\\frac{\\partial}{\\partial \\varphi}+2R^3 \n\\frac{\\partial g(\\varphi)}{\\partial\\varphi} \\right] \\psi_4 &=& 0 \\, .\n\\label{scalar}\n\\end{eqnarray}\n\nWe get as solution for (\\ref{factor}) and (\\ref{scalar})\n\n\\begin{equation}\n\\Psi\\equiv \\psi_4 (R,\\varphi)=C_o\\,R^{3\/4} e^{(-2g(\\varphi)\\,R^3+\n3\\sqrt{k} M^2_{pl} R^2)} \\, .\n\\label{solution}\n\\end{equation}\n\nThe scalar product  for the solution (\\ref{solution}) is normalizable \nwith the measure $R^{\\frac{1}{2}}\\, dR \\,d\\varphi$ and for the superpotential \n$g(\\varphi \\to\\pm\\infty )\\to \\infty$. \n\nConsequently the solution \n(\\ref{solution}) is the eigenstate of the Hamiltonian (\\ref{hamil}) with \n zero energy  and also with zero fermionic number.\n\nThe superpotential for the fluctuating scalar fields \n$\\varphi =\\varphi_0 + \\tilde\\varphi$ near the minimum of the potential \n$V(\\varphi_0)=0$ has the following form  \n(see (\\ref{uno},\\ref{dos},\\ref{tres}, \\ref{mass})).\n\n\\begin{eqnarray}\ng(\\tilde\\varphi)&=&g(\\varphi_0)+ \n\\frac{\\partial g(\\varphi_0)}{\\partial\\varphi} \\tilde\\varphi \n+\\frac{1}{2} \\frac{\\partial^2 g(\\varphi_0)}{\\partial\\varphi^2}\\tilde\\varphi^2 \n \\nonumber\\\\\n&=& m_{3\/2} M^2_{pl}\\left[1 + \\sqrt{\\frac{3}{2}} \\frac{\\tilde \\varphi}{M_{pl}}+\n\\frac{3}{4} \\,\\frac{\\tilde \\varphi^2}{M^2_{pl}}  \\right]=\nm_{3\/2} M^2_{pl} \\, f(x) \\, ,\n\\end{eqnarray}\nwhere $f(x)=1 + \\sqrt{\\frac{3}{2}} x+ \\frac{3}{4} \\,x^2$, with \n$x=\\frac{\\tilde \\varphi}{M_{pl}}$.\n\nSo, as an example we can consider the case $k=0$ \n\\begin{eqnarray}\n 1&=& C_o^2 \\int^{\\infty}_{-\\infty}\n\\int^\\infty_0 R^{\\frac{3}{2}} e^{-4g( \\tilde\\varphi)R^3} \\,\nR^{\\frac{1}{2}}\\, dR \\,d\\tilde\\varphi \\nonumber\\\\\n&=&\\frac{C_o^2}{12} \\int^{\\infty}_{-\\infty}\n\\frac{d\\tilde\\varphi}{g(\\tilde\\varphi)}= \\frac{C_o^2}{12 m_{3\/2} M_{pl}}\n\\int^{\\infty}_{-\\infty} \\frac{d x}{f(x)} = \n\\frac{C_o^2 \\sqrt {2} \\pi}{12 \\sqrt{3} m_{3\/2} M_{pl}} \\, ,\n\\end{eqnarray}\nthus, the normalization constant has the following value\n\\begin{equation}\nC_o=  \\left( \\frac{3}{2} \\right)^{\\frac{1}{4}} \n\\sqrt{\\frac{6 m_{3\/2}\\,M_{pl}}{\\pi}}.\n\\end{equation}\nThe behaviour for the wave-function $\\Psi$ in the $k=0$ case is shown in  \nFigure 1.\n\nThe spectation value for the scalar factor $R$ with the chosen measure is\n\n\\begin{eqnarray}\n\\overline R &=&\n<\\Psi|R|\\Psi> = C_o^2 \\int^\\infty_{-\\infty} d \\tilde\\varphi \\int^\\infty_{0}\n\\left[ R^3 e^{-4g(\\tilde\\varphi)\\, R^3} dR  \\right] \\nonumber\\\\\n&=&\\left( \\frac{ \\sqrt{3}\\Gamma\\left( \\frac{4}{3}\\right)}{4\\pi 2^{1\/6} } \n\\int^\\infty_{-\\infty} \\frac{dx}{\\left[ f(x) \\right]^{4\/3}} \\right)\n\\left( \\frac{M_{pl}}{m_{3\/2}} \\right)^{\\frac{1}{3}} \\frac{1}{M_{pl}}\\, ,\n\\end{eqnarray}\nwhere $\\Gamma(\\frac{4}{3})$ is the Gamma function.\n\nThe size of the universe in the supersymmetry breaking state is of the \norder of\n\\begin{equation}\n\\overline R = C_1 \\left(\\frac{M_{pl}}{m_{3\/2}} \\right)^{\\frac{1}{3}} \n\\ell_{pl}\\, ,\n\\end{equation}\nwhere $\\ell_{pl}= \\frac{1}{M_{pl}}=\\sqrt{8 \\pi G_N}$ is the Planck length,\nand $C_1$ has the following value\n\\begin{equation}\nC_1= \\frac{2^{1\/6} \\sqrt{3} \\left(\\frac{3}{4} \\right)^{\\frac{1}{3}}}{2 \n\\left( \\frac{3}{8}\\right)^{\\frac{5}{6}}} \n\\frac{\\Gamma\\left(\\frac{4}{3} \\right) \n\\left[\\Gamma\\left(\\frac{2}{3} \\right)\\right]^2 }{\n\\Gamma\\left(\\frac{5}{3}\\right) \n\\left[\\Gamma\\left(\\frac{1}{3} \\right)\\right]^2   } \\approx 0.505468 \\, .\n\\end{equation}\n\n\\section{Conclusion}\nUsing the ideas of the supersymmetry breaking selection rules under local n=2 \nconformal supersymmetry, we have presented  a solution to the wave function\n$\\Psi$ of the universe for the FRW cosmological model $(k=0)$, that is \nnormalizable. For this proposal it was necessary to define one ``weighted'' \ninner product with the factor $R^{1\/2}$ and  we constructed the corresponding\nhermitian operators too.\nTherefore, it  was  straighforward  to find the behaviour of the\n spectation value \nfor the scalar factor like \n$\\overline R = C_1 \\left(M_{pl}\/m_{3\/2} \\right)^{1\/3} \n\\ell_{pl}$, giving us the size of our universe.\n\nIn the proposed framework it is also possible to include the potential of \nhybrid inflation scenario \\cite{LS,KS}, since the corresponding potential term \ncan be easily introduced in the supersymmetric case. This subject will be \nreported elsewhere. \n\n\n\\section{\\bf Acknowledgments}\nThanks to I. Lyanzuridi, E. Ivanov, S. Krivonos, \nL. Marsheva and A. Pashnev for their interest in this paper. \nThis work was supported in part by CONACyT grant 3898P-E9608.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlthough the initial mass function of metal-free stars formed out of primordial matter has not yet been determined, there is evidence that metal-free stars of low and intermediate mass were formed. \n   For example, the fact that the frequency of stars with strong carbon enhancements is much larger than in the case of population I and II stars \\citep[see also the review by Beers \\& Christlieb 2005] {ros98,luc05} is consistent with theoretical predictions. \n   One of the most prominent characteristics of models of low-mass and intermediate-mass extremely metal-poor stars (EMPSs) is that they become carbon stars at an earlier stage of evolution than the stars of younger populations \\citep[hereinafter FII00]{fuj00}.  \n  Recently, it is argued that EMP stars survived to date were formed as a secondary component in the binary systems of massive companions \\citep{Komiya07}. \n   As another example, the peculiar abundance characteristics of the most metal poor stars yet discovered \\citep{chr02} can be understood in terms of the evolutionary properties of $Z = 0$ or extremely iron-poor models in an intermediate-mass interacting binary \\citep{sud04}. \n   Thus, the evolutionary characteristics of $Z = 0$ models have direct relevance to discussions of star formation in the early universe.\n\nEMP stars may alter their initial surface abundances by bringing to the surface products of internal nuclear transformations which occur in unique ways.  \n   For initial CNO abundances $\\textrm{Z}_{\\rm CNO} \\lesssim 10^{-7}$, the outer edge of the convective zone, generated by the first off-center helium flash in the hydrogen-exhausted core extends into the hydrogen-rich layers, eventually leading to the enrichment of carbon and nitrogen in the surface (Fujimoto et al.~1990, hereinafter FIH90; Hollowell et al.~1990). \n   This ``He-flash driven deep mixing (He-FDDM)'' mechanism is distinguished from the third dredge-up in asymptotic giant branch (AGB) stars of Populations I and II which enriches surface material in $^{12}$C \\citep{ibe75}.  \n   Even if we take into account the possible surface pollution by the accretion of metal-rich gas after birth, the interior evolution of metal-poor stellar models is not altered as long as the original CNO abundance satisfies $\\textrm{Z}_{\\rm CNO} \\leq 10^{-8}$ \\citep{fuj95}.  \n\nMany calculations of Population III (hereinafter Pop.III) star evolution have been published \\citep[see FIH90; ][and references therein]{sud04}.\nBased on a top heavy initial mass function for the primordial cloud \\citet{eze71} first computed sets of massive metal-free models. \nComputations of the evolution of low mass metal-deficient stars were first carried out by \\cite{wag74} in order to provide a set of stellar lifetimes for use in calculations of galactic chemical evolution.\nComputations of low-mass, zero-metallicity star evolution were performed by \\citet{cas75} up to the exhaustion of hydrogen at the center.\n\\cite{dan82} pursued the evolution of metal-free models of mass 1 $\\ensuremath{M_\\odot}$ to the red giant branch (RGB) phase, but because of a large time step in her computation, did not find any abnormal behavior. \n\\cite{gue83} suggested that a convective instability occurs during the central hydrogen-burning phase and FIH90 found this to be the case. \nThe ``peculiar'' evolution of $Z=0$ model stars and its interpretation is revealed by the calculations of FIH90 and \\cite{hol90}, who followed the evolution of a $Z=0$ star of mass $1 \\ensuremath{M_\\odot}$ and gave explanations for the core He-H flash, a shell He-H flash, and the He-FDDM phenomenon.\nBy computing in detail the progress of the hydrogen-mixing event and the subsequent evolution, \\citet{hol90} first demonstrated that metal-free stars become CN rich carbon stars at the red giant stage.\n\\citet{cas93} and \\citet{cas96} found the core He-H flash in models of $Z=10^{-10}$, but, since their computations did not continue beyond the initiation of the helium core flash, they found no evidence for the additional events just described. \n\\cite{wei00} (hereinafter W00) also made computations of low mass metal-free stars, but terminated their computations just after the ignition of the helium core flash while the flash-driven convective zone was still growing in mass.\n\nComputations by \\citet[hereinafter S01]{sch01} support the reality\nof the He-FDDM phenomenon for low-mass zero metallicity stars, whereas computations by \\cite{sie02} (hereinafter SLL02) do not.\nWhile they found the mixing of hydrogen into the helium-flash driven convective zone, the failure of SLL02 to reproduce the He-FDDM phenomenon may be ascribed to their assumption of instantaneous mixing of elements in the helium flash convective zone; in consequence, nuclear energy released from the mixed hydrogen is distributed throughout the entire convective zone and the entropy of the zone is not built up sufficiently for the edge of the convective zone to reach deeply into the hydrogen profile.\nBecause mixing and burning of hydrogen occur on a very short timescale in the middle of the helium convective zone, following the He-FDDM event properly requires a treatment of time-dependent mixing \\citep{hol90}. \n\\citet{hol90} first derived the surface composition caused by the He-FDDM phenomenon for a $Z = 0$ model of mass $M = 1 \\ensuremath{M_\\odot}$. \n\\citet{sch02} also derived the surface composition for a model of mass $M = 0.82 \\ensuremath{M_\\odot}$ and compared the results with the observations of EMPSs.\n\nAfter the discovery of HE0107-5240 \\citep{chr02}, which held the distinction of being the most metal-poor giant known before the discovery of HE1327-2326 \\citep{fre05}, models of EMPSs were computed in an effort to determine the evolutionary history of the observed star. \nBoth \\citet[][ hereinafter P04]{pic04} and \\citet{wei04} found the He-FDDM phenomenon for $Z=0$ models of mass $M = 0.8 \\ensuremath{M_\\odot}$ and $M = 0.82 \\ensuremath{M_\\odot}$, respectively.\n\nAll but one of the works just cited find the core He-H flash; thus far, only Fujimoto and his coworkers have found the shell He-H flash driven by a violent CN-cycle burning at the base of the hydrogen-burning shell. \nWhy various groups find different evolutionary characteristics for similar masses and compositions has not yet been fully understood. \nOne reason for this is that the computation of $Z=0$ models requires not only a careful treatment of the input physics but also a careful numerical solution.\nIn particular, the treatment of convective regions requires consideration of many complicated factors including uncertain parameters. \nS01 made comparisons among models by changing parameters related to abundances, diffusion, stellar mass, and the mixing length algorithm for convection, and suggestions have been made for the causes of differences such as radiative and\/or conductive opacities and neutrino energy-loss rates (W00; S01; SLL02). \nHowever, we are not satisfied that the basic causes for differences have been pinpointed satisfactorily.\n\nIn this paper, we examine the differences in the evolutionary characteristics of low-mass $Z = 0$ models by taking into account the choice not only of the radiative and conductive opacity and of neutrino energy-loss rates, but also of nuclear reaction rates. \nThanks to the many efforts until today, useful sets of input physics are available and they have been more and more accurate for numerical computations. \n  But we think it is worth trying to check the dependence on the input physics and to elucidate the structural characteristics in detail because these efforts are still going on and because the evolution of low-mass Popualtion III stars has not been established yet. \n  In particular, as for the three alpha reaction rates, the determination of resonance states in \\nuc{12}C is still controversial \\citep{fyn05}; \n   the simulations of three body interactions are challenging, and yet, become feasible task in near future thanks to the improved computer resources \\citep[see e.g., ][and the references there]{kur05}. \n\nIn the next section, we elaborate the input physics adopted in the stellar evolution program. \nIn \\S 3, we present the result of computations of evolution of low mass $Z=0$ model stars with updated input physics, and discuss the model characteristics and their dependences on the input physics including the resonant and non-resonant reaction rates of 3 $\\alpha$ reactions. \n  In section 4, comparisons with other works with the different input physics taken into account, and, in \\S 5, we summarize conclusions.\n\n\\section{The Computational Program}\n\nThe original program to compute stellar evolution was constructed by \\citet{ibe65} and has been modified periodically \\citep[e.g., see][]{ibe75,ibe92}. \nThe size of each mass shell and the time step are regulated, respectively, by the gradients with respect to space and time of structure and composition variables.\nTypically, 200-300 mesh points are required for main sequence models and 700-800 for core helium-burning models with hydrogen-burning shells.\nThe number of equilibrium models required to follow evolution from the zero-age main sequence to the end of RGB or AGB phase varies from 5000 and 30000, with the exact number depending on initial mass. \nThe stellar structure equations are typically satisfied to better than one part in $10^{5}$.\n\nIn the program, the radiative opacity is obtained by interpolation in OPAL tables \\citep{igl96} and in tables by \\citet[][ hereinafter, AF94 tables]{ale94} and the conductive opacity is from \\citet{ito83}. \nThe equation of state involves fits by \\cite{ibe92} to work by \\cite{abe59}, \\cite{bow60}, \\cite{sla80}, \\cite{sla82}, \\cite{iye82}, \\cite{han73}, \\cite{han78}, \\cite{coh55}, and \\cite{car61}.\nNeutrino energy-loss rates are from \\citet[][ in the following I96]{ito96} for plasma, photo, pair and bremsstrahlung processes.\nIn order to compare with other works, we make use of two sets of nuclear reaction rates: those given by \\citet[][ in the following CF88]{cau88} and those given in the latest NACRE compilation \\citep{ang99}.\nNuclear screening factors for weak and strong screening are also taken into consideration using standard prescriptions \\citep[see, e.g.,][]{bohm58}, but only weak screening is dominant in the actual computational range in this work. \nNine nuclear species are considered: \\nuc{1}H, \\nuc{3}He, \\nuc{4}He, \\nuc{12}C, \\nuc{14}N, \\nuc{16}O, \\nuc{18}O, \\nuc{22}Ne, and \\nuc{25}Mg. \nAbundances of these isotopes are determined by 16 nuclear reactions which include proton, electron, and alpha captures.\n\nIn regions of high temperature and high density which are not covered by OPAL tables, we use the analytical radiative opacities from the fits by \\cite{ibe75} to \\cite{cox70a,cox70b} ones.\nAt table boundaries, no interpolation is made between table values and analytical values. \nHence, jumps in the radiative opacity occur at table boundaries, but jumps are normally smaller than a factor of 2 and do not seriously affect model convergence, primarily because, in regions not covered by the tables, the overall opacity is dominated by electron conductivity. \nAt low temperature and low density, interpolation between OPAL and AF94 tables is accomplished by setting\n$\\kappa=\\kappa_{\\rm OPAL}\\ (1-\\Theta)+\\kappa_{\\rm AF}\\ \\Theta$, where\n$\\Theta=\\sin^2{(\\pi\/2)(T-T_1)\/(T_2-T_1)}$ for the temperature $T$\nbetween $T_1=8000$ K and $T_2=10000$ K, and $\\kappa_{\\rm OPAL}$ and $\\kappa_{\\rm AF}$ are OPAL and AF94 opacities, respectively.\nIn surface layers of all models discussed here, AF94 tables provide opacities for all densities and temperatures encountered. \nIn regions of overlap, OPAL and AF94 opacities agree rather well, so switching from one table to the other introduces little uncertainty.\n\nAlthough the quantum correction to the conductivity at low temperatures has been calculated \\citep{mit84}, it is now believed that the approximation employed is not appropriate at the temperatures considered. \nTherefore, we use the analytic fits devised by I83 for the conductive opacity at low temperature; \nthe liquid metal phase for various elemental compositions as well as strong degeneracy are taken into consideration.\nStrong degeneracy prevails if $T \\ll T_{\\rm F}$, where $T_{\\rm F}(K) = 5.930 \\times 10^{9}\\\n((1+1.018(\\rho_{6}\/\\mu_e)^{2\/3} )^{1\/2} -1 )$, $\\rho_{6}$ is the\ndensity in units of $10^{6}$ g cm$^{-3}$, and $\\mu_e$ is the electron molecular weight. For the actual computations, since their approximations are good for $T \\lesssim T_{\\rm F}$, we adopt the analytical approximations of \\citet{ito83} for $T \\leq 0.5~T_{\\rm F}$.\nFor $T \\geq 2~T_{\\rm F}$, we adopt the conductive opacity used in the fits by \\citet{ibe75} to calculations by \\cite{hub69} for non-relativistic electrons and to \\cite{can70} for relativistic electrons.\nFor intermediate temperatures, interpolation between the two approximations is achieved by ``sine square'' weighting, analogous to that used in interpolating between OPAL and AF tabular opacities.\nFinally, the conductive opacity is subjected to a limit described in the Appendix.\nInterpolated conductive opacities are given in Figure~\\ref{fig:cop}, which covers the overall range of application for stellar evolution.\n\nFor neutrino energy-loss rates, we adopt the fitting formulae presented by I96, who include pair, photo, plasma, and bremsstrahlung with weak degeneracy, the liquid-metal phase with low-temperature quantum corrections and the crystalline lattice phase, and the recombination neutrino processes. \nAmong these latter processes, we do not adopt the low-temperature correction to the liquid-metal phase as calculated by \\cite{ito84} since the corrections are generally small, as stated in I96.\nFor $T \\leq 10^{7}$ K, we assume that neutrino energy-losses of all kinds can be neglected. \nThe treatment of a multicomponent gas, which is important for bremsstrahlung, is considered in applying the neutrino energy-loss rates of I96, as described in the Appendix.\n\nMass loss is neglected in our work because we are interested primarily in the stellar interior and the evolution of the helium core is not affected by modest surface mass loss. \nTo determine the temperature gradient in convective regions, we use the standard mixing length recipe by \\cite{bohm58} \\citep[see, e.g., ][]{cox68} and the mixing length is taken to be 1.5 times the local pressure scale height.\nNeither overshooting nor semi-convection are considered.\n\nIn our calculations we focus on the evolutionary trajectory up to the\nHe-FDDM event and on the thermal structure of a star at stages of interest along its path to this event; \nour computations are terminated at the onset of hydrogen mixing into the convective zone driven by a helium flash.\nThis event can occur during an off-center helium flash, \nand\/or at the beginning of the thermally pulsing AGB (TPAGB) phase, depending on the initial mass and input physics.\nSubsequent evolution after the mixing event is beyond the scope of thispaper and will be discussed in detail in a separate paper (Suda, Fujimoto, \\& Iben, in preparation).\n\n\\section{Evolution of Low-Mass Pop.III\\ Stars}\n\nThe evolution of $Z = 0$ stars has been computed from the zero-age main sequence (ZAMS) through the RGB and\/or through the TPAGB phase for model masses of $0.8 - {1.2} \\ensuremath{M_\\odot}$.\n     For all models, the initial chemical composition is $X = 0.767$, $Y = 0.233$, and $Z=0$, and the initial abundance by mass of \\nuc{3}He is $2 \\times 10^{-5}$. \nThese abundances are based on models of big bang nucleosynthesis and are the same as those chosen by FII00. \n\nTable~\\ref{tab:model} lists the models computed with all the input physics updated for this paper. \n  The first two columns give, respectively, the model identifier and the initial model mass;  the labels ``nac'' and ``cf'' mean that the model has been computed with the choice of nuclear reaction rates of NACRE and CF88, respectively.\n   The third to the ninth columns give the effective temperature and surface luminosity at the turn-off point and at the tip of the RGB, the times to reach these two stages, respectively.  \n\nFigures~\\ref{fig:hrd} and \\ref{fig:rhot} show, respectively, evolutionary tracks in the H-R diagram and in the central-density and central-temperature plane for ''cf'' and ''nac'' models of mass in the range $0.8 -1.2 \\ensuremath{M_\\odot}$. \n   All models are terminated at the onset of hydrogen mixing episodes.\n   We found that $M \\leq 1.1 \\ensuremath{M_\\odot}$ models ignite the helium burning in the off-center shell and the engulfment of hydrogen by the flash-driven convection occurs just after the peak of the major core helium-flash at the tip of RGB. \n   On the other hand, $1.2 \\ensuremath{M_\\odot}$ models undergo the helium core flash at the center without the hydrogen mixing, and encounter the hydrogen mixing in the helium-flash convection during the helium shell flash at the beginning of thermally pulsating AGB (TP-AGB) phase. \n   The two models display almost the same trajectory irrespective of the choice of nuclear reactions, with the small differences stemming from the difference in the 3$\\alpha$ reaction\\ rates, less than a factor of 2 in the temperature range $\\log T = 7.8 - 7.9$.  \n\nThe evolutionary characteristics of the core helium flash and the hydrogen mixing event are summarized in Table~\\ref{tab:he-flash}. \n   Each column gives the model identifier defined in Table~\\ref{tab:model}, the helium core mass $M_{1}\\ups{max}$ and the helium burning rate $L\\lows{He}\\ups{max}$ when the helium burning rate reaches maximum, the mass coordinate $M\\lows{BCS}$ and the maximum temperature  $T\\lows{BCS}\\ups{max}$ at the base of the convective shell, driven by the core helium flash, the helium burning rate $L\\lows{He}\\ups{mix}$ at the onset of hydrogen mixing at the RGB, and the time intervals, $\\Delta t^\\prime$ and $\\Delta t\\lows{mix}$, to it from the appearance of helium flash-driven convection and from the stage of maximum helium burning, respectively. \n   The mass $M_1$ of helium core is defined as the mass coordinate where the abundance of hydrogen is half of the surface abundance of hydrogen.\n\nIn this section, we discuss in detail the evolutionary behavior of Pop.III\\ models with respect to the differences in initial mass and input physics.\n\n\\subsection{Hydrogen Burning Phase}\n\nVariations in several quantities characterizing model stars are summarized in Figure~\\ref{fig:phys} as a function of the hydrogen abundance at the center.  \n   In a Pop.III\\ model, due to the absence of CNO catalysts, p-p chain reactions are initially the only mode of energy generation by hydrogen burning. \n   Because of the weak temperature dependence of the energy-generation rate, the central temperature keeps rising as the hydrogen abundance decreases (top panel of Fig. \\ref{fig:phys}). \n   As the central temperature increases, the 3$\\alpha$ reaction\\ gradually becomes active and CNO-cycle reactions begin to occur.  \n   Eventually, CNO-cycle reactions dominate the p-p chains with regard to total energy production. \n   When this first occurs, the central abundance by mass $X \\lows{CNO}$ of catalysts is $10^{-11} - 10^{-9}$ at $\\log T_{c} \\simeq 7.8 - 7.9$ (top and middle panels of Fig. \\ref{fig:phys}). \n   Because of lower central temperatures, the less massive the star, the later is the evolutionary stage (and the smaller is the central hydrogen abundance) at which the transition from burning dominated by the p-p chains to burning dominated by the CNO-cycle reactions takes place. \n   For $M \\leq 0.8 \\ensuremath{M_\\odot}$, hydrogen is depleted in the center before a transition can take place. \n   After the CNO cycle takes over as the main source of energy generation, because of the strong temperature dependence of CNO-cycle reactions, the central temperature remains nearly constant. \n   At the same time, the core expands because of the central concentration of energy generation and the rate of production of catalysts slows down; \n   the abundances of catalysts saturate at $X \\lows{CNO} \\simeq 10^{-10} - 10^{-8}$.\n\nThe transition to the CNO-cycle dominated phase is accompanied by the formation of a convective zone which develops outward from the center as described in the lower panel of Fig.~\\ref{fig:phys}. \n   The growth of the convective core is also evident in the middle panel of Fig.~\\ref{fig:phys}, which shows that the hydrogen abundance at the center, $X_{c}$, stops decreasing monotonically and increases for a time as evolution progresses. \n   Central convection is caused by a thermonuclear runaway and persists after the transition; due to the larger temperature dependence of the CNO-cycle energy-generation rate, energy generation is highly concentrated toward the center. \n   Both the inward mixing of hydrogen from outer hydrogen-rich layers and the outward mixing of CNO elements generated near the center during the thermonuclear runaway amplify the average hydrogen-burning rate over the region encompassed by convection relative to the average rate in the absence of convection.\n\nFor stars of mass $0.9 \\leq M \/ \\ensuremath{M_\\odot} \\leq 1.2$, the transition from the p-p chain dominated phase to the CNO-cycle dominated phase is delayed until the core has already begun to contract rapidly and electrons have begun to become degenerate at the center;  \n   the maximum energy generation has already shifted away from the center and core contraction has initiated the expansion of the envelope. \n   A thermonuclear runaway called the core helium-hydrogen (He-H) flash takes place \\citep{fuj90}. \n   In Fig.~\\ref{fig:rhot}, a first increase in the central temperature with decreasing central density indicates that the electron degeneracy is lifted; \n   then, the temperature turns to decrease with the density so as to settle in the thermal equilibrium state where the nuclear energy generation balances the energy loss from the core.   \n   In this runaway, helium burning plays a key role through the production of CNO-cycle catalysts. \n   Because of the strong temperature dependence of CNO-cycle reaction rates, the flash starts even when the contribution of the CNO cycle to the total energy-generation rate is smaller than that of p-p chain reactions in central regions.  \n   Smaller mass stars experience stronger flashes since the central entropy is smaller and the electron degeneracy is stronger at the onset of the thermonuclear runaway. \n   Thus, the temperature reaches larger, leading to a greater production of CNO catalysts and a greater extension of convection. \n   For stars of mass $M \\leq 0.8 \\ensuremath{M_\\odot}$, the central temperature does not become large enough for the production of sufficient carbon to activate the CNO cycle, irrespective of the choice of reaction rates. \n   We see a small hump along the trajectories of $M = 0.8 \\ensuremath{M_\\odot}$ models in Fig.~\\ref{fig:rhot}, which marks where the exhaustion of hydrogen at the center quenches a thermonuclear runaway.\n\nThese evolutionary characteristics during core hydrogen burning are common to all of the models, regardless of the adopted nuclear reaction rates since the evolutionary tracks prior to the transition are dominated by the p-p chain reactions with relatively small temperature dependences.  \n   All the models of masses in the range $0.9 \\leq M \/ \\ensuremath{M_\\odot} \\leq 1.2$ experience the core He-H flash.  \n   We may notice that the differences in the ignition temperatures are rather small because of the strong temperature dependence of the 3$\\alpha$ reaction\\ rate, the flash grows weaker for models of larger mass.\n\nCore convection lasts until hydrogen is almost exhausted at the center ($X_{c} \\lesssim 10^{-4}$).  \n   After the model settles in the thermal equilibrium, the central density increases again as the hydrogen abundance decreases. \n   The central temperature at first rises, and then, decreases as the nuclear energy-generation rate decreases at the center (when $X_{c} < 2 - 3 \\times 10^{-4}$). \n   Note that since the nuclear energy generation is dominated by contribution from off-center burning, the model at this phase tends to have an isothermal core with the central temperature directly reflecting the temperature in the H-burning shell. \n   In Fig.~\\ref{fig:rhot}, we can see the rise in central temperature with the growth of core, a sudden jump in temperature occurring when the hydrogen burning shell passes the shells occupied by the flash convection, and hence, have larger CNO abundances.  \n   \nThe resultant loop, as seen in Fig.~\\ref{fig:rhot}, is characteristics of flash, implying that He-H core flash eventually exerts work through the expansion and contraction of core, during which the envelope first contacts and then expands again, as seen in Fig.~\\ref{fig:hrd}. \n   In the subsequent evolution, our models do not encounter the phenomena, so called ``shell He-H flash'' at the base of hydrogen burning phase, as FIH90 discovered. \n   FIH90 discuss the behavior of hydrogen burning shell in the $Z=0$ environment and found the convective instability by the production of carbon and of nuclear energy by CN cycles. \n   This is the counterpart of core He-H flash, i.e., electrons are degenerate at the base of hydrogen burning shell, which causes He-H flash. \n   However, no other groups than Fujimoto and his collaborators find such a event. \n   The different consequence is due to the consideration of non-resonant effect in 3$\\alpha$ reaction\\ rate as we will discuss in \\S \\ref{sec:tal}. \n\n\\subsection{Helium Burning Phase}\\label{sec:heburn}\n\nCharacteristic properties of the helium core flash are given in Table~\\ref{tab:he-flash}. \n   The thermal structure of the core when the helium core flash is ignited are determined by (1) neutrino energy losses which produce cooling, (2) the radiative and conductive opacity which controls heat flow, and (3) the temperature and energy generation in the hydrogen-burning shell which affects the heating due to gravitational compression. \n   If neutrino cooling in the central region is effective enough to produce a strong positive temperature gradient, the helium core flash is ignited off center. \n   Model mass must be larger than $M \\geq 1.2 \\ensuremath{M_\\odot}$ for central helium ignition to occur (Fig.~\\ref{fig:rhot}). \n   The mass $M_{1}$ of the helium core when the helium core flash occurs differs greatly between the off-center and central ignition cases, as seen from the third column of Table~\\ref{tab:he-flash}. \n   Central ignition occurs before neutrino cooling becomes appreciable, and, hence, core masses at ignition are small:  $M_{1} \\lesssim 0.4 \\ensuremath{M_\\odot}$.  \n   Once neutrino cooling becomes effective in the central region, off-center ignition is delayed until the core mass is much larger, namely, $M_1 \\gtrsim 0.5 \\ensuremath{M_\\odot}$.\n\nFigure~\\ref{fig:str} shows the interior temperature as a function of density for models of mass 0.8 $\\ensuremath{M_\\odot}$ and core mass $M_{1} = 0.49 \\ensuremath{M_\\odot}$. \n   A constant temperature ``plateau'' spreads inward from the base of the hydrogen-burning shell. \n   This implies that the thermal structure of the helium core results mainly from the temperature in the hydrogen burning shell and neutrino energy losses in central regions \\citep{fuj84}. \n   Along with a decrease in the initial abundance of CNO elements, the hydrogen-burning rate decreases, and, in order to compensate for this, the temperature in the hydrogen-burning shell increases, which makes the contribution of compressional heating smaller. \n   Accordingly, an isothermal plateau develops in the region where the radiative heat transport dominates over electron conduction for the metallicity of $[{\\rm Fe}\/{\\rm H}] \\lesssim -5$ \\citep[see ][]{fuj95}. \n   In our models, the compressional heating plays a small part to raise the maximum temperature in the helium zone slightly higher than the temperature in the hydrogen burning shell, as seen from the lowest mass model;  \n   note that for the massive models, the helium burning already contributes appreciably to increase the temperatures in the right shoulder of structure lines in this figure.  \n   This behavior contrasts with that of model stars of younger population in which the compressional heating play the dominant part in determining the maximum temperature in the helium core and makes it much larger than the temperature in the hydrogen-burning shell.  \n   Because of high temperature in the hydrogen burning shell, therefore, $Z=0$ model stars experience central ignition at smaller initial masses than do model stars of younger populations for which the minimum initial mass for central helium ignition is $\\sim 2.5 \\ensuremath{M_\\odot}$. \n   While the density and temperature of the hydrogen burning shell are in local maximum just before the helium ignition, hot CNO-cycle is not still effective.\n   At this stage, the nuclear timescale of $\\nucm{13}{N}$ against proton capture reaction ($\\sim 10000$ sec) is much larger than that against $\\beta$-decay reaction (863 sec) and is negligible in the outcome of neither the nucleosynthesis nor the nuclear energy output.\n\nIf neutrino energy losses are sufficiently effective before helium is ignited, the central region cools, and helium is ignited off-center as is the case in low-mass stars of younger populations. \n   In the central region of the model, plasma neutrinos are the dominant neutrino energy-loss mechanism and produce a steep gradient in the rate of released energy. \n   On the other hand, the conduction, which is important in the electron-degenerate core, transport the energy towards the center where the neutrino loss works. \n   Consequently, both conductivity and neutrino energy-loss rates promote cooling of the core and, thus, delay the off-center ignition of helium until a larger core mass gives rise to a larger hydrogen-shell burning rate and a larger temperature in the hydrogen-burning shell, as seen from Table~\\ref{tab:he-flash}.   \n\nIn all of the models which experience off-center ignition, convection driven by helium burning extends into the upper hydrogen-rich layers during the decay phase of the core helium flash, because of smaller entropy in the hydrogen burning shell, as shown by \\cite{fuj90,fuj95}. \n   The ingestion of hydrogen into the helium convective zone begins a sequence of events that leads to the enrichment of the surface with carbon and nitrogen \\citep{hol90,fuj00,sch02,pic04,wei04}. \n   Characteristics of hydrogen mixing are also given in Table~\\ref{tab:he-flash}. \n   The ingestion of hydrogen occurs within a matter of days after the helium-burning luminosity reaches its peak.\n   In the models undergoing central helium burning, we do not find a hydrogen-mixing event;  \n   The core helium flash is rather weak (see $L_{\\rm He}^{\\rm max}$ in Table~\\ref{tab:he-flash}), which makes it difficult for the outer edge of the convective region driven by helium burning to reach the hydrogen-containing layer \\citep{fuj77}. \n   For the models of $M=1.2 \\ensuremath{M_\\odot}$, we follow the evolution through the thermally pulsing AGB phase to find that the He-FDDM is triggered during the helium shell flashes. \n\n\\section{Discussion and Comparison with Other Works}\n\nIn this section, we compare the models in the literature with our models adopting similar input physics to see which cause the dominant effect on the differences in the evolution and to confirm the correctness of numerical computations.\n   In addition, we compare the models without non-resonant effect for $\\alpha$-capture reactions to see the influence of uncertainty in 3$\\alpha$ reaction\\ rates on the stellar structure at $Z=0$. \n\n\\subsection{Comparison with $0.8 \\ensuremath{M_\\odot}$ Models}\n\nA model comparable with our $0.8 \\ensuremath{M_\\odot}$ model is Model~1 of \\citet[][P04]{pic04}, of composition $Y = 0.23$ and $Z = 0$.\nIn their computations, release 4.98 of FRANEC was used and time-dependent convective mixing was calculated; neutrino energy-loss rates by plasma-neutrino emission were modified, with consequences being reported as minimal.\nReaction rates and conductive opacities are, respectively, common with our models.\nThe neutrino energy-loss rates are common with our model for photo- and pair-neutrino processes, but they use energy-loss rate of \\citet{dic76} for bremsstrahlung and of \\citet{bea67} for recombination processes.  \n   For plasma neutrino energy losses, they adopt an energy-loss rate \\citep{esp03} which differs only slightly from I96 in the temperature and density ranges relevant to the ignition of the helium flash.\n\nAlthough there are some differences in adopted input physics, the evolution of the helium core flash of the P04 model is similar to that of our 08cf model. \n     The helium core mass at the onset of the core helium flash is the same in both models, namely, $M_{1} = 0.52 \\ensuremath{M_\\odot}$. \nWhen CNO-cycle reactions are the dominant contributors to the hydrogen-burning luminosity, so that the hydrogen profile is very steep, the quantity $M_{\\rm{He}}$ defined by P04 as the mass of the helium core when the maximum hydrogen-burning luminosity is reached is nearly the same as the quantity $M_{1}$ we have defined as the mass of the helium core when the core helium flash begins.  \nThe maximum helium-burning luminosities differ by less than a factor of two, being $L_{\\rm He} = \\pow{1.2}{10} L_{\\sun}$ in the P04 model and $L_{\\rm He} = \\pow{7.6}{9} L_{\\sun}$ in our model 08cf. \nThe mass at the outer edge of the convective shell at the onset of hydrogen mixing is the same in both cases, namely, $0.506 \\ensuremath{M_\\odot}$.\nThe values of $\\Delta t_{\\rm{mix}}$ (see Table \\ref{tab:he-flash}) and $X_{\\rm C}$, the mass fraction of carbon in the helium convective shell, are also comparable: \n$\\Delta t_{\\rm{mix}} = \\pow{2.1}{5}$ sec in the P04 model versus $\\pow{1.87}{5}$ sec in model 08cf, and $X_{\\rm C} = \\pow{4.15}{-2}$ in the P04 model versus $X_{\\rm C} = \\pow{4.28}{-2}$ in model 08cf.\nThere is a large difference in model characteristics when the helium convective shell first appears;  \nin the P04 model, the convective shell driven by helium burning appears at a mass shell $M_{\\rm BCS} = 0.348 \\ensuremath{M_\\odot}$ when $L_{\\rm{He}} = 0.658 L_{\\sun}$, while, in our model 08cf, $M_{\\rm BCS} = 0.3825 \\ensuremath{M_\\odot}$ when $L_{\\rm He} = \\pow{1.57}{2} L_{\\sun}$.  \n    The factor of 200 difference in the helium-burning luminosity when shell convection begins is probably simply a typographical error in P04, an interpretation reinforced by the fact that the time for the helium-burning luminosity to reach its maximum value is almost the same in the P04 model (723 yr) and in ours (715 yr).\nIt takes more than $\\sim 3\\times 10^4$ yr for the helium-burning rate to increase by a factor of 200 in this range. \n\nThe model of $0.8 \\ensuremath{M_\\odot}$ and $Z=0$ in FII00 is based on those of \\citet{fuj95} (hereafter F95) and the comparable results are given in their Table 1. \n   In F95, the values of  $M_{1} \\ups{max}$, $M_{\\rm BCS}$, and $\\ensuremath{\\log L_{\\textrm{He}}} \\ups{max} (L_\\odot) $ are $0.5116 \\ensuremath{M_\\odot}$, $0.3705 \\ensuremath{M_\\odot}$, and 9.983, respectively.  \n   This core mass $M_1$ coincides with our 08nac very closely despite the differences in the input physics;  \n   F95 took into account only the resonant 3$\\alpha$ reaction\\ reactions \\citep{aus71}, which is smaller by a factor of $\\sim 2.4$ than the NACRE rate at the relevant temperature range ($\\log T \\simeq 7.94$). \n   The smaller helium burning rate tends to delay the ignition of helium core flash.  \n   On the other hand, the I83 formulae adopted here give larger conductivity than the Iben's fitting formulae used by F95 in the region of coulomb-liquid regime where the maximum temperature in the helium zone occurs (see fig.~\\ref{fig:cop}), which works to delay the helium ignition due to the enhanced cooling of helium zone through the inward heat conduction in our model.  \n   These two effects compensate for each other, while the effect of  larger conduction is manifest in inner ignition (or in smaller  $M_{\\rm BCS}$) in the 08nac model.  \n   In actuality, the 08cf model with the same input physics as the 08nac model except for the nuclear reaction rates results in a larger core mass than the 08nac model. This is because the cf88 3$\\alpha$ reaction\\ rate is smaller by $20 \\%$ than the NACRE 3$\\alpha$ reaction\\ rate around the temperature relevant here.  \n   In any case, the dependence of $M_1$ on the nuclear reaction rate is very small because of strong temperature dependence of 3$\\alpha$ reaction\\ rate. \n   It is also worth noting that the non-resonant reaction \\citep{nom85} has little to do with the ignition of helium core flash because of rather high temperatures in the helium zone $\\log T > 7.9$, although it takes a major part in the later phase of core hydrogen burning and the omission delays the depletion of hydrogen until higher temperature is reached in the center ($\\Delta T_{\\rm c} \\simeq  0.04$ and $\\Delta \\rho_{\\rm c} \\simeq  0.25$).  \n\nSince our calculation does not follow the burning of mixed-in hydrogen, we cannot assess the results of FII00 and P04 with regard to the occurrence of a hydrogen-burning flash in the middle of helium convective zone, the splitting into two convective shells, and the merging of the upper convective shell with the surface convective zone. \n   This remains for further investigation and will be discussed in a separate paper (Suda, Fujimoto, \\& Iben, in preparation).\n\n\\subsection{Comparison with $1 \\ensuremath{M_\\odot}$ Models}\n\nIn this section, we compare our results with those of FIH90, W00, S01, and SLL02 for the evolution to the beginning of the core helium flash of models of mass $1 \\ensuremath{M_\\odot}$ and initial composition $Z=0$. \n   In all of the cited calculations, the core He-H flash occurs, although the size of the blue loop differs among the different works.\n\nWe first compare our results with the model of FIH90.  \n   The distinctive feature of FIH90 model is the He-H shell flash during the hydrogen shell-burning, as stated in the introduction. \n   We will show in the following subsection that the instability of the hydrogen shell burning and the resultant shell flash are solely attributable to the exclusion of non-resonant rate of 3$\\alpha$ reaction\\ reactions \\citep{nom85}\\footnote{The computation of FIH90 was done at the University of Tokyo Observatory in 1984 when one of authors (I.I., jr.) visited it as JSPS fellow.}.  \n   Furthermore, the FIH90 model ended in a significantly larger core mass at the ignition of helium core flash ($M_{1} = 0.528 \\ensuremath{M_\\odot}$). \n   It is even larger as compared with the $0.8 \\ensuremath{M_\\odot}$ model of FII00 with the same input physics except for the equation of state (EOS), despite the general tendency of decrease for larger stellar masses as seen in Table.~{\\ref{tab:he-flash};   \n  Since these two computations differ only in the equation of state (EOS) for a Coulomb liquid and solid among the input physics \\citep[see][for the adopted the EOS]{ibe92}, the main reason for the larger core mass may be the larger radius of the helium core in the FIH90 model (see their Table 1 in FIH90), resulting from the difference in the adopted Coulomb corrections in the EOS.  \n    A larger core radius implies a smaller gravity of the core, and hence, a smaller temperature in the hydrogen-burning shell for a given core mass, to defer the ignition of helium core flash until a larger core mass is achieved.   \n   In actuality, in FIH90 model, a He-H shell flash is postponed until the core mass grows as large as $M_1 = 0.505 \\ensuremath{M_\\odot}$ and ignited at a low density in the bottom of hydrogen burning shell under a flat configuration ($V \\gg 4$) but under non electron-degenerate conditions \\citep[see their Fig.~2 in][]{fuj82}. \n   Because of a large core mass, the He-H shell flash has little effect on the thermal state of the inner core and FIH90 find off-center ignition of the helium core flash.  \n\nNext, we compare with the work by \\citet[][ W00]{wei00} who use a code which differs from the one used by \\citet[][ S01]{sch01} with regard to the EOS \\citep{str88}, reaction rates \\citep[][which include the non-resonant term in the 3$\\alpha$ reaction\\ rate]{thi87}, and the radiative opacity \\citep[old version of OPAL,][]{rog92,igl92}. \n   The conductive opacity and the neutrino energy-loss rates are not described in their paper. \n   Since they do not follow the progress of the helium core flash, they do not find the He-FDDM event while helium is ignited off center. \n   Their value of $t \\lows{TO} = 6.31$ Gyr is close to that of our model 10cf (6.54 Gyr);  \n   the small difference may be due to the use of different versions of OPAL opacities. \n   As for the core He-H flash, their values for the location of the outer edge of the central convective zone $M_{ECS} = 0.11 \\ensuremath{M_\\odot}$ and for the helium-burning luminosity $L \\lows{He} \\ups{max} = 2.57 \\times 10^{-7} L_{\\sun}$ are similar to our values of $M_{ECS} = 0.115 \\ensuremath{M_\\odot}$ and $L \\lows{He} \\ups{max} = 3.27 \\times 10^{-7} L_{\\sun}$ for our model 10cf. \n   Element abundances at maximum nuclear burning luminosity are also comparable; \n   W00 find $X_{12} = \\pow{6.50}{-12}$, $X_{14} = \\pow{2.19}{-10}$, and $X_{16} = \\pow{2.77}{-12}$, and we find $X_{12} = \\pow{7.90}{-12}$, $X_{14} = \\pow{1.20}{-10}$, and $X_{16} = \\pow{8.44}{-13}$. \n   These abundances are influenced slightly by the choice of time step. \n   At the tip of the RGB, we can compare with their ``canonical'' model that element diffusion is not considered in their work;  \n   their $M_{1} \\ups{tip} = 0.497 \\ensuremath{M_\\odot}$ and $\\log (L_s^{\\rm tip} \/L_{\\sun}) = 2.357$ differ slightly from our 10cf model: $M_1^{\\rm tip}  = 0.5054 \\ensuremath{M_\\odot}$ and $\\log (L_s^{\\rm tip}\/L_{\\sun}) = 2.413$ (See Tables~\\ref{tab:model} and \\ref{tab:he-flash}). \n\nIn the S01 calculations, evolution is followed from the main sequence to the TPAGB phase.\n   The S01 input physics differs from ours with regard to the EOS in regions of electron degeneracy \\citep{kip90}, the energy-loss rates for photo, pair, and plasma neutrinos \\citep{mun85}, and weak screening for nuclear reactions (they use the Salpeter formula). \n   For the EOS in core regions, they adopt a simplified equation of state for a degenerate electron gas \\citep{kip90} but they provide no\nexplicit statement as to their treatment of the ion gas.\n   Their nuclear reaction rates and conductive opacities are the same as those which we have used\n\nS01 give numerical results only for the helium core flash phase.\nTheir value of $\\log (L\/L_{\\sun}) = 2.314$ at the start of the\ncore helium flash is slightly smaller than our model 10cf value and\ntheir value of $M_{1} \\ups{max} = 0.482 \\ensuremath{M_\\odot}$ is also smaller than ours. \nHowever, they find $M \\lows{BCS} = 0.151 \\ensuremath{M_\\odot}$, while our model\n10cf gives $M \\lows{BCS} = 0.3390 \\ensuremath{M_\\odot}$ and none of our other models\nignite a helium core flash with $M_{\\rm BCS}$ smaller than $0.3 \\ensuremath{M_\\odot}$,\nexcept in the case of central ignition (Table~\\ref{tab:he-flash}). \nBecause of the much larger mass between the base of\nthe convective shell and the location of the base of the hydrogen-rich layer,\nthe time required for the outer edge of the convective shell to\nreach hydrogen-rich material is much larger in the S01 model than in our models:\n$\\Delta t \\lows{mix} = 10$ yr for S01 versus ${\\Delta t}_{\\rm mix} \n= 10^{-3}$ - $10^{-2}$ yr for all our models, irrespective of the input\nphysics. \n  We note that it takes more than 1000 yr for convection generated in the center to reach the maximum extension in mass and ${\\Delta t}_{\\rm mix}$ for S01 falls in the middle of our two cases.\n\nIn an effort to reproduce the S01 results, we constructed a $1.0 \\ensuremath{M_\\odot}$ model 10cf$^\\prime$, using the fitting formula for neutrino energy-loss rates given by \\citet{mun85} which does not include neutrino bremsstrahlung. \n   The contribution of neutrino loss is rather small as compared with the gravitational energy release in the core, and yet, affects the internal structure of helium core significantly. \n    In Fig.~\\ref{fig:str}, we compare the structure line at core mass $M_1 = 0.49 \\ensuremath{M_\\odot}$ in which the neutrino energy loss rate ($L_\\nu = 0.70 L_\\odot$) is smaller than the rate of gravitational energy release ($L_g = 2.3 L_\\odot$) and the helium burning rate is still small ($L_{\\rm He} = 0.35 L_\\odot$). \n   We see that the maximum temperature shifts to the inner shell as much as $\\Delta \\log \\rho \\simeq 0.13$. \n   The reason for this inward shift is that, near the stellar center, the energy-loss rate due to neutrino bremsstrahlung is comparable to the energy-loss rate due to other neutrino processes; \n   neglect of the neutrino bremstrahlung contribution means that the cooling rate in central regions is reduced from what it would otherwise have been. \n   Accordingly, the initiation of a helium-burning thermonuclear runaway occurs for a smaller core mass than would otherwise be the case. \n   We find that, when $M_{1} = 0.4968 \\ensuremath{M_\\odot}$, helium is ignited at a mass point $M \\lows{BCS} = 0.2933 \\ensuremath{M_\\odot}$, which is about 10\\% smaller than we find for our 10cf model when neutrino bremstrahlung is included. \n   The 10\\% reduction we have found is, however, far too small to account for the S01 result but we suspect that differences in neutrino energy-loss rates are in part responsible for the small value of $M \\lows{BCS}$ found by S01. \n   Since radiative opacities, conductive opacities, and nuclear reaction rates are presumably also not responsible, differences in the EOS may be another source of the discrepancy. \n   In dense stellar matter, Coulomb corrections reduce the pressure, leading to an increase in the density and to a reduction of core radius.  \n   The increase in the gravity entails the higher temperature in the hydrogen burning shell and heats up the core.  \n   To explore this point more quantitatively, we examine conditions in the 10cf$^\\prime$ model for the same core mass as the S01 model ignites helium ($M_1 = 0.482 \\ensuremath{M_\\odot}$). \n   In Fig.~\\ref{fig:str}, we locate by filled circles the density and temperature of the hydrogen-burning shell and the density and temperature at the mass point $M_{r} \\approx 0.151 \\ensuremath{M_\\odot}$ when $M_1 = 0.482 \\ensuremath{M_\\odot}$. \n   We conclude that the S01 structure curve must be fairly different from ours in the sense that their internal core might be kept much hotter despite the neutrino energy loss, presumably due to larger Coulomb corrections. \n\nFinally, we compare the $1 \\ensuremath{M_\\odot}$ model of SLL02 with our model 10nac. \n   SLL02 provide many evolutionary tracks of zero metallicity models covering a large mass range. \n   They do not encounter the He-FDDM phenomenon and their evolutionary calculations for low mass stars extend to the AGB phase.  \n   Much of the input physics they adopt is the same as we have used to construct model ``nac''. \n   The conductive opacities, radiative opacities, nuclear reaction rates, and the mixing length parameter ($\\alpha = 1.5$) are presumably the same. \n   However, they use a different EOS and a different treatment of the nuclear screening factor \\citep{gra73}. \n   They do not comment on the choice of neutrino energy-loss rates. \n\nWith respect to the core He-H flash, the minimum carbon abundance by mass for the appearance of convection at the center in their models is $\\log X_{12C} \\simeq -11.5$ for various model masses and this agrees well with our results for lower mass models. \n   The main sequence lifetime, measured by the age at the turn-off point, is $t \\lows{TO} = 6.56$ Gyr in excellent agreement with our result of $6.53$ Gyr (see Table~\\ref{tab:model}); \n   this agreement is to be expected since both opacities and nuclear reaction rates are the same in both cases. \n   For the same reason, the CN-cycle takes over as the main energy-production mechanism at essentially the same abundance of carbon at the center: \n   $X_{c} = \\pow{5.8}{-4}$ in the SLL02 model and $\\pow{5.71}{-4}$ in ours.\n   The maximum mass of the convective core and the maximum helium-burning luminosity are very similar in the two cases: \n   $M \\lows{ECS} \\simeq 0.095 \\ensuremath{M_\\odot}$ and $L \\lows {He} \\ups{max} \\simeq 10^{-7} L_{\\sun}$ in the SLL02 model, compared with $M \\lows{ECS} = 0.104 \\ensuremath{M_\\odot}$ and $L \\lows {He} \\ups{max} \\simeq = \\pow{2.51}{-7} L_{\\sun}$ in ours. \n   At the maximum helium-burning luminosity, element abundances are $X_{12} = \\pow{6.50}{-12}$, $X_{14} = \\pow{2.19}{-10}$, and $X_{16} = \\pow{2.77}{-12}$ in the SLL02 model, compared with $X_{12} = \\pow{6.64}{-12}$, $X_{14} = \\pow{1.19}{-10}$, and $X_{16} = \\pow{8.97}{-13}$ in our model. \n\nSLL02 do not find a shell He-H flash, consistent with our results. \n   Very similar results are obtained for stellar luminosity and the mass of the helium core at the RGB tip; \n   they find $\\log L \\ups{tip} = 2.357$ and $M_{1} \\ups{tip} = 0.497 \\ensuremath{M_\\odot}$, compared with $\\log L \\ups{tip} = 2.372$ and $M_{1} \\ups{tip} = 0.4922 \\ensuremath{M_\\odot}$ for our model. \n   Core helium burning begins when $M \\lows{BCS} = 0.31 \\ensuremath{M_\\odot}$ for SLL02 and $M \\lows{BCS} = 0.3320 \\ensuremath{M_\\odot}$ in model 10nac (see Table~\\ref{tab:he-flash}). \n   SLL02 actually find that hydrogen mixes into the convective zone driven by core helium flash, but, because the hydrogen abundance that appears in the zone is so small ($X < 10^{-8}$), the luminosity due to hydrogen burning is relatively small ($L \\lows{H} \\lesssim 10^{4} L_{\\sun}$), so they neglect the effects. \n   As mentioned earlier, their result may be an artifact occasioned by their assumption that convective mixing is instantaneous.\n\nFor completeness, we discuss additional properties of our $1 M_\\odot$ model 10nac and, when possible, compare with properties of the other models. \nAt the turn-off point, our model has $M_{1} = 0.2032 \\ensuremath{M_\\odot}$ and\nthe abundance of hydrogen at the center is finite with the value\n$X_{c} = \\pow{1.57}{-2}$. \nAfter 553 Myr of evolution beyond the turnoff point, a convective core\nis formed, driven by the He-H core flash. \nStill 3.06 Myr later, the hydrogen-burning luminosity reaches a maximum\nof $L \\lows{H} \\ups{max}=24.6 L_{\\sun}$, with $L_{\\rm pp}=18.3 L_{\\sun}$\nand $L_{\\rm CN} = 6.29 L_{\\sun}$ being, respectively, the contributions of\nthe pp-chain reactions and of the CN-cycle reactions. \nThe blue loop in the H-R diagram is characterized by\n$3.795 \\leq \\log \\ensuremath{T_{\\textrm{eff}}}$ (K) $\\leq$ 3.863, and 1.27 $\\leq \\log L_{s} \\leq 1.33$. \nDuring the flash, the convective core grows to a maximum mass of\n$M \\lows{ECS} = 0.104 \\ensuremath{M_\\odot}$, which is between the masses found by\nW00 and SLL02.\nThe maximum carbon abundance is achieved at nearly the same time that\nthe convective shell achieves its maximum mass and has the value\n$X_{12} \\ups{max} = \\pow{7.146}{-12}$. \nAs evident from Fig.~\\ref{fig:rhot}, the inner CN-burning shell passes through\nthe site of this central convective zone when, at the center, \n$\\log \\rho_{c} = 5.017$ and $\\log T_{c} = 7.801$.\nA temperature plateau develops in the outer core, as is characteristic\nof all zero metallicity models, and as the core mass continues to increase,\na temperature inversion is formed \\citep{fuj84}.\nThereafter, energy transport into the center plays a crucial role\nin determining the maximum temperature in the core. \nOur model center evolves into a region of strong degeneracy, reaching the maximum central density $\\log \\rho_{c} \\ups{max} =6.074$ at $\\log T_{c} = 7.840$. \nWhen the helium flash ignites off-center, the core mass is\n$M_{1} \\ups{max} = 0.5028 \\ensuremath{M_\\odot}$ (see Table~\\ref{tab:model}) and\nthe luminosity is $L \\ups{max} = 251.6 L_\\odot$. \nAt maximum helium luminosity, the central abundances of CNO elements are\n$X_{12}=\\pow{1.18}{-5}$, $X_{14}=\\pow{5.35}{-10}$, and $X_{16}=\\pow{9.21}{-8}$.\nIn our model, $T_{c}=\\pow{5.622}{7}$ K and  $\\rho_{c}=\\pow{8.498}{5}$ g cm$^{-3}$ at this time.\n\n\\subsection{Influence of 3$\\alpha$ reaction\\ rates}}\\label{sec:tal}\n\nIn this subsection, we discuss the relevance of characteristics of 3$\\alpha$ reaction\\ rates to the evolution of zero-metallicity stars.  \n  Recently, the properties of resonances other than the so-called Hoyle resonance in \\nuc{12}C and their effects on 3$\\alpha$ reaction\\ rates have been discussed experimentally and theoretically since the compilation of NACRE \\citep[see for example,][]{ito04b,fyn05,kur05}.  \n   On the other hand, the non-resonant term of 3$\\alpha$ reaction\\ rate may also be subject to uncertainty in the cross section because the nuclear theory hardly determines the behavior of three body interactions (K. Kat$\\bar{\\rm o}$ 2007, private communication). \n   Considering these improvement in the field of nuclear physics, it is important to examine the effects of resonant and non-resonant terms of 3$\\alpha$ reaction s, separately.  \n   Indeed, for $Z=0$ models, the non-resonant term of 3$\\alpha$ reaction\\ rate can change the evolutionary behavior drastically at the hydrogen shell burning phase.\n\nAs a test for exploring the contribution of non-resonant term, we compute the models of 1.0 and $1.1 \\ensuremath{M_\\odot}$ by adopting the nuclear reaction rates of \\citet{fow75} (hereafter FCZ75) with the same other input physics as the models in this work. \n   The main difference of FCZ75 rates from the NACRE rates is the consideration of non-resonant term in 3$\\alpha$ reaction.  \n   Since FCZ75 have not yet taken into account the effect, the 3$\\alpha$ reaction\\ rate drops by far more rapidly than the NACRE rates below $\\log T \\lesssim 7.89$. \n   Other cross sections may differ by within factor of 2 or 3 \\citep{sud03} and do not affect the qualitative results. \n\nFigures~\\ref{fig:shellHeH} shows the evolutionary tracks and interior structures in the density-temperature plane for these models with the FCZ75 rates (refereed as models 10fcz and 11fcz, respectively, in the following) and compare them with those of our models with the NACRE rates.  \n   The effect of different nuclear reaction rates is apparent before the depletion of hydrogen in the center. \n   Both models with and without the non-resonant term experience the core He-H flash, the FCZ models postpone it until higher central temperature than the NACRE models; \n   in the latter models, it is ignited at the low temperatures where the non-resonant term is effective ($\\log T < 7.89$). \n   Because of stronger electron degeneracy, the FCZ75 models undergo much stronger flashes ($L_{\\rm He} = 1.06 \\times 10^4$ and $78 L_\\odot$ for models 10fcz and 11fcz, respectively), as seen from larger loops in this figure, than our models 10nac and 11nuc ($L_{\\rm He} = 24$ and $29 L_\\odot$, respectively), which entails larger extension of flash convection; \n   for models 10fcz and 11fcz, the flash convection reaches to the shells of $M_{\\rm conv} = 0.181$ and $0.149 \\ensuremath{M_\\odot}$ at the maximum extension, respectively, about twice as large as compared to our models 10nac and 11nac, with the CN abundance of $X_{\\rm CN} = 6.0 \\times 10^{-10}$ and $ 2.6 \\times 10^{-10}$, respectively. \n\nThere is also a remarkable difference in the evolutionary tracks during the hydrogen-shell burning between the models with and without the non-resonant terms. \n   In the hydrogen burning-shell, the CN-burning with carbon produced by 3$\\alpha$ reaction\\ contributes considerably to the total energy generation though the contribution to the total energy is still smaller than that of p-p chain reactions for the stars of mass $1.2 > M\/ \\ensuremath{M_\\odot} > 0.9 $, because of lower entropy in the hydrogen burning shell as compared with the stars of younger populations. \n   In the models without the non-resonant term, in particular, there form two local maxima in the energy generation rates, a narrow one associated with CN burning and localized at the base of small hydrogen abundance ($X < 0.01$), and a broad one associated with pp-chain reactions in the middle of large hydrogen abundance (nearly half of the surface hydrogen abundance). \n   When the hydrogen burning shell passes across the inner sphere, occupied by the convection during the core He-H flash, a thermonuclear runaway is triggered at the base of hydrogen-burning shell under electron degeneracy, which is a similar situation as the ignition of a core He-H flash in the center; \n   in addition, at the base of hydrogen burning shell, the pressure scale height is much smaller than the radial distance to the center, i.e., $V = r \/ \\vert d r \/ d \\ln P \\vert = G M_r \\rho \/ r P \\gg 1$, and the flat configuration also contributes to the instability \\citep{sug78}. \n   It occurs in the shell of mass $M_r = 0.212 \\ensuremath{M_\\odot}$ with $M_1 = 0.340 \\ensuremath{M_\\odot}$ for mode 10fcz and $M_r = 0.170 \\ensuremath{M_\\odot}$ with $M_1 = 0.352 \\ensuremath{M_\\odot}$ for the first shell flash of model 11fcz.\n\nOn the other hand, the models with the non-resonant term stably burn hydrogen, although electrons are degenerate ($\\Psi \\simeq 8$) at the base of the hydrogen burning shell. \n   This different behavior stems mainly from the difference in the temperature dependence of the resonant and non-resonant terms, rather than from the difference in the burning rate itself, as can be seen from the analysis of thin shell burning by \\citet[see also Fujimoto 1982]{sch65}. \n   At temperatures of $\\log T < 7.9$, the 3$\\alpha$ reaction\\ rates by FCZ75 and NACRE give a large difference in their temperature dependences when compared at the same burning rates. \n   For less massive FCZ75 models of $M \\le 0.9 \\ensuremath{M_\\odot}$, the entropy is smaller and hydrogen is burnt before carbon production becomes appreciable. \n   For FCZ75 models of mass $M \\ge 1.2 \\ensuremath{M_\\odot}$, higher temperature as well as weaker electron degeneracy tend to stabilize hydrogen shell burning; \n   in addition, as the contribution of CN burning with products of 3$\\alpha$ reaction\\ reaction as catalysis overweighs the p-p chain reactions, the pressure scaleheight grows comparable to the radial distance to the center, whicl also stabilize the shell burning by making the heat capacity negative with hydrostatic readjustment \\citep[e.g., see ][]{fuj82}.  \n   It is noted that the temperature dependence of resonant reaction rate decreases with increase in the temperature, which also stabilizes the hydrogen shell-burning in combination with the reduction in electron degeneracy when the core grows more massive than $M_1 \\gtrsim 0.4 \\ensuremath{M_\\odot}$, as is the case for 10fcz model.  \n\nDuring the shell flash, the CN-cycle burning shell expands, and the hydrogen exhausted core also expands due to the reduction in the weight of overlying layers; \n   the central temperature and density decrease almost adiabatically (Label ``A'' in Fig.~\\ref{fig:shellHeH}). \n  During the decay phase of the first shell flash, after the flash-driven convective zone disappears, the core is heated by the flow of energy from the burning shell. \n   For model 11fcz, the second shell flash is ignited at the shell of mass $M_r = 0.287 \\ensuremath{M_\\odot}$ with $M_1 = 0.374 \\ensuremath{M_\\odot}$, when the hydrogen shell-burning passes across the shells, incorporated into the convection and enriched in CN elements during the first shell flash. \n   This flash grows so strong as to drive the convection deep into the hydrogen-rich envelope up to the shell of $M_1 = 0.404 \\ensuremath{M_\\odot}$ (Label ``B''), which expedites the growth of helium core to cause the increase in central density.  \n  Since the matter in the convective zone driven by the shell flash is enriched in CNO elements, the CN-cycle reactions dominate the energy generation rate while the burning shell traverse in the site of convective zone. \n   During this stable burning phase, the growth rate of core is large because of small hydrogen abundance therein. \n   Accordingly, concomitant rapid compression of helium core increases the central temperature. \n   This enhanced growth of helium core after the shell flashes leads directly to the ignition at the center of a helium core flash (Label ``C'') with a small core masses.  \n  This is the case if the convective zone is sufficiently large, as in the case for the second shell flashes of 11fcz model, while it is barely missed in 10fcz model.  \n\nThe most important point by this test is that the contribution of the non-resonant term is discernible in the circumstance of $Z=0$, although it may be difficult to detect differences by the observations. \n   There is a difference in the mass range that the off-center helium core flashes occur. \n   If the non-resonant term is included in the 3$\\alpha$ reaction\\ rate, the central helium burning occurs at $\\geq 1.2 \\ensuremath{M_\\odot}$. \n   Otherwise, it occurs at $\\geq 1.0 \\ensuremath{M_\\odot}$ if we use the smaller conductivity by Iben's approximates to \\citet{hub69} and by \\citet{can70} and the neutrino loss rates by \\citet{bea67} as in FII00, both slightly smaller than ours. \n   Since the lifetime of $1.0 \\ensuremath{M_\\odot}$ model is $\\sim$ 7 Gyr (Table~\\ref{tab:model}), such stars cannot be seen in the present halo if they were born. \n   Only clue to those objects will be binary mass transfer between giants of mass $\\ge1.0 \\ensuremath{M_\\odot}$ and dwarfs of mass $\\le 0.8 \\ensuremath{M_\\odot}$. \n   Since FII00 predict the abundance ratio of C\/N $\\sim 1$ for He-FDDM at RGB, while C\/N $\\gtrsim 5$ for He-FDDM at AGB, it will be crucial for the constraint on the estimate of nuclear reaction rate to determine the abundance ratio of C\/N and the mass of the primary for $Z=0$.\n\n\\section{Conclusions}\n\nWe have explored the evolution of low-mass, zero-metallicity stars with the most recent input physics.   \n  The mass range is $0.8 \\ensuremath{M_\\odot} \\leq M \\leq 1.2 \\ensuremath{M_\\odot}$ in step of $0.1 \\ensuremath{M_\\odot}$ and the initial composition is $X = 0.767$, $Y=0.233$, and $Z=0$.  \n  Calculations extend from the zero-age main sequence to the beginning of hydrogen mixing into the helium convective region on the RGB or at the start of the TPAGB phase, depending on the characteristics of the helium core flash.\n\n\\begin{enumerate}\n\n\\item The emergence of CN-cycle reactions as important contributors to nuclear energy production occurs during the core hydrogen-burning phase in models of mass $M \\geq 0.9 \\ensuremath{M_\\odot}$ in consequence of the formation of carbon by the 3$\\alpha$ reaction. This phenomenon is independent of the adopted physics and its importance is a function only of the initial mass and metallicity. \n\n\\item The models of $M \\leq 1.1 \\ensuremath{M_\\odot}$ undergo an off-center helium flash and hydrogen-mixing into helium flash-convection, leading to helium-flash driven deep mixing at the tip of red giant branch.  \n   On the other hand, the models of mass $M \\ge 1.2\\ensuremath{M_\\odot}$ ignite the core helium flash at the center and postpone the He-FDDM until the helium shell flashes occur during the early phase of thermal pulsation at the asymptotic giant branch. \n   For models of mass $0.8 \\ensuremath{M_\\odot}$ and $0.9 \\ensuremath{M_\\odot}$, our results coincide qualitatively with those first found by \\cite{fuj90} and \\citet{fuj00}. \n   For $1.0 \\ensuremath{M_\\odot} \\leq M \\leq 1.1 \\ensuremath{M_\\odot}$, whether or not a helium core flash is ignited off the center and hydrogen is mixed inward into the convective zone driven by it depend on the adopted nuclear reaction rates.   \n\\end{enumerate}\n\nWe have also compared our results with those of other investigations. Our models made with the most up-to-date input physics agree well with the models of \\citet{pic04}, \\citet{wei00} and \\cite{sie02} during evolution on the main sequence and the RGB. \n   In particular, we obtain nearly the same results as do \\cite{sie02}, although we do not follow evolution after the mixing of hydrogen into the helium flash driven convective zone at the beginning of the helium core flash. \n   To check the behavior of the He-FDDM event, we need to treat mixing with a time-dependent algorithm. \n   On the other hand, even after adopting the same radiative opacities, conductive opacities, nuclear reaction rates, and neutrino energy-loss rates as \\cite{sch01}, we are not able to obtain the inner ignition at the onset of core helium burning which they find; \n  we suspect that the discrepancy may be due to differences in the EOS for the Coulomb corrections in the liquid and solid states and due to the neglect in their work of neutrino energy losses associated with neutrino bremsstrahlung.\n\nBy treating the resonant and non-resonant rates of 3$\\alpha$ reaction s separately, we demonstrate that the non-resonant term plays a critical role in the low-mass, zero-metal stars. \n   The neglect of non-resonant term causes the lower border in mass of central helium burning, i.e., $\\geq 1.0 \\ensuremath{M_\\odot}$. \n   This explains the discrepancy of the results between ours and \\citet{fuj00}, which stems mainly from the difference in the temperature dependence rather than in the energy generation rates themselves.  \n   We first point out the possibility of discerning the effect of non-resonant term of 3$\\alpha$ reaction\\ from the evolution of stars other than at low temperature regime in the accreting degenerate stars \\citep{nom85}.  \n   It is important to precisely determine the abundances and the properties of extremely metal-poor stars to constrain the nuclear reaction rates. \n\n\\acknowledgments\n\nWe are grateful to I. Iben Jr. for improving and revising our manuscript. \n   We wish to thank A. Ohnishi and K. Kat$\\bar{\\rm o}$ for valuable comments on uncertainties on nuclear reaction rates. \n   This work is part of a PhD. thesis constructed at Hokkaido University and is in part supported by a Grant-in-Aid for Science Research from the Japanese Society for the Promotion of Science (15204010, 18104003).   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplemental Material }\n\n\n\n\\maketitle\n\\renewcommand{\\theequation}{S\\arabic{equation}}\n\\setcounter{equation}{0}\n\\renewcommand{\\thefigure}{S\\arabic{figure}}\n\\renewcommand{\\figurename}{Supplementary Fig.}\n\n\\setcounter{figure}{0}\n\\renewcommand{\\thesection}{S\\arabic{section}}\n\\setcounter{section}{0}\n\nIn this Supplemental Material, we present additional details and calculations regarding: 1) derivation of the evolution operator for the left and right state vectors in the non-Hermitian case; 2) Fourier transformation of the anomalous Green's functions; 3) electron-electron interaction mediated by phonons from two bands with strong asymmetric interaction. \n\n\\section{S1. Derivation of evolution operators for left and right state vectors in non-Hermitian case}\nIn this section, we derive the evolution operators for the left and right state vectors. As we consider the non-Hermitian case, it is not obvious how they look like.\n\nThe $\\eta$-pseudo-Hermiticity is defined as \\cite{mostafazadeh:jmp02}\n\\begin{eqnarray}\n\\label{eq:etaH}\nH^\\dagger=\\eta H\\eta^{-1},\n\\end{eqnarray}\nwhere $\\eta$ is a Hermitian and invertible linear operator. Then the $\\eta$-product is defined as\n\\begin{eqnarray}\n\\langle\\Phi|\\Phi'\\rangle_\\eta=\\langle\\Phi|\\eta|\\Phi'\\rangle.\n\\end{eqnarray}\nThe $\\eta$-product can be derived from the definition of pseudo-Hermiticity and Schr\\\"odinger equation as follows. The Schr\\\"odinger equation for the right state vector is\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t}|\\Phi_R\\rangle=H|\\Phi_R\\rangle.\n\\end{eqnarray}\nIf we take Hermitian conjugation of it and substitute the definition of $H^\\dagger$ from Eq. (\\ref{eq:etaH}), we obtain\n\\begin{eqnarray}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_R|\\eta=\\langle\\Phi_R|\\eta H,\n\\end{eqnarray}\nthat gives\n\\begin{eqnarray}\n\\label{eq:SchroedPhiL}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_L|=\\langle\\Phi_L|H.\n\\end{eqnarray}\nNow let's derive the evolution operator in the interaction representation. For that, we represent the Hamiltonian in free and perturbation parts, $H=H_0+V$. Then, we use the Schr\\\"odinger equation following the standard procedure:\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t}[e^{-iH_0t}|\\Phi^I_R\\rangle]=i[-iH_0e^{-iH_0t}|\\Phi^I_R\\rangle+e^{-iH_0t}\\partial_t|\\Phi^I_R\\rangle ]=(H_0+V)|\\Phi_R\\rangle.\n\\end{eqnarray}\nThe terms with $H_0$ cancel each other and we obtain\n\\begin{eqnarray}\ni\\partial_t|\\Phi^I_R\\rangle =e^{iH_0t}Ve^{-iH_0t}e^{iH_0t}|\\Phi_R\\rangle=V^I|\\Phi^I_R\\rangle.\n\\end{eqnarray}\nThis is a standard expression of the Schr\\\"odinger equation in the interaction picture. We can do exactly the same with Eq. (\\ref{eq:SchroedPhiL}) with the only change that $\\langle\\Phi_L^I|=\\langle\\Phi_L|e^{-iH_0t}$ and obtain\n\\begin{eqnarray}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_L^I|=\\langle\\Phi_L|e^{-iH_0t}e^{iH_0t} Ve^{-iH_0t}=\\langle\\Phi_L^I|V^I.\n\\end{eqnarray}\nthe evolution operator for $\\langle \\Phi_L^I|$ is\n\\begin{eqnarray}\n\\mathcal{S}_L=\\bar{T}e^{i\\int_0^td t'V^I(t')},\n\\end{eqnarray}\nand for $|\\Phi_R^I\\rangle$ is\n\\begin{eqnarray}\n\\mathcal{S}_R=Te^{-i\\int_0^td t'V^I(t')}.\n\\end{eqnarray}\n\n\\section{S2. Fourier transformation of the anomalous Green's functions}\nThe poles of the Green's functions derived in Eq. (8) of the main text are $\\omega_{1,2}=\\pm\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}$. We perform the corresponding integration following the standard procedure and firstly add $\\pm i\\eta$ to the denominators of the Green's functions with $\\eta$ being infinitesimally small, $\\eta\\rightarrow+0$. As the electron Green's function usually has $i\\eta$ and the hole Green's function has $-i\\eta$, we put $i\\eta$ to the denominator of $\\bar{F}$ and $-i\\eta$ to the denominator of $F$. If we express denominators in terms of products and expand the square roots in $\\eta$, we obtain\n\\begin{eqnarray}\n\t\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(\\omega)&=&\\frac{\\Delta_{\\bf k}}{[\\omega-\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}+i\\eta][\\omega+\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}-i\\eta]}, \\\\\n\tF_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(\\omega)&=&\\frac{-\\Delta_{\\bf k}}{[\\omega-\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}-i\\eta][\\omega+\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}+i\\eta]}.\n\\end{eqnarray}\nThis implies that $\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}$ has a pole $\\omega_2+i\\eta$ and $F_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}$ has a pole $\\omega_1+i\\eta$ in the upper half plane. Building a contour of half-circle shape with the radius $R\\rightarrow\\infty$ and employing residues (see e.g. Ref. \\onlinecite{kornich:prr21}), we obtain for the anomalous Green's functions in the time domain\n\\begin{eqnarray}\n\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(t)&=&-\\frac{i\\Delta_{\\bf k}e^{-it\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}}{2\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}},\\\\\nF_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(t)&=&-\\frac{i\\Delta_{\\bf k}e^{it\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}}{2\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}.\n\\end{eqnarray}\nWe can see that $\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(t)\\neq (F_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(t))^\\dagger$.\n\n\n\n\n\\section{S3. Derivation of Electron-electron interaction mediated by phonons from two interacting bands}\nLet's consider electrons interacting with phonons. Then, the partition function is given by\n\\begin{eqnarray}\n\\mathcal{Z}=\\int \\mathcal{D}[\\bar{\\psi},\\psi,\\bar{a},a]\\exp{[i(S_{ph}+S_e+S_{e-ph})]},\n\\end{eqnarray}\nwhere $S_e$ is the action of electrons with $\\psi$ and $\\bar{\\psi}$ being Grassmann fields of electrons and $S_{ph}$ is the action of phonons with $\\bar{a}$ and $a$ being bosonic fields of phonons. We make a common assumption that $\\langle a\\rangle=\\langle\\bar{a}\\rangle=0$. The action $S_{e-ph}$ describes electron-phonon interaction.\n\nLet's consider $S_{e-ph}$ as a perturbation, and expand the exponent in it. Then, we average over phononic degrees of freedom assuming that $S_{e-ph}$ is linear in $a$ and $\\bar{a}$, and thus all odd powers of $S_{e-ph}$ are zero. In such a way, we obtain\n\\begin{eqnarray}\n\\mathcal{Z}=\\int \\mathcal{D}[\\bar{\\psi},\\psi,\\bar{a},a]\\exp{[i(S_{ph}+S_e)]}(1+iS_{e-ph}-\\frac{1}{2}S_{e-ph}S_{e-ph}+...)=\\int \\mathcal{D}[\\bar{\\psi},\\psi]\\exp{[iS_e-\\frac{1}{2}\\langle S_{e-ph}S_{e-ph}\\rangle]}.\\ \\ \\\n\\end{eqnarray}\nThe last term is the action for electron-electron interaction mediated by phonons $S_{e-e}=i\\langle S_{e-ph}S_{e-ph}\\rangle\/2$.\n\nNow we consider the phonon action in detail. We assume that there are two phonon bands $\\omega_{1,{\\bf q}}$ and $\\omega_{2,{\\bf q}}$ that interact with the off-diagonal element $\\delta_{\\omega,{\\bf q}}$:\n\\begin{eqnarray}\nS_{ph}=\\int \\begin{pmatrix}\\bar{a}_{1,\\omega,{\\bf q}} & \\bar{a}_{2,\\omega,{\\bf q}}\\end{pmatrix}\\begin{pmatrix}\\omega-\\omega_{1,{\\bf q}} & -\\delta_{\\omega,{\\bf q}}\\\\ -\\delta_{\\omega,{\\bf q}} & \\omega-\\omega_{2,{\\bf q}}\\end{pmatrix}\\begin{pmatrix}a_{1,\\omega,{\\bf q}} \\\\ a_{2,\\omega,{\\bf q}}\\end{pmatrix}d\\omega d{\\bf q}.\n\\end{eqnarray}\nThe electron-phonon interaction is described as\n\\begin{eqnarray}\nS_{e-ph}=\\int d\\omega d{\\bf q}F_{\\omega,{\\bf q}}\\sum_{j=1,2}(a_{j,\\omega,{\\bf q}}+\\bar{a}_{j,-\\omega,-{\\bf q}})\\int d\\Xi d{\\bf K}\\bar{\\psi}_{\\omega+\\Xi,{\\bf K}+{\\bf q}}\\psi_{\\Xi,{\\bf K}}=\\int d\\omega d{\\bf q}F_{\\omega,{\\bf q}}\\sum_{j=1,2}(a_{j,\\omega,{\\bf q}}+\\bar{a}_{j,-\\omega,-{\\bf q}})\\rho_{\\omega,{\\bf q}}.\\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{eqnarray}\nHere, $F_{\\omega,{\\bf q}}$ is the prefactor that usually contains material characteristics and constants, for instance, elastic constants. This gives\n\\begin{eqnarray}\n\\langle S_{e-ph}S_{e-ph}\\rangle=\\int \\mathcal{D}[\\bar{a},a]e^{iS_{ph}}S_{e-ph}S_{e-ph}=\\\\ \\nonumber=\\int d{\\bf q}d{\\bf q'}d\\omega d\\omega'F_{\\omega,{\\bf q}}F_{\\omega',{\\bf q'}}\\rho_{\\omega,{\\bf q}}\\rho_{\\omega',{\\bf q'}}\\int \\mathcal{D}[\\bar{a},a]\\sum_{i,j=1,2}[a_{i,\\omega,{\\bf q}}\\bar{a}_{j,-\\omega',-{\\bf q}'}+\\bar{a}_{i,-\\omega,-{\\bf q}}a_{j,\\omega',{\\bf q}'}] e^{iS_{ph}}.\n\\end{eqnarray}\nNow we will use the property that\n\\begin{eqnarray}\n\\langle\\bar{a}_{\\bf q_1}a_{\\bf q_2}\\rangle=\\int \\mathcal{D}[\\bar{a},a]\\bar{a}_{\\bf q_1}a_{\\bf q_2}e^{-\\sum_q\\bar{a}_{\\bf q}A_{\\bf q}a_{\\bf q}}=\\frac{\\delta_{{\\bf q}_1,{\\bf q}_2}}{A_{{\\bf q}_1}},\n\\end{eqnarray}\nin order to perform averaging. After averaging, we obtain\n\\begin{eqnarray}\n\\langle S_{e-ph}S_{e-ph}\\rangle&=&i\\int d{\\bf q}d\\omega F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\rho_{\\omega,\\bf q}\\rho_{-\\omega,-{\\bf q}}\\times \\\\ \\nonumber&&\\left[\\frac{\\omega_{1,{\\bf q}}+\\omega_{2,{\\bf q}}-2\\omega-2\\delta_{\\omega,{\\bf q}}}{\\delta_{\\omega,{\\bf q}}^2-(\\omega-\\omega_{1,{\\bf q}})(\\omega-\\omega_{2,{\\bf q}})}+\\frac{\\omega_{1,-{\\bf q}}+\\omega_{2,-{\\bf q}}+2\\omega-2\\delta_{-\\omega,-{\\bf q}}}{\\delta_{-\\omega,-{\\bf q}}^2-(\\omega+\\omega_{1,-{\\bf q}})(\\omega+\\omega_{2,-{\\bf q}})}\\right].\\ \\ \\ \n\\end{eqnarray}\nThis yields the expression for the electron-electron interaction potential that we state in the main text in Eq. (17):\n\\begin{eqnarray}\nV(\\omega,{\\bf q})=-\\frac{F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}}{2}\\left[\\frac{\\omega_{1,{\\bf q}}+\\omega_{2,{\\bf q}}-2\\omega-2\\delta_{\\omega,{\\bf q}}}{\\delta_{\\omega,{\\bf q}}^2-(\\omega-\\omega_{1,{\\bf q}})(\\omega-\\omega_{2,{\\bf q}})}+\\frac{\\omega_{1,-{\\bf q}}+\\omega_{2,-{\\bf q}}+2\\omega-2\\delta_{-\\omega,-{\\bf q}}}{\\delta_{-\\omega,-{\\bf q}}^2-(\\omega+\\omega_{1,-{\\bf q}})(\\omega+\\omega_{2,-{\\bf q}})}\\right].\n\\end{eqnarray}\nNow let's assume that $\\omega_{1,\\pm{\\bf q}},\\omega_{2,\\pm{\\bf q}}\\ll \\delta_{\\omega,{\\bf q}},\\delta_{-\\omega,-{\\bf q}},\\omega$. Then we obtain\n\\begin{eqnarray}\nV(\\omega,{\\bf q})\\simeq F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\left[\\frac{1}{\\delta_{\\omega,{\\bf q}}-\\omega}+\\frac{1}{\\delta_{-\\omega,-{\\bf q}}+\\omega}\\right]=F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\left[\\frac{\\delta_{-\\omega,-{\\bf q}}+\\delta_{\\omega,{\\bf q}}}{(\\delta_{\\omega,{\\bf q}}-\\omega)(\\delta_{-\\omega,-{\\bf q}}+\\omega)}\\right].\n\\end{eqnarray}\nIf the phonon interaction is strongly asymmetric in momentum space, $\\delta_{\\pm\\omega,{\\bf q}},\\omega\\gg\\delta_{\\pm\\omega,-{\\bf q}}$, then\n\\begin{eqnarray}\nV(\\omega,{\\bf q})&\\simeq& F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\frac{\\delta_{\\omega,{\\bf q}}}{\\omega(\\delta_{\\omega,{\\bf q}}-\\omega)},\\\\\nV(\\omega,-{\\bf q})&\\simeq& F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\frac{\\delta_{-\\omega,{\\bf q}}}{-\\omega(\\delta_{-\\omega,{\\bf q}}+\\omega)}.\n\\end{eqnarray}\nIf we further assume, $\\delta_{\\pm\\omega,{\\bf q}}\\gg \\omega$, we obtain Eq. (18) from the main text,\n\\begin{eqnarray}\nV(\\omega,{\\bf q})\\simeq -V(\\omega,-{\\bf q})\\simeq\\frac{F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}}{\\omega}.\n\\end{eqnarray}\nThus we obtain odd or asymmetric electron-electron interaction potential due to an externally-induced asymmetry in the phonon system, namely, the phonon-phonon interaction. The band structure of phonons, $\\omega_{1,{\\bf q}}$ and $\\omega_{2,{\\bf q}}$, can also be asymmetric, but here we have not used this property.\n\n\n\n\n\\end{widetext}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nNon-radial pulsations (NRPs) are commonly found in isolated white dwarfs (WDs) of DA type, so called ZZ Ceti stars. These stars have hydrogen-rich atmospheres, and pulsations occur as the WD cools and passed through a phase of pulsational instability, detected mainly as g-modes (\\citealt{2006AJ....132..831G}).         \n\nDuring the last decennium, similar signals, generally interpreted as non-radial WD pulsations, have also been detected in faint cataclysmic variables (CVs). A CV is a close binary system where a late-type main-sequence star looses mass to a primary white dwarf. The first CV proposed to harbour a pulsating white dwarf was GW Librae.~\\cite{1998IAUS..185..321W} found rapid, periodic, and non-commensurate signals in its light curve, suggesting non-radial pulsations of the underlying white dwarf. In most CVs, the accretion energy tends to dominate the luminosity, and the white dwarf itself, shining with $M_{V} \\sim$ 10 -- 13, is seldom seen. However, for some of the most intrinsically faint CVs, spectroscopy and time-series photometry can reveal signatures of the underlying white dwarf, such as broad absorption features in the spectrum, sharp eclipses, and sometimes non-radial pulsations in the light curve. These signals have now been detected in about a dozen CVs, all quiescent systems of low luminosity. Here, we call these systems \\emph{GW Lib stars}, after the first discovery.~\\cite{2010ApJ...710...64S} and~\\cite{2009JPhCS.172a2069M} present recent reviews of this group of stars. \n\nAccreting WDs are different from isolated ones since they are being exposed to mass transfer, giving them atmospheres of solar-composition. The white dwarfs in CVs are therefore hotter and are also found to be spinning faster compared to isolated ones (\\citealt{2009ASPC..404..229S}). Studying these systems will provide important information of how the process of accretion is affecting the evolution of the white dwarf. In isolated WDs, pulsations are only observed in stars with temperatures located within a so-called \\emph{instability strip} in the $\\log g$ -- $T_{\\text{eff}}$ plane, spanning the temperature range $T_{\\text{eff}}$ = 10900 K -- 12200 K (see Figure 3 of~\\citealt{2006AJ....132..831G}). However, there is no clear instability strip for the GW Lib stars (see Figure 13 of~\\citealt{2010ApJ...710...64S}), and pulsations are found in systems with WD effective temperatures up to at least 15000 K. \n\nOur theoretical understanding of the mechanism for exciting non-radial pulsations in CVs is still limited. The observed pulse amplitudes are quite variable, and in some ZZ Ceti stars this is known to be the result of the beating of two signals closely spaced in frequency, a classic and highly informative signature of non-radial pulsations. But the GW Lib stars have not yet clearly revealed this kind of behaviour (although a hint of it emerged in the V386 Ser campaign reported by~\\citealt{2010ApJ...714.1702M}).  \n\nWe here present time-series photometry of two more CVs which are probably members of the GW Lib class. Both systems have very low accretion luminosity, and show signatures from the white dwarf in their optical spectra. Also, both systems show double-humped orbital signals, and non-commensurate periodic signals, suggesting non-radial pulsations. One is SDSS J1457+51 which has a photometric wave suggesting an orbital period of $77.885 \\pm 0.007$ minutes. The other is BW Sculptoris, with $P_{\\text{orb}} = 78.22639 \\pm 0.00003$ minutes. Both stars show main pulsations near 10 and 20 minutes. These rapid signals drift slightly in frequency, and may consist of several, finely spaced components. BW Sculptoris also shows a remarkable photometric variation at 87 minutes, which could be explained as a \\emph{quiescent superhump}, possibly arising from a 2:1 orbital resonance in the accretion disc.\n\n\\section{SDSS J1457+51}\n\nSDSS J1457+51 (hereafter J1457) was first identified in the Sloan Digital Sky Survey by~\\cite{2005AJ....129.2386S}. They obtained spectroscopy that showed the broad absorption characteristics of a white dwarf, indicating a system of low accretion rate. The source was found to be faint ($g \\approx 19.5$). Due to the double-peaked nature of the emission lines, they suggested the system to be of high inclination.\n\n\n\n\\subsection{Observations, Data Reduction and Analysis} \\label{obs}\n\nTime-resolved photometry of J1457 was obtained with the 1.3 m and 2.4 m MDM telescopes at the Kitt Peak observatory, Arizona, during April and May 2010. The star was observed during 14 nights in total, spread over 47 days. With all data coming from the same terrestrial longitude, we were not immune from aliasing problems and therefore strove to obtain the longest possible nightly time series (generally $\\sim$ 4 - 7 hours). Weather conditions such as clouds, snow and full moon prevented us from obtaining times series over more than 4 consecutive nights at a time. The time resolution was in the range of 20 s -- 30 s. Table~\\ref{tab:obs1} presents a log of the observations. A clear filter with a blue cutoff was used to minimise differential-extinction effects and allow for a good throughput.\n\nThe data reduction was done in real time during the observations, using standard \\textsc{Iraf} routines. The data consisted of differential photometry with respect to the field star USNO A2.0:1350-08528847. The search for periodic signals was initially done for single nights separately, and Lomb-Scargle periodograms (\\citealt{1982ApJ...263..835S}) were constructed. Formal flux errors were rescaled so that the $\\chi^{2}_{\\nu} \\approx$ 1. This was done by fitting a fake light curve to the original data, composed of multiple sine waves with periods corresponding to the strongest signals found in the single-night power spectrum. Monte-Carlo simulations were performed on every peak of interest in the power spectrum to find the period and its error. In this method, the peak errors are found by randomly re-distributing the points in the light curve within their errors, a repeated number of times, and constructing a Lomb-Scargle periodogram each time. The 1-$\\sigma$ error is then found by fitting a Gaussian to the output distribution of the peaks found in the periodograms. Data from several nights were then combined to allow the search for signals with lower amplitude, and also to improve the frequency resolution. Bootstrap analysis was performed to distinguish between the most likely alias, and was also used to find errors on the peaks. In this method, the sampling pattern of the data is changed by creating a mock dataset where every point from the original dataset is randomised. This is done to efficiently destroy or weaken the aliases pattern.\n\n\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{llllll}\n  \\hline\n  \\hline\n\\textbf{Date} & \\textbf{HJD} & \\textbf{Telescope} & \\textbf{Length} \\\\\n& (-2455000) &  & (hours) \\\\\n  \\hline\n100414 & 301 &  1.3m &  2.30 \\\\\n100416 & 303   & 1.3m & 7.89 \\\\\n100417 & 304   & 1.3m & 4.32 \\\\\n100419 & 306   & 1.3m   & 4.72 \\\\\n100424 & 311    & 1.3m & 5.42 \\\\\n100425 & 312    & 1.3m  & 7.82 \\\\\n100503 & 320  &  1.3m  & 7.92 \\\\\n100504 & 321   & 1.3m  & 8.26 \\\\\n100506 & 323   & 1.3m  & 8.40 \\\\\n100507 & 324   & 1.3m  & 3.50 \\\\\n100508 & 325   & 1.3m  & 5.47 \\\\\n100528 & 345   & 2.4m  & 4.97 \\\\\n100529 & 346   & 2.4m  & 8.11 \\\\\n\\hline\n  \\hline\n\\end{tabular}\n \\caption{Summary observing log for SDSS J1457+51. Data was \\newline obtained at the MDM observatory during April and May 2010.}\n \\label{tab:obs1} \n\\end{center}\n\\end{table}\n \n\n\n    \n\\begin{figure*}\n\\centering\n\\subfigure[The mean Lomb-Scargle periodogram for J1457, from the six nights of best quality. The orbital period at 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$, 77.9 min) and its first harmonic ($2\\omega_{\\text{o}}$) are plotted as solid lines.]\n{\n    \\label{fig:s_avpow}\n    \\includegraphics[width=7.75cm]{pow_best2_mean.eps}\n}\n\\hspace{0.2cm}\n\\subfigure[Normalised and smoothed light curve of J1547 from one sample night. A model light-curve constructed from the four strongest periods (including $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$) found in Figure~\\ref{fig:s_avpow}, is plotted together with the data.]\n{\n    \\label{fig:s_light}\n    \\includegraphics[width=7.12cm]{light_fit_320.eps}\n}\n\\caption{}\n\\label{fig:sub}\n\\end{figure*}\n    \n    \n\n\\subsection{Light Curve and average Power Spectrum}\n\nThe mean power spectrum, averaged over the six best nights is shown in Figure~\\ref{fig:s_avpow}. The 18.48 c\\,d$^{-1}$ (77.9 min) and 36.97 c\\,d$^{-1}$ (38.9 min) signals are almost certainly the orbital frequency ($\\omega_{\\text{o}}$) and its first harmonic ($2\\omega_{\\text{o}}$). This kind of variation at $2\\omega_{\\text{o}}$ is commonly seen in the orbital light curves of CVs with low accretion rates (for instance in WZ Sge; see~\\citealt{2002PASP..114..721P}). We find that the signals in the range 142 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$ ($\\approx$ 10 min) are non-commensurate with the orbital frequency. These peaks vary slightly in period and amplitude, when present at all in the nightly power spectrum. The complex structure around them indicate either an unresolved fine structure, or periods varying from night to night, and is discussed in detail below. Also, during a few nights, peaks were found at 135 c\\,d$^{-1}$ (10.7 min) and 72 c\\,d$^{-1}$ (20 min), which are also non-commensurate with the orbital frequency. The lower region of the power spectrum show strong peaks at 4 c\\,d$^{-1}$ -- 6 c\\,d$^{-1}$ (4 -- 6 hours), corresponding to the typical length of a nightly observing run. The unit c\\,d$^{-1}$ is used throughout the paper since it is the natural unit for the sampling pattern of the multi-day light curves. Also, it clearly shows the natural daily alias pattern.  \n\nThe average power spectrum in Figure~\\ref{fig:s_avpow} has low resolution since each night is less than 8 hours long. The power spectrum of a spliced light curve spanning several nights (the coherent power spectrum) is in principle better, since the resolution is always near 0.1\\,N$^{-1}$ c\\,d$^{-1}$, where N is the duration in days. It does, however, make the assumption that a candidate periodic signal is constant in period, phase, and amplitude over the duration of the observation. Power spectra will be difficult or impossible to interpret correctly when this assumption is grossly violated.\n\nFigure~\\ref{fig:s_light} shows the normalised and smoothed light curve for a sample night. A model light curve constructed from the four strongest peaks found in the power spectrum that night (including both $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$), is plotted on top of the smoothed light curve. Peaks found at higher frequencies than 40 c\\,d$^{-1}$ are not represented in the model light curve. During one of the observing nights, we obtained multicolour data and found the brightness of of the star to be V = 19.2 $\\pm$ 0.2. The mean brightness on each night was constant within the measurement error of 0.03 mag. Flickering, an essentially universal feature of CV light-curves, was very low. This together with the fact that absorption lines are seen in the optical spectrum implies that the total light, 4000\\,\\AA\\, -- \\,7000\\,\\AA\\, is dominated by the white dwarf. Measurements of the absorption line depth of the hydrogen lines indicate that probably no more than half of the total light comes from accretion processes.\n\n\n\n\\begin{figure*}\n\\centering\n\\subfigure[The power spectrum of J1457 from 11 nights, showing the orbital signal $\\omega_{\\text{o}}$ and its first harmonic, $2\\omega_{\\text{o}}$. A zoom of the region around $\\omega_{\\text{o}}$ is plotted on top.]\n{\n    \\label{fig:s_poworb}\n    \\includegraphics[width=7.15cm]{pow_orb_text.eps}\n}\n\\hspace{0.2cm}\n\\subfigure[A model power spectrum of J1457 constructed from two sine waves at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$, using the same sampling pattern as in Figure~\\ref{fig:s_poworb}.]\n{\n    \\label{fig:s_poworbmod}\n    \\includegraphics[width=7.15cm]{pow_model_orb.eps}\n}\n\\caption{}\n\\label{fig:sub}\n\\end{figure*}\n\n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{fold18.eps}\n\\caption{Light curve of J1457 folded on the $\\omega_{\\text{o}}$ frequency, showing a double-humped orbital wave.} \n \\label{fig:fold18} \n\\end{figure}\n\n\n\\subsection{The Orbital Signal}\n\nA dominant, stable feature at $\\approx$ 37 c\\,d$^{-1}$ (38 min) is always present in the nightly power spectra. In four of the nights, a weaker but stable signal is also present at half that frequency,  $\\approx$ 18.5 c\\,d$^{-1}$  (78 min). We interpret these two signals as the orbital frequency and its first harmonic, $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$. As mentioned above, double-humped orbital waves are quite common among CVs, especially those of very low luminosity (for instance in WZ Sge and AL Com). A power spectrum composed of 11 nights, spanning 45 days, yields $\\omega_{\\text{o}} = 18.4888 \\pm 0.0017$ c\\,d$^{-1}$ and $2\\omega_{\\text{o}} = 36.9740 \\pm 0.0005$ c\\,d$^{-1}$. Errors are calculated from bootstrap simulations as described in Section~\\ref{obs}.\n\nFigure~\\ref{fig:s_poworb} shows the low-frequency portion of the full 11 night power spectrum. A zoom of the region around the orbital frequency is plotted on top. A model power spectrum constructed from two artificial sinusoids at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$, using the exact same sampling as for the original dataset, is shown in Figure~\\ref{fig:s_poworbmod}. When comparing model versus data, we find the surrounding picket-fence pattern similar in structure and height. This implies that the orbital signal indeed maintains an essentially constant amplitude and phase. In Figure~\\ref{fig:fold18}, data from all 11 nights are folded onto the orbital frequency, showing the double-humped orbital wave at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$.  \n \nThe lower frequency range of the power spectrum was further investigated to rule out the possibility of signals hiding in the noise (see Section~\\ref{superh} for the case of BW Sculptoris). The power spectrum was cleaned from the strongest signals at $\\omega_{\\text{o}}$, $2\\omega_{\\text{o}}$ and also from the high-amplitude peaks between 4 c\\,d$^{-1}$ -- 6 c\\,d$^{-1}$. However, no additional peak was found in this region, or in the vicinity of the orbital period.\n\n \\begin{table*}\n\\begin{center}\n \\begin{tabular}{lllllll}\n  \\hline\n  \\hline\n\\textbf{Star} & \\textbf{Frequency} & \\textbf{Period} & \\textbf{Amplitude} & \\textbf{Comments}  \\\\\n & (c\\,d$^{-1}$) & (min) & (mmag) &   \\\\\n \\hline\n J1457 & $18.4888 \\pm 0.0017$  & 77.92 & 12.0 & Orbital period ($\\omega_{\\text{o}}$) \\\\      \n& $36.9740 \\pm 0.0005$ & 38.95 & 23.9 & $2\\omega_{\\text{o}}$ \\\\\n& 71.9 (nightly mean error: 0.5) & 20.0 & 12.0 & NRP, low amplitude, non-stable \\\\ \n& 135.2\/144.3\/147.9 (nightly mean error: 0.8) & 10.7\/9.9\/9.7 & 3.9\/12.7\/7.6 & NRP, non-stable \\\\\n\\hline\nBW Scl & $\\sim$ 16.5 & 87.27 & $\\sim$ 50 & Quiescent superhump, non-stable \\\\\n & 18.40811 $ \\pm$ 0.03  & 78.23 & 16& $\\omega_{\\text{o}}$  \\\\ \n& 32.98 $\\pm$ 0.01 & 43.66 & $\\sim$ 15& Harmonic of quiescent superhump \\\\\n& 36.81622 $ \\pm$ 0.03  & 39.11 & 55& $2\\omega_{\\text{o}}$  \\\\\n&  69.55 $\\pm$ 0.03 & 20.70 &$\\sim$ 25& NRP, $\\omega_{1}$, non-stable  (amplitude from 2009)\\\\\n& 103.55 $\\pm$ 0.03  & 13.90 & $\\sim$ 20& $\\omega_{2}$ - $2\\omega_{\\text{o}}$, non-stable (amplitude from 2009) \\\\\n& 140.37 $\\pm$ 0.03 & 10.26 & $\\sim$ 26& NRP, $\\omega_{2}$, non-stable (amplitude from 2009) \\\\\n& 121.99 $\\pm$ 0.03  & 11.80 &$\\sim$ 23.5 & $\\omega_{2}$ - $\\omega_{\\text{o}}$, non-stable (amplitude from 2009) \\\\\n& 153.0 $\\pm$ 0.5 & 9.4 & 9.5& periodic signal (amplitude from 2001) \\\\\n& 307.0 $\\pm$ 0.5 & 4.7 & 8.5 & Harmonic of signal at 153 c\\,d$^{-1}$, transient \\\\\n  \\hline\n\\hline\n\\end{tabular}\n \\caption{Summary of frequencies found in SDSS J1457+51 and BW Sculptoris.}\n\\label{tab:freq} \n\\end{center}\n\\end{table*}\n\n\\subsection{High-Frequency Power Excess}\n\nThe complex range of signals spanning 142 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$ ($\\approx$ 10 min) moves slightly in frequency and is non-commensurate with the orbital frequency. In addition, during five of the observing nights a broad, low-amplitude peak appeared at 135 c\\,d$^{-1}$ (10.7 min) along with a signal at 72 c\\,d$^{-1}$ (20.0 min), neither of which are of orbital origin.  \n\nWith the aim to study these signals in more detail, a coherent power spectrum was constructed from adding consecutive nights together. However, this did not produce clean signals, even though the nightly power-spectrum windows were always clean. The broadness and complexity of the signals indicate a slight shift in amplitude and frequency on the time scale of a few nights, and\/or an internal fine structure unresolved by our observations. Therefore, analysis was performed on power spectra from separate nights in comparison with the overall mean power spectrum.  \n\nWhen combining our sparse data collected over 43 nights, there is a broad power excess around the frequencies, 135 c\\,d$^{-1}$, 144 c\\,d$^{-1}$ and 148 c\\,d$^{-1}$. This splitting is evident also in the mean spectrum shown in Figure~\\ref{fig:s_avpow}, and is always seen when combining nights from the start and the end of the campaign. When studying the nightly power-spectrum in this range, there was (in general) only one peak present at the time. However, during one observing night, three peaks were seen simultaneously at approximately these frequencies (Figure~\\ref{fig:pow320}). The nightly mean error for any peak appearing at 135 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$, is about 0.8 c\\,d$^{-1}$.\n \n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{pow_320_zoom145.eps}\n\\caption{Power spectrum of J1457 from one single night (HJD=2455320). Peaks were seen simultaneously at 135.6 c\\,d$^{-1}$, 144.6 c\\,d$^{-1}$ and 148.9 c\\,d$^{-1}$.} \n \\label{fig:pow320} \n\\end{figure}\n\n\nThe upper two frames in Figure~\\ref{fig:s_pow_best2} show the combined power spectra of the six nights of best quality. The bottom two panels show model power-spectra constructed from sine waves at the peak frequencies found in the combined six-night dataset, at 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$), 36.97 c\\,d$^{-1}$ ($2\\omega_{\\text{o}}$), 71.9 c\\,d$^{-1}$, 135.2 c\\,d$^{-1}$, 144.3 c\\,d$^{-1}$ and 147.9 c\\,d$^{-1}$, using the same sampling pattern as the data. The model is able to re-construct the overall appearance and widths of the signals seen in the data reasonably well, indicating that there is power excess at these frequencies. If excluding any of the frequencies from the model, the data power spectrum cannot be reproduced. We note that 71.9 c\\,d$^{-1}$ is about half that of 144.3 c\\,d$^{-1}$, but not exactly in a 2:1 ratio, indicating that the signals are not constant in amplitude and phase (see Section~\\ref{2009} for the case of BW Sculptoris). For a complete summary of the frequency analysis, see Table~\\ref{tab:freq}. \n\n\n\n    \n\\begin{figure}\n\\includegraphics[width=8.5cm]{pow_highfreq.eps}\n\\caption{The two top panels show the power spectrum of J1457 from the same set of nights shown in the mean spectrum (see Figure~\\ref{fig:s_avpow}). The two bottom frames show models that was constructed from the same sampling pattern as the data. The following frequencies was included in the model; 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$), 36.97 c\\,d$^{-1}$ ($2\\omega_{\\text{o}}$), 71.9 c\\,d$^{-1}$, 135.2 c\\,d$^{-1}$, 144.3 c\\,d$^{-1}$ and 147.9 c\\,d$^{-1}$. All main signals along with their line widths seen in the data can be reproduced fairly well by the model, indicating that there is power excess at these frequencies.} \n \\label{fig:s_pow_best2} \n\\end{figure}\n\n    \n    \n\n\\section{BW Sculptoris}\n \nBW Sculptoris (hereafter BW Scl) is a 16th magnitude blue star which was found to\ncoincide with RXJ2353-0-3852 in the Rosat bright-source catalogue, and\nthen identified as a cataclysmic variable by~\\cite{1997A&A...318..134A}.~\\cite{1997A&A...324L..57A} independently discovered the star as a blue object in the Hamburg\/ESO\nsurvey for bright QSOs. These two studies established the very short\norbital period of 78 minutes. In addition to broad and doubled H and He\nemission lines, BW Scl also shows very broad Balmer and Lyman absorptions,\nsignifying the presence of a white dwarf of modest temperature ($\\sim 15000$ K;\n~\\citealt{2005ApJ...629..451G}). If roughly half of the visual light comes\nfrom such a white dwarf, then the white dwarf has $V=17.3$ and $M_{V}\\sim 12$,\nimplying a distance of only $\\sim 110$ pc. This also agrees with the large\nproper motion found in the USNO catalogue (105 ms$\\,$yr$^{-1}$, \\citealt{2004AJ....127.3060G}). These considerations (a nearby star of very short $P_{\\textbf{orb}}$), and the possibility to study the underlying white dwarf, motivated us to carry out campaigns of time-series photometry nearly every year since 1999.\n\n  \n\\subsection{Observations}\n \nIn total, BW Scl was observed for about 1000 hours spread over about 200 nights, mainly using the globally distributed telescopes of the Center for Backyard Astrophysics (CBA: \\citealt{1993ApJ...417..298S, patterson_1998}). A summary observing log is presented in Table~\\ref{tab:obs2}. To maximise the signal and optimise the search for periodic features, usually no filter or a very broad filter, 4000\\,\\AA\\, --\\, 7000\\,\\AA\\,was used. Occasional runs were obtained in V and I bandpasses to provide a rough calibration, and to verify that the periodic features in the light curve are indeed broadband signals. The smaller (25 cm -- 35 cm) telescopes generally used the star GSC 8015-671 as a comparison, while the larger (91 cm) telescopes used USNO 0450-40780391, a nearby 16th magnitude star. These comparison stars can be considered to be constant. The clear and broadband filters permit only a rough calibration, but BW Scl remained within $\\sim 0.3$ mag of $V=16.6 $ throughout the campaign. In terms of instrumental magnitude, limits on night-to-night variability within each season are more stringent: typically $< 0.05$ mag, and always $< 0.1$ mag. This degree of constancy is remarkable for a cataclysmic variable, and is probably due to the WD's large contribution to the light in the optical. \n    \nIn order to study the periodic behaviour, we always tried to obtain photometry densely distributed in time, preferably with contribution from telescopes widely spaced in longitude (in order to solve problems associated with daily aliases). Most of the analysis below is based on long time series from stations in New Zealand, South Africa, and Chile, and hence not afflicted by aliasing problems.\n\n \\begin{table}\n  \\begin{center}\n \\begin{tabular}{llllll}\n  \\hline\n  \\hline\n\\textbf{Year} & \\textbf{Spanned}  & \\textbf{Observer} & \\textbf{Telescope} & \\textbf{Nights} \\\\\n& (days) & &  & (hours) \\\\\n \\hline\n1999 &  13  &  Kemp & CTIO 91cm & 12\/61\\\\      \n2000 & 38 & McCormick & Farm Cove 25cm & 7\/35\\\\\n         &      & Rea    &     Nelson 35cm   &  2\/9\\\\\n2001  & 77  &    Rea   &           \"     &         16\/87\\\\\n        &     & Kemp     &   CTIO 91cm       &   14\/84\\\\\n           &  & Woudt     &  SAAO 76cm           & 1\/4\\\\\n2002   & 0   &   Kemp      &       \"               & 1\/4\\\\\n2004   & 7    &  Monard   &   Pretoria 25cm  & 7\/38\\\\\n2005  & 55    &  Rea       &  Nelson 35cm    & 16\/66\\\\\n          &  & Christie   & Auckland 35cm      & 12\/48\\\\\n        &  &  Retter\/Liu   &                    & 6\/32\\\\\n    &   &  Monard     & Pretoria 25cm &  4\/22\\\\\n   &   &  Moorhouse   &                    & 3\/10\\\\\n2006  & 90   &   Rea        & Nelson 35cm  &  26\/112\\\\\n          &  & Monard     & Pretoria 35cm  & 3\/21\\\\\n&         &   McCormick  & Farm Cove 25cm & 5\/15\\\\\n2007  & 21  &    Rea        & Nelson 35cm   &  7\/28\\\\\n          &   & Richards  &  Melbourne 35cm     & 1\/3\\\\\n&         & Allen     &  Blenhem 41cm   & 1\/3\\\\\n2008  & 71   &   Monard   &   Pretoria 35cm & 22\/88\\\\\n&          &  Rea        & Nelson 35cm   & 13\/90\\\\\n2009  & 50  &    Rea         &     \"             & 15\/94\\\\\n          &   & Monard      & Pretoria 35cm & 11\/62\\\\\n\\hline\n  \\hline\n\\end{tabular}\n \\caption{Summary observing log for BW Sculptoris.}\n \\label{tab:obs2} \n\\end{center}\n\\end{table}\n\n\n\\subsection{The 1999 Campaign}\n\nThe upper frame of Figure~\\ref{fig:1} shows the light curve from one night during 1999. The general appearance is typical of all nights, as well as seen in previously published light curves (\\citealt{1997A&A...324L..57A, 1997A&A...318..134A}). As those papers demonstrated, a 39-minute wave is always present, with a small even-odd asymmetry suggesting that the fundamental period is actually the double, 78 minutes (confirmed by spectroscopy). The middle frame shows the 1999 double-humped orbital light curve, similar to that of J1457 (see Figure~\\ref{fig:fold18} above). Study of the seasonal timings yielded a 1995 -- 2009 ephemeris:\n\\\\\\\\\nOrbital maximum = HJD 2,450,032.182(3) + 0.05432392(2) E. \n\\\\\\\\\nThe average nightly power spectrum, the incoherent sum of the 11 nights of best quality, is shown in the bottom frame of Figure~\\ref{fig:1}. Power excesses near 72 c\\,d$^{-1}$ (20 min) and 143 c\\,d$^{-1}$  (10 min) are evident. In order to study these higher-frequency signals in more detail, adjacent nights were added together, and the coherent power spectrum was constructed. However, even though the 72 c\\,d$^{-1}$ and the 143 c\\,d$^{-1}$ signals were always present, they were always broad, complex, and slightly variable in frequency (similar to J1457). This is usually a sign that the actual signals violate the assumptions of Fourier analysis: constancy in period, amplitude and phase. \n \n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\caption{BW Scl in 1999. The upper frame shows an 8-hour light curve, dominated by the orbital wave. The middle frame presents the season's mean orbital light curve, showing the double-humped waveform (the feature at $2\\omega_{\\text{o}}$ dominates all power spectra). The lowest frame shows the mean nightly power spectrum, averaged over the 11 best nights. Significant features are labelled with their frequency in c\\,d$^{-1}$ (with an average error of $\\pm$ 0.7).} \n \\label{fig:1} \n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig2.eps}\n\\caption{BW Scl in 2001. The upper two frames show the mean nightly power spectrum, averaged over the 15 best nights. Features are labelled with their frequency in c\\,d$^{-1}$ (with an average error of $\\pm$ 0.7). The signal at 724 c\\,d$^{-1}$, is likely to be caused by instrumental effects. The lowest frame shows the power spectrum of a particularly dense 12-night segment, after subtraction of the orbital waveform. A powerful signal near 16.5 c\\,d$^{-1}$ is obvious, with a strong first harmonic. The formal error is $\\sim$ 0.01 c\\,d$^{-1}$, and each peak is about that wide. However, the power excess shows more structure and width than seen in the power-spectrum window (see Figure~\\ref{fig:4} below), indicating some amplitude and\/or phase modulation.} \n \\label{fig:2} \n\\end{figure}\n\n\n\\subsection{The 2001 Campaign}\n \nDuring 2001, observatories at three longitudes in Chile, New Zealand \nand South Africa, contributed with good nightly coverage of BW Scl. A very low flickering background is seen in the light curves of BW Scl, enabling detection of periodic signals as weak as $\\sim 0.002$ magnitudes.\n\nThe upper frames of Figure~\\ref{fig:2} shows the nightly mean power spectrum, with significant signals marked to an accuracy of 0.5 c\\,d$^{-1}$. Both the orbital period and the signal at 72 c\\,d$^{-1}$  (20 min) were present, along with two other signals at low frequency: a powerful signal at 16.6 c\\,d$^{-1}$  (86.7 min), and a weak signal at 50.5 c\\,d$^{-1}$ (28.5 min). At higher frequency, signals are detected at 153 c\\,d$^{-1}$ (9.4 min), 307 c\\,d$^{-1}$ (4.7 min), and 724 c\\,d$^{-1}$ (2 min), though the latter is likely to be caused by instrumental effects. Many telescopes have worm gears which turn with a period of exactly 120 sidereal seconds, and this period was reported in research on many types of stars during 1960 -- 1990, i.e. during the photolectric-photometer era. CCDs are much less prone to this error. However, since 724 $\\pm$ 2 c\\,d$^{-1}$ corresponds to 120 sidereal seconds (to within the measurement error), we interpret the signal to be caused by this instrumental effect.   \n \nWe interpret the signal at 153 c\\,d$^{-1}$ as a pulsation frequency, with a significant first harmonic. Examination of individual nights showed this signal to be somewhat transient, at least in amplitude. This behaviour was seen on 7 of the 15 good-quality nights. Since the orbital frequency is known precisely, and since its photometric signature is powerful and constant, we subtracted its first harmonic (and also the second harmonic when detected) from the central 12-night time-series, prior to analysis. The resultant power spectrum at low frequency is seen in the bottom panel of Figure~\\ref{fig:2}. A weak signal appears at the orbital frequency, and stronger signals at 16.34\/16.64 c\\,d$^{-1}$ and 32.98 c\\,d$^{-1}$ (with an accuracy of $\\pm\\,\\,0.01$ c\\,d$^{-1}$). It seems likely that these are superhump signals in the quiescent light curve. Specifically, we interpret the 16.34\/16.64 pair as signifying an underlying frequency of $\\sim 16.5$ c\\,d$^{-1}$, as such splitting can be produced by amplitude and\/or phase changes. The precession frequency $\\Omega$ can be expressed as $16.50 = \\omega_{o} - \\Omega$, where $\\omega_{o}$ is the orbital frequency (implying that $32.98 = 2(\\omega_{o} - \\Omega)$). The precession frequency $\\Omega$ is then equal to 1.9 c\\,d$^{-1}$.\n\nThe primary signal near 16.5 c\\,d$^{-1}$ has been seen before. It was the dominant signal reported by~\\cite{1997A&A...318..134A}, but was then discounted by~\\cite{1997A&A...324L..57A} as probably the result of cycle-count error. The data presented here certainly has no ambiguity in cycle count, and reveals this signal very clearly. \n\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption{ BW Scl in 2009. The top frame shows the mean nightly power spectrum, averaged over the 19 best nights. Features are labelled with their frequency in c\\,d$^{-1}$ ($\\pm$ 0.6). Other frames show power spectra of $\\sim$ 5 day intervals with particularly dense coverage, with significant features labelled ($\\pm$ 0.03). The most interesting features are the trio of weak satellites of $\\omega_{2}$=140.37 c\\,d$^{-1}$ (103.55 c\\,d$^{-1}$ and 121.99 c\\,d$^{-1}$ are displaced by exactly $\\omega_{2}$ - $2\\omega_{\\text{o}}$ and $\\omega_{2}$ - $\\omega_{\\text{o}}$).} \n \\label{fig:3} \n\\end{figure}\n\n \n\\subsection{The 2009 Campaign}  \\label{2009}\n\nThe year 2009 saw another intensive observing campaign on BW Scl. The upper frame of Figure~\\ref{fig:3} shows the nightly power spectrum, averaged over the 19 nights of coverage. Signals near 70 c\\,d$^{-1}$  (20.6 min) and 140 c\\,d$^{-1}$  (10.3 min) are evident, with weaker signals near 104 c\\,d$^{-1}$  (13.9 min) and 122 c\\,d$^{-1}$  (11.8 min). In the three lower frames, we show intervals of particularly dense coverage, each with a nominal frequency resolution of $\\pm\\,\\,0.04$ c\\,d$^{-1}$. In these frames, the 70 c\\,d$^{-1}$ and 140 c\\,d$^{-1}$ signals show their variability in frequency and amplitude (similar to J1457). Such variability is characteristic of all our data. The second frame (JD 2,455,068 - 074) illustrates the following:\n\n\\begin{enumerate}\n\\item [1.] Although the 70 c\\,d$^{-1}$ and 140 c\\,d$^{-1}$ signal are related (when one moves\nto slightly lower frequency, so does the other), the frequencies do not appear to be exactly in the ratio 2:1. Here, these frequencies are described as $\\omega_{1}$ and $\\omega_{2}$, with $\\omega_{2} \\sim 2\\omega_{1}$.\n\n\\item [2.] The weaker signals flanking the 140 c\\,d$^{-1}$ feature are displaced by exact\ninteger multiples of the orbital frequency. Thus the signals seen in JD 068-74 are $\\omega_{1}$, $\\omega_{2}$, ($\\omega_{2} - \\omega_{\\text{o}}$) and ($\\omega_{2} - 2\\omega_{\\text{o}}$). These orbital sidebands of $\\omega_{2}$ are also visible in Figure~\\ref{fig:1}, however such orbital sidebands was never seen of $\\omega_{1}$. This reproduces the properties of the pulsations seen in SDSS J1507+52 (\\citealt{2008PASP..120..510P}).\n\\end{enumerate} \n\n  \n\\subsection{Power-Spectrum Window}\n  \nFinally, in Figure~\\ref{fig:4} we show the power spectrum in the vicinity of $2\\omega_{\\text{o}}$, for each season. The $2\\omega_{\\text{o}}$ signal maintains constancy in phase and amplitude, and therefore acts effectively like a test signal whose power spectrum should be reproduced in structure by any coherent signal (constant in phase and amplitude). The central peak at exactly 36.816 c\\,d$^{-1}$ (39.113 min) always dominates over its neighbours in the picket-fence pattern because of very long runs, and\/or observations at several longitudes. This demonstrates that these campaigns are free from aliasing. Also, it is worth noting that peaks at frequencies greater than $2\\omega_{\\text{o}}$, are always slightly wider (for the single-night time series) than expected for a truly periodic signal, and always more complex (for the multiple-night time series). This is true for all our data. The underlying $\\omega_{1}$ and $\\omega_{2}$ signals either have an intrinsic complex structure, or vary strongly in amplitude and\/or phase on short timescales (or both). In addition, the entire complex of signals near $\\omega_{1}$ and $\\omega_{2}$ moves in frequency by a few percent on timescales as short as $\\sim10$ days.\n  \n\\subsection{Superhumps in BW Scl?} \\label{superh}\n\nThe obvious signal at 87 minutes in BW Scl, displaced by $\\sim 11$ \\% from $P_{\\text{orb}}$ seen in the 2001 power spectrum (Figure~\\ref{fig:2}), is a transient but repeating feature in the light curve. In our ten years of coverage, six campaigns were sufficiently extensive to reveal such a signal. This feature was always strong when it was detected, and moved slightly in frequency, even on timescales of a few days. This qualitatively describes a common superhump, which is a well-known feature in CV light-curves. However, there are three substantial differences:\n\\begin{enumerate}\n\\item [1.] Common superhumps are found in outburst states, typically near $M_{V} \\sim 5$, not at quiescence when $M_{V} \\sim 12$.\n\\item [2.] Common superhumps occur with fractional period excesses, $(P_{\\text{sh}} - P_{\\text{orb}})\/P_{\\text{orb}}$, near 3\\%, not 11\\%.\n\\item [3.] Common superhumps are more stable, wandering from a constant-period ephemeris on a timescale of 100 -- 200 cycles. The power-spectrum signal in BW Scl appears complex and broad, suggesting much lower coherence.\n\\end{enumerate}\n\nA system with quiescent superhumps is a rare beast in the CV kingdom, but it is not unprecedented. In AL Com, a nearly identical signal was reported and extensively discussed by \\cite{1996PASP..108..748P}. Something similar has also been reported in V455 And (\\citealt{2005A&A...430..629A}), SDSS J0745+45 (\\citealt{2010ApJ...710...64S}) and possibly SDSS J1238-03 (\\citealt{2010ApJ...711..389A}). All these stars are period-bouncer candidates (systems that have already passed the minimum period; see Tables 3 and 5 of \\citealt{2011MNRAS.tmp...27P}). These candidates were chosen due to their low donor-star mass (or low $q = M_{2}\/M_{1}$). A possible account of how stars of very low $q$ might be able to manufacture quiescent superhumps has been given in~\\cite{1996PASP..108..748P}, and includes the idea that a low donor mass implies a larger Roche lobe surrounding the disc. Weak tidal torques could then allow the quiescent disc to extend to the 2:1 orbital resonance, where an eccentric instability could drive a fast prograde precession (viz., at $\\Omega$ = 1.9 c\\,d$^{-1}$), resulting in a superhump with $\\omega = \\omega_{o} - \\Omega$. \n\n\n\n\\begin{figure*}\n\\includegraphics[width=13cm]{fig4.eps}\n\\caption{BW Scl. Power spectra in the vicinity of $2\\omega_{\\text{o}}$ for the lowest frame in Figure~\\ref{fig:2}, and the three lower frames in Figure~\\ref{fig:3}. Since the $2\\omega_{\\text{o}}$ signal is essentially constant in frequency and amplitude, this is effectively the power-spectrum window for each observation. This implies that the nightly length of run and span of longitude is sufficient to exclude confusion by aliases.} \n \\label{fig:4} \n\\end{figure*}\n\n\n\\section{Interpretations as Non-Radial Pulsations}\n\nBoth BW Scl and J1457 are quiescent CVs of very low accretion luminosity. Their spectrum show evidence of the primary WD, as do their light curves which contain rapid, non-commensurate signals. This is the general signature of the GW Lib stars, where the periodic signals are believed to represent the non-radial pulsations of the underlying WD. However, no proof of this has ever been found, not for GW Lib or any other of the 10 -- 15 members of the class. In order to explain these signals, two other hypotheses deserve consideration: first, quasi-periodic oscillations (QPOs) arising from the accretion disc, and second, the spin frequency of a magnetic WD (intermediate-polar model).\n\nQPOs were first discovered, and named, for a broad excess of power seen during a dwarf-nova eruption of RU Pegasi (\\citealt{1977ApJ...214..144P}). Since then, they have been commonly found with periods of 10 -- 20 minutes in the high-state light-curves of many nova-like variables (reviewed by \\citealt{2004PASP..116..115W}). QPOs are typically seen as very broad peaks in the power spectrum ($\\delta v\/v \\sim 0.5$; see for instance Figure 11 and 12 in~\\citealt{2002PASP..114.1364P}), and typically have modulations of 1\\% -- 3\\% of an accretion disc in full outburst (for a light source with $M_{V}\\sim 4$). The most famous example is the 20-minute signal in TT Arietis (which belongs to the class of VY Sculptoris stars). However, VY Sculptoris stars appear to be the among the most luminous CVs. This can be compared to the faint BW Scl, where the 70\/140 c\\,d$^{-1}$ signals have $\\delta v\/v \\sim 0.01$ and are $\\sim 1$\\% modulations of a light source with $M_{V}\\sim 12$. The peaks in J1457 have slightly higher $\\delta v\/v$, but on both counts, the QPO hypothesis earns no applause. It would have to be an essentially new kind of accretion-disc QPO.\n\nA white dwarf spinning with a period of 20 minutes could explain the features of the signal seen at $\\sim 70\/140$ c\\,d$^{-1}$ in both objects, the common appearance of the first harmonic, the occasional switch to a pure first harmonic (two-pole accretion), and the orbital sidebands seen in BW Scl (from amplitude modulations, and\/or reprocessing from structures fixed in the orbital frame). But this hypothesis fails to account for the shifts in frequency exhibited by the $\\omega_{1}$ and $\\omega_{2}$ signals (the $\\sim 5$\\% wandering, e.g. in the range 68 c\\,d$^{-1}$ -- 73 c\\,d$^{-1}$ and 135 c\\,d$^{-1}$ -- 153 c\\,d$^{-1}$), the fact that $\\omega_{2}$ is not exactly $2\\omega_{1}$, and the intrinsic breadth (or fine structure) of both signals. These are profound inconsistencies.\n\nSuch considerations drive us back to the GW Lib model. Actually, we have studied all of the 10 -- 15 known class members with time-series photometry, and the resemblances to BW Scl and J1457 are substantial:\n\\begin{enumerate}\n\\item [1.] Low-amplitude and non-commensurate periodic signals in low-$\\dot{M}$ CVs. \n\\item [2.] Signals roughly constant over a few days, but somewhat transient in\n    frequency and amplitude on longer timescales.\n\\item [3.] Signals frequently with strong first harmonics or quasi harmonics.\n\\item [4.] Signals sometimes with known or suspected fine structure.\n\\end{enumerate}\nThese resemblances, and the difficulties with alternative models, seem\nsufficient to accept both BW Scl and J1457 as new members of the GW Lib class.     \n     \n\\section{Summary}\n\n\\begin{enumerate}\n\\item [1.] Two more CVs of very low accretion-rates have shown rapid non-commensurate signals in quiescence, which makes them likely new members of the GW Lib class. Both J1457 and BW Scl show a complex spectrum with the main signals near 10 and 20 minutes.\n\n\\item [2.] The pulsation frequencies in both stars wander by a few percent on a timescale of days. In addition, the power-spectrum constructed from multiple-night light curves, show broad peaks, which might be due to the frequency wandering, or from intrinsic fine structure not resolved by our data (or due to a strong amplitude modulation).\n\n\\item [3.] BW Scl shows several peaks displaced from the main pulsation frequency $\\omega_{2}$, by $\\omega_{2} \\pm \\omega_{\\text{o}}$ and $\\omega_{2} \\pm 2\\omega_{\\text{o}}$. Similar behaviour is evident in other GW Lib stars (SDSS J1507+52, V386 Ser and SDSS J1339+48). The origin of this phenomenon is still unknown. The rich pulsation spectrum makes BW Scl a good candidate for an intensive round-the-world time-series campaign with larger telescopes.\n\n\\item [4.] The orbital light curves of both stars show double-humped waves. From these waves, precise periods are found at $P_{\\text{orb}} = 78.22639 \\pm 0.00003$ minutes for BW Scl, and $77.885 \\pm 0.007$ minutes for J1457. Similar systems displaying non-radial pulsations, such as SDSS J1339+4847, SDSS J0131-0901 and SDSS J0919+0857, all have orbital periods right at the same orbital period ($\\approx$ 80 minutes). \n\n\\item [5.] BW Scl sometimes shows a transient wave with a period, $P_{\\text{sh}} = 87.27$ minutes, which is interpreted as a quiescent superhump. It thereby joins a small group of stars who manage a superhump at quiescence, all of which are likely to have very low mass ratios. This might arise from an eccentric instability at the 2:1 resonance in the disc.\n\n\\item [6.] The white-dwarf domination of the spectra in these stars suggests great faintness of the accretion light. This signifies a very low accretion rate, and both stars are likely to be very old CVs.\n\\end{enumerate}\n   \n   \n\\section*{Acknowledgments}   \n\nJoe Patterson would like to acknowledge the support from NASA grant GO11621.03A, the Mt Cuba Astronomical Foundation and the NSF grant AST 0908363. \n   \n   \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMotivated from the microscopic understanding of black holes in string\ntheory, brane configuration and microscopic proposals \\cite{hep-th\/0508023,\nhep-th\/0107119}, we focus on investigating the possibilities of a covariant\nthermodynamic geometric study of a class of coarse grained configurations.\nThese solutions are described in terms of the chemical parameters and a set\nof arbitrarily  excited boxes of the random Young tableaux. In particular,\nwe intend to study the thermodynamic geometry arising from the canonical\nenergy and counting degeneracy for the two parameter finite temperature\nsolutions. Ref. \\cite{07073601v2} offers an interesting motivation for\nthe case of near-extremal black holes in $AdS_5$. The main idea is first\nto develop a geometric notion for black hole thermodynamics and then to\nrelate it with the existing microscopic quantities already known from CFT\nconfigurations \\cite{hep-th\/0508023, hep-th\/0107119}. Furthermore,\nwe incorporate all order fluctuations in the exact canonical energy\n\\cite{SM} and entropy associated with the counting of excited boxes\nin random Young tableaux and analyze their respective contributions\nto the covariant thermodynamic geometries.\nThe phenomenon of the statistical fluctuation in the bubbling solutions\ninvolves the same asymptotic charges as that of the considered giant black holes.\nTo be precise, we consider an ensemble of horizonless geometries and the black hole\narises as a coarse graining of such states.\n\nLunin and Mathur have connoted a conjecture \\cite{lu-ma} that the\nblack hole microstates may be characterized by string theory\nbackgrounds with no horizons. Moreover, these solutions saturate\nan exact bound on their angular momenta, and have the same set of\nasymptotic charges as the respective black hole whose horizon area\nclassically vanishes. Some finite temperature supersymmetric \\cite{0411001v2}\nand associated non-supersymmetric generalizations appear as well,\nwhich correspond to definite black holes with finite size horizons. \nAs a consequence, we are interested in finding state-spaces for the\nBPS black holes which are already known to have a finite size horizon\nat zero temperature \\cite{mathur1, mathur2}. Furthermore, the complete\nclass of two parameter\\footnote{The two parameters of the interest\nare the charge and effective canonical temperature of the solution\n\\cite{hep-th\/0508023, hep-th\/0107119}. It is worth mentioning that \nthis temperature is not the physical temperature of the black holes.\nHowever, it arises as the canonical averaging e.g. Eqn(\\ref{effTemp}).}\nsupersymmetric black hole solutions in five dimensions are well-known\nsince the invention of \\cite{LLM}. Ref. \\cite{09103225v2} brings out\nan interesting issue of the horizonless solutions in AdS space and thereby\nconstructs its relation for the case of the small black hole solutions.\nThe associated finite temperature solutions \\cite{0411001v2,09103225v2}\ncould be thought as a deviation from the extremality condition. Importantly,\nthe near-extremal configurations \\cite{07073601v2} may further indicate a\nset of interesting issues for the present discussion thermodynamic geometry. \nThus, it would be inspiring to analyze the thermodynamic intrinsic manifold\nfor the giants and superstars configurations having two parameters, which\ncharacterize fluctuation about an ensemble of equilibrium CFT microstates\n\\cite{hep-th\/0508023, hep-th\/0107119}. From the perspective of the thermodynamic\ngeometry, this sets a good motivation to study the issue of statistical\ncorrelations for the black holes in string theory.\n\nInterestingly, the existence of such a geometric structure in\nequilibrium thermodynamics was introduced by Weinhold \\cite{wien1,wien2}\nthrough an inner product in the space of equilibrium chemical\npotentials defined by the minima of the internal energy function\n$U=U( \\lambda_i, V, S, T)$ as the Hessian function\n$h_{ij}= \\partial_i \\partial_j U $.  In this description, the\nquantities $ \\lbrace \\lambda_i, V, S, T \\rbrace $ are defined as\nthe chemical potentials $ \\lambda_i$, phase-space volume $V$,\nentropy $S$ and the temperature $T$ of the underlying equilibrium\nstatistical configuration. In order to provide a physical scale,\nwe need to restrict negative eigenvalues of the metric tensor,\nwhich may be accomplished by requiring that the volume or any other\nexternal parameter of the given fluctuating giant configurations\nbe held fixed.\n\nLet us recall that the Riemannian geometric structure thus\ninvolved indicates a certain physical significance ascribed to\ngiants and dual giants. The associated inner product structure\non the intrinsic space may either be formulated in the entropy\nrepresentation, as a negative Hessian matrix of the counting\nentropy with respect to the extensive variables, where the total\nnumber of boxes and number of excited boxes in a random Young\nTableaux define the degeneracy, or be formulated in the energy\nrepresentation, as the Hessian matrix of the associated energy\nwith respect to the intensive chemical potentials\n\\cite{hep-th\/0508023,hep-th\/0107119}. Such geometric considerations\nhave been studied earlier in the literature, as the covariant\nthermodynamics whose application in the energy representation\nhas first been considered by Weinhold \\cite{wien1,wien2}. Whilst,\nthe entropy representation in which we shall investigate fluctuating\nYoung tableaux has been considered in the view-points of thermodynamic\nfluctuation theory \\cite{RuppeinerRMP,math-ph\/0507026,RuppeinerA20,\nRuppeinerPRL,RuppeinerA27,RuppeinerA41}.\n\nThe present paper is thus intended to analyze both pictures,\nand indeed our study strikingly provides a set of exact covariant\nthermodynamic geometric quantities ascribed to the giants and to the\nsuperstars \\cite{hep-th\/0508023}. Recall that the thermodynamic\nconfiguration of the $1\/2$-BPS (dual) giants may be parameterized\nby the underlying chemical potentials, $\\lambda_1, \\lambda_2$.\nThus, fluctuations in the associated ensemble may be described by\nthe minima of the energy function $E= E(\\lambda_1, \\lambda_2)$.\nExplicitly, the Weinhold metric tensor of an intrinsic Riemannian\nspace spanned by the chemical potentials may be given as $g_{ij}=\n\\partial_i \\partial_j E(\\lambda_1, \\lambda_2)$ and turns out to be\nconformal to the Ruppenier metric\\footnote{The present paper\nassumes that the terms (Ruppenier, Weinhold) geometry and\n(State-space, Chemical) geometry are synonyms and can be used\ninterchangeably. It is worth mentioning that the geometry associated\nwith the Weinhold metric tensor is said to be the chemical geometry\nand the other whose metric tensor is defined by the Ruppenier metric\nis said to be the state-space geometry. The general definitions are\ngiven by the Eqns(\\ref{Wienmetricgen},\\ref{Ruppmetricgen}). Subsequently,\nThe Eqns(\\ref{Wienmetric},\\ref{Ruppmetric}) explain the precise\nconsideration of the analysis of the present paper.}\nwith the temperature as the conformal factor. Thus, the metric as an inner\nproduct on the intrinsic surface of chemical potentials is necessarily\nensured to be symmetric, positive definite and satisfies the triangle inequality,\nsince the energy function has a minimum configuration in the equilibrium.\n\nAs a matter of fact, a few simple manipulations illustrate that\nthe conformally related state-space curvature is inversely\nproportional to the singular part of the free energy associated\nwith long range correlation(s), whose regularity consequently\nsignifies the fact that underlying giant configurations have no\nphase transition(s) over the range of admissible boxes. It is\nworth to mention that the Ruppeiner formalism, which has further\nbeen applied to diverse condensed matter systems with two dimensional\nintrinsic Riemannian spaces have been found to be completely\nconsistent with the scaling and hyper scaling relations involving\ncertain critical phenomena and in turn reproduce their corresponding\ncritical indices, see for review \\cite{RuppeinerRMP,math-ph\/0507026}.\nIn general, such geometric notions remain well-defined and correspond\nto an interacting statistical basis for diverse extremal and non-extremal\nblack brane configurations \\cite{bnt,SST,0606084v1}.\n\nThere exists a large class of black holes which have the\nspherical horizon topology in the physical regime, whose thermodynamic\nconfigurations are quite energizing in their own respect, as we shall exhibit\nin the next section. Subsequently, we shall analyze the behavior\nof collective state-space correlations, and in particular, we wish to\nillustrate certain peculiarities of the state-space geometry which in\nthe large charge limit provides the correlation length for the associated\nmicroscopic configurations. \nThe purpose of this paper is to investigate the relation between\nthe covariant thermodynamic geometries and the microscopic counting\nperspective from the total and excited boxes of random Young tableaux.\nBoth of these notions have received considerable recent attention from\nthe perspective of black hole physics and the corresponding thermodynamics.\nThus, the goal here is to explore underlying microscopic arguments behind\nthe macroscopic notion of the covariant thermodynamic geometries arising\nfrom the energy\/ degeneracy of fluctuating (dual) giant configurations.\nFurthermore, the underlying state-space curvature for higher dimensional\nblack holes appears to be intertwined with vacuum fluctuations existing\nin the possible brane configurations, and thus one may revel certain\nintriguing informations of the possible microscopic CFTs \\cite{fuzzball}.\n\nIn this connection,\nwhat follows here shows that the thermodynamic intrinsic geometries and\ntheir respective microscopic resolutions may however be pursued by\nan intriguing involvement of the framework of Mathur's fuzzball\nsolutions \\cite{LLM,fuzzball,LM}. Such a perspective thus discloses\nan indispensable ground with the statements that the thermodynamic\nscalar curvatures of interest not only has exiguous microscopic\nknowledge of the black hole configuration, but also it has rich\nintrinsic Riemannian geometric structures. Specifically, we procure\nthat the configurations under the present analysis are effectively\nattractive in general, while they are stable only if at least one\nof the parameters remain fixed. Our hope is that finding statistical\nmechanical models with like behavior might yield further insight\ninto the microscopic properties of black holes and a conclusive\nphysical interpretation of their state-space curvatures and\nrelated intrinsic geometric invariants.\n\nWithin the well understood framework of the microscopic counting technique,\nwhat fundamentally follows is that the present geometric notions\nimplicitly assume an infinite set of subensembles of the most\ngeneral statistical system having an integrated ensemble \\cite{LM}.\nWe thus explore the intriguing role of chemical fluctuations in\nthe united canonical energy, or box counting statistical entropy,\nof the underlying black brane configurations. Consequently, we\nfind that the associated intrinsic Riemannian geometries being\nLegendre transforms of each other, indeed describe corresponding\nfluctuations of the excited droplets or fuzzballs. For example,\nthe canonical energy corrected by the contributions coming from\nthe fluctuations of an ensemble of excited boxes may be regarded\nas a closer approximation to that of the simple energy in the\ncorresponding subensembles. We shall show that such considerations\napply as well to two parameter fuzzballs \\cite{LLM,fuzzball,LM}\nor the liquid droplets \\cite{liquid}, when considered as an union\nsubensembles. The specific forms of the infinite summation and\nspecial polynomial contributions may thus be calculated for a\nclass of supergravity black holes \\cite{LLM}.\n\nIt is noteworthy that the applicability of this analysis\npresupposes that the underlying net ensemble is thermodynamically\nstable, which requires a positive specific heat, or correspondingly\nthat the Hessian matrix of the energy function or the negative Hessian\nmatrix of the counting entropy, be positive definite. The energy\nfunction and degeneracy for the (dual) giant or superstar systems\nincorporating such contributions \\cite{LLM,fuzzball,LM,liquid} may\neasily be written to an appropriate form, without any approximation.\nThe present article discovers that such contributions to the canonical\nenergy\/ box counting entropy induce certain bumps in the scalar curvature\nof underlying intrinsic thermodynamic surfaces spanned by the chemical\npotentials, or the total and excited boxes of the Young tableaux.\n\nThe present paper is organized as follows. In the first section,\nwe have presented the motivations to study the intrinsic Riemannian\nsurfaces obtained from the Gaussian fluctuations of the giants or\nsuperstars canonical energy or counting entropy. In particular,\nwe have outlined some microscopic implications of the thermodynamic\ngeometry for the giants and superstars, in the lieu of the underlying\nmicroscopic configurations, and the remaining significations have\nbeen summarized in the final section. In section $2$, we shall\nbriefly explain what are the intrinsic thermodynamic geometries\nbased on a large number of equilibrium parameters characterizing\nthe giant and superstar black hole configurations. In section $3$,\nwe obtain the two parameter giants (superstars) canonical\nenergy and counting entropy as a function of chemical potentials\nand in terms of the number of excited and total boxes of the Young\ntableaux. In section $4$, we investigate the chemical geometry\nfor the giant (\/superstar) black holes, and examine two parameter\nfluctuating canonical configurations.\n\nIn section $5$, we focus our attention on the state-space geometry\nof the aforementioned configurations, with some excited boxes and\nan arbitrarily random Young tableaux. Explicitly, we investigate\nthe nature of the state-space geometry thus defined, as an intrinsic\nRiemannian manifold obtained from the entropies, with all possible\ncontributions being considered in the most general canonical ensemble.\nWe have explained that such an intrinsic geometric configuration,\nobtained from an effective canonical energy or degeneracy of the\ngiants\/ superstars, results to be well-defined and pertains to an\ninteracting statistical system. Finally, section $6$ contains a set\nof concluding issues and a possible discussion of the thermodynamic\nchemical and state-space geometries. The general implications thus\nobtained may thus divulge the geometry of both the chemical and the\nequilibrium microstate acquisitions which are the matter of an AdS\/CFT\ncorrespondence. Furthermore, our results are in close connection\nwith the microscopic implications arising from the underlying CFTs,\nand thus offer definite physically sound explanations of the finite\nchemical potential\/ finitely many excited boxes configurations.\n\\section{Thermodynamic Geometries}\nThe present section presents a brief review of the essential\nfeatures of thermodynamic geometries from the perspective of the\napplication to the two parameter configuration\nAs mentioned in the introduction, in order to illustrate the\nconnection between thermodynamic aspects and microscopic counting,\nwe focus our attention on the geometric nature of underlying\nfluctuating chemical parameters and number of excited (unexcited)\nboxes in an ensemble of finitely  random Young tableaux. The respective\nanalysis is divulged in the neighborhood of small chemical potentials\nor in an arbitrarily large number of excited boxes. In the present\ncontext, these limits essentially provide an equilibrium configuration.\nIn the next section, we shall motivate the thermodynamic geometric\nnature  for the (dual) giants and superstars.\n\nIn order to illustrate the basic notion of thermodynamic fluctuations,\nlet us consider an intrinsic Riemannian geometric model, whose\ncovariant metric tensor may be defined as the Hessian matrix of the\ncanonical energy, with respect to a finite number of arbitrary\nchemical potentials carried by the giant gravitons and superstar,\nconsidered in a given equilibrium configuration at fixed volume, or\nany other parameters of the solution. Specifically, let us define a\nrepresentation $ E( \\lambda_i, T) $ for given canonical energy, chemical\npotentials, and temperature $ \\lbrace E,  \\lambda_i, T \\rbrace $.\nIn the present consideration, the energy, as defined below, may\nfurther be shown to be solely a function of the chemical potentials,\nwhich in fact accompanies an intriguing set of expressions, as we have\nindicated in general by Eqs. \\ref{energy}, \\ref{entropy}. The underlying\nmethod, in general, results to be in an intrinsic Riemannian geometric\nconfiguration, which is solely based on nothing but the giant\nchemical potentials. Such a general consideration in fact yields an\nintrinsic space spanned by the $n$ chemical potentials of\nthe theory under examination, and thus it exhibits a $n$-dimensional\nintrinsic Riemannian manifold $ M_n $. The components of the\ncovariant metric tensor of the so called chemical geometry\n\\cite{wien1,wien2} may be defined as\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wienmetricgen}\ng_{ij}:=\\frac{\\partial^2 E(\\vec{x})}{\\partial x^j \\partial x^i}\n\\end{eqnarray}\n\\end{scriptsize}\nwhere the vector $\\vec{x} =(\\lambda_i, V, S, T) \\in M_n $.\nAs anticipated in the appendix A, it is worth to mention that\nthe thermodynamic curvature corresponds to the nature of the intrinsic\ncorrelation present in the statistical system. This will altogether\nimply that the scalar curvature for two component systems can be\nthought of as the square of the correlation length, at some given\nnon-zero configuration temperature, to be $R( \\lambda_1,\\lambda_2)\n\\sim \\chi^2 $, where we may identify the $\\chi(\\lambda_1,\\lambda_2)$\nto be the correlation length of the concerned chemical system.\nConsecutively, it is not surprising that the geometric analysis,\nbased on the Gaussian approximation to the classical fluctuation\ntheory, precisely yields the correlation volume of the chosen configuration.\nThis strongly suggests that, even in the context of chemical reactions\nor in any closed system, a non-zero scalar curvature might provide\nuseful information regarding the range of microscopic phase-space\ncorrelations between various components of the underlying statistical\nconfiguration.\n\nNote that such Riemannian structures, defined by the metric tensor\nin a chosen representation, are in fact closely related to the\nclassical thermodynamic fluctuation theory \\cite{RuppeinerA20,\nRuppeinerPRL, RuppeinerA27, RuppeinerA41} and existing critical\nphenomena. The probability distribution $\\Omega(x)$ of thermodynamic\nfluctuations over an equilibrium intrinsic surface naturally\ncharacterizes an invariant interval of the corresponding\nthermodynamic geometry, which in the Gaussian approximation reads\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Omega(x)= A \\ \\ exp [-\\frac {1}{2} g_{ij}(x) dx^i \\otimes dx^j],\n\\end{eqnarray}\n\\end{scriptsize}\nwhere the pre-factor $A$ is the normalization constant of the\nGaussian distribution and $\\otimes$ denote a symmetric product.\nThe associated inverse metric may easily be shown to be the second\nmoment of the fluctuations or the pair correlation functions, and\nthus it may be given as $g^{ij}= < x^i \\vert x^j>$, where \n$\\lbrace x_i \\rbrace $'s are the intensive chemical variables\nconjugated to the charges $\\lbrace X^i \\rbrace$ of the Legendre\ntransformed entropy representation. Moreover, such Riemannian\nstructures may further be expressed in terms of a suitable\nthermodynamic potential, obtained by certain Legendre transforms,\nwhich correspond to certain general coordinate transformations\non the equilibrium thermodynamic manifold.\n\nFollowing \\cite{RuppeinerRMP, 0606084v1, 0510139v3}, it turns out\nthat the natural inner product on the state-space manifold may\neasily be ascertained for an arbitrary finite parameter black\nbrane configuration, and the concerned state-space turns out to\nbe a $n$-dimensional intrinsic manifold $ M_n $. Typically, the\nassociated entropy $S(X^i)$ as an embedding function defines\nthe covariant components of the metric tensor of the thermodynamic\nstate-space geometry, which has originally been anticipated by\nRuppeiner in the related articles \\cite{RuppeinerRMP,RuppeinerA20,\nRuppeinerPRL, RuppeinerA27, RuppeinerA41}. Here, we shall take\nthis representation of the intrinsic geometry, and thus find\nthat the covariant components of state-space metric tensor\nmay be defined to be\n\\begin{scriptsize} \n\\begin{eqnarray} \\label{Ruppmetricgen}\ng_{ij}:=-\\frac{\\partial^2 S(\\vec{X})}{\\partial X^i \\partial X^j}\n\\end{eqnarray}\n\\end{scriptsize}\nWe may thus explicitly describe the present state-space geometric\nquantities as simply an intrinsic two dimensional Riemannian\nmanifold for $1\/2$-BPS configurations. Furthermore, the underlying\nstate-space geometry may parametrically be defined by the two invariant\nparameters, {\\it viz.}, $\\vec{X}= (n,M) \\in M_2$.\nWe may therefore notice that the components of the state-space\nmetric tensor are related to the statistical pair correlation\nfunctions, which may as well be defined in terms of the parameters\ndescribing the dual microscopic conformal field theory living on the\nboundary. This is because of the fact that the underlying metric\ntensor comprising Gaussian fluctuations of the entropy defines the\nstate-space manifold for the rotating black brane configuration.\nWe may thus easily perceive, in the present consideration, that the\nlocal stability of the underlying statistical configuration\nrequires that the principle components of state-space metric\ntensor $\\{ g_{ii} \\ | \\ i:= n, M \\}$ signifying heat capacities\nshould be positive definite\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}(n,M) &>& 0 \\nonumber \\\\\ng_{MM}(n,M) &>& 0\n\\end{eqnarray}\n\\end{scriptsize}\nMoreover, the positivity of the state-space metric tensor imposes\na stability condition on the Gaussian fluctuations of the underlying\nstatistical configuration, which requires that the determinant and\nhyper-determinant of the metric tensor must be positive definite.\nIn order to have a positive definite metric tensor $\\Vert g(n,M)\\Vert$\non the two dimensional state-space geometry, one thus demands that\nthe determinant of metric tensor must satisfy $\\Vert g \\Vert >0$,\nwhich in turn defines a positive definite volume form on the\nconcerned state-space manifold. Furthermore, it is not difficult\nto calculate the Christoffel connection $\\Gamma_{ijk}$, Riemann\ncurvature tensor $R_{ijkl}$, Ricci tensor $R_{ij}$, and the scalar\ncurvature $ R $ for the two dimensional state-space intrinsic\nReimannian manifold $(M_2,g)$. Remarkably, it turns out that\nthe above two dimensional state-space scalar curvature appears\nas the inverse exponent of the inner product defining the pair\ncorrelation functions between arbitrary two equilibrium microstates\ncharacterizing the black brane statistical configuration. \n\nNotice that there exists an intriguing relation of the scalar\ncurvature of the state-space intrinsic Riemannian geometry,\ncharacterized by the parameters of the equilibrium microstates,\nwith the correlation volume of the corresponding black brane\nphase-space configuration. Additionally, Ruppeiner has revived \nthe subject with the fact that the state-space scalar curvature\nremains proportional to the correlation volume $\\tilde{\\chi}^d$,\nwhere $d$ is the system spatial dimensionality and $\\tilde{\\chi}(n,M)$\nis its correlation length, which reveals related information residing\nin the microscopic models \\cite{RuppeinerA20}. It is worth mentioning\nfurther that the state-space scalar curvature in general signifies\npossible interaction in the underlying statistical configuration. \nFurthermore, one may appreciate that the general coordinate\ntransformations on the state-space manifold thus considered\nexpound to certain microscopic duality relations associated with\nthe fundamental invariant charges of the configurations.\n\nFrom the perspective of an intrinsic Riemannian geometry, it seems\nthat there exists an obvious mechanism on the black brane side, and\nthat it would be interesting to illuminate an associated stringy notion\nfor the statistical correlations to the microstates of the giant and\nsuperstar solutions or vice-versa. In this concern, it turns out\nthat the state-space constructions so described might elucidate\ncertain fundamental issues, such as statistical interactions and\nstability of the underlying brane configurations with spins and\nnon-equal $R$ charges. Nevertheless, one may arrive to a definite\npossible realization of the equilibrium statistical structures,\nwhich is possible to determine in terms of the parameters of an\nensemble of microstates describing the equilibrium configurations.\n\nThe relation of a non-zero scalar curvature with an underlying\ninteracting statistical system remains valid even for higher\ndimensional intrinsic Riemannian manifolds, and the connection of a\ndivergent scalar curvature with phase transitions may accordingly\nbe divulged from the Hessian matrix of the considered energy\/\ncounting entropy. It is significant to remark that our analysis\ntakes an intriguing account of the scales that are larger than the\ncorrelation length and considers that only a few microstates do not\ndominate the whole macroscopic equilibrium intrinsic quantities.\nSpecifically, we shall focus on the interpretation that the\nunderlying energy includes all order contributions from a large\nnumber of subensembles of the fluctuating microstates, and thus\nit characterizes our description of the geometric thermodynamics\nfor (dual) giants, superstars and fuzzballs.\n\nOur geometric formulations thus tacitly involves an unified\nstatistical basis, in terms of the chosen union of subensembles.\nAlthough the analysis has only been considered in the limit of\nsmall fluctuations, however the underlying correlation length takes\nan intriguing account upon the quartic corrections of the canonical\nenergy or the counting entropy. With this general introduction to\nthe thermodynamic geometries defined as the (negative) Hessian\nfunction of the canonical energy (or counting entropy), let us now\nproceed to investigate the energy\/ entropy of two parameter giants,\nsuperstars and their thermo-geometric structures. In the present\ninvestigation, we shall focus our attention on the interpretation\nthat the underlying energy includes contributions from a large number\nof excited boxes, and thus our description of the geometric thermodynamics\nextends itself to the spinless $1\/2$ BPS supergravity configurations.\n\\section{The Fuzzballs and Liquid Droplets}\nIn this section, we shall first review the giant and superstar\nconfigurations \\cite{hep-th\/0508023, hep-th\/0107119} and subsequently\ncompute the canonical energy and box counting degeneracy in the\nrequired form. Thereafter, we explore the thermodynamic geometries of\nthe giants and superstars arising from the consideration of type IIB\nstring theory. For obtaining the energy as a function of an effective\ntemperature and chemical potential, we consider the liquid droplet model\nof giants and superstars. The present paper considers, {\\it viz.} (i)\nthe canonical energy with two distinct parameters $ (T, \\lambda)$,\nwith $T$ as an effective canonical temperature and $\\lambda$ as\nthe chemical potential dual to the underlying $R$-charge of the theory\nand (ii) the box counting entropy with two distinct large integers\n$(n,M)$, ($n$ corresponds to the number of excited boxes and $M^2$\ncorresponds to the total number of possible boxes). In order to obtain\nthe desired expression for the energy and counting entropy, we consider\nthe dual $\\mathcal{N}=4$ super Yang Mills theory on $AdS_5 \\times S^5$\nfor type IIB string theory.\n\\subsection{Chemical Description}\nIn order to understand the very basic picture of the giant and\nsuperstar configurations \\cite{hep-th\/0508023, hep-th\/0107119},\nwe shall restrict ourselves to the field theory description defined\non $ \\mathrm{R} \\times S^3$ and focus our attention by recalling the\n$1\/2$ BPS sector of $\\mathcal{N}=4 \\ SYM$ with conformal dimension\n$\\Delta=R$. It is known \\cite{hep-th\/0508023, hep-th\/0107119} that the\nisomorphism group involves an $SO(6)$ symmetry, and thus we shall consider\nan embedding $SO(6) \\subset SO(2) \\times SO(2) \\times  SO(2)$, which may in \ngeneral have various states $\\{ X, \\overline{X}, Y, \\overline{Y},Z, \\overline{Z} \\}$\ncorresponding to either of the $SO(2)$. However, for spinless black holes,\nas there is no spin structure in $SO(2)$, one obtains $ J=0=T$. Thus an\nappropriate dimensional reduction of the above configuration yields to a\nsimple quantum mechanical system involving only $\\{ X, \\overline{X} \\}$+\na bunch of BPS states. Following \\cite{0111222, 0403110}, the problem under\nconsideration may thus be mapped to a real matrix model with a real orthogonal\nmatrix $X$.\n\nThe problem under consideration may be viewed as a $N$ fermion configuration\nlying in a one dimensional harmonic oscillator. One may in principle arrive\nat the solution and thus find eigenvalues and eigen vectors of the\nequations of motion. For the present purpose, let the eigen values of\nthe real matrix wave function be $\\{\\lambda_i\\}_{i=1}^N$, then the\ncomplete problem may be described by considering the Van der monde determinant\nof the function $X$. Surprisingly, one recognizes that the product\n$\\prod_{k+j} (Tr X^k)^j$ may be expressed in terms of the eigen\nvalues $\\{\\lambda_i\\}$. The fermions under consideration thus\nacquire a ``Fermi sea'', which may physically be understood as\nfollows. We observe that the individual energies $\\epsilon_i= e_i\n\\hbar + \\hbar \/2$ acquire a gap of\n\\begin{scriptsize}\n\\begin{eqnarray}\nr_i&=& \\frac{1}{\\hbar} (E_i-E_i^g)= e_i-i+1, \\ \\forall \\\ni=1,2,..., N\n\\end{eqnarray}\n\\end{scriptsize}\nIn order to explicate the dual gravity picture, we shall consider the\nwell-known LLM description, see for details \\cite{LLM}. It thus\nfollows from type $IIB$ string theory that the classical moduli\nmay be described with a symmetric $AdS_5 \\times S^5$ background,\nwhich has the symmetry of $\\mathrm{R} \\times SO(4) \\times SO(4)\n\\times U(1)$, having generators $ \\partial_t $, $ S^3 \\subset AdS_5\n$, $\\widetilde{S}^3 \\subset S^3$ and $ \\partial_{\\phi}$, respectively,\nfor the concerned symmetry components. It has been pointed out by the authors\nof \\cite{hep-th\/0508023} that almost all such states have an underlying\nstructure of the ``quantum foam'', whose universal effective description\nin supergravity is a certain singular spacetime that dubs the ``hyperstar''.\nThey have further argued that the singularity arises because of the fact\nthat the classical description integrates out the microscopic details of\nthe quantum mechanical wavefunction.\n\nBubbling AdS space geometries \\cite{LLM} illustrate that the\nten-dimensional type IIB metric corresponding to the half-BPS\nsupergravity may be described as\n\\begin{scriptsize}\n\\begin{eqnarray}\nds^2 &=& - h^{-2} (dt + V_i dx^i)^2 + h^2 (d\\eta^2 + \\sum_i\ndx^idx^i) + \\eta\\ e^{G} ds_{S^2}^2 + \\eta\\ e^{ - G} ds^2_ {\\tilde\nS^3} \\\\ \\nonumber h^{-2} &=& 2 \\eta \\cosh G , \\\\ \\nonumber  \\eta\n\\partial_\\eta V_i &=& \\epsilon_{ij} \\partial_j z,\\qquad \\eta\n(\\partial_i V_j-\\partial_j V_i) = \\epsilon_{ij} \\partial_\\eta z\n\\\\ \\nonumber z &=&{ 1 \\over 2} \\tanh G\n\\end{eqnarray}\n\\end{scriptsize}\nThe complete solution may thus be determined by the single\nfunction $z$, which satisfies the linear differential equation\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\partial_i \\partial_i z + \\eta \\partial_\\eta (\\frac{ \\partial_\\eta z}{ \\eta}) =0\n\\end{eqnarray}\n\\end{scriptsize}\n\nThis is simply an electrostatic problem in $\\mathrm{R}^6=\n\\mathrm{R}^4 \\times \\mathrm{R}^2$ with potential $\\Phi(\\eta,\nx^1,x^2)= 2 \\eta^{-2}$. Let the coordinates $(x^1,x^2)\\in\n\\mathrm{R}^2$, then the smoothness of the moduli space thus\nobtained may be determined from the symmetry of the background.\nHere, one reveals that the appropriate boundary conditions may be\ncharacterized by\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(\\eta;x_1,x_2) = \\frac{\\eta^2}{\\pi}\\int dx_1' dx_2' \\frac{z(0;\nx_1', x_2')}{[(x_2'-x_1')^2 + \\eta^2]^2}\n\\end{eqnarray}\n\\end{scriptsize}\nThus, the smoothness requires an absence of singularity and, at\n$\\eta =0$, it reads that $z(0; x_1,x_2)= 1\/2$ for $S^3 \\subset\nS^5$, which in this limit shrinks to zero, while the other\nadmissible boundary condition comes with  $z(0; x_1,x_2)= -1\/2$\nfor $S^3 \\subset AdS_5$. The $R$-charge associated with the\nunderlying configuration may thus be expressed via the conformal\ndimension $\\Delta$ of a given state, which may be computed as\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Delta = \\int_{ \\mathcal{D}} \\frac{d x^1 dx^2 }{2\\pi\\hbar}\n\\frac{1}{2}(\\frac{x_1^2+x_2^2}{\\hbar} - \\frac{1}{2}N^2),\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $\\mathcal{D}$ is the region with $\\eta =0$, known as the\ndroplet, which satisfies $z(\\eta;x_1,x_2)=0$ \\cite{hep-th\/0508023}.\nIt is also immediate that the phase-space velocity which may be\ndefined as $u(0,x^1,x^2)= \\frac{1}{2}- z(0;x_1,x_2)$, makes the\nunderlying configuration smooth. Specifically, one finds that the\n$u(0;x^1,x^2)$ satisfies\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0; x^1, x^2) &=& 1, \\ if \\ \\eta =1\n\\\\ \\nonumber &=& 0, \\ otherwise\n\\end{eqnarray}\n\\end{scriptsize}\nIn order to have a proper comparison, we may consider typical\nstates in the boundary field theory and thus explain the matching\nbetween the supergravity configuration and the underlying ensemble of\ntypical microstates. What follows next shows that the pure states with\nrelative energies $\\{ r_i\\}$ may be described by considering a\nYoung diagram having $N$ rows and $N_c$ columns. Such a Young\ndiagram may in general be depicted as\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N,N_c) &=& \\Box \\ \\Box \\ \\Box \\ \\ \\ \\ldots \\ \\Box \\\\ \\nonumber && \\vdots \\\\\n\\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\\\\n\\nonumber && \\Box\n\\end{eqnarray}\n\\end{scriptsize}\nNotice that the conformal dimension $\\Delta \\sim N^2$ puts a\nconstraint on the underlying ensemble, namely that, at most, it is\nallowed to have $N$ number of rows. Similarly, the number of column\ncan be at most $N_c$, which may be treated as some sort of cutoff in\nthe underlying quantum theory. This is due to the fact that the\ngravity singular black hole solutions may be sourced by a certain\ndistribution of giant gravitons. In this case, the singularity\ncan be described by considering spherical $D_3$-branes having\nconformal dimension  $\\Delta =J \\sim N$, while the $1\/2$\nBPS condition allows for sixteen charges in the theory. Furthermore,\nthe existence of a RR-flux in $AdS_5$ sources certain\nnon-abelian interactions, which ascribes bound states of $N$ giant\ngravitons. The concerned microstates of the underlying ensemble has\nprecisely been described in \\cite{hep-th\/0107119}. Subsequently,\nit follows \\cite{hep-th\/0107119} that the number of giants on\naverage remains the same as the number of giants sourcing the\ngravity singular solutions.\n\nThus, having a striking microscopic picture of the gravity singular\nsolutions, we shall now move on to describe the Weinhold geometric\nframework for the statistical mechanics of a large number of random\nYoung tableaux whose (rows, columns) may respectively vary from\n$(1,1)$ to $(N, N_c)$. Let us consider an ensemble of Young diagrams\nin which the total number of boxes is $\\Delta \\in [N_c,NN_c]$. This\nis because in any random Young diagram, we can have $N_c$ columns,\nand thus the maximum number of admissible row would be $NN_c$, see\nfor instance\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N_c,1) &=& \\Box \\ \\Box \\ \\Box \\ \\ \\ \\ldots \\ \\Box \\\\ \\nonumber\n&& 1 \\ \\ 2\\ \\ 3 \\ \\ \\ \\ldots \\ \\ N_c\n\\end{eqnarray}\n\\end{scriptsize}\nFor any other state with $\\Delta= N_c+ \\delta \\Delta$, the system would\nheat up and thus would acquire a non-vanishing effective temperature.\nConsequently, the entropy would increase with $S>S_0$, and the\nsystem would get randomized. This certainly does not happen for the\n$1\/2$-BPS spinless configurations, and the vacuum entropy at $T=0$\ngets maximized when half of the Young diagram is full.\n\nTo be concrete, let us consider the microstates having relative\nenergies $\\{ r_i\\}_{i=1}^N$ and compute the statistical quantities\nassociated with the promising giant configurations. Let us define\nthe difference of relative fermion energies as\n\\begin{scriptsize}\n\\begin{eqnarray}\nc_N &=& r_1 \\\\ \\nonumber && \\vdots \\\\ \\nonumber c_{N-i}&=&\nr_{i+1}-r_i \\\\ \\nonumber && \\vdots  \\\\ \\nonumber c_1&=& r_N-\nr_{N-1}\n\\end{eqnarray}\n\\end{scriptsize}\nWe thus see that the quantity $c_j$ describes the number of columns\nof length $j$ in the diagram associated to $\\{r_N,r_{N-1}, \\ldots,\nr_1\\}$. As mentioned earlier, we are interested in the statistical\nfluctuations of typical half-BPS states of large charge $\\Delta= J\n= N^2$, and thus the excitation energy of fermions is comparable\nto the energy of the Fermi sea.\n\nIn order to analyze the chemical correlation in the excited states of \na large number of free fermions, we would consider Weinhold geometry\nto study the typical correlation of statistical states being\ncharacterized by certain arbitrary Young diagrams. In order to\nsimplify the notations, we shall define $q= e^{-\\beta}, \\xi=\ne^{-\\lambda}, \\beta = 1\/T$. Then, the canonical partition\nfunction reads\n\\begin{scriptsize}\n\\begin{eqnarray}\nZ(\\beta, \\lambda)= \\xi^{-N_c}\\prod_j \\frac{1}{1-\\xi q}\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the very definition of the canonical ensemble\n\\cite{hep-th\/0508023}, the average canonical energy may be\nexpressed by\n\\begin{scriptsize}\n\\begin{eqnarray}\n<E(\\beta, \\lambda)>= q \\partial_q \\ln Z (q)= \\sum_j \\frac{j \\xi\nq^j}{1-\\xi q^j}\n\\end{eqnarray}\n\\end{scriptsize}\nWe thus find that the average occupation number of the theory,\nwhich measures the maximum number of allowed columns in any chosen\nYoung diagram, may be given to be\n\\begin{scriptsize}\n\\begin{eqnarray}\nN_c= \\sum_j <c_j>= \\sum_j \\frac{\\xi q^j}{1-\\xi q}\n\\end{eqnarray}\n\\end{scriptsize}\nAs mentioned in  the previous section, we may now divulge the\nnotion of chemical fluctuations without any approximation. For\nthis purpose, we shall redefine the canonical energy in terms\nof the effective canonical temperature $T$, which leads to the\nfollowing average energy\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{energy}\n<E(T,\\lambda)>= \\sum_{j=0}^{\\infty} \\frac{j e^{-(\\lambda +\nj\/T)}}{1- e^{-(\\lambda + j\/T)}}\n\\end{eqnarray}\n\\end{scriptsize}\nFor investigating the chemical fluctuations, we shall consider\ntwo neighboring statistical states characterized by the chemical\npotentials $(T, \\lambda)$ and $(T+ \\delta T , \\lambda +\\delta\n\\lambda )$.\nThe chemical pair correlation functions may thus be defined as the\ncomponents of the Weinhold metric\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wienmetric}\ng_{ij}^{(E)}= \\partial_i \\partial_j E(T, \\lambda), \\ i,j= T,\n\\lambda\n\\end{eqnarray}\n\\end{scriptsize}\nThe stability of the underlying canonical chemical configuration may thus\nbe determined as the positivity of the determinant of the Weinhold\nmetric tensor, $g^{(E)}(T,\\lambda) = det(g_{ij}^{(E)})$. The\nfundamental configuration is just an ordinary two dimensional intrinsic\nsurface $(M_2,g)$ spanned by the chemical potentials $(\\beta,\\lambda)$.\nThus, the global correlation length may be defined as the $M_2$ scalar\ncurvature invariant. Explicitly, one can determine the intrinsic\ncovariant Riemann curvature tensor $ R_{T \\lambda, T, \\lambda}^{(E)}$.\nHence, the Ricci scalar may, as in Eqs. \\ref{Wiencur}, be read off for\nEqs. \\ref{energy} to be\n\\begin{scriptsize}\n\\begin{eqnarray}\nR^{(E)}= \\frac{R_{T \\lambda T \\lambda}^{(E)}}{g^{(E)}(T, \\lambda)}\n\\end{eqnarray}\n\\end{scriptsize}\n\\subsection{State-space Description}\nIn order to have  a test of our state-space formulation, the possible\nexplication of the correlations associated with the box counting\nentropy may be divulged as follows. Let us first consider the case\nof the large $N$ limit, with $N \\rightarrow \\infty$, $\\hbar \\rightarrow 0$\nand $ N\\hbar$ fixed. Then, the expectation values of relative energies\n$<r_i$ define a curve $<y(x)>= \\sum_{i=N-x}^N <c_i>$. This essentially\nleads to the evaluation of an integral $\\int_{N-x}^N di <c_i>$. The\nalgebraic curve associated with this problem may thus be defined as\na set of energies\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\varepsilon:= \\{ \\alpha q^{N-x}+\\gamma q^y=1 \\ | \\ \\alpha=\n\\frac{1-q^{N_c}}{1-q^{N_c+N}}, \\gamma=\n\\frac{1-q^{N}}{1-q^{N_c+N}}\\}\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the extremization of the curve $f(x,y)=  \\alpha\nq^{N-x}+\\gamma q^y-1$, it is not difficult to see that the entropy\ngets maximized when $q\\rightarrow 1$. In this limit, we have a\n``limit curve''\n\\begin{scriptsize}\n\\begin{eqnarray}\ny(x)= \\frac{N_c}{N}x= wx\n\\end{eqnarray}\n\\end{scriptsize}\nThis is clearly a straight line with slope $N_c\/N$, and in fact it\ncorresponds to the triangular Young diagrams. For an illustration,\nwe may depict such a Young diagram for $N_c=4$ as\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(4,4) &=& \\Box \\ \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\\\\n\\nonumber && \\Box\n\\end{eqnarray}\n\\end{scriptsize}\nThe liquid droplet model thus suggests that the involved probes\nmay be resolved by considering a large number of particles between\nthe phase-space shells $\\sigma$ and $\\sigma +d\\sigma$. Thus the\ndifferential curve may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\frac{u(0;\\sigma^2)}{2 \\hbar^2}d\\sigma^2= dx\n\\end{eqnarray}\n\\end{scriptsize}\nHere, the droplet $u(0; x^1, x^2)$ has been expressed as $u(0;\n\\sigma^2)$ in terms of the radial coordinate $\\sigma$, which from\nthe view-points of the underlying $U(1)$ symmetry presents single\nYoung diagram states in the $(x^1, x^2)$ plane. We thus procure an\nalgebraic curve satisfying previously mentioned droplet boundary\nconditions with\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\frac{\\sigma^2(x)}{2 \\hbar}= y(x)+x\n\\end{eqnarray}\n\\end{scriptsize}\nThe consistency may easily be checked and, in particular, one finds,\nby differentiating the algebraic curve, that it describes the\nnatural boundary conditions associated with the giant black holes\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0;\\sigma^2)= \\frac{1}{1+y'(x)}= \\frac{1}{1+w}\n\\end{eqnarray}\n\\end{scriptsize}\nThe present analysis thus describes $1\/2$-BPS $R$-charged black\nholes in $AdS_5 \\times S^5$, which are sometime called superstars\n\\cite{0109127,0411145}. It is worth to note further that the metric\nof the superstar may precisely be defined by the three harmonic\nfunctions involving the $R$-charge $q_1$ and the length $L$ of the AdS.\nIn this perspective, the number of allowed giants may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nN_c= w N; \\ \\forall \\ w:= \\frac{q_1}{L^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt follows \\cite{0109127} that the phase-space density function\n$z$ may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(\\eta;x^1,x^2) = \\frac{1}{2} \\frac{r^2\\gamma - L^2 \\sin^2\n\\theta}{r^2\\gamma + L^2 \\sin ^2 \\theta},\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $\\gamma$ may be defined to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\gamma(r)= 1+ \\frac{q_1^2 \\sin^2 \\theta}{r^2}\n\\end{eqnarray}\n\\end{scriptsize}\nFor $\\eta =0$, we have $r^2 \\sin^2 \\theta =0$, which implies that\neither $r =0$ or $ \\sin \\theta =0$. Thus, we obtain that the\ncondition $ \\sin \\theta =0$ leads to the realization of $z(\\eta;\nx^1,x^2)=1\/2$. This is precisely the condition that there are no\n$D$-brane excitations in the ensemble of vacuum microstates. It is\nthus immediate to see that the phase-space density function may\nnow be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0;x^1,x^2) = \\frac{1}{2}- z(0;x^1,x^2)\n\\end{eqnarray}\n\\end{scriptsize}\nIt is worth to note for $r=0$ that the function $\\gamma(r)$ leads\nto the condition that the function $z(0;x^1,x^2)$ must satisfy\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(0;x^1,x^2)= \\frac{1}{2} \\frac{w-1}{w+1}\n\\end{eqnarray}\n\\end{scriptsize}\nThis, in turn, is consistent with the previously defined boundary\ncondition of the droplet $\\mathcal{D}$ and provides an intriguing\nmicroscopic picture for our present consideration. We may thus\npronounce that the Young tableaux description provides an\nappropriate framework for the microscopic understanding of the\nthermodynamic geometries of $1\/2$ BPS supergravity configurations.\n\nWe now offer an interesting origin of the box counting entropy in\nwhich the state-space correlations may, without any approximation, be\ndescribed. Here, an important ingredient follows from the non-trivial\nnotion of the coarse graining picture of the phase-space density $u_{c,g}$\nof the ground state configuration, which conjointly provides an appropriate\ndefinition of the canonical ensemble with fixed temperature, rather than\nfixing the energy \\cite{hep-th\/0508023}. In order to be concrete, let us consider\na thin shell in the phase-space $(x^1,x^2)$ of thickness $(dx^1,dx^2)$.\nThen the average density may be defined as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{\\int_E^{E+dE} u_{g,m}dx^2}{\\int_E^{E+dE}\ndx^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt is not difficult to ascertain that the density $u_{c,g}$ may easily\nbe expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{\\Delta_X}{\\Delta_E}\\hbar\n\\end{eqnarray}\n\\end{scriptsize}\nWe may however re-express the phase-space density $u_{c,g}$ and\nappreciate that it satisfies the familiar form\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{1}{1+y'(x)},\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $y(x)= \\frac{\\sigma^2}{2\\hbar^2}-x$ is the aforementioned\nrelative energy curve for the $1\/2$-BPS giant configurations at\nvery large $N$. The origin of the entropy may thus be ascribed as\nfollows. Let us consider the phase-space density and choose a set of coherent\nbasis which picks up the least uncertainty, and thus turns out to be\nappropriate for the present discussion, following from the coarse\ngraining phenomenon. In order to divulge the intrinsic geometric notion of\ndegeneracy, we may focus our attention on an arbitrary phase-space,\nhaving $M^2$ cells with at most random $n$ excited cells. The\npictorial view of this may be given as follows:\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N,N_c) &=& \\Box \\ {\\color{blue} \\maltese} \\ \\Box  \\ \\ldots \\\\\n\\nonumber && {\\color{blue} \\maltese} \\ \\Box \\ {\\color{blue}\n\\maltese} \\\\ \\nonumber && \\Box \\ \\Box  \\ \\Box \\\\ \\nonumber && \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ddots \\\\ \\nonumber && \\ \\ \\vdots \\ \\ \\ \\  \\\\\n\\nonumber && \\downarrow_{(RG \\ trans.)} \\\\ \\nonumber && \\downarrow\n\\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\ \\ldots  \\\\ \\nonumber && \\Box\n\\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ddots \\\\ \\nonumber && \\ \\vdots \\ \\ \\ \\  \\\\\n\\nonumber &=& Phase \\ Space \\ (N,N_c)\n\\end{eqnarray}\n\\end{scriptsize}\nIn an arbitrary Young diagram $Y(N,N_c)$, we thus see that there\nare $n$ boxes filled with maltese and the rest $(M^2-n)$ of them are\nempty boxes. Therefore, the existing degeneracy in choosing random\n$n$ maltese (excited boxes) out of the total $M^2$ boxes may  be\ngiven by\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Omega(n,M)= \\ ^{M^2}C_n = \\frac{(M^2)! }{(n)! (M^2-n)! }\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the first principle of statistical mechanics, we thus find\nthat the canonical counting entropy may be defined to be\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{entropy}\nS(n,M)= \\ln (M^2)! - \\ln (n)! - \\ln (M^2-n)!\n\\end{eqnarray}\n\\end{scriptsize}\nIt is nevertheless important to emphasize that the subsequent analysis\ndoes not exploit any approximation, such as Stirling's approximation or\nthe thermodynamic limit, and thus we shall work in the full picture of the\ncanonical ensemble. Consequently, the present analysis bequeaths an\nexact expression for the state-space pair correlation functions and\ncorrelation length.\n\nIt is now immediate to divulge an appropriate appraisal for the\nstate-space geometry associated with the random Young tableaux\n$Y(N, N_c)$ thus described with arbitrary $n$ excited droplets\namong the $M^2$ fundamental cells in an ensemble. We observe that\nthe state-space interactions allow us to analyze the correlation\nbetween two neighboring statistical states $(n,M)$ and $(n+ \\delta n,\nM+ \\delta M)$, and thus the fluctuations in the droplets with $(n,M)$\nmay be defined via the well-defined Ruppenier metric as\n\\begin{scriptsize}\n\\begin{eqnarray}\\label{Ruppmetric}\ng_{ij}^{(S)}= - \\partial_i \\partial_j S(n,M), \\ i,j= n, M\n\\end{eqnarray}\n\\end{scriptsize}\nAs we have outlined in the case of chemical fluctuations from the\nWeinhold geometry, a very similar analysis may thus easily be\nperformed to reveal the state-space correlations existing among\nthe fluctuating microstates of the $1\/2$ BPS configuration\ndescribing certain giants or the superstars.\n\nThe conformal relation of the two geometries may formally be\nderived via the temperature as a conformal factor, which may be\naccomplished by taking the derivative of the entropy with respect\nto the energy $ \\Delta $. One may however easily expose that\n$q= exp(-\\frac{\\alpha}{N})$, and it comes as no surprise that this yields\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{effTemp}\nT^{-1}= \\frac{\\partial S}{\\partial \\Delta}= \\frac{ \\alpha}{N}\n\\end{eqnarray}\n\\end{scriptsize}\nNevertheless, it is important to note that there is no physical\ntemperature involved with half-BPS states. However, $T$ should be\nunderstood as an effective temperature of a canonical ensemble\nthat sums up the half-BPS states with the canonical energy sharply\npeaked at $\\Delta $. More generally, if $\\Delta $ scales as $N^{\\xi}$,\nthen the Ref. \\cite{hep-th\/0508023} shows that the effective temperature\ngrows as $N^{\\xi \/2}$.\n\nOur thermodynamic geometries, thus explained in either the\ndescription of chemical potentials or the number of boxes in an\narbitrarily excited Young tableau, entail remarkably simple\nexpressions. Furthermore, we discover that there exists an\nappropriate microscopic\/macroscopic explanation for the\nstatistical correlations of the $1\/2$ BPS supergravity\nconfigurations. From the viewpoint of $AdS_2\/CFT_1$,\nit could be interesting to explore the case of more charged\nblack holes, e.g. extreme Reissner-Nordstr\\\"om black hole\n\\cite{SB2}, non-BPS supergravity black holes \\cite{07104967v3}\nor a further addition of electric-magnetic charges. \nIn the next section, we shall intimately discuss the above\noutlined study for the two parameter giants and superstar\nsolutions. In particular, our investigation shows that there\nexists an exact account for the thermodynamic correlations of\nthe spinless black holes arising from the coarse grained\nLLM geometries \\cite{LLM}.\n\n\\section{Canonical Energy Fluctuations}\nIn this section, we shall present the essential features of\nthe chemical geometry and thereby put them into effect for the 2-\nparameter giants and superstars. Here, we focus our attention\non the geometric nature of a large number of gravitons, within the\nneighborhood of small fluctuations with given chemical potentials\nintroduced in the framework of $1\/2$-BPS configurations. As stated\nearlier, the thermodynamic metric in the chemical potential space is\ngiven by the Hessian matrix of the canonical energy, with respect to\nthe intensive variables, which in this case are the two distinct chemical\npotentials carried by the giants and superstars. We find, in this framework,\nthat the exact canonical energy as the function of chemical potentials\ntakes an intriguing expression\n\\begin{scriptsize}\n\\begin{equation}\nE(T, \\lambda)= \\sum_{j=0}^{\\infty} \\frac{j \\ exp(-( \\lambda\n+j\/T))}{1-exp(-( \\lambda +j\/T))}\n\\end{equation}\n\\end{scriptsize}\nFor to obtain the thermodynamic metric tensor in the chemical\npotential space, we may employ the formula in Eq. \\ref{Wienmetric},\nwhich leads to the following series expression for the components\nof the metric tensor:\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{TT}(T, \\lambda)&=&\\sum_{j=0}^{\\infty}\\big( -2 \\frac{j^2}{T^3}\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T))}+ \\frac{j^3}{T^4}\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T))}\\\\ \\nonumber && +\n3\\frac{j^3}{T^4}\\frac{exp(- \\lambda-j\/T)^2}{(1-exp(-\n\\lambda-j\/T))^2}\n+2\\frac{j^3}{T^4}\\frac{exp(- \\lambda-j\/T)^3}{(1-exp(- \\lambda-j\/T))^3}\\\\\n\\nonumber && -2\\frac{j^2}{T^3}\\frac{exp(- \\lambda-j\/T)^2}\n{(1-exp(- \\lambda-j\/T))^2}\\big)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{T \\lambda}(T, \\lambda)&=& \\sum_{j=0}^{\\infty}\n\\big(-\\frac{j^2}{T^2} \\frac{exp(- \\lambda-j\/T)}{(1-exp(-\n\\lambda-j\/T))}-3\n\\frac{j^2}{T^2} \\frac{exp(- \\lambda-j\/T)^2}{(1-exp(- \\lambda-j\/T))^2} \\\\\n\\nonumber && -2\\frac{j^2}{T^2}\\frac{exp(- \\lambda-j\/T)^3}\n{(1-exp(- \\lambda-j\/T))^3}\\big)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{ \\lambda \\lambda}(T, \\lambda)&=& \\sum_{j=0}^{\\infty} \\big( j\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T)}\n+3 j \\frac{exp(- \\lambda-j\/T)^2} {(1-exp(- \\lambda-j\/T))^2} \\\\\n\\nonumber && +2 j \\frac{exp(- \\lambda-j\/T)^3} {(1-exp(-\n\\lambda-j\/T))^3} \\big)\n\\end{eqnarray}\n\\end{scriptsize}\nFurthermore, it turns out that the determinant of the metric\ntensor takes an inelegant polynomial form and thus, in order\nto simplify the notations, we may define a level function\n\\begin{scriptsize}\n\\begin{eqnarray}\nb_j(T, \\lambda):= exp(- \\lambda-j\/T)\n\\end{eqnarray}\n\\end{scriptsize}\nIt is however evident that the local stability of the full\nphase-space configurations may be determined by computing the\ndeterminant of the concerned thermodynamic metric tensor. Here,\nwe may likewise provide a compact formula for the determinant of\nthe metric tensor, and exclusively, the intrinsic geometric analysis\nassigns a compact expression to the determinant of the metric tensor.\nA straightforward computation thus shows that the determinant of the\nmetric tensor as the function of arbitrary chemical potentials is\n\\begin{scriptsize}\n\\begin{eqnarray}\n||g(T, \\lambda)|| &=& \\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\n\\frac{(-2 T+2 T b_j+j+ jb_j)}{(b_j-1)^3)}) \\times \\\\ \\nonumber &&\n\\sum_{j=0}^{\\infty}(-jb_j \\frac{(1+b_j)}{(b_j-1)^3})-\n(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\n\\frac{(1+b_j)}{(b_j-1)^3)})^2\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{figure}[htb]\n\\vspace*{-1cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Wiendet.ps,width=12cm}\n\\end{center}\n\\vspace*{0.01cm}\n\\caption{The Wienhold determinant of the metric tensor as a function of temperature,\nT; and chemical potential, $ \\lambda=L$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\nWe thus see from Fig. 1 that the determinant of the Weinhold metric develops\nan instability when $T>0.15$. This is also expected, since the coarse graining\nphenomenon should break down near small effective canonical temperatures. The\nsystem is well-defined and physically sound for all $ T \\in [0, 0.15]$.\n\nIn order to conclusively analyze the nature of chemical correlations\nand the concerned properties of the statistical configuration, one needs\nto determine certain globally invariant quantities on the intrinsic manifold\n$(M_2,g)$ of the  parameters. In fact, one may easily ascertain that\nthe simplest of such invariants is the scalar curvature of $(M_2,g)$. In order to\ndo so, one may first compute the Riemann curvature tensor $R_{T \\lambda T\n\\lambda}(T,\\lambda)$ and then, from the previously defined expression, obtain\nthe scalar curvature. We see that it is not difficult to compute the\ncovariant Riemann tensor $R_{T\\lambda T \\lambda}(T, \\lambda)$. The\nexact expression for the $R_{T\\lambda T \\lambda} (T, \\lambda)$ is quite\ninvolved and thus we relegate it to the Appendix B.\n\nIn a straightforward fashion we can show, by applying our previously advertised\nintrinsic geometric technology, that one may easily obtain the scalar\ncurvature simply via the relation\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(T, \\lambda)= \\frac{R_{T  \\lambda T \\lambda}(T, \\lambda)}{||g(T,\n\\lambda)||}\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{figure}[htb]\n\\vspace*{0cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Wiencur.ps,width=12cm}\n\\end{center}\n\\vspace*{0.01cm}\n\\caption{The Wienhold scalar curvature as a function of temperature, T; and\nchemical potential, $ \\lambda=L$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\nWe observe from Fig. 2 that the system becomes strongly correlated\nfor $T<0.05$, and for $ T \\in [0.01, 0.02]$ it acquires the chemical\ncorrelation length of $8 \\times 10^3$. Interestingly, we do not see\nany chemical fluctuations for $T>0.08$.\nIn the domain of small chemical potentials $T, \\lambda \\in [0,0.2] $,\nthe pictorial views of the stability may be perceived from the\ndeterminant of the Weinhold metric tensor $g^{(E)}(T, \\lambda)$\nand the concerned correlation length from the corresponding scalar\ncurvature $R^{(E)}(T, \\lambda)$. We notice that both the plot of\nthe determinant and that of the scalar curvature of the Wienhold \ngeometry are nice regular 3D hyper-surfaces.\n\nWe thence realize that the microcanonical representation of the black\nholes under consideration posits that there exists a large number\nof correlated degenerate microscopic states of the $1\/2$ BPS giants\nand superstars. The non-vanishing scalar curvature suggests that\na constituent microstate may as well avoid that the initial pure\nstates collapses to a particular pure black hole microstate. This\nis because of the fact that the intrinsic geometric structures of\nAdS black hole can be deduced by certain suitably defined subtle\nmeasurements \\cite{hep-th\/0508023}, and thus there is no loss of\ninformation.\n\\section{The Fluctuating Young Tableaux}\nAs in the previous section, the present section is devoted to investigate\nthe notion of large number of boxes into our state-space geometry, and thereby\nwe shall analyze the possible role of concerned large integers in the veritable\ngiant black hole configurations, being considered in the framework of liquid\ndroplet model or fuzzball description. In this concern, we shall precisely\ncompute the statistical pair correlations under this consideration, which\nascribes a set of self convincing physical meaning to the state-space\nquantities, for the simplest excited giant configurations\n\\begin{scriptsize}\n\\begin{eqnarray}\nS(n,M)= ln((M^2)!)-ln(n!)-ln((M^2-n)!)\n\\end{eqnarray}\n\\end{scriptsize}\nEmploying the previously proclaimed formulation, one may easily read\noff the components of the covariant state-space metric tensor to be\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}(n,M)= \\Psi(1,n+1)+ \\Psi(1,M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nM}(n,M)=  -2 M \\Psi(1,M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{MM}(n,M)= 4 M^2 \\Psi(1,M^2-n+1) -4 M^2 \\Psi(1,M^2+1)-2\n\\Psi(M^2+1)+2 \\Psi(M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\nIn the above expressions, the $\\Psi(n,x)$ is the $n^{th}$\npolygamma function, which is the $n^{th}$ derivative of the usual\ndigamma function. Specifically, it turns out that the $\\Psi(x)$\nmay defined to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Psi(x)= \\frac{\\partial }{ \\partial x} \\ln( \\Gamma(x))\n\\end{eqnarray}\n\\end{scriptsize}\nIn this framework, we observe that there exists a very simple\ndescription which divulges the geometric nature of the statistical\npair correlations. The fluctuating extremal black holes may thus\neasily be determined in terms of the mass and angular momentum of\nthe underlying configurations. Moreover, it is evident that the\nprinciple components of the statistical pair correlations are\npositive definite, for a range of the parameters of the concerned black\nholes, which physically signifies a certain self-interaction of a\nfictitious particle moving on an intrinsic surface $(M_2(R),g)$.\nSignificantly, it is clear in this case that the following\nstate-space metric constraints hold:\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn} &>& 0, \\ \\ \\forall \\ (n,M)  \\ |  \\ \\Psi(1,n+1)+ \\Psi(1,M^2-n+1)>0  \\\\\n\\nonumber  g_{MM} &>& 0, \\ \\ \\forall \\ (n,M)  \\ |  \\\n\\Psi(1,M^2-n+1)-\\Psi(1,M^2+1)> \\\\ \\nonumber &&\n\\frac{1}{2M^2}(\\Psi(M^2+1)- \\Psi(M^2-n+1))\n\\end{eqnarray}\n\\end{scriptsize}\nConsequently, we may easily reveal that the common domain of the above\nstate-space constraints defines the range of physically sensible\nvalues of the chosen number of boxes and total number of boxes, such\nthat the giants may remain into certain locally stable statistical\nconfigurations. We may also notice that the total boxes component\n$g_{MM}$ of the state-space metric tensor is asymmetrical, in comparison\nto the $g_{nM}$ and $g_{nn}$. This is physically well-accepted, because\nof the fact that the component associated with the large number of boxes\nis somewhat like the heavy head-on collision of two equal particles, which\nalternate more energy, in contrast to the other excitations, either involving\nthe single particle or the excited-unexcited particles. It is pertinent to\nmention that the relative pair correlation function determines the selection\nparameter for a chosen black hole, which may be defined as the modulus of the\nratio of excited-excited to excited-unexcited statistical correlations.\nImportantly, we procure that the parameter thus apprised may be given\nas the ratio of two digamma functions\n\\begin{scriptsize}\n\\begin{eqnarray}\na: = \\frac{1}{2M} \\vert \\frac{\\Psi(1,n+1)}{\\Psi(1,M^2-n+1)}\\vert\n\\end{eqnarray}\n\\end{scriptsize}\nIt is worth to mention that when $n>1$, one has $\\Psi(n,x)= \\Psi(\nn ) + \\gamma$, which is an ordinary rational number, where the\n$\\gamma$ is the standard Euler's constant. For small values of $n$,\n$\\Psi(n)$ is computed as a sum of gamma functions, which is again a rational\nnumber. To force this computation to be performed for larger values\nof the $n$, we may use\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Psi(n,x) = \\frac{\\partial^n \\Psi(x)}{\\partial x^n},\n\\end{eqnarray}\n\\end{scriptsize}\nwith $\\Psi(0,x) = \\Psi(x)$. Moreover, the stability of the\nunderlying statistical configurations may earnestly be analyzed by\ncomputing the degeneracy of the associated two dimensional state-space\nmanifold. In fact, we may easily ascertain that the determinant of\nthe state-space metric tensor may be given to be\n\\begin{scriptsize}\n\\begin{eqnarray}\ng(n,M) &=& -4M^2 \\Psi(1,n+1)\\Psi(1,M^2+1)-2\\Psi(1,n+1)\\Psi(M^2+1) \\\\\n\\nonumber && +4M^2 \\Psi(1,n+1)\\Psi(1,M^2-n+1)\n+2\\Psi(1,n+1)\\Psi(M^2-n+1) \\\\ \\nonumber &&\n-4M^2 \\Psi(1,M^2-n+1)\\Psi(1,M^2+1) -2\\Psi(1,M^2-n+1)\\Psi(M^2+1) \\\\\n\\nonumber && +2\\Psi(1,M^2-n+1)\\Psi(M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{center}\n \\begin{figure}[htb]\n\\vspace*{-3.5cm}\n\\begin{center}\n\\hspace{-7.5cm}\n\\epsfig{file=rupp.ps,width=14cm,angle= 90}\n\\end{center}\n\\vspace*{-4cm}\n\\caption{The Ruppenier determinant of the state-space metric tensor\nas a function of excited droplets, $n$; and total number of fundamental\ncells, $M^2$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\n\\end{center}\nFig. 3 indicates that the canonical configurations become unstable\nand ill-defined when both of the $n$ and $M$ take small values. Up\nto $10^5$ boxes, we discover that the same qualitative features\nhold. Particularly, it is worth to mention that the fluctuation\nbumps are aligned towards the $n^{th}$ boundary, when $n$ and $M$\nare of the same order.\n\nThe determinant of the metric tensor thus calculated is non-zero\nfor any set of given non-zero total boxes and excited boxes, and\nthus, for the set of proper choices, it provides a non-degenerate\nstate-space geometry for this configuration. In turn, one may\nillustrate the order of statistical correlations between the\nequilibrium microstates of the giant black hole system. Besides\nthe fact that the principle component constraints $\\lbrace g_{ii}\n> \\ 0 \\ \\vert \\ i= n, M \\rbrace$ imply that this system may\naccomplish certain locally stable statistical configurations,\nhowever the negativity of the determinant of the state-space\nmetric tensor indicates that the underlying systems may globally\nendure a certain instability, as well for the bad choice of boxes,\nwhich correspond to certain unphysical macroscopic configurations.\n\nThis significantly connotes that there exists a positive definite\nvolume form on the $(M_2,g)$, for the good choice of boxes and\ntheir excitations. One may thus conclude that this system may\nremain in the nice chosen configuration or, for certain\nunacceptable choices, might move to some more stable brane\nconfigurations. Furthermore, one may easily examine, in general,\nthat the covariant Riemann correlation tensor may be given by the\ntwo large integers characterizing the droplets. However, an\nexplicit expression is given by Eq. \\ref{countentcovcur} for the\n$R_{nMnM}(n,M)$, which is rather involved, and thus we have\nrelegated it to the Appendix B.\n\nFurthermore, in order to examine certain global properties of\nsuch black holes phase-space configurations, one is required to\ndetermine the associated geometric invariants of the underlying\nstate-space manifold. For the giant and superstar black holes,\nthe simplest invariant turns out to be the state-space scalar\ncurvature, which may easily be computed by using the intrinsic\ngeometric technology defined as the negative Hessian matrix of the\nentropy captured by the excited contributions. Indeed, we discover\nthat the state-space curvature scalar for the fluctuating giant\n(and superstar) configurations may easily be depicted.\nNevertheless, the exact involved expression for the scalar\ncurvature $R(n,M)$ has been depicted in Eq. \\ref{countentscacur}\nand is shifted to the Appendix B.\n\\begin{figure}[htb]\n\\vspace*{.30cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Rupcur.ps,width=12cm}\n\\end{center}\n\\vspace*{-0.1cm} \\caption{The Ruppenier scalar curvature as a\nfunction of excited droplets, $n$, and the total number of\nfundamental cells, $M^2$, in the $1\/2$-BPS supergravity\nconfigurations.} \\vspace*{0.50cm}\n\\end{figure}\nWe numerically find the following conclusions for the state-space\nscalar curvature correlations.\n\n\\begin{itemize}\n\\item For $n, M \\in [1, 10^{10}]$, it turns out from Fig. 4 that\npositive, as well as negative correlations exist only along the\n$M^{th}$ boundary of the state-space configuration. It is also\nexpected from the general consideration of statistical fluctuation\ntheory that the system would possess certain attractions and\nrepulsions.\n\\item We notice, from the respective plots of the determinant and scalar\ncurvature of the state-space configuration, that the system\nremains stable and regular along the $n^{th}$ boundary of the\nstate-space configuration. We need not mention in this range that\nthe system becomes ill-defined near the origin $(0, 0)$ of the\nstate-space manifold.\n\\item We further observe for $10^4$ boxes that there exist definite small\nattraction and repulsion along the $n^{th}$ boundary of the $R$-$n$\nsurface. Interestingly, the same conclusion remains true for the $g$-$n$\nsurface of the underlying state-space configuration.\n\\end{itemize}\nWe recognize, in the framework of our state-space geometry, that\nthe negative sign of the curvature scalar signifies that this\nsystem is effectively an attractive configuration, under the\nGaussian fluctuations. Thus, the giants and superstars appear to\nbe manifestly stable configurations, with nice combinatorial\nproperties of the Young tableaux. Furthermore, we observe that the\ncurvature scalar thus considered is inverse squarely proportional\nto the determinant of the underlying state-space metric tensor,\nand thus there are no genuine state-space instability in the\nunderlying microscopic configurations. In turn, we discover that\nthe underlying statistical correlations remain definite, non-zero,\nfinite, regular functions of total boxes and excited boxes carried\nby the giant black holes.\n\nIt is important to note that the state-space geometric quantities\nmay become ill-defined, if the state-space co-ordinates, being\ndefined as the space-time parameters, jump from one existing\ndomain to another domain of the solution. This indicates that the\nmicroscopic configurations may correspond to certain interacting\nstatistical systems, in the chosen branch of the full black hole\nsolutions. In an admissible domain of physically acceptable boxes,\nwe discover that the giant black holes have no phase transition,\nand thus the fundamental statistical configurations are completely\nfree from critical phenomena. Note that the absence of divergences\nin the scalar curvature indicates that the BPS black brane\nsolutions endure an everywhere thermodynamically stable systems,\non their respective state-space configurations.\n\nMore generally, we find that the regular state-space scalar\ncurvature seems to be comprehensively universal for the given\nnumber of parameters of the configuration. In fact, the concerned\nidea turns out to be related with the typical form of the\nstate-space geometry, arising from the negative Hessian matrix of\nthe duality invariant expression of the black hole entropy. As in\nthe standard interpretation, the state-space scalar curvature\ndescribes the nature of underlying statistical interactions of the\npossible microscopic configurations, which characteristically turn\nout to be non-zero, for the $1\/2$-BPS black holes. In fact, we may\neasily appreciate that the constant entropy curve is a standard\ncurve, which may be given by\n\\begin{eqnarray}\n(M^2)!= c\\ n!(M^2-n)!\n\\end{eqnarray}\nHere, the real constant $c$ can be determined from the given\nexpression of the vacuum entropy. This determines the giant black\nhole embedding in the view-points of the state-space geometry.\nMoreover, we may also disclose in the present case that the curve\nof constant scalar curvature is, however, a complicated curve on\nthe space of total boxes and excited boxes, but the the concerned\nfundamental nature may easily be fixed by the given number of the\nboxes. Nonetheless, it is not difficult to enunciate the\nquantization condition existing on the charges, as we have an\ninteger number of boxes, which in turn signifies a general\ncoordinate transformation in the large number of boxes on the\nstate-space manifold, and may thus be presented in terms of the\nnet number of respective branes.\n\\section{Remarks and Conclusion}\nThis paper provides an exact intrinsic thermodynamic geometric\naccount of the canonical energy fluctuations and box counting\nentropy for the giants and superstars.  The arguments required\nhave been accomplished from an ensemble of Young tableaux with\ncertain randomly excited boxes, while the remaining boxes were\nkept intact. We notice that the microcanonical representations of\nthe black holes under consideration posit a large number of\ncorrelated degenerate microscopic states. These microstates may be\nevaded by showing that an initial pure state collapses to a\nparticular pure black hole microstate, whose exact structure can\nbe deduced by suitably subtle measurements \\cite{hep-th\/0508023},\nalong with the loss of information, if any. Here, our focus has\nbeen to investigate the implications of correlated states for the\n$1\/2$-BPS supergravity black holes. Our analysis clarifies the\nnature of underlying equilibrium configurations over the Gaussian\nfluctuations.\n\nGiven the importance of the canonical correlations, we have\naddressed the following questions: (i) how do the Gaussian\ncorrelations of pure microstates look like in the view-points of\nthermodynamic geometries, (ii) what sorts of correlation areas may\ndistinguish the giants or superstars from each other, and (iii)\nwhat is the typical nature of statistical fluctuations over an\nequilibrium ensemble of CFT microstates? Interestingly, we have\nexplicitly demonstrated that the underlying chemical fluctuations\ninvolve ordinary summations, while the state-space fluctuations\nmay simply be depicted by standard polygamma functions. Hereby, we\nnotice that the large black holes in AdS spacetimes, whose horizon\nsizes are bigger than the scale of the AdS curvature, are stable,\nand exclusively there is no thermodynamic phase transition. We\nherewith find a precise matching with the fact that these black\nholes come into an equilibrium configuration, because the AdS\ngeometries create an effective confining potential \\cite{HP}, and\nthus yield the stability of the thermodynamic system.\n\nImportantly, our framework exploits the fact that the classical\ngeneral relativity large black holes come with the basic property\nthat their horizon area satisfies the Bekenstein-Hawking entropy\nrelation. With an understanding of the limiting microscopic\nconfigurations, an appropriate intrinsic geometric notion has in\neffect been offered to the counting problem of the microstates of\nthe giants and superstars. Following the discussion of the Appendix\nA, this provides an account to the degeneracy of microscopic states\nin the limit $G_N\/l_P>>1$. The semi-classical approximation thus \nuncovers that the Boltzmann statistical entropy entails certain \nwell-defined equilibrium microstate systems, upon the inclusion\nof a quadratic fluctuation on the phase-space configurations.\nOur analysis thus shows that the $1\/2$-BPS giants and superstars\nare thermodynamically stable objects and do not emit Hawking radiation.\n\nThe state-space description shows that a set of exact expressions\nmay easily be given for the quadratic fluctuations, over an\nequilibrium canonical configuration characterized by possible\nexcited and unexcited droplets, in an ensemble of arbitrary random\nYoung tableaux. In turn, the state-space fluctuations, stability\ncriterion and state-space correlation length have precisely been\ndetermined, without any approximation, for a large number of giant\ngravitons and superstars. Following the standard Riemannian\ngeometric notions, we expose that the chemical configuration\nyields in general that the chemical pair correlation functions,\nstability condition and correlation length, for an arbitrary value\nof the effective canonical temperature and the chemical potential,\nmay be determined over an intrinsic Weinhold manifold. Our\nintrinsic geometric study, thus, exemplifies that there exists an\nexact fluctuating statistical configuration, which involves an\nensemble of fuzzballs, or a number of liquid droplets.\n\nFurthermore, we can ask how the underlying highly degenerate\nequilibrium microstates get correlated with the chemical\npotential, carried by a set of given excited droplets. We have\nshown, from the perspective of string theory on $AdS_5 \\times\nS^5$, that the five dimensional $R$ charged AdS black holes admit\nan exact Legendre transformed dual chemical configuration, which\nmay be described in terms of the  chemical potential and an\neffective canonical temperature. Thus, the origin of gravitational\nthermodynamics comes with the existence of a non-zero\nthermodynamic curvature, under the coarse graining mechanism of\nalike ``quantum information geometry'', associated with the wave\nfunctions of underlying BPS black holes. Notice, further, that the\nphysical meaning of the respective curvatures is that they\ndescribe the correlation, in the concerned ensemble of black\nholes. Characteristically, it is worth to mention that the\nsemi-classical gravitational description, arising with the\nnon-vanishing scalar curvature, signifies the existence of the\nthroat of the underlying AdS background.\n\nWe notice an intriguing support, from the Mathur's fuzzball\nproposal, that the leading order entropy should come from those\nfuzzs, whose radius varies as the fuzzy throat of the black hole\nhorizon size. Furthermore, the liquid droplet model suggests that\nthe AdS length scales like $L\\sim l_P f(N_i)$, where $N_i$ is the\nnumber of bound states \\cite{hep-th\/0508023, hep-th\/0107119}. It\nhas further been suggested that the classical description is\nachieved, when $N \\rightarrow \\infty$ while $\\hbar \\rightarrow 0$,\nsuch that the Fermi level $N\/ \\hbar$ remains constant. Most\nimportantly, we have investigated the role of thermodynamic\nfluctuations in the two parameter giants and superstars, whose\nchemical and state-space configurations are being respectively\ncharacterized by the chemical variables of the effective canonical\nensemble, and by the number of both excited and total boxes,\nconstituting an ensemble of arbitrary shaped CFT microstates. The\npresent analysis thus explicates that the underlying chemical and\nstate-space configurations are well-defined, non-degenerate and\nregular, for all physically admissible domains of the statistical\nparameters, defining an ensemble of arbitrary fuzzballs, or a set\nof indiscriminate liquid droplets.\n\nNow, we enlist a number of attributes arising from our study of\nthe thermodynamic intrinsic geometry of (dual) giants and\nsuperstars, divulged in the framework of liquid droplets or\nfuzzball solutions. The intriguing nature of the chemical and the\nstate-space correlations, existing in an underlying statistical\nbasis, ascribes that the local and global thermodynamic\nstructures, thus revealed, may in either case be summarized as\nfollows.\n\\subsection{Chemical Description:}\nFollowing our specific notions, we exclusively observe that the\nchemical configurations of the $1\/2$ BPS giants and superstars\nrealize the following general properties:\n\\begin{itemize}\n\\item There exists an exact expression for the chemical pair\ncorrelation functions, stability condition and correlation length,\nfor arbitrary values of $T, \\lambda \\in M_2$.\n\\item Explicit plots displayed in the Figs. 1 and 2 show that\nthe determinant and scalar curvature are non-trivially curved, and\nsurprisingly the results remain the same, even for a single\ncomponent with $j=1$.\n\\item We further reveal, from the numerical conclusions displayed in\nFigs. 1 and 2, that the Weinhold metric tensor, and the\ncorresponding correlation length, show that the statements of\nstability and regularity hold for all $\\lambda \\in M_2$.\n\\end{itemize}\n\\subsection{State-space Description:}\nIn this case, the findings obtained from the intrinsic state-space\ngeometry from an ensemble of Young tableaux may be summarized as\n\\begin{itemize}\n\\item A set of exact precise expressions may easily be given for the\nstate-space fluctuations, over an equilibrium canonical ensemble,\ncharacterized by possible arbitrary random Young tableaux.\n\\item The state-space fluctuations, stability criterion and\nstate-space correlation length may easily be determined without\nany approximation.\n\\item We can express the $g^{(S)}_{ij}(n,M)$, $g^{(S)}(n,M)$\nand $R^{(S)}(n,M)$ in terms of nice, well-behaved digamma\n$\\Psi(n)$ and poly-gamma $\\Psi(n,M)$ functions. Notice that the\nmixing between excited and non-excited droplets or fuzzs may\nprecisely be caused by these standard polygamma functions.\n\\item The concerned state-space geometry corresponds to a non-degenerate,\nlocally stable and attractive statistical configuration.\n\\item The underlying numerical computations plotted in Figs. 3 and 4\ndisclose that the state-space correlation exists along the\nboundary of $n$ and $M$, for non-large $n$, $M$, when we take\napproximately $1$ to $10^5$ boxes in the Young diagrams.\n\\end{itemize}\nIt is worth to mention that the present analysis takes into\naccount the scales, that are larger than the respective\ncorrelation length, and contemplates that just a few giant or\nsuperstar microstates cannot dominate the entire macroscopic\nsolution. Most importantly, we have procured that the\nthermodynamic intrinsic geometric structures of the canonical\nenergy and box counting entropy provide a coherent framework, to\nfurther study the thermodynamic geometries, arising from the large\nnumber of microstates of the chosen giants, superstars and the\nother superconformal field theory and supergravity configurations\n\\cite{9711200v3}.\n\nFinally, it is worth to mention that the interpretation of a\nnon-zero intrinsic scalar curvature, with an underlying\ninteracting statistical system, remains valid even for higher\ndimensional intrinsic Riemannian manifolds. The implication of a\ndivergent intrinsic covariant curvature may accordingly be\ndivulged, from the Hessian matrix of the canonical energy or the\nbox counting entropy, irrespectively whether, or not, there exists\na phase transition in the $1\/2$ BPS black hole configurations.\n\\section*{Acknowledgements}\nThis work has been supported in part by the European Research\nCouncil grant n.~226455, ``SUPERSYMMETRY, QUANTUM GRAVITY AND\nGAUGE FIELDS (SUPERFIELDS)''. We would like to thank Prof. J.\nSim\\'on for useful discussions and view-points provided during the\n``School on Attractor Mechanism SAM-2009, INFN- Laboratori\nNazionali di Frascati, Roma, Italy''. BNT would like to thank\nProf. S. Mathur and Prof. A. Sen for useful discussions offered\nduring the ``Indian String Meeting, ISM-2006, Puri, India''; Prof.\nJ. de Boer during the ``Spring School on Superstring Theory and\nRelated Topics-2007 and 2008, ICTP, Trieste, Italy''; and Yogesh\nSrivastava during the Indian String Meeting-2007, Harish-Chandra\nResearch Institute, Allahabad, India; and Mohd. A. Bhat and V.\nChandra for reading the manuscript and making interesting\nsuggestions. BNT especially thanks Prof. V. Ravishankar for\nencouragements and necessary supports provided during this work.\nBNT would like to acknowledge nice hospitality of the\n``INFN-Laboratori Nazionali di Frascati, Roma, Italy'' being\noffered during the ``School on Attractor Mechanism: SAM-2009''\nwhere part of this work was performed. The research of BNT has\npartially been supported by the CSIR, New Delhi, India and Indian\nInstitute of Technology Kanpur, Kanpur-208016, Uttar Pradesh,\nIndia.\n\\section*{Appendix A}\nIn this appendix, we provide precise expression for the case of two\nparameter thermodynamic configuration. This review set-up is offered from\nthe viewpoint of two parameter family giant and superstar solutions.\nIn order to do so, we recall important recent studies of the thermodynamic\nproperties of diverse (rotating) black holes have elucidated interesting\naspects of phase transitions, if any, in the state-space geometric framework\nand their associated relations with the extremal black hole solutions in the\ncontext of $\\mathcal N \\geq 2$ compactifications \\cite{sfm1, sfm2}. It may\nbe argued however that the connection of such a geometric formulation to\nthe thermodynamic fluctuation theory of black holes requires several\nmodifications \\cite{rup3}. The geometric formulation thus involved has\nfirst been applied to $\\mathcal  N\\geq 2$ supergravity extremal black\nholes in $D=4$, which arise as low energy effective field theories from\nthe compactifications of Type II string theories on Calabi-Yau manifolds\n\\cite{fgk}. Since then, several authors have attempted to understand\nthis connection \\cite{cai1,gr-qc\/0304015v1,0510139v3,Arcioni}, for both\nthe supersymmetric as well as non-supersymmetric four dimensional black\nholes and five dimensional rotating black string and ring solutions.\nInteresting discussions on the vacuum phase transitions, if any,\nexist in the literature, which involves some change of the black hole\nhorizon topology, \\cite{cai1,gr-qc\/0304015v1,0510139v3,Arcioni}.\n\nRuppenier has conjointly advocated the assumption ``that\nall the statistical degrees of freedom of black hole live on the\nblack hole event horizon'', and thus the scalar curvature signifies\nthe average number of correlated Planck areas on the event horizon\nof the black hole \\cite{RuppeinerPRD78}. Specifically, the zero\nscalar curvature indicates certain bits of information on the\nevent horizon fluctuating independently of each other, while the\ndiverging scalar curvature signals a phase transition indicating\nhighly correlated pixels of the informations. Moreover, Bekenstein\nhas introduced an elegant picture for the quantization of the area of\nthe event horizon, being defined in terms of Planck areas\n\\cite{Bekenstein}. Recently, the state-space geometry of the\nequilibrium configurations thus described has extensively been\napplied to study the thermodynamics of a class of rotating black\nhole configurations \\cite{bnt,BNTBull,bntSb, BSBR}.\n\nFrom the viewpoint of the present consideration,\nthe underlying moduli configuration appears to be horizonless and\nsmooth. However, in the classical limit in which the Planck length and\nthe AdS throat scale as, respectively $l_P\\rightarrow 0$ and $L\\rightarrow 0$,\nin such a way that their ratio diverges $l_P\/L \\rightarrow \\infty$,\nthe underlying moduli configuration acquires an entropy which may\nbe assumed to be associated with the average horizon area of the \nblack hole. In general, one wishes to compare the quantization of a\nclassical moduli space from the known perspective of the AdS\/CFT\ncorrespondence \\cite{LLM, SB}. In this concern, it is worth to\nmention that we have \n\\begin{scriptsize}\n\\begin{eqnarray}\nAdS\/CFT: \\ \\{\\mathcal{N}=4 \\ SYM \\}\\ \\leftrightarrow \\ \\{Type \\\nIIB \\ String \\ Theory \\ on \\ AdS_5 \\times S^5 \\}\n\\end{eqnarray}\n\\end{scriptsize}\nFollowing \\cite{hep-th\/0508023,hep-th\/0107119}, it is important to mention\nthat the supergravity description emerges in the strong coupling limit with\n\\begin{scriptsize} $g^2_{YM}N>>$ \\end{scriptsize}\nwhile, the dual CFT emerges in the weak coupling limit \\begin{scriptsize}\n$g^2_{YM}N<<1$\\end{scriptsize}. Thus, one finds the matching of entropies\nin the $\\frac{1}{2}$ BPS sector of the configuration \\cite{hep-th\/0508023,\nhep-th\/0107119}. In this sector, the authors of \\cite{hep-th\/0508023,hep-th\/0107119}\nhave shown  that nearly all states look alike and they belong to the same chiral\nring quantization of the classical moduli space. It is however intriguing to note \nthat the dual CFT formalism faces problems with the wave functional renormalization,\nmixing of states, and in general it may feature certain other technical difficulties\nas well \\cite{hep-th\/0508023, hep-th\/0107119}. Nevertheless, none of these concerns\nthe present analysis, and thus we may safely divulge the probable fluctuations\nover chosen equilibrium giants configurations. \n\nAs mentioned in the introduction, the $1\/2$-BPS (dual) giant configurations\nare parameterized by two chemical potentials, $\\lambda_1, \\lambda_2$. Thus,\nthe fluctuations around the minima energy configuration are describe an\nintrinsic Wienhold surface. \nExplicitly, let us consider the case of the two dimensional intrinsic chemical geometry,\nsuch that the components of the metric tensor are given by \\\\\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{ \\lambda_1  \\lambda_1}&=& \\frac{\\partial^2 E}{\\partial  \\lambda_1^2} \\nonumber  \\\\\ng_{ \\lambda_1  \\lambda_2}&=& \\frac{\\partial^2 E}{{\\partial  \\lambda_1}{\\partial  \\lambda_2}}\\nonumber  \\\\\ng_{ \\lambda_2  \\lambda_2}&=& \\frac{\\partial^2 E}{\\partial\n\\lambda_2^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIn this case, it follows that the determinant of the metric tensor is \n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Vert g(\\lambda_1,\\lambda_2) \\Vert &= &S_{\\lambda_1 \\lambda_1}S_{\\lambda_2 \\lambda_2}- S_{\\lambda_1 \\lambda_2}^2\n\\end{eqnarray}\n\\end{scriptsize}\nNow, we can calculate the $\\Gamma_{ijk}$, $R_{ijkl}$, $R_{ij}$ and\n$ R $ for the above two dimensional thermodynamic geometry $(M_2,g)$.\nOne may easily inspect that the scalar curvature is given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(\\lambda_1, \\lambda_2)&=& -\\frac{1}{2} (E_{ \\lambda_1 \\lambda_1}E_{ \\lambda_2\n\\lambda_2}- E_{ \\lambda_1  \\lambda_2}^2)^{-2} (E_{ \\lambda_2\n \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_1}E_{ \\lambda_1  \\lambda_2\n \\lambda_2}\\nonumber\n\\\\ &&+ E_{ \\lambda_1  \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_2}E_{\n\\lambda_1 \\lambda_2 \\lambda_2}+\nE_{ \\lambda_1 \\lambda_1}E_{ \\lambda_1 \\lambda_1 \\lambda_2}E_{ \\lambda_2 \\lambda_2 \\lambda_2}\\nonumber  \\\\\n&&- E_{ \\lambda_1 \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_1}E_{\n\\lambda_2 \\lambda_2 \\lambda_2}- E_{ \\lambda_1 \\lambda_1}E_{\n\\lambda_1 \\lambda_2 \\lambda_2}^2 - E_{ \\lambda_2 \\lambda_2}E_{\n\\lambda_1 \\lambda_1 \\lambda_2}^2 )\n\\end{eqnarray}\n\\end{scriptsize}\nFurthermore, the relation between the chemical scalar curvature\nand the Riemann covariant curvature tensor for any two dimensional\nintrinsic geometry is given (see for details \\cite{bnt}) by\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wiencur}\nR=\\frac{2}{\\Vert g \\Vert}R_{ \\lambda_1 \\lambda_2 \\lambda_1 \\lambda_2}\n\\end{eqnarray}\n\\end{scriptsize}\nThe relation Eq. \\ref{Wiencur} is quite usual for an arbitrary\nintrinsic Riemannian surface $(M_2(R),g)$. \nCorrespondingly, the Legendre transformed version of the Wienhold manifold\nis known as state-space manifold. In fact, the associated configuration\nis parameterized by the number of excited and total boxes, i.e. $ \\{n,M\\}$. \nThus, the Gaussian fluctuation of the entropy of $1\/2$-BPS configurations\nform a two dimensional state-space surface $M_2$.\nExplicitly, the components of covariant state-space metric tensor may be given as\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}&=&- \\frac{\\partial^2 S(n,M)}{\\partial n^2} \\nonumber \\\\\ng_{nM}&=&- \\frac{\\partial^2 S(n,M)}{{\\partial n}{\\partial M}}  \\nonumber  \\\\\ng_{MM}&=&- \\frac{\\partial^2 S(n,M)}{\\partial M^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt is easy to express, in this simplest case, that the\ndeterminant of the metric tensor turns out to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Vert g(n,M) \\Vert &= &S_{nn}S_{MM}- S_{nM}^2\n\\end{eqnarray}\n\\end{scriptsize}\nAs in the case of Wienhold geometry, we find that the state-space scalar curvature is given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(n,M)&=& \\frac{1}{2} (S_{nn}S_{MM}- S_{nM}^2)^{-2}\n(S_{MM}S_{nnn}S_{nMM} + S_{nM}S_{nnM}S_{nMM} \\nonumber\\\\ &+&\nS_{nn}S_{nnM}S_{MMM} -S_{nM}S_{nnn}S_{MMM}- S_{nn}S_{nMM}^2-\nS_{MM}S_{nnM}^2 )\n\\end{eqnarray}\n\\end{scriptsize}\nFollowing the observations of \\cite{BNTBull}, it is essentially evident\nthat the scalar curvature and the corresponding Riemann curvature tensor\nof an arbitrary two dimensional intrinsic state-space manifold\n$(M_2(R),g)$ may be given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(n,M)=\\frac{2}{\\Vert g \\Vert}R_{nMnM}(n,M)\n\\end{eqnarray}\n\\end{scriptsize}\n\\section*{Appendix B}\nIn this appendix, we provide the explicit form of the most general\nthermodynamic scalar curvatures describing the family of two\ncharged giant and superstars. Our analysis illustrates that the\nphysical properties of the specific scalar curvatures may exactly\nbe exploited, without any approximation. The definite behavior of\ncurvatures, as accounted in section three, suggests that the\nvarious intriguing chemical and state-space examples of $1\/2$ BPS\nsolutions include the nice property that they do not diverge,\nexcept for the determinant singularity. As mentioned in the main\nsections, these configurations are an interacting statistical\nsystem. We discover that their thermodynamic geometries indicate\nthe possible nature of general two parameter equilibrium\nconfigurations. Significantly, one may notice, from the very\ndefinition of intrinsic metric tensors, that the relevant Riemann\ncovariant curvature tensors and scalar curvatures may be thus\npresented as follows.\n\\subsection*{(i) Canonical Energy Fluctuations:}\nHere, we shall explicitly supply the exact Riemann covariant\ntensor of the Weinhold geometry for the 2- parameter giants and\nsuperstars. It turns out that the functional nature of a large\nnumber of gravitons, within the neighborhood of small chemical\nfluctuations introduced in the canonical ensemble of $1\/2$ BPS\nconfigurations, may precisely be divulged. Surprisingly, we expose\nin this framework, that the intrinsic covariant curvature tensor\ntakes the exact and simple expression\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{canenfluc}\nR_{T \\lambda T \\lambda}(T, \\lambda)&=& -\\frac{1}{4}\n\\{(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j+j+j\nb_j)} {(b_j-1)^3}))\\times\n(\\sum_{j=0}^{\\infty}(-j b_j\\frac{(1+b_j)}{(b_j-1)^3}) \\\\\n\\nonumber && -(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\n\\frac{(1+b_j)}{(b_j-1)^3}))^2\\}^{-1} \\{-(\\sum_{j=0}^{\\infty}(-j\nb_j\\frac{(1+b_j)}{(b_j-1)^3})) \\times \\\\ \\nonumber &&\n(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j^2+j+4\nj b_j+j b_j^2)}{(b_j-1)^4}))^2 +(\\sum_{j=0}^{\\infty}(-j\nb_j\\frac{(1+b_j)}{(b_j-1)^3})) \\times\n\\\\ \\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^6}\n\\frac{(j^2+4 j^2 b_j+j^2 b_j^2-6 j T+6 T^2+6 j T b_j^2-12 T^2\nb_j+6 T^2 b_j^2)}{(b_j-1)^4})) \\\\ \\nonumber && \\times\n(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2} \\frac{(1+4\nb_j+b_j^2)}{(b_j-1)^4})) +(\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+b_j)} {(b_j-1)^3})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2\nT+2 T b_j^2+j+4 j b_j+j b_j^2)}{(b_j-1)^4})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\\frac{(1+4\nb_j+b_j^2)}{(b_j-1)^4})) -(\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+b_j)}{(b_j-1)^3})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^6} \\frac{(j^2+4\nj^2 b_j+j^2 b_j^2-6 j T+6 T^2+6 j T b_j^2-12 T^2 b_j+6 T^2\nb_j^2)}{(b_j-1)^4})) \\\\ \\nonumber && \\times\n(\\sum_{j=0}^{\\infty}(-j b_j \\frac{(1+4 b_j+b_j^2)} {(b_j-1)^4}))\n-(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j+j+j\nb_j)}{(b_j-1)^3})) \\times \\\\ \\nonumber && (\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+4 b_j+b_j^2)}{(b_j-1)^4}))^2\n+(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4} \\frac{(-2 T+2 T b_j+j+j\nb_j)} {(b_j-1)^3})) \\times \\\\ \\nonumber &&\n(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4} \\frac{(-2 T+2 T\nb_j^2+j+4 j b_j+j b_j^2)}{(b_j-1)^4})) \\times (\\sum_{j=0}^{\\infty}\n(-j b_j\\frac{(1+4 b_j+b_j^2)} {(b_j-1)^4})) \\}\n\\end{eqnarray}\n\\end{scriptsize}\n\\subsection*{(ii) State-space Fluctuations:}\nAs stated earlier, the state-space metric in the excited and\nunexcited droplets is given by the negative Hessian matrix of the\nbox counting entropy. Here, the numbers of excited and empty boxes\nin a given Young diagram are respected to be extensive variables.\nIn this case,  the two distinct large integers characterize the\nintrinsic state-space correlation length, carried by the giants\nand superstars. Our computation shows that the exact Riemann\ncovariant curvature tensor is given by\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{countentcovcur}\nR_{nMnM}(n,M) &=&\n1\/2\\{(-2M^2\\Psi(1,n+1)\\Psi(1,M^2+1)-\\Psi(1,n+1)\\Psi(M^2+1)\n\\\\ \\nonumber && +2M^2\\Psi(1,n+1)\\Psi(1,M^2-n+1) +\\Psi(1,n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && -2M^2\\Psi(1,M^2-n+1)\n\\Psi(1,M^2+1)-\\Psi(1,M^2-n+1)\\Psi(M^2+1) \\\\ \\nonumber &&\n+\\Psi(1,M^2-n+1)\\Psi(M^2-n+1))\\}^{-1} [\n\\Psi(1,n+1)\\Psi(1,M^2-n+1)^2 \\\\ \\nonumber && -\\Psi(1,M^2-n+1)\n\\Psi(2,n+1)\\Psi(M^2+1) \\\\ \\nonumber && +\\Psi(1,M^2-n+1)\\Psi(2,n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && +\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\n\\Psi(M^2+1)\\\\ \\nonumber &&-\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && +2 M^2 \\{ 3\\Psi(2,M^2-n+1)\n\\Psi(1,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber && +2\\Psi(1,M^2-n+1)\n\\Psi(2,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber &&\n-2\\Psi(1,M^2-n+1)^2\\Psi(2,n+1) +\\Psi(1,M^2-n+1)^3\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1) \\Psi(2,n+1)\\Psi(M^2+1)\\\\ \\nonumber &&\n+\\Psi(2,M^2-n+1)\\Psi(2,n+1) \\Psi(M^2-n+1)\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1) \\Psi(1,n+1)\\Psi(1,M^2-n+1) \\\\ \\nonumber &&\n+\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\\Psi(1,M^2+1)\\} \\\\ \\nonumber &&\n+4M^4 \\{ \\Psi(2,M^2-n+1)\\Psi(1,n+1)\\Psi(2,M^2+1)\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1)\\Psi(2,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber &&\n+\\Psi(1,M^2-n+1)\\Psi(2,n+1) \\Psi(2,M^2+1)\\}]\n\\end{eqnarray}\n\\end{scriptsize}\nFinally, let us turn our attention to the state-space scalar\ncurvature for the two parameter droplet configurations. A\nsystematic examination demonstrates that the scalar curvature is\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{countentscacur}\nR(n,M) &=& \\frac{1}{2} \\{-2 \\Psi(1,n+1) \\Psi(1,M^2+1)\nM^2-\\Psi(1,n+1) \\Psi(M^2+1) \\\\ \\nonumber && +2 M^2 \\Psi(1,n+1)\n\\Psi(1,M^2-n+1) +\\Psi(1,n+1) \\Psi(M^2-n+1) \\\\ \\nonumber && -2 M^2\n\\Psi(1,M^2-n+1) \\Psi(1,M^2+1)- \\Psi(1,M^2-n+1) \\Psi(M^2+1)\\\\\n\\nonumber && +\\Psi(1,M^2-n+1) \\Psi(M^2-n+1)\\}^{-2} \\\\ \\nonumber\n&&[ \\Psi(1,M^2-n+1)^3 + \\Psi(1,n+1) \\Psi(1,M^2-n+1)^2 \\\\ \\nonumber\n&& -\\Psi(1,M^2-n+1) \\Psi(2,n+1) \\Psi(M^2+1) \\\\\n\\nonumber && +\\Psi(1,M^2-n+1) \\Psi(2,n+1) \\Psi(M^2-n+1) \\\\\n\\nonumber && +\\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1) \\Psi(M^2+1) \\\\\n\\nonumber && -\\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1) \\Psi(M^2-n+1) \\\\\n\\nonumber && +2 M^2 \\{ \\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1)\n\\Psi(1,M^2+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1) \\Psi(1,n+1)\n\\Psi(1,M^2-n+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1) \\Psi(2,n+1)\n\\Psi(M^2+1) \\\\ \\nonumber && + \\Psi(2,M^2-n+1)\n\\Psi(2,n+1)\\Psi(M^2-n+1)\\\\ \\nonumber && - 2 \\Psi(1,M^2-n+1)^2\n\\Psi(2,n+1) \\\\ \\nonumber && + 2 \\Psi(1,M^2-n+1) \\Psi(2,n+1)\n\\Psi(1,M^2+1) \\\\ \\nonumber && +3 \\Psi(2,M^2-n+1) \\Psi(1,n+1)\n\\Psi(1,M^2+1)\\} \\\\ \\nonumber && +4 M^4 \\{ \\Psi(2,M^2-n+1)\n\\Psi(1,n+1) \\Psi(2,M^2+1)\\\\ \\nonumber && + \\Psi(1,M^2-n+1)\n\\Psi(2,n+1) \\Psi(2,M^2+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1)\n\\Psi(2,n+1) \\Psi(1,M^2+1) \\}]\n\\end{eqnarray}\n\\end{scriptsize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}