Global anomaly and a family of structures on fold product of complex two-cycles

We propose a new set of IIB type and eleven-dimensional supergravity solutions which consists of the $n$-fold product of two-spaces ${\bf H}^n/\Gamma$ (where ${\bf H}^n$ denotes the product of $n$ upper half-planes $H^2$ equipped with the co-compact action of $\Gamma \subset SL(2, {\mathbb R})^n$) and $({\bf H}^n)^*/\Gamma$ (where $(H^2)^* = H^2\cup \{{\rm cusp of} \Gamma\}$ and $\Gamma$ is a congruence subgroup of $SL(2, {\mathbb R})^n$). The Freed-Witten global anomaly condition have been analyzed. We argue that the torsion part of the cuspidal cohomology involves in the global anomaly condition. Infinitisimal deformations of generalized complex (and K\"ahler) structures also has been analyzed and stability theorem in the case of a discrete subgroup of $SL(2, {\mathbb R})^n$ with the compact quotient ${\bf H}^n/\Gamma$ was verified.Comment: 7 pages, no figures. To appear in the Proceedings of XXVIII Workshop on Geometrical Methods in Physics, Bialowieza (Poland), 28.06 - 04.07.200

Asymptotic Density of Open p-brane States with Zero-modes included

We obtain the asymptotic density of open p-brane states with zero-modes included. The resulting logarithmic correction to the p-brane entropy has a coefficient - \frac{p + 2}{2 p}, and is independent of the dimension of the embedding spacetime. Such logarithmic corrections to the entropy, with precisely this coefficient, appear in two other contexts also: a gas of massless particles in p-dimensional space, and a Schwarzschild black hole in (p + 2)-dimensional anti de Sitter spacetime.Comment: 9 pages, Latex. V 2: Results are for open p-branes only; Title modified; a few references and an acknowledgement adde

Quantum States, Thermodynamic Limits and Entropy in M-Theory

We discuss the matching of the BPS part of the spectrum for (super)membrane, which gives the possibility of getting membrane's results via string calculations. In the small coupling limit of M--theory the entropy of the system coincides with the standard entropy of type IIB string theory (including the logarithmic correction term). The thermodynamic behavior at large coupling constant is computed by considering M--theory on a manifold with topology ${\mathbb T}^2\times{\mathbb R}^9$. We argue that the finite temperature partition functions (brane Laurent series for $p \neq 1$) associated with BPS $p-$brane spectrum can be analytically continued to well--defined functionals. It means that a finite temperature can be introduced in brane theory, which behaves like finite temperature field theory. In the limit $p \to 0$ (point particle limit) it gives rise to the standard behavior of thermodynamic quantities.Comment: 7 pages, no figures, Revtex style. To be published in the Physical Review

Thermodynamics of Abelian Gauge Fields in Real Hyperbolic Spaces

We work with $N-$dimensional compact real hyperbolic space $X_{\Gamma}$ with universal covering $M$ and fundamental group $\Gamma$. Therefore, $M$ is the symmetric space $G/K$, where $G=SO_1(N,1)$ and $K=SO(N)$ is a maximal compact subgroup of $G$. We regard $\Gamma$ as a discrete subgroup of $G$ acting isometrically on $M$, and we take $X_{\Gamma}$ to be the quotient space by that action: $X_{\Gamma}=\Gamma\backslash M = \Gamma\backslash G/K$. The natural Riemannian structure on $M$ (therefore on $X$) induced by the Killing form of $G$ gives rise to a connection $p-$form Laplacian ${\frak L}_p$ on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on abelian $p-$forms on the real compact hyperbolic manifold $X_{\Gamma}$. The spectral zeta function related to the operator ${\frak L}_p$, considering only the co-exact part of the $p-$forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic fuctions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.Comment: Six pages, Revtex4 style, no figures; small typo correcte

Orbifold Compactification and Solutions of M--Theory from Milne Spaces

In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications $SL(2,{\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$, where $Id$ is the identity matrix, is analyzed in detail. The spectrum of a eleven-dimensional field theory can be obtained with the help of the theory of harmonic functions in the fundamental domain of this group and it is associated with the cusp forms and the Eisenstein series. The supersymmetry surviving for supergravity solutions involving real hyperbolic space factors is briefly discussed.Comment: 14 pages, no figures. To appear in The European Physical Journal C - Particles and Field

Quantum State Density and Critical Temperature in M-theory

We discuss the asymptotic properties of quantum states density for fundamental $p-$branes which can yield a microscopic interpretation of the thermodynamic quantities in M-theory. The matching of BPS part of spectrum for superstring and supermembrane gives the possibility of getting membrane's results via string calculations. In the weak coupling limit of M-theory the critical behavior coincides with the first order phase transition in standard string theory at temperature less than the Hagedorn's temperature $T_H$. The critical temperature at large coupling constant is computed by considering M-theory on manifold with topology ${\mathbb R}^9\otimes{mathbb T}^2$. Alternatively we argue that any finite temperature can be introduced in the framework of membrane thermodynamics.Comment: 16 pages, published in Mod. Phys. Lett. A16(2001)224

Statistical entropy of near-extremal and fundamental black p-branes

The problem of asymptotic density of quantum states of fundamental extended objects is revised in detail. We argue that in the near-extremal regime the fundamental $p$-brane approach can yield a microscopic interpretation of the black hole entropy. The asymptotic behavior of partition functions, associated with the $p$-branes, and the near-extremal entropy of five-dimensional black holes are explicitly calculated.Comment: 15 pages, LateX file.Minor changes,refs added, version to appear in Progr.Theor.Phy

Are p-Branes Asymptotically Black Holes?

An attempt is made to compare the asymptotic state density of twisted $p$-branes and the related state density of mass level $M$ of a $D$-dimensional neutral black hole. To this aim, the explicit form of the twisted $p$-brane total level degeneracy is calculated. The prefactor of the degeneracy, in contrast to the leading behaviour, is found to depend on the winding number of the $p$-brane.Comment: 8 pages, LaTex, UB-ECM-PF 93/2