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Heat-Kernel Asymptotics of Locally Symmetric Spaces of Rank One and Chern-Simons Invariants
A. A. Bytsenko
Physics , 2001, DOI: 10.1016/S0920-5632(01)01599-7
Abstract: The asymptotic expansion of the heat kernel associated with Laplace operators is considered for general irreducible rank one locally symmetric spaces. Invariants of the Chern-Simons theory of irreducible U(n)- flat connections on real compact hyperbolic 3-manifolds are derived
Quantum Corrections to the Entropy for Higher Spin Fields in Hyperbolic Space
M. Bordag,A. A. Bytsenko
Physics , 1994,
Abstract: We calculate the one-loop corrections to the free energy and to the entropy for fields with arbitrary spins in the space $S^1\otimes H^N$. For conformally invariant fields by means of a conformal transformation of the metric the results are valid in Rindler space with $D=N+1$ dimensions. We use the zeta regularization technique which yields an ultraviolet finite result for the entropy per unit area. The problem of the infinite area factor in the entropy which arises equally in Rindler space and in the black hole background is addressed in the light of a factor space $H^N/\Gamma$.
Casimir Energy and Thermodynamic Properties of the Relativistic Piecewise Uniform String
I. Brevik,A. A. Bytsenko
Physics , 2000,
Abstract: The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The great adaptibility of this string model with respect to various regularization methods is pointed out. We survey several regularization methods: the cutoff method, the complex contour integration method, and the zeta-function method. The most powerful method in the present case is the contour integration method. The Casimir energy turns out to be negative, and more so the larger is the number of pieces in the string. The thermodynamic free energy F is calculated for a two-piece string in the limit when the tension ratio x=T_{I}/T_{II} approaches zero.
On Partition Functions of Hyperbolic Three-Geometry and Associated Hilbert Schemes
A. A. Bytsenko,E. Elizalde
Physics , 2013,
Abstract: Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated $q$-series are discussed. Examples of these connection in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS$_{3}$ case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincar\'e and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of $SL(2, {\mathbb Z})$) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.
Quantum Corrections to Bekenstein-Hawking Black Hole Entropy and Gravity Partition Functions
A. A. Bytsenko,A. Tureanu
Physics , 2013, DOI: 10.1016/j.nuclphysb.2013.05.001
Abstract: Algebraic aspects of the computation of partition functions for quantum gravity and black holes in $AdS_3$ are discussed. We compute the sub-leading quantum corrections to the Bekenstein-Hawking entropy. It is shown that the quantum corrections to the classical result can be included systematically by making use of the comparison with conformal field theory partition functions, via the $AdS_3/CFT_2$ correspondence. This leads to a better understanding of the role of modular and spectral functions, from the point of view of the representation theory of infinite-dimensional Lie algebras. Besides, the sum of known quantum contributions to the partition function can be presented in a closed form, involving the Patterson-Selberg spectral function. These contributions can be reproduced in a holomorphically factorized theory whose partition functions are associated with the formal characters of the Virasoro modules. We propose a spectral function formulation for quantum corrections to the elliptic genus from supergravity states.
AdS Solutions in Gauge Supergravities and the Global Anomaly for the Product of Complex Two-Cycles
A. A. Bytsenko,E. Elizalde
Mathematics , 2011, DOI: 10.1140/epjc/s10052-011-1592-7
Abstract: Cohomological methods are applied for the special set of solutions corresponding to rotating branes in arbitrary dimensions, AdS black holes (which can be embedded in ten or eleven dimensions), and gauge supergravities. A new class of solutions is proposed, the Hilbert modular varieties, which consist of the $2n$-fold product of the 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 cohomology groups of the Hilbert variety, which inherit a Hodge structure (in the sense of Deligne), are analyzed, as well as bifiltered sequences, weight and Hodge filtrations, and it is argued that the torsion part of the cuspidal cohomology is involved in the global anomaly condition. Indeed, in presence of the cuspidal part, all cohomology classes can be mapped to the boundary of the space and the cuspidal contribution can be involved in the global anomaly condition.
Finite Temperature Effects for Massive Fields in D-dimensional Rindler-like Spaces
Andrei A. Bytsenko,Guido Cognola,Sergio Zerbini
Physics , 1995, DOI: 10.1016/0550-3213(95)00585-4
Abstract: The first quantum corrections to the free energy for massive fields in $D$-dimensional space-times of the form $\R\times\R^+\times\M^{N-1}$, where $D=N+1$ and $\M^{N-1}$ is a constant curvature manifold, is investigated by means of the $\zeta$-function regularization. It is suggested that the nature of the divergences, which are present in the thermodynamical quantities, might be better understood making use of the conformal related optical metric and associated techniques. The general form of the horizon divercences of the free energy is obtained as a function of free energy densities of fields having negative square masses (absence of the gap in the Laplace operator spectrum) on ultrastatic manifolds with hyperbolic spatial section $H^{N-2n}$ and of the Seeley-DeWitt coefficients of the Laplace operator on the manifold $\M^{N-1}$. Furthermore, recurrence relations are found relating higher and lower dimensions. The cases of Rindler space, where $\M^{N-1}=\R^{N-1}$ and very massive $D$-dimensional black holes, where $\M^{N-1}=S^{N-1}$ are treated as examples. The renormalization of the internal energy is also discussed.
Are p-Branes Asymptotically Black Holes?
Andrei Bytsenko,Klaus Kirsten,Sergio Zerbini
Physics , 1994, DOI: 10.1142/S0217732394001404
Abstract: 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.
Determinant of Laplacian on a non-compact 3-dimensional hyperbolic manifold with finite volume
A. A. Bytsenko,Guido Cognola,Sergio Zerbini
Physics , 1996, DOI: 10.1088/0305-4470/30/10/028
Abstract: The functional determinant of Laplace-type operators on the 3-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is computed by quadratures and making use of the related terms of the Selberg trace formula.
Statistical entropy of near-extremal and fundamental black p-branes
A. A. Bytsenko,S. D. Odintsov
Physics , 1996, DOI: 10.1143/PTP.98.987
Abstract: 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.
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