Dmitri V. Fursaev

Distributional Geometry of Squashed Cones

A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface

Statistical Origin of Black Hole Entropy in Induced Gravity

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Thermal fields, entropy and black holes

so that the surface is allowed to have extrinsic curvatures. A new feature of the squashed conical singularities is that the surface terms in the integral invariants, in the limit of small angle deficit, now depend also on the extrinsic curvatures of

Cones, spins and heat kernels

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Heat kernel coefficients and spectra of the vector Laplacians on spherical domains with conical singularities

. A case of invariants which are quadratic polynomials of the Riemann curvature is elaborated in different dimensions and applied to several problems related to entanglement entropy. The results are in complete agreement with computations of the logarithmic terms in entanglement entropy of 4D conformal theories [2]. Among other applications of the suggested method are logarithmic terms in entanglement entropy of non-conformal theories and a holographic formula for entanglement entropy in theories with gravity duals.

Anomalies, entropy, and boundaries

The statistical-mechanical origin of the Bekenstein-Hawking entropy SBH in the induced gravity is discussed. In the framework of the induced gravity models the Einstein action arises as the low energy limit of the effective action of quantum fields. The induced gravitational constant is determined by the masses of the heavy constituents. We established the explicit relation between the statistical entropy of the constituent fields and the black hole entropy SBH.

Black hole entropy in induced gravity: Reduction to 2D quantum field theory on the horizon

In this review we describe the statistical mechanics of quantum systems in the presence of a Killing horizon and compare statistical-mechanical and 1-loop contributions to black-hole entropy. The study of these questions was motivated by attempts to explain the entropy of black holes as a statistical-mechanical entropy of quantum fields propagating near the black-hole horizon. We provide an introduction to this field of research and review its results. In particular, we discuss the relation between the statistical-mechanical entropy of quantum fields and the Bekenstein-Hawking entropy in the standard scheme with renormalization of gravitational coupling constants and in the theories of induced gravity.

EUCLIDEAN AND CANONICAL FORMULATIONS OF STATISTICAL MECHANICS IN THE PRESENCE OF KILLING HORIZONS

Abstract The heat kernels of Laplacians for spin - 1 2 , spin-1, spin - 3 2 and spin-2 fields, and the asymptotic expansion of their traces are studied on manifolds with conical singularities. The exact mode-bymode analysis is carried out for 2-dimensional domains and then extended to arbitrary dimensions. The corrections to the first Schwinger-DeWitt coefficients in the trace expansion, due to conical singularities, are found for all the above spins. The results for spins 1 2 and 1 resemble the scalar case. However, the heat kernels of the Lichnerowicz spin-2 operator and the spin - 3 2 Laplacian show a new feature. When the conical angle deficit vanishes the limiting values of their traces differ from the corresponding values computed on the smooth manifold. The reason for the discrepancy is breaking of the local translational isometries near a conical singularity. As an application, the results are used to find the ultraviolet divergences in the quantum corrections to the black hole entropy for all these spins.

Black hole statistical mechanics and induced gravity

The spherical domains with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with this type of singular point. In this paper the vector Laplacian on is considered and its spectrum is calculated exactly for any dimension d. This enables one to find the Schwinger - DeWitt coefficients of this operator by using the residues of the -function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the 1-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to first order in the conical deficit angle.

Holographic calculation of entanglement entropy in the presence of boundaries

A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFTs in even dimensions. In odd dimensions the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a non-trivial integrated anomaly if an odd-dimensional spacetime has boundaries. We show that, similarly, there exists a logarithmic term in the entanglement entropy when the entangling surface crosses the boundary of spacetime. The relation of the entanglement entropy to the integrated conformal anomaly is elaborated for three-dimensional theories. Distributional properties of intrinsic and extrinsic geometries of the boundary in the presence of conical singularities in the bulk are established. This allows one to find contributions to the entropy that depend on the relative angle between the boundary and the entangling surface.