A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds M α with conical defects (or singularities) of the topology C α × Σ is developed. According to the proposed prescription M α are considered as limits of the converging sequences of smooth spaces. This enables one to give a strict mathematical meaning to a number of invariant integral quantities on M α and make use of them in applications. In particular, an explicit representation for the Euler numbers and Hirtzebruch signature in the presence of conical singularities is found. Also, higher dimensional Lovelock gravity on M α is shown to be well-defined and the gravitational action in this theory is evaluated. Other series of applications is related to computation of black hole entropy in the higher derivative gravity and in quantum 2-dimensional models. This is based on its direct statistical-mechanical derivation in the Gibbons-Hawking approach, generalized to the singular manifolds M α , and gives the same results as in the other methods.

Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is payed to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed.1 First indications on a possible relation between quantum entanglement and gravity phenomena should be attributed to authors of [3],[4] who suggested to interpret the Bekenstein-Hawking entropy as the entanglement entropy.

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 Σ 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 Σ. 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.

Attention is paid to the fact that temperature of a classical black hole can be derived from the extremality condition of its free energy with respect to variation of the mass of a hole. For a quantum Schwarzschild black hole evaporating massless particles the same condition is shown to result in the following one-loop temperature T = (8πM ) −1 1 + σ(8πM 2 ) −1 and entropy S = 4πM 2 − σ log M expressed in terms of the effective mass M of a hole together with its radiation and the integral of the conformal anomaly σ that depends on the field species. Thus, in the given case quantum corrections to T and S turn out to be completely provided by the anomaly. When it is absent (σ = 0), which happens in a number of supersymmetric models, the one-loop expressions of T and S preserve the classical form. On the other hand, if the anomaly is negative (σ < 0) an evaporating quantum hole seems to cease to heat up when its mass reaches the Planck scales.