Distributional Geometry of Squashed Cones

TL;DR

The authors develop a geometric regularization framework for squashed cones, where the entangling surface Σ can possess nonzero extrinsic curvature. They show that, in the replica limit, curvature invariants acquire universal surface terms depending on the extrinsic geometry of Σ, derive explicit expressions for quadratic invariants, and verify consistency with known entanglement-entropy logarithms and holographic formulas. The approach extends to higher dimensions, clarifies the structure of heat-kernel coefficients and conformal anomalies with surface contributions, and yields a holographic entanglement-entropy prescription for gravity theories with quadratic curvature terms. They also propose a classical gravitational entropy for non-Killing horizons, linking entanglement, holography, and generalized gravitational entropy in a coherent framework.

Abstract

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.