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 $\Sigma$ 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 $\Sigma$. 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.