# Generalizing the entanglement entropy of singular regions in conformal field theories

## TL;DR

This paper investigates the universal and divergent structures of entanglement and Rényi entropies in singular regions of 3+1 dimensional conformal field theories, revealing new insights into geometric and CFT-dependent contributions.

## Contribution

It generalizes the understanding of entanglement entropy divergences for singular regions, including vertices and wedges, across different CFTs, and clarifies when universal terms appear or vanish.

## Key findings

- Universal logarithmic contributions depend on geometry and CFT type.
- For polyhedral corners, universal terms are non-local and CFT-dependent.
- Sharp corners can lead to finite mutual information despite singularities.

## Abstract

We study the structure of divergences and universal terms of the entanglement and R\'enyi entropies for singular regions. First, we show that for $(3+1)$-dimensional free conformal field theories (CFTs), entangling regions emanating from vertices give rise to a universal contribution $S_n^{\rm univ}= -\frac{1}{8\pi}f_b(n) \int_{\gamma} k^2 \log^2(R/\delta)$, where $\gamma$ is the curve formed by the intersection of the entangling surface with a unit sphere centered at the vertex, and $k$ the trace of its extrinsic curvature. While for circular and elliptic cones this term reproduces the general-CFT result, it vanishes for polyhedral corners. For those, we argue that the universal contribution, which is logarithmic, is not controlled by a local integral, but rather it depends on details of the CFT in a complicated way. We also study the angle dependence for the entanglement entropy of wedge singularities in 3+1 dimensions. This is done for general CFTs in the smooth limit, and using free and holographic CFTs at generic angles. In the latter case, we show that the wedge contribution is not proportional to the entanglement entropy of a corner region in the $(2+1)$-dimensional holographic CFT. Finally, we show that the mutual information of two regions that touch at a point is not necessarily divergent, as long as the contact is through a sufficiently sharp corner. Similarly, we provide examples of singular entangling regions which do not modify the structure of divergences of the entanglement entropy compared with smooth surfaces.