Generalizing the entanglement entropy of singular regions in conformal field theories

Understand

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.