Anomalies, entropy and boundaries

TL;DR

The paper investigates how boundaries in odd-dimensional spacetimes induce nontrivial infrared structure in conformal field theories. Using the replica method and conical singularities, it shows that a logarithmic term in entanglement entropy arises when the entangling surface intersects the boundary, governed in 3D by boundary invariants $\chi_2$ and $j$ with theory-dependent charges $a$ and $q$, and by angle-dependent functions at intersection points. A key finding is that the entanglement entropy $s$ and the integrated conformal anomaly $s_{anom}$ need not coincide in 3D due to non-minimal couplings of scalars, with the relation $s_{anom} = s + s_{nc}$ and $s_{nc}$ deriving from delta-function curvature terms. The work also analyzes the distributional geometry induced on the boundary by conical bulk singularities, deriving explicit forms for delta-function contributions to intrinsic and extrinsic curvature, and provides a general expression for $s_{anom}$ in terms of crossing data. These results illuminate boundary effects on entanglement in odd dimensions and point to further connections with boundary CFT data and holographic descriptions.

Abstract

A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFT's in even dimensions. In odd dimensions the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a non-trivial integrated anomaly if an odd-dimensional spacetime has boundaries. We show that, similarly, there exists a logarithmic term in the entanglement entropy when the entangling surface crosses the boundary of spacetime. The relation of the entanglement entropy to the integrated conformal anomaly is elaborated for three-dimensional theories. Distributional properties of intrinsic and extrinsic geometries of the boundary in the presence of conical singularities in the bulk are established. This allows one to find contributions to the entropy that depend on the relative angle between the boundary and the entangling surface.