A note on the resolution of the entropy discrepancy

TL;DR

The paper analyzes the entropy discrepancy (HMS puzzle) between field-theoretic and holographic entanglement entropy for 6d CFTs and compares two proposed resolutions. Using the Lewkowycz-Maldacena regularization, it shows that entropies from total derivative terms vanish and that the total entropy, including Wald-like and anomaly-like contributions plus a generalized Wald term, matches the holographic result when gravity is Einstein. It then demonstrates that the Astaneh1Astaneh2 approach, which attributes nonzero entropy to total derivative terms via Fursaev regularization, fails to reproduce the holographic entropy, while the Miao1 approach succeeds, thereby favoring covariant total derivative arguments and clarifying the regularization dependence. The work also argues that Wald entropy is generally ill-defined for horizons with extrinsic curvature and that only the combined total entropy is robust, with a stationary spacetime reducing to the total entropy.

Abstract

It was found by Hung, Myers and Smolkin that there is entropy discrepancy for the CFTs in 6-dimensional space-time, between the field theoretical and the holographic analysis. Recently, two different resolutions to this puzzle have been proposed. One of them suggests to utilize the anomaly-like entropy and the generalized Wald entropy to resolve the HMS puzzle, while the other one initiates to use the entanglement entropy which arises from total derivative terms in the Weyl anomaly to explain the HMS mismatch. We investigate these two proposals carefully in this note. By studying the CFTs dual to Einstein gravity, we find that the second proposal can not solve the HMS puzzle. Moreover, the Wald entropy formula is not well-defined on horizon with extrinsic curvatures, in the sense that, in general, it gives different results for equivalent actions.