Holographic R\'enyi entropy in AdS$_3$/LCFT$_2$ correspondence

TL;DR

This paper extends the study of holographic R'enyi entanglement entropy in AdS$_3$/LCFT$_2$ to massive gravity theories, showing exact matches with CFT results up to order $x^6$ and exploring 2-loop corrections.

Contribution

It generalizes holographic R'enyi entropy calculations to massive gravity theories at critical points, including CTMG and CNMG, and compares classical, 1-loop, and 2-loop results with CFT predictions.

Findings

Classical and 1-loop HRE match CFT results up to order $x^6$.

Difference between massless graviton and logarithmic mode is evident.

2-loop corrections show distinct features from pure AdS$_3$ gravity.

Abstract

The recent study in AdS$_3$/CFT$_2$ correspondence shows that the tree level contribution and 1-loop correction of holographic R\'enyi entanglement entropy (HRE) exactly match the direct CFT computation in the large central charge limit. This allows the R\'enyi entanglement entropy to be a new window to study the AdS/CFT correspondence. In this paper we generalize the study of R\'enyi entanglement entropy in pure AdS$_3$ gravity to the massive gravity theories at the critical points. For the cosmological topological massive gravity (CTMG), the dual conformal field theory (CFT) could be a chiral conformal field theory or a logarithmic conformal field theory (LCFT), depending on the asymptotic boundary conditions imposed. In both cases, by studying the short interval expansion of the R\'enyi entanglement entropy of two disjoint intervals with small cross ratio $x$, we find that the classical and 1-loop HRE are in exact match with the CFT results, up to order $x^6$. To this order, the difference between the massless graviton and logarithmic mode can be seen clearly. Moreover, for the cosmological new massive gravity (CNMG) at critical point, which could be dual to a logarithmic CFT as well, we find the similar agreement in the CNMG/LCFT correspondence. Furthermore we read the 2-loop correction of graviton and logarithmic mode to HRE from CFT computation. It has distinct feature from the one in pure AdS$_3$ gravity.