Note on non-vacuum conformal family contributions to R\'enyi entropy in two-dimensional CFT

TL;DR

This paper calculates how non-vacuum conformal families contribute to R9 entropy in 2D CFT, providing expansions for different geometries and confirming results with gravity duals in the large central charge limit.

Contribution

It extends R9 entropy calculations to include non-vacuum conformal families and matches these with gravity results in the large central charge limit.

Findings

Derived expansions for R9 mutual information and entropy involving non-vacuum conformal families.

Established conditions on the conformal dimension  for meaningful results.

Confirmed consistency between CFT calculations and gravity duals.

Abstract

We calculate the contributions of a general non-vacuum conformal family to R\'enyi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by $\Delta$. For the case of two short intervals on complex plane, we expand the R\'enyi mutual information by the cross ratio $x$ to order $x^{2\Delta+2}$. For the case of one interval on torus with the temperature being low, we expand the R\'enyi entropy by $q=\exp(-2\pi\beta/L)$, with $\beta$ being the inverse temperature and $L$ being the spatial period, to order $q^{\Delta+2}$. To make the result meaningful, we require that the scaling dimension $\Delta$ cannot be too small. For two intervals on complex plane we need $\Delta>1$, and for one interval on torus we need $\Delta>2$. We work in small Newton constant limit in gravity side and so large central charge limit in CFT side, and find matches of gravity and CFT results.