Short interval expansion of Rényi entropy on torus

TL;DR

This work develops a short-interval expansion for the Rényi entropy $S_n$ of a 2D CFT on a torus by applying the twist-operator OPE, recasting the problem into torus one-point functions of quasiprimary operators in ${\rm CFT}^n$. The authors systematically compute the vacuum-conformal-family contributions up to order $\ell^6$ and organize results in a large-$c$ expansion, comparing with holographic expectations from AdS$_3$ gravity and its rotating BTZ generalizations; they also extend the analysis to theories with $\mathcal{W}(2,3)$ symmetry and to $\mathcal{N}=(1,1)$ SCFTs, obtaining 1-loop and higher corrections and showing how $1/c$ terms are accessible. The framework handles arbitrary torus moduli, enables low- and high-temperature limits via modular transformations, and naturally incorporates chemical potentials, yielding a unified picture that matches known gravity results and clarifies the structure of quantum corrections in holographic entanglement. Overall, the short-interval OPE approach provides a practical and versatile route to extract finite-size and thermal corrections to Rényi entropy, including nontrivial $1/c$ effects, with clear connections to AdS$_3$/CFT$_2$ holography.

Abstract

We investigate the short interval expansion of the Rényi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval $\ell$ to be small with respect to the spatial and temporal sizes of the torus. The operator product expansion of the twist operators allows us to compute the short interval expansion of the Rényi entropy at any temperature. In particular, we pay special attention to the large $c$ CFTs dual to the AdS$_3$ gravity and its cousins. At both low and high temperature limits, we read the Rényi entropies to order $\ell^6$, and find good agreements with holographic results. Moreover, the expansion allows us to read $1/c$ contribution, which is hard to get by expanding the thermal density matrix. We generalize the study to the case with the chemical potential as well.