Genus Two Partition Functions and Renyi Entropies of Large c CFTs

TL;DR

This work analyzes genus-two partition functions in large-$c$ 2d CFTs to extract the third Rényi entropy for two disjoint intervals, comparing generalized free theories and symmetric orbifolds to pure gravity. By exploiting large-$c$ factorization, the authors reduce the problem to counting Wick contractions and OPE coefficients, demonstrating that the genus-two partition function $Z_2(y)$ accumulates a non-universal contribution from the growth of OPE data, encoded in $D(\Delta)$. They find a critical lightest operator dimension around $\Delta_{\min} \approx 0.19$ below which a new phase appears at some $y_c<1/2$, breaking universality of the third Rényi entropy. The results imply that while $I^{(2)}$ can be universal at large $c$, $I^{(3)}$ can depend sensitively on the spectrum and OPE coefficients, with nucleation of new bulk saddles tied to light scalar fields. The study provides a framework to assess higher-genus entanglement features across moduli space and motivates further exploration of gravitational saddles in holography with light matter content.

Abstract

We compute genus two partition functions in two dimensional conformal field theories at large central charge, focusing on surfaces that give the third Renyi entropy of two intervals. We compute this for generalized free theories and for symmetric orbifolds, and compare it to the result in pure gravity. We find a new phase transition if the theory contains a light operator of dimension $Δ\leq0.19$. This means in particular that unlike the second Renyi entropy, the third one is no longer universal.

Figures (1)

• Figure 1: The two different phases of entanglement entropy of two intervals, corresponding to different minimal surfaces in the bulk. The phase a) is dominant for $y<1/2$ and has $\mathcal{O}(1)$ mutual information. The phase b) dominates for $y>1/2$ and has $\mathcal{O}(c)$ mutual information