Analyticity of replica correlators and Modular ETH

TL;DR

This paper develops a method to analytically continue replica correlators in conformal field theories using the Renyi transform, linking it to modular ETH and off-diagonal OPE coefficients, with applications in holographic and 2D CFTs.

Contribution

It introduces a novel approach to analytically continue replica correlators via the Renyi transform and relates its discontinuity to modular ETH and off-diagonal OPE coefficients.

Findings

Derived a formula for analytic continuation of replica correlators.

Connected the discontinuity of the Renyi transform to modular ETH.

Validated the approach with explicit calculations in holographic and 2D CFTs.

Abstract

We study the two point correlation function of a local operator on an $n$-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable $w$, dual to the Renyi parameter $n$. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform $f(w)$ has bounded behavior at infinity in the complex $w$ plane, the discontinuity of the Renyi transform disc $f(w)$ provides the unique analytic continuation in $n$ of the original replica correlation function. We check our formula by explicitly calculating the Renyi transform of a particular replica correlator in a large $N$ holographic CFT$_d$ in dimensions $d>2$. We also discover that the discontinuity of the Renyi transform is related to the matrix element of local operators between two distinct eigenstates of the modular Hamiltonian. We calculate the Renyi transform in $2d$ conformal field theories, and use it to extract the off-diagonal elements of (modular) ETH. We argue that in $2d$, this is equivalent to the off-diagonal OPE coefficients of a CFT and show that our technique exactly reproduces recent results in the literature.