Entanglement Entropy and Cauchy-Hadamard Renormalization

TL;DR

This paper offers a geometric and purely mathematical approach to understanding entanglement entropy in conformal field theory by relating partition functions on singular surfaces to correlation functions of primary fields.

Contribution

It introduces a Cauchy-Hadamard renormalization method for Polyakov anomaly integrals and connects partition functions on branched covers to primary field correlations in CFT.

Findings

Partition functions on conical singularities are related to primary field correlations.

The approach provides a mathematical interpretation of entanglement entropy results.

The method generalizes the replica trick using geometric constructions.

Abstract

This note presents a purely geometric construction of the so-called twist-field correlation functions in Conformal Field Theory (CFT), derived from conical singularities. This approach provides a purely mathematical interpretation of the seminal results in physics by Cardy and Calabrese on the entanglement entropy of quantum systems. Specifically, we begin by defining CFT partition functions on surfaces with conical singularities, using a ``Cauchy-Hadamard renormalization'' of the Polyakov anomaly integral. Next, we demonstrate that for a branched cover $f:\Sigma_d\to \Sigma$ with $d$ sheets, where the cover inherits the pullback of a smooth metric from the base, a specific ratio of partition functions on the cover to the base transforms under conformal changes of the base metric in the same way as a correlation function of CFT primary fields with specific conformal weights. We also provide a discussion of the physical background and motivation for entanglement entropy, focusing on path integrals and the replica trick, which serves as an introduction to these ideas for a mathematical audience.