Twist operator correlators and isomonodromic tau functions from modular Hamiltonians

TL;DR

This paper develops a new method to compute twist operator correlators and isomonodromic tau functions in 2d CFTs, using modular Hamiltonians and correlation matrix techniques, with applications to holographic CFTs.

Contribution

It introduces a novel approach based on modular Hamiltonians for calculating TOC and tau functions, including determinantal and integral representations for genus-zero cases.

Findings

Derived a determinantal representation for free fermion cases.

Presented an integral representation from universal modular Hamiltonians.

Argued an approximate factorization property for large-c holographic CFTs.

Abstract

We introduce a novel approach for computing the twist operator correlators (TOC) in two-dimensional conformal field theories (2d CFT) and the closely related isomonodromic tau functions. The method stems from the formal path integral representation of the ground state reduced density matrix in 2d CFT, and exploits properties of the associated modular Hamiltonians. For a class of genus-zero TOC/tau functions associated with branched covers with non-abelian monodromy group, we present: i) a determinantal representation derived from the correlation matrix method for free fermions, and ii) a formal integral representation derived from the universal single-interval modular Hamiltonians. For the class of genus-zero TOC/tau functions, we also argue an approximate factorization property, utilizing the known ground state correlation structure of large-$c$ holographic CFT and the universality of genus-zero TOCs. We provide explicit examples for verifying the determinantal representation and the approximate factorization property.