# On the modular operator of mutli-component regions in chiral CFT

## TL;DR

This paper presents a novel method to compute the modular flow for multi-component regions in chiral conformal field theory, utilizing locality, analyticity, and the KMS condition to transform the problem into a Riemann-Hilbert problem.

## Contribution

It introduces a new approach based on locality and analyticity to determine the modular flow in multi-region chiral CFT, connecting it to Riemann-Hilbert problems.

## Key findings

- The method successfully transforms the modular flow problem into a Riemann-Hilbert problem.
- Examples demonstrate the applicability of the approach to specific cases.
- The approach offers a new perspective on modular operators in chiral CFT.

## Abstract

We introduce a new approach to find the Tomita-Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo-Martin-Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann-Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.