Torus one-point functions in critical loop models

TL;DR

This paper expresses torus 1-point functions in critical loop models via sphere 4-point functions, establishing a link between crossing symmetry and modular covariance, and computes these functions numerically.

Contribution

It systematically computes torus 1-point functions in critical loop models using a numerical bootstrap, revealing their structure as linear combinations of conformal blocks.

Findings

Torus 1-point functions are expressed as linear combinations of conformal blocks.

Coefficients involve double Gamma functions and polynomial functions of loop weights.

Identified 10 solutions to modular covariance equations for primary fields.

Abstract

We show that in critical loop models, torus 1-point functions can be expressed in terms of sphere 4-point functions at a different central charge. Unlike in the Moore--Seiberg formalism, crossing symmetry on the sphere therefore implies modular covariance on the torus. We systematically compute torus 1-point functions in critical loop models, using a numerical bootstrap approach. We focus on the 1-point functions of the 6 simplest primary fields, which give rise to 10 solutions of modular covariance equations. Such 1-point functions are infinite linear combinations of conformal blocks. The coefficients are products of double Gamma functions, times polynomial functions of loop weights. For each solution, we determine the first 6 to 12 polynomials.