Fusion rules and macroscopic loops from discretized approach to two-dimensional gravity

TL;DR

This paper explores the structure of multi-loop correlators and fusion rules in two-dimensional quantum gravity coupled with minimal conformal models, revealing new insights into loop interactions and boundary conditions.

Contribution

It introduces a unified framework for fusion rules of loops and boundary operators in two-dimensional gravity, derived from a discretized two-matrix model approach.

Findings

Fusion rules for loops are summarized in a compact form.

A general formula for n-resolvent correlators is derived.

The connection between boundary conditions and loop touching is clarified.

Abstract

We investigate the multi-loop correlators and the multi-point functions for all of the scaling operators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for these scaling operators exist, and these are summarized in a compact form as fusion rules for loops. We clarify the role of the boundary operators and discuss its connection to how loops touch each other. We derive a general formula for the n-resolvent correlators, and point out the structure similar to the crossing symmetry of underlying conformal field theory. We discuss the connection of the boundary conditions of the loop correlators to the touching of loops for the case of the four-loop correlators.