Macroscopoic Three-Loop Amplitudes and the Fusion Rules from the Two-Matrix Model

TL;DR

This paper derives explicit three-loop amplitudes in a two-matrix model coupled to 2D gravity, revealing their structure aligns with conformal field theory fusion rules and involves Bessel functions, with potential generalizations.

Contribution

It provides the first explicit expression for macroscopic three-loop amplitudes in the two-matrix model, linking them to CFT fusion rules and Bessel functions.

Findings

Three-loop amplitudes expressed via Bessel functions

Fusion rules emerge from the sum structure in amplitudes

Potential extension to n-loop amplitudes discussed

Abstract

From the computation of three-point singlet correlators in the two-matrix model, we obtain an explicit expression for the macroscopic three-loop amplitudes having boundary lengths $\ell_{i}$ $(i = 1\sim 3)$ in the case of the unitary series $(p,q)= (m+1,m)$ coupled to two-dimensional gravity. The sum appearing in this expression is found to conform to the structure of the CFT fusion rules while the summand factorizes through a product of three modified Bessel functions. We briefly discuss a possible generalization of these features to macroscopic $n$-loop amplitudes.