Macroscopic $n$-Loop Amplitude for Minimal Models Coupled to Two-Dimensional Gravity: Fusion Rules and Interactions

TL;DR

This paper derives a general formula for macroscopic $n$-loop amplitudes in minimal models coupled to 2D gravity, revealing fusion rules, interactions, and recursion relations in the continuum limit.

Contribution

It provides a novel explicit formula for $n$-resolvent correlators, incorporating fusion rules, interactions, and recursion relations in the context of minimal models and 2D gravity.

Findings

Derived a general formula for $n$-resolvent correlators.

Identified fusion rule conforming terms and residual interactions.

Established recursion relations linking $n$- and $(n-1)$-resolvents.

Abstract

We investigate the structure of the macroscopic $n$-loop amplitude obtained from the two-matrix model at the unitary minimal critical point $(m+1,m)$. We derive a general formula for the $n$-resolvent correlator at the continuum planar limit whose inverse Laplace transform provides the amplitude in terms of the boundary lengths $\ell_{i}$ and the renormalized cosmological constant $t$. The amplitude is found to contain a term consisting of $\left( \frac{\partial} {\partial t} \right)^{n-3}$ multiplied by the product of modified Bessel functions summed over their degrees which conform to the fusion rules and the crossing symmetry. This is found to be supplemented by an increasing number of other terms with $n$ which represent residual interactions of loops. We reveal the nature of these interactions by explicitly determining them as the convolution of modified Bessel functions and their derivatives for the case $n=4$ and the case $n=5$. We derive a set of recursion relations which relate the terms in the $n$-resolvents to those in the $(n-1)$-resolvents.