Higher Genus Correlators from the Hermitian One-Matrix Model

TL;DR

This paper introduces an iterative method to compute higher genus correlators in the hermitian one-matrix model, revealing dependence on lower moments and providing explicit genus one results, with connections to conformal field theory.

Contribution

The authors develop a new iterative algorithm for genus expansion in the hermitian one-matrix model, explicitly calculating genus one correlators and relating them to conformal field theory results.

Findings

Genus $g$ correlators depend only on $3g-2+m$ lower moments.

Explicit genus one partition function and one-loop correlator results.

Agreement of genus zero correlators with $c=1$ CFT at zero momenta.

Abstract

We develop an iterative algorithm for the genus expansion of the hermitian $N\times N$ one-matrix model ( = the Penner model in an external field). By introducing moments of the external field, we prove that the genus $g$ contribution to the $m$-loop correlator depends only on $3g-2+m$ lower moments ($3g-2$ for the partition function). We present the explicit results for the partition function and the one-loop correlator in genus one. We compare the correlators for the hermitian one-matrix model with those at zero momenta for $c=1$ CFT and show an agreement of the one-loop correlators for genus zero.