A Hint on the External Field Problem for Matrix Models

TL;DR

This paper investigates the external field problem in hermitian matrix models, establishing an equivalence between different potential forms and providing a method to compute partition functions and correlators, supporting a connection to the $c=1$ conformal field theory.

Contribution

It proves an equivalence between two matrix models with different potentials and introduces a systematic method to compute partition functions via Schwinger--Dyson equations.

Findings

Explicit genus-one calculations of partition functions and correlators.

Support for the conjecture linking the models to the $c=1$ conformal field theory.

Method for calculating partition functions order by order in genus.

Abstract

We reexamine the external field problem for $N\times N$ hermitian one-matrix models. We prove an equivalence of the models with the potentials $\tr{({1/over2N}X^2 + \log X - \Lambda X)}$ and $\sum_{k=1}^\infty t_k\tr{X^k}$ providing the matrix $\Lambda$ is related to $\{t_k\}$ by $t_k=\fr 1k \tr{\Lambda^{-k}}-\frac N2 \delta_{k2}$. Based on this equivalence we formulate a method for calculating the partition function by solving the Schwinger--Dyson equations order by order of genus expansion. Explicit calculations of the partition function and of correlators of conformal operators with the puncture operator are presented in genus one. These results support the conjecture that our models are associated with the $c=1$ case in the same sense as the Kontsevich model describes $c=0$.