A Hint on the External Field Problem for Matrix Models

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 $\tr{({1/over2N}X^2 + \log X - ΛX)}$ and $\sum_{k=1}^\infty t_k\tr{X^k}$ providing the matrix $Λ$ is related to $\{t_k\}$ by $t_k=\fr 1k \tr{Λ^{-k}}-\frac N2 δ_{k2}$. 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 $c=0$.