A Hint on the External Field Problem for Matrix Models

Details A Hint on the External Field Problem for Matrix Models A Hint on the External Field Problem for Matrix Models

1992-02-03

arxiv.org/abs/hep-th/9202006v1

Phys.Lett. B278 (1992) 271-278

Physics

High Energy Physics

High Energy Physics - Theory

12 pages

Scientific paper

10.1016/0370-2693(92)90192-7

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 $c=0$.

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