Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model

Understand

In arXiv:hep-th/0310113 we started a program of creating a reference-book on matrix-model tau-functions, the new generation of special functions, which are going to play an important role in string theory calculations.

• The main focus of that paper was on the one-matrix Hermitian model tau-functions.
• The present paper is devoted to a direct counterpart for the Kontsevich and Generalized Kontsevich Model (GKM) tau-functions.
• We mostly focus on calculating resolvents (=loop operator averages) in the Kontsevich model, with a special emphasis on its simplest (Gaussian) phase, where exists a surprising integral formula, and the expressions for the resolvents in the genus zero and one are especially simple (in particular, we generalize the known genus zero result to genus one).