Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model

TL;DR

This paper explores matrix model partition functions as fundamental special functions in string theory, focusing on the finite-size Hermitean 1-matrix model and analyzing its phase structure, correlators, and potential as a building block for quantum gravity.

Contribution

It introduces the perspective of matrix model partition functions as new special functions and investigates their properties and structures in the context of the Hermitean 1-matrix model.

Findings

Analysis of phase/branch structure as a D-module.

Evaluation of multiloop correlators.

Discussion of the role of the CIV-DV prepotential.

Abstract

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators.