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

摘要

Even though matrix model partition functions do not exhaust the entire set oftau-functions relevant for string theory, they seem to be elementary buildingblocks for many others and they seem to properly capture the fundamentalsymplicial nature of quantum gravity and string theory. We propose to considermatrix model partition functions as new special functions. This means theyshould be investigated and put into some standard form, with no reference toparticular applications. At the same time, the tables and lists of propertiesshould be full enough to avoid discoveries of unexpected peculiarities in newapplications. This is a big job, and the present paper is just a step in thisdirection. Here we restrict our consideration to the finite-size Hermitean1-matrix model and concentrate mostly on its phase/branch structure arisingwhen the partition function is considered as a D-module. We discuss the role ofthe CIV-DV prepotential (as generating a possible basis in the linear space ofsolutions to the Virasoro constraints, but with a lack of understanding of whyand how this basis is distinguished) and evaluate first few multiloopcorrelators, which generalize semicircular distribution to the case ofmultitrace and non-planar correlators.