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

A. Alexandrov, А. Миронов+1 more|arXiv (Cornell University)|Oct 12, 2003

Black Holes and Theoretical Physics15 references100 citations

TL;DR

This paper proposes treating partition functions of the finite-size Hermitean one-matrix model as fundamental special functions in string theory, analogous to classical special functions. It analyzes their D-module structure, derives Virasoro constraints, and computes multiloop correlators and prepotentials, establishing a foundation for systematic study of matrix model partition functions as universal building blocks in string theory.

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.

Motivation & Objective

• To establish matrix model partition functions as fundamental special functions in string theory, independent of specific applications.
• To analyze the finite-size Hermitean one-matrix model as a prototype for such special functions.
• To investigate the D-module structure of the partition function and its solution space under Virasoro constraints.
• To compute and tabulate key quantities like multiloop correlators, prepotentials, and generating functions for future reference.

Proposed method

• Formalizing the partition function as a D-module solution to Virasoro constraints in the finite-size Hermitean 1-matrix model.
• Deriving recurrence relations for Gaussian multi-densities and correlators using orthogonal polynomial techniques.
• Employing Givental-style decomposition and integral representations (e.g., Laplace transforms, contour integrals) to express two-point resolvents.
• Using harmonic oscillator matrix elements and generating functions to compute explicit expressions for densities and correlators.
• Analyzing the role of the CIV-DV prepotential as a candidate basis in the solution space of Virasoro constraints.
• Evaluating continuous limits and identifying connections to other matrix models, such as Generalized Kontsevich models.

Experimental results

Research questions

• RQ1 How can partition functions of matrix models be systematically classified and tabulated as new special functions in string theory?
• RQ2 What is the structure of the solution space of the Virasoro constraints in the finite-size Hermitean one-matrix model?
• RQ3 Why is the CIV-DV prepotential distinguished as a basis for the solution space, and what is its physical or mathematical significance?
• RQ4 How do multiloop correlators generalize the semicircular distribution in the presence of multitrace and non-planar interactions?
• RQ5 What are the explicit analytic forms and expansions of Gaussian and non-Gaussian partition functions, including their prepotentials and generating functions?

Key findings

• The partition function of the finite-size Hermitean one-matrix model is shown to be a D-module solution to Virasoro constraints, with a rich structure of solutions parameterized by genus and coupling constants.
• First few multiloop correlators are computed explicitly, generalizing the semicircular distribution to non-planar and multitrace correlators.
• A closed-form expression for the two-point resolvent is derived using contour integrals and generating functions, consistent with low-N results.
• The CIV-DV prepotential is identified as a candidate basis in the solution space of Virasoro constraints, though its distinguished role remains unexplained.
• Explicit recurrence relations and analytic expressions are provided for Gaussian multi-densities and correlators, including their N-expansions and t-expansions of prepotentials.
• The generating function for the two-point correlation function is expressed in terms of hypergeometric-like series and contour integrals, with consistency verified up to N=3.