Holomorphic matrix models

TL;DR

This paper explores holomorphic matrix models related to the Dijkgraaf-Vafa conjecture, demonstrating their advantages in resolving perturbative puzzles, probing moduli space, and constructing smooth algebraic curves, especially for holomorphic $ADE$ models.

Contribution

It provides a systematic analysis of holomorphic matrix models, showing their consistency with the conjecture and their ability to produce smooth algebraic curves where Hermitian models fail.

Findings

Holomorphic models resolve perturbative expansion puzzles.

Planar solutions probe entire moduli space.

Regularization yields smooth Riemann surfaces.

Abstract

This is a study of holomorphic matrix models, the matrix models which underlie the conjecture of Dijkgraaf and Vafa. I first give a systematic description of the holomorphic one-matrix model. After discussing its convergence sectors, I show that certain puzzles related to its perturbative expansion admit a simple resolution in the holomorphic set-up. Constructing a `complex' microcanonical ensemble, I check that the basic requirements of the conjecture (in particular, the special geometry relations involving chemical potentials) hold in the absence of the hermicity constraint. I also show that planar solutions of the holomorphic model probe the entire moduli space of the associated algebraic curve. Finally, I give a brief discussion of holomorphic $ADE$ models, focusing on the example of the $A_2$ quiver, for which I extract explicitly the relevant Riemann surface. In this case, use of the holomorphic model is crucial, since the Hermitian approach and its attending regularization would lead to a singular algebraic curve, thus contradicting the requirements of the conjecture. In particular, I show how an appropriate regularization of the holomorphic $A_2$ model produces the desired smooth Riemann surface in the limit when the regulator is removed, and that this limit can be described as a statistical ensemble of `reduced' holomorphic models.