Holomorphic matrix models

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 hermiticity 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 A2 quiver, for which I extract explicitly the relevant Riemann surface. In this case, use of the holomorphic model is crucial, since the hermitean 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 A2 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. © SISSA/ISAS 2003.