Holomorphic matrix models

TL;DR

The paper develops holomorphic matrix models as a natural framework for the Dijkgraaf-Vafa conjecture, resolving issues that arise with Hermitian models for odd-degree potentials and broadening the planar limit to the entire moduli space of associated algebraic curves. It introduces a complex microcanonical ensemble with chemical potentials, proves that the special geometry relations hold at finite $N$ in this holomorphic setting, and shows that planar solutions sample the full curve moduli via a reconstruction theorem. The work extends to holomorphic ADE models, detailing the A2 case where a regulator is removed to yield a smooth limiting Riemann surface corresponding to physically relevant gaugino condensates. Overall, holomorphic models provide a consistent, regulator-free route to DV1 predictions and illuminate the geometry behind the gauge/string correspondence.

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.

Figures (7)

• Figure 1: Convergence sectors of the holomorphic matrix model. We show the case $n=deg W=4$, with $\theta_4=0$ and a contour belonging to the sector $(k_-,k_+)=(1,0)$.
• Figure 2: Convergence sectors for an entire potential $W=e^z$, and a good choice of contour for such a model. The filled-in regions are forbidden sectors for $\mu_\pm$.
• Figure 3: The normal vector field $n(s)=i t(s)$, and the tangent vector field $t(s)={\dot \lambda}(s)$ of $\gamma$. Note that $|n(s)|=|t(s)|=1$ since $s$ is the length coordinate along $\gamma$.
• Figure 4: Cuts of $\omega_0$. We also show a closed contour $\Gamma$ surrounding all cuts and a point $z$ in its exterior.
• Figure 5: A choice of strip domains in the complex plane. We also show two of the bounding contours (namely $\Gamma_3$ and $\Gamma_5$). In this example, we take the domains to be infinite strips; this assures us that $\gamma$ and its deformations will cut each such domain along a non-void interval.