Emergent Geometry Through Holomorphic Matrix Models

TL;DR

This paper explores how emergent elliptic curves and generalized resolvents in holomorphic matrix models reveal deep geometric structures underlying supersymmetric gauge theories, connecting N=1* and N=2* models.

Contribution

It explicitly constructs the map from elliptic curves to eigenvalue planes and shows how these geometric objects encode eigenvalue densities and critical points in supersymmetric gauge theories.

Findings

Exact eigenvalue densities can be recovered from the elliptic curve and resolvent.

Weak coupling reproduces known spectral densities like Wigner semi-circle.

Evidence of parabolic density at strong coupling in N=1* theory.

Abstract

Over the years, deep insights into string theory and supersymmetric gauge theories have come from studying geometry emerging from matrix models. In this thesis, I study the N=1* and N=2* theories from which an elliptic curve is known to emerge, alongside an elliptic function called the generalized resolvent into which the physics is encoded. This is indicative of the common origin of the two theories in N=4 SYM. The N=1* Dijkgraaf-Vafa matrix model is intrinsically holomorphic with parameter space corresponding to the upper-half plane. The Dijkgraaf-Vafa matrix model 't Hooft coupling S has been previously shown to be holomorphic on the upper-half plane and quasi-modular with respect to SL(2,Z). The allowed N=2* coupling is constrained to a Hermitian slice through the enlarged moduli space of the holomorphic N=1* model. After explicitly constructing the map from the elliptic curve to the eigenvalue plane, I argue that the N=1* coupling S encodes data reminiscent of N=2*. A collection of extrema (saddle-points) of S behave curiously like the quantum critical points of N=2* theory. For the first critical point, the match is exact. This collection of points lie on the line of degeneration which behaves in a sense, like a boundary at infinity. I also show explicitly that the emergent elliptic curve along with the generalized resolvent allow one to recover exact eigenvalue densities. At weak coupling, my method reproduces the inverse square root of N=2* as well as the Wigner semi-circle in N=1*. At strong coupling in N=1*, I provide encouraging evidence of the parabolic density arising in the neighborhood of the line of degeneration. To my knowledge, the parabolic density has only been observed asymptotically. It is interesting to see evidence that it may be exactly encoded in the other form of emergent geometry: the elliptic curve with the generalized resolvent.