Penner Type Matrix Model and Seiberg-Witten Theory

TL;DR

This paper explores the Penner type matrix model's connection to Seiberg-Witten theory and Liouville theory, analyzing gauge coupling relations, modular transformations, and decoupling of flavors to reproduce known supersymmetric gauge theory results.

Contribution

It demonstrates how the Penner type matrix model can describe N=2 supersymmetric gauge theories and their dualities, extending previous models to include asymptotically free cases.

Findings

Reproduces standard N=2 gauge theory results

Analyzes modular transformations on the matrix model

Derives matrix models for asymptotically free theories

Abstract

We discuss the Penner type matrix model recently proposed by Dijkgraaf and Vafa for a possible explanation of the relation between four-dimensional gauge theory and Liouville theory by making use of the connection of the matrix model to two-dimensional CFT. We first consider the relation of gauge couplings defined in UV and IR regimes of N_f = 4, N = 2 supersymmetric gauge theory being related as $q_{{\rm UV}}={\vartheta_2(q_{{\rm IR}})^4/\vartheta_3(q_{{\rm IR}})^4}$. We then use this relation to discuss the action of modular transformation on the matrix model and determine its spectral curve. We also discuss the decoupling of massive flavors from the N_f = 4 matrix model and derive matrix models describing asymptotically free N = 2 gauge theories. We find that the Penner type matrix theory reproduces correctly the standard results of N = 2 supersymmetric gauge theories.