Penner Type Matrix Model and Seiberg-Witten Theory

TL;DR

The paper demonstrates that a Penner-type matrix model, as proposed by Dijkgraaf and Vafa, reproduces the Seiberg-Witten data for ${\cal N}=2$ SU(2) gauge theories with four flavors and its asymptotically free limits. It derives the UV/IR coupling relation ${q_{UV} = {\vartheta_2(q_{IR})^4}/{\vartheta_3(q_{IR})^4}}$ from the M-theory/Seiberg-Witten framework and shows the matrix-model spectral curve matches the SW curve while preserving modular invariance under duality transformations. The matrix-model free energy $F_m$ is shown to coincide with the SW prepotential up to moduli-independent terms, and the decoupling limits to $N_f=3$ and $N_f=2$ reproduce the expected discriminants and beta-function coefficients, confirming the DV proposal in this setting. The work highlights a robust bridge between Penner-type matrix models and 4d ${\cal N}=2$ gauge theories, with potential implications for the AGT correspondence and extensions to quiver 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.