Seiberg-Witten theory, matrix model and AGT relation
Tohru Eguchi, Kazunobu Maruyoshi
TL;DR
The authors investigate how Penner-type matrix models realize the AGT relation between Liouville conformal blocks and 4d ${ m N}=2$ SU(2) gauge theories by deriving spectral curves and computing the planar free energy. They explicitly evaluate $F_m$ for $N_f=2,3,4$ and show, up to a conventional factor, that $4F_m$ reproduces the Seiberg-Witten prepotential including one-loop and instanton contributions, validating the matrix-model realization. The work further extends the framework to linear quiver gauge theories and discusses decoupling limits that yield asymptotically free theories, demonstrating a consistent mapping between matrix-model data and multi-node SW data. These results strengthen the connection between Liouville theory, AGT, and SW amplitudes and pave the way toward incorporating full Nekrasov partition functions and broader gauge groups in this matrix-model approach.
Abstract
We discuss the Penner-type matrix model which has been proposed to explain the AGT relation between the 2-dimensional Liouville theory and 4-dimensional N=2 superconformal gauge theories. In our previous communication we have obtained the spectral curve of the matrix model and showed that it agrees with that derived from M-theory. We have also discussed the decoupling limit of massive flavors and proposed new matrix models which describe Seiberg-Witten theory with flavors N_f=2,3. In this article we explicitly evaluate the free energy of these matrix models and show that they in fact reproduce the amplitudes of Seiberg-Witten theory.