On Gauge Theories as Matrix Models

TL;DR

This paper explores the deep connections between gauge theories, matrix models, and conformal field theories, providing new insights into instanton contributions, quasiclassical solutions, and their relations to integrable systems and Teichmüller spaces.

Contribution

It derives matrix models for supersymmetric gauge theories, studies their solutions, and links them to conformal theories and quantum integrable systems, revealing novel relations.

Findings

Derived Eguchi-Yang type matrix models for instanton contributions.

Provided exact quasiclassical solutions for conformal theories.

Suggested links between Nekrasov functions, conformal blocks, and Teichmüller spaces.

Abstract

The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. We derive quasiclassically the matrix models of Eguchi-Yang type, describing the instantonic contribution to the deformed partition functions of supersymmetric gauge theories. The exact quasiclassical solution for the case of conformal four-dimensional theory is studied in detail, and some aspects of its relation with the recently proposed logarithmic beta-ensembles are considered. We discuss also the "quantization" of this picture in terms of two-dimensional conformal theory with extended symmetry, and stress its difference from common picture of perturbative expansion a la matrix models. Instead, the representation for Nekrasov functions in terms of conformal blocks or Whittaker vector suggests some nontrivial relation with Teichmueller spaces and quantum integrable systems.