Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems

TL;DR

This paper explores the deep connections between topological string theory, matrix models, Toda systems, and N=2 gauge theories, revealing dualities and holographic interpretations that unify these frameworks.

Contribution

It establishes a novel correspondence between topological string partition functions, Toda systems, and matrix models in the context of N=2 gauge theories with large N dualities.

Findings

Partition function captured by A_{n-1} Toda chiral blocks.

Realization of systems via Penner-like matrix models for genus zero.

Holographic interpretation of the spectral curve as the Seiberg-Witten curve.

Abstract

We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an A_{n-1} singularity over a Riemann surface. These models realize the N=2 SU(n) superconformal gauge systems recently studied by Gaiotto and collaborators. Employing large N dualities we show why the partition function of topological strings in these backgrounds is captured by the chiral blocks of A_{n-1} Toda systems and derive the dictionary recently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zero Riemann surfaces, we show how these systems can also be realized by Penner-like matrix models with logarithmic potentials. The Seiberg-Witten curve can be understood as the spectral curve of these matrix models which arises holographically at large N. In this context the Nekrasov deformation maps to the beta-ensemble of generalized matrix models, that in turn maps to the Toda system with general background charge. We also point out the notion of a double holography for this system, when both n and N are large.