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

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.

41 pages, 11 figures

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• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Matrix Models for Random Partitions
• AGT conjecture and Integrable structure of Conformal field theory for c=1
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• CFT and topological recursion
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Direct integration for general Omega backgrounds
• AGT conjecture and AFLT states: a complete construction
• Logarithmic potential beta-ensembles and Feynman graphs
• Mirror of the refined topological vertex from a matrix model
• An(1) Affine Quiver Matrix Model
• Refined Cigar and Omega-deformed Conifold
• Note on refined topological vertex, Jack polynomials and instanton counting
• A note on the automorphism group of the root lattice of the U-dual modular group