S-duality and 2d Topological QFT

Abstract

We study the superconformal index for the class of N=2 4d superconformal field theories recently introduced by Gaiotto. These theories are defined by compactifying the (2,0) 6d theory on a Riemann surface with punctures. We interpret the index of the 4d theory associated to an n-punctured Riemann surface as the n-point correlation function of a 2d topological QFT living on the surface. Invariance of the index under generalized S-duality transformations (the mapping class group of the Riemann surface) translates into associativity of the operator algebra of the 2d TQFT. In the A_1 case, for which the 4d SCFTs have a Lagrangian realization, the structure constants and metric of the 2d TQFT can be calculated explicitly in terms of elliptic gamma functions. Associativity then holds thanks to a remarkable symmetry of an elliptic hypergeometric beta integral, proved very recently by van de Bult.

25 pages, 11 figures

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• CFT4 as SO(4,2)-invariant TFT2
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• Wilson punctured network defects in 2D q-deformed Yang-Mills theory
• A Family of Vertex Algebras from Argyres-Douglas Theory
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• On 'categories' of quantum field theories
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• Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing
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