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S-duality and 2d topological QFT

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Abstract

We study the superconformal index for the class of \( \mathcal{N} = 2 \) 4d superconformal field theories recently introduced by Gaiotto [1]. 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 [2].

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Authors and Affiliations

  1. C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794-3840, U.S.A.

    Abhijit Gadde, Elli Pomoni, Leonardo Rastelli & Shlomo S. Razamat

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  1. Abhijit Gadde
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  2. Elli Pomoni
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  3. Leonardo Rastelli
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  4. Shlomo S. Razamat
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Correspondence to Shlomo S. Razamat.

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ArXiv ePrint: 0910.2225

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Gadde, A., Pomoni, E., Rastelli, L. et al. S-duality and 2d topological QFT. J. High Energ. Phys. 2010, 32 (2010). https://doi.org/10.1007/JHEP03(2010)032

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