S-duality and 2d Topological QFT

TL;DR

This paper explores the relationship between 4d N=2 superconformal field theories from Gaiotto's class and 2d topological quantum field theories, showing how S-duality invariance corresponds to algebraic associativity in the 2d TQFT.

Contribution

It establishes a correspondence between the superconformal index of 4d theories and correlation functions in 2d TQFT, with explicit calculations for the A_1 case.

Findings

Superconformal index interpreted as 2d TQFT correlation functions

Associativity of the 2d TQFT algebra linked to S-duality invariance

Explicit structure constants computed using elliptic gamma functions

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.