S-duality and 2d Topological QFT

TL;DR

The paper studies the 4d ${\cal N}=2$ superconformal index for Gaiotto's class of theories arising from the twisted $(2,0)$ theory compactified on a punctured Riemann surface. It proposes that the index is the $n$-point function of a 2d TQFT on the surface, with generalized S-duality corresponding to associativity of the 2d algebra, and provides explicit constructions in the ${\cal A}_1$ case. The authors derive the index as a matrix integral, identify the 2d TQFT data $C_{\alpha\beta\gamma}$ and $\eta^{\alpha\beta}$, and prove associativity by showing that crossing symmetry reduces to a nontrivial elliptic Beta integral identity expressed through elliptic Gamma functions. The associativity proof leverages a Weyl symmetry of the $E^{(5)}$ elliptic Beta integral (van de Bult), revealing a deep connection between elliptic hypergeometric identities and 4d S-duality. The results illuminate a precise 4d–2d correspondence and suggest broader implications for dualities and topological field theories tied to elliptic integrals.

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.

Figures (11)

• Figure 1: (a) Generalized quiver diagrams representing ${\cal N} = 2$ superconformal theories with gauge group $SU(2)^6$ and no flavor symmetries ($N_G =6$, $N_F = 0$). There are five different theories of this kind. The internal lines of a diagram represent and $SU(2)$ gauge group and the trivalent vertices the trifundamental chiral matter. (b) Generalized quiver diagrams for $N_G =3$, $N_F = 3$. Each external leg represents an $SU(2)$ flavor group. The upper left diagram corresponds the ${\mathcal{N}}=2$${\mathbb Z}_3$ orbifold of ${\mathcal{N}}=4$ SYM with gauge group $SU(2)$.
• Figure 2: An example of a degeneration of a graph and appearance of flavour punctures. As one of the gauge coupling is taken to zero the corresponding edge becomes very long. Cutting the edge, each of the two resulting semi-infinite open legs will be associated to chiral matter in an $SU(2)$ flavor representation. In this picture setting the coupling of the middle legs in (a) to zero gives two copies of the theory represented in (b), namely an $SU(2)$ gauge theory with a chiral field in the bifundamental representation of the gauge group and in the fundamental of a flavour $SU(2)$.
• Figure 3: (a) Topological interpretation of the structure constants ${C_{\alpha\beta\gamma}} \equiv \langle C | \, |\alpha \rangle |\beta \rangle |\gamma \rangle$. The path integral over the sphere with three boundaries defines $\langle C | \in {\cal H}^* \otimes {\cal H}^* \otimes {\cal H}^*$. (b) Analogous interpretation of the metric $\eta_{\alpha\beta} \equiv \langle \eta | |\alpha \rangle |\beta \rangle$, with $\langle \eta | \in {\cal H}^* \otimes {\cal H}^*$, in terms of the sphere with two boundaries.
• Figure 4: Topological interpretation of (a) the inverse metric ${\eta^{\alpha\beta}}$, (b) the relation $\eta_{\alpha\beta} \eta^{\beta\gamma}=\delta^{\gamma}_{\alpha}$. By convention, we draw the boundaries associated with upper indices facing left and the boundaries associated with the lower indices facing right.
• Figure 5: Pictorial rendering of the associativity of the algebra.