Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks

TL;DR

This paper establishes a connection between super topological recursion, super Airy structures, and Gaiotto vectors for superconformal blocks, enabling new computational methods for supersymmetric gauge theories.

Contribution

It introduces super topological recursion and super Airy structures to compute Gaiotto vectors in superconformal blocks, extending the AGT correspondence to supersymmetric theories.

Findings

Gaiotto vectors can be obtained via super topological recursion.

Partition functions of super Airy structures match Gaiotto vectors.

Framework applies to Nekrasov partition functions for supersymmetric gauge theories.

Abstract

We investigate a relation between the super topological recursion and Gaiotto vectors for $\mathcal{N}=1$ superconformal blocks. Concretely, we introduce the notion of the untwisted and $\mu$-twisted super topological recursion, and construct a dual algebraic description in terms of super Airy structures. We then show that the partition function of an appropriate super Airy structure coincides with the Gaiotto vector for $\mathcal{N}=1$ superconformal blocks in the Neveu-Schwarz or Ramond sector. Equivalently, the Gaiotto vector can be computed by the untwisted or $\mu$-twisted super topological recursion. This implies that the framework of the super topological recursion -- equivalently super Airy structures -- can be applied to compute the Nekrasov partition function of $\mathcal{N}=2$ pure $U(2)$ supersymmetric gauge theory on $\mathbb{C}^2/\mathbb{Z}_2$ via a conjectural extension of the Alday-Gaiotto-Tachikawa correspondence.