Super Quantum Airy Structures

Abstract

We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Furthermore, we present various examples of super quantum Airy structures, both finite-dimensional -- which include well known superalgebras and super Frobenius algebras, and whose classification scheme we also discuss -- as well as infinite-dimensional, that arise in the realm of vertex operator super algebras.