Abstract
We introduce the notion of {mathcal {N}}=1 abstract super loop equations and provide two equivalent ways of solving them. The first approach is a recursive formalism that can be thought of as a supersymmetric generalization of the Eynard–Orantin topological recursion, based on the geometry of a local super spectral curve. The second approach is based on the framework of super Airy structures. The resulting recursive formalism can be applied to compute correlation functions for a variety of examples related to 2d supergravity.
Highlights
The Eynard–Orantin topological recursion introduced in [17,28,29] can be used to compute various kinds of enumerative invariants, such as Gromov–Witten invariants, Hurwitz numbers, knot invariants, and more
The topological recursion does not come out of nowhere. It can be obtained as a unique solution of a set of equations known as abstract loop equations, which were formalized in [6]. (The well-known loop equations for Hermitian matrix models fit into this abstract framework.) Concretely, the Eynard– Orantin topological recursion solves the loop equations through residue analysis at the poles of the correlation functions
The abstract loop equations are reformulated as Virasoro constraints, and the framework of Airy structures guarantees that these Virasoro constraints have a unique solution
Summary
The Eynard–Orantin topological recursion introduced in [17,28,29] can be used to compute various kinds of enumerative invariants, such as Gromov–Witten invariants, Hurwitz numbers, knot invariants, and more (see [11,12,15,25,30,31,32,33,37,41] and references therein). In the simple context of a local spectral curve with one component, these differential operators form a (suitably polarized) representation of the Virasoro algebra In this way, the abstract loop equations are reformulated as Virasoro constraints, and the framework of Airy structures guarantees that these Virasoro constraints have a unique solution. In the context of a local super spectral curve with one component (which is what we mainly focus on in this paper), these differential operators form a (suitably polarized) representation of the N = 1 super Virasoro algebra in the Neveu–Schwarz sector. 4, we transform the abstract super loop equations into differential constraints, and show that they form a super Airy structure, which comes with a unique partition function (Theorem 4.4). For the sake of brevity, the proofs of all theorems and propositions are given in “Appendix 1”
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