Superconformal topological recursion
Nezhla Aghaei, Reinier Kramer, Nicolas Orantin, Kento Osuga
TL;DR
This work develops a superconformal generalization of topological recursion by uniting geometric data on a super Riemann surface with algebraic structures from the $\mathcal{N}=1$ super Virasoro algebra. It introduces partial super Airy structures and associated $F_{g,n}$ that satisfy super Virasoro constraints, and constructs a super spectral curve $\mathcal{S}_\Sigma$ together with multidifferentials $\omega_{g,n}$ that encode these invariants. A recursive framework is established via a detailed super loop equation analysis, with a novel expansion in an odd parameter $\gamma$ that yields a blobbed-like recursion for $\omega_{g,n}^{(k)}$, reflecting zero-mode and Ramond-structure effects. Explicit examples, including the super Airy and super Weber curves, illustrate the theory and support conjectures about variational structures and potential connections to super enumerative geometries and supergravity-inspired moduli. The results lay a foundation for a comprehensive super topological recursion and its links to supersymmetric invariants, deformations of super Riemann surfaces, and possible super-quantisation of spectral curves.
Abstract
We investigate a supersymmetric generalisation of topological recursion from two perspectives: algebraic and geometric. The algebraic side concerns a recursive structure encoded in modules of a super Virasoro algebra, and the geometric counterpart is what we call superconformal topological recursion defined on a super Riemann surface. Superconformal topological recursion indicates that odd holomorphic one-forms on a super Riemann surface are related to zero modes of copies of the Clifford algebras, and it also provides a tool to study deformation of non-split super Riemann surfaces, e.g. certain families of super Riemann surfaces carrying odd parameters. On a super Riemann surface over a non-reduced base, the formalism is recursive not only in terms of pants-decomposition but also in terms of odd parameters in a suitable sense.