Super-quantum curves from super-eigenvalue models

TL;DR

This paper introduces super-quantum curves by combining supersymmetric and quantum generalizations of algebraic curves within super-eigenvalue models, revealing an infinite hierarchy of differential equations linked to super-Virasoro structures.

Contribution

It constructs super-quantum curves in super-eigenvalue models, establishing their connection to super-Virasoro singular vectors and expanding the framework of quantum algebraic geometry.

Findings

Existence of infinite super-quantum curves for each model

Super-quantum curves correspond to super-Virasoro singular vectors

Introduction of $eta$-deformed super-eigenvalue models

Abstract

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce $\beta$-deformed version of those models, and derive differential equations for associated $\alpha/\beta$-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.