From CFT to Ramond super-quantum curves

TL;DR

This paper reformulates the connection between classical algebraic curves and quantum curves within conformal field theory, specifically for the Ramond super-Virasoro algebra, revealing new supersymmetric structures and models.

Contribution

It introduces a CFT-based approach to identify quantum curves and eigenvalue models for the Ramond super-Virasoro algebra, generalizing previous methods.

Findings

Derived Ramond super-eigenvalue models

Constructed Ramond super-quantum curves as singular vectors

Identified supersymmetric generalizations of BPZ equations

Abstract

As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that have the structure of relevant singular vectors, and identify underlying Ramond super-spectral curves. We also analyze Ramond multi-Penner models and show that they lead to supersymmetric generalizations of BPZ equations.