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 β-deformed version of those models, and derive differential equations for associated α/β-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, superVirasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.

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. CALT-2017-070 7. Ramond-R super-eigenvalue model and super-quantum curves 56 A. Proofs and computations 63 A.1 Computations in the Ramond-NS sector: the supercurrent S(y) 63 A.2 Computations in the Ramond-NS sector: the energy-momentum tensor T (y) 66 A.3 Computations in the Ramond-NS super-eigenvalue model 67 -1 -The above results have been generalized to a supersymmetric case in [24], by considering (β-deformed) super-eigenvalue models for the Neveu-Schwarz sector [25][26][27][28][29][30]. These models generalize eigenvalue representation of hermitian matrix models in such a way, that the underlying algebra takes form of the Neveu-Schwarz version of the super-Virasoro algebra; in particular corresponding loop equations can be rewritten as super-Virasoro constraints. Consequently, to a super-eigenvalue model one can associate an infinite family of super-quantum curves, which have the structure of Neveu-Schwarz singular vectors of the super-Virasoro algebra. In the classical limit, such super-quantum curves reduce to supersymmetric algebraic curves, which are interesting in their own right [31,32].To sum up, to a given classical (possibly supersymmetric) curve one can associate an infinite family of quantum curves, which have the structure of singular vectors of the underlying algebra. This result was found in [20,24] upon the analysis of eigenvalue models, which provide a representation (or generalization) of matrix models; for a summary see also [33].The aim of the present paper is twofold. First, we clarify the role of conformal field theory in the description of quantum curves. In particular, we rederive (in Virasoro and Neveu-Schwarz case) quantum curves using only conformal field theory techniques (instead of eigenvalue models). The main feature of this approach is the fact, that the singular vector structure of qu...

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.

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We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.