5 research outputs found
Singular vector structure of quantum curves
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.Comment: 33 pages; proceedings of the 2016 AMS von Neumann Symposiu
From CFT to Ramond super-quantum curves
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.Comment: 72 page
Super-quantum curves from super-eigenvalue models
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, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations
Super-quantum curves from super-eigenvalue models
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, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations