Super-quantum curves from super-eigenvalue models

  • Paweł Ciosmak
  • Leszek Hadasz
  • Masahide Manabe
  • Piotr Sułkowski
Open Access
Regular Article - Theoretical Physics


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.


Matrix Models Conformal and W Symmetry 2D Gravity Topological Strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Paweł Ciosmak
    • 1
  • Leszek Hadasz
    • 2
  • Masahide Manabe
    • 3
  • Piotr Sułkowski
    • 3
    • 4
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.M. Smoluchowski Institute of PhysicsJagiellonian UniversityKrakówPoland
  3. 3.Faculty of PhysicsUniversity of WarsawWarsawPoland
  4. 4.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.

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