Super Quantum Airy Structures
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Communications in
Mathematical Physics
Super Quantum Airy Structures Vincent Bouchard1 , Paweł Ciosmak2 , Leszek Hadasz3 , Kento Osuga4 , Bła˙zej Ruba3 , Piotr Sułkowski5,6 1 Department of Mathematical and Statistical Sciences, University of Alberta, 632 CAB, Edmonton,
AB T6G 2G1, Canada. E-mail: [email protected]
2 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw,
Poland. E-mail: [email protected]
3 M. Smoluchowski Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland.
E-mail: [email protected]; [email protected]
4 School of Mathematics and Statistics, University of Sheffield, The Hicks Building, Hounsfield Road,
Sheffield S3 7RH, UK. E-mail: [email protected]
5 Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland.
E-mail: [email protected]
6 Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125,
USA. E-mail: [email protected] Received: 21 September 2019 / Accepted: 19 August 2020 Published online: 13 October 2020 – © The Author(s) 2020
Abstract: 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. Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . Super Quantum Airy Structures Finite-Dimensional Examples . Infinite-Dimensional Examples Conclusion and Open Questions
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1. Introduction In recent years we have learnt that solutions to a plethora of problems in physics and mathematics that involve some form of quantization arise from a universal system of recursive equations, referred to as the Chekhov–Eynard–Orantin topological recursion [24,37,38]. The topological recursion was originally discovered in the realm of matrix models as a way of solving loop equations, which enables computation of the free energy to all orders in the large N expansion, based on the information encoded in the
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V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba, P. Sułkowski
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