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A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}(\mathfrak{gl}_r)$-Algebras
by Vincent Bouchard, Kieran Mastel
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Submission summary
| Authors (as registered SciPost users): | Vincent Bouchard |
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| Preprint Link: | https://arxiv.org/abs/2009.13047v1 (pdf) |
| Date submitted: | 2021-01-13 22:18 |
| Submitted by: | Bouchard, Vincent |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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Abstract
Quantum $r$-Airy structures can be constructed as modules of $\mathcal{W}(\mathfrak{gl}_r)$-algebras via restriction of twisted modules for the underlying Heisenberg algebra. In this paper we classify all such higher quantum Airy structures that arise from modules twisted by automorphisms of the Cartan subalgebra that have repeated cycles of the same length. An interesting feature of these higher quantum Airy structures is that the dilaton shifts must be chosen carefully to satisfy a matrix invertibility condition, with a natural choice being roots of unity. We explore how these higher quantum Airy structures may provide a definition of the Chekhov, Eynard, and Orantin topological recursion for reducible algebraic spectral curves. We also study under which conditions quantum $r$-Airy structures that come from modules twisted by arbitrary automorphisms can be extended to new quantum $(r+1)$-Airy structures by appending a trivial one-cycle to the twist without changing the dilaton shifts.
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Report
Please see the attached report.
Report #1 by Anonymous (Referee 2) on 2022-4-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2009.13047v1, delivered 2022-04-03, doi: 10.21468/SciPost.Report.4849
Report
This paper is a follow-up of a previous paper co-authored by one of the authors, in which the concept of higher quantum Airy structures was introduced. This quadratic quantum Airy structures introduced by Kontsevich and Soibelman generalize the Virasoro constraints on the tau-functions related to matrix models and to the theory of intersections, while the higher quantum Airy structures generalize the W-constraints. In the previous paper, the quantum Airy structure was associated with a W-algebra for the Lie-algebra gl(n) and an automorphism induced by the Coxeter element of the Weyl group. Such quantum Airy structure describes the critical points of the two-matrix model. In the present paper, a more general automorphism with several cycles of the same length is considered. A classification of the higher quantum Airy structures obtained in this way is presented and some examples are given. The relation to the topological recursion is discussed, and a conjecture is advanced that the generalisation considered corresponds to topological recursion associated with a reducible algebraic curve. An interesting point not discussed in the paper is which type large-N matrix integrals are described by reducible algebraic curves.
I think that this paper contains original results and contributes to a larger program of finding a complete classification of the higher quantum Airy structures and their relation to the topological recursion. I recommend publication.