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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
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-06-07
Date submitted: 2023-05-31 17:22
Submitted by: Bouchard, Vincent
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


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 consisting of products of disjoint 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.

Published as SciPost Phys. 14, 169 (2023)

Author comments upon resubmission

The authors thank the referees for constructive comments and suggestions.

List of changes

- Minor revisions throughout the paper.
- Remark (with reference) added in the introduction.
- Minor rewriting of the proofs of Theorems 3.5 and 4.1.
- Remark 4.2 added.

Submission & Refereeing History

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Resubmission 2009.13047v2 on 31 May 2023

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