A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}(\mathfrak{gl}_r)$-Algebras

TL;DR

This paper classifies higher quantum Airy structures derived from twisted modules of $ ext{W}( ext{gl}_r)$-algebras, explores their relation to topological recursion, and investigates their extension properties.

Contribution

It provides a classification of higher quantum Airy structures from specific twisted modules and examines their role in defining topological recursion for spectral curves.

Findings

Classified all higher quantum Airy structures from certain automorphisms.

Identified conditions for dilaton shifts involving roots of unity.

Showed how these structures relate to topological recursion and extendability.

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 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.