Highest weight vectors, shifted topological recursion and quantum curves

TL;DR

This paper generalizes topological recursion by incorporating highest weight vectors from $ ext{W}$-algebra representations, leading to a shifted recursion that produces quantum curves with $ ext{WKB}$ solutions.

Contribution

It introduces a shifted topological recursion framework linked to highest weight vectors of $ ext{W}$-algebras, connecting spectral curves, quantum curves, and $ ext{WKB}$ solutions.

Findings

Shifted topological recursion extends classical formulas.

Wave-functions satisfy $ ext{WKB}$ solutions of quantized spectral curves.

Conditions for highest weight vectors determine the structure of the recursion.

Abstract

We extend the theory of topological recursion by considering Airy structures whose partition functions are highest weight vectors of particular $\mathcal{W}$-algebra representations. Such highest weight vectors arise as partition functions of Airy structures only under certain conditions on the representations. In the spectral curve formulation of topological recursion, we show that this generalization amounts to adding specific terms to the correlators $ \omega_{g,1}$, which leads to a ``shifted topological recursion'' formula. We then prove that the wave-functions constructed from this shifted version of topological recursion are WKB solutions of families of quantizations of the spectral curve with $ \hbar$-dependent terms. In the reverse direction, starting from an $\hbar$-connection, we find that it is of topological type if the exact same conditions that we found for the Airy structures are satisfied. When this happens, the resulting shifted loop equations can be solved by the shifted topological recursion obtained earlier.