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Highest weight vectors, shifted topological recursion and quantum curves

Raphaël Belliard, Vincent Bouchard, Reinier Kramer, Tanner Nelson

TL;DR

The paper develops a shifted extension of topological recursion by constructing shifted rs-Airy structures whose partition functions are highest weight vectors for $\mathcal{W}(\mathfrak{gl}_r)$ at self-dual or deformed levels. This framework yields shifted loop equations and a corresponding shifted topological recursion that produces symmetric correlators and a WKB-type quantum curve with $\hbar$-dependent terms; it also establishes a converse direction: an $\hbar$-connection of topological type must satisfy the same shift conditions. The results demonstrate that under the modular constraint $r=\pm1\pmod{s}$, the shifted construction recovers the full quantization spectrum of the $(r,s)$-spectral curve, including all operator orderings in special cases $s=1$ and $s=r-1$, and connect the algebraic, geometric, and analytic viewpoints via loop equations, wave-functions, and determinantal formulations. Overall, the work broadens the scope of topological recursion to access a wider class of quantum curves and clarifies the role of highest-weight shifts in the correspondence between Airy structures, WKB analysis, and quantum curves.

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 $ ω_{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.

Highest weight vectors, shifted topological recursion and quantum curves

TL;DR

The paper develops a shifted extension of topological recursion by constructing shifted rs-Airy structures whose partition functions are highest weight vectors for at self-dual or deformed levels. This framework yields shifted loop equations and a corresponding shifted topological recursion that produces symmetric correlators and a WKB-type quantum curve with -dependent terms; it also establishes a converse direction: an -connection of topological type must satisfy the same shift conditions. The results demonstrate that under the modular constraint , the shifted construction recovers the full quantization spectrum of the -spectral curve, including all operator orderings in special cases and , and connect the algebraic, geometric, and analytic viewpoints via loop equations, wave-functions, and determinantal formulations. Overall, the work broadens the scope of topological recursion to access a wider class of quantum curves and clarifies the role of highest-weight shifts in the correspondence between Airy structures, WKB analysis, and quantum curves.

Abstract

We extend the theory of topological recursion by considering Airy structures whose partition functions are highest weight vectors of particular -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 , 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 -dependent terms. In the reverse direction, starting from an -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.

Paper Structure

This paper contains 39 sections, 54 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation: quantum curves
  3. A trifecta of viewpoints
  4. Contributions of this paper
  5. Open questions
  6. Notation
  7. Shifted rs-Airy structures and highest weight vectors
  8. Airy structures
  9. The Rees Weyl algebra
  10. Airy ideals
  11. Partition function
  12. Airy ideals in universal enveloping algebras
  13. rs-Airy structures
  14. The $\mathcal{W}(\mathfrak{gl}_r)$-algebra at self-dual level
  15. A few preliminary lemmas
  16. ...and 24 more sections

Key Result

Theorem 2.6

Let $\mathcal{I} \subset \mathcal{D}_A^{\hslash}$ be an Airy ideal. Then there exists a unique partition function $Z$ of the form eq:pfFgncoeff such that $\mathcal{I}$ is the annihilator ideal of $Z$ in $\mathcal{D}_A^{\hslash}$. Moreover, $Z$ is semistable, and if $p_a(x_A) = 0$ for all $a \in A$,

Figures (1)

  • Figure 1: A trifecta of viewpoints.

Theorems & Definitions (135)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 125 more