[Paper Review] Higher Airy structures, W algebras and topological recursion

Gaëtan Borot, Vincent Bouchard+3 more|arXiv (Cornell University)|Dec 20, 2018

Algebraic structures and combinatorial models54 references20 citations

TL;DR

This paper introduces higher quantum Airy structures as generalizations of Kontsevich–Soibelman's quantum Airy structures, allowing differential operators of arbitrary order. It constructs these structures from $χ$-algebras at self-dual level for $χ = τ_{N+1}, σ_{2N}, ε_N$, and establishes their equivalence to topological recursion via $χ$-algebra constraints, providing a Givental-like decomposition and a new characterization of admissible spectral curves for well-defined recursion.

ABSTRACT

We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{g})$ algebras at self-dual level, with $\mathfrak{g}= \mathfrak{gl}_{N+1}$, $\mathfrak{so}_{2 N }$ or $\mathfrak{e}_N$. We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these $\mathcal{W}$ constraints have already appeared in the literature, but we find many new ones. For $\mathfrak{gl}_{N+1}$ our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard-Eynard topological recursion gives symmetric $ω_{g,n}$s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to $\mathcal{W}(\mathfrak{gl})$ constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

Motivation & Objective

• To generalize quantum Airy structures by allowing differential operators of arbitrary order, beyond the quadratic case.
• To construct higher quantum Airy structures as modules of $χ(τ_{N+1})$, $χ(σ_{2N})$, and exceptional $χ(ε_N)$ algebras at self-dual level.
• To establish a correspondence between these higher quantum Airy structures and the Bouchard–Eynard topological recursion for symmetric $ω_{g,n}$ forms.
• To provide a new characterization of spectral curves for which topological recursion is well-defined, including arbitrary ramification.
• To derive a Givental-like decomposition for partition functions associated with these structures.

Proposed method

• Define higher quantum Airy structures as collections of differential operators $H_k = θ\partial_{x_k} - P_k$, where $P_k$ is homogeneous of degree 2 in the $θ$-graded algebra of differential operators.
• Construct examples from $χ(τ_{N+1})$, $χ(σ_{2N})$, and $χ(ε_N)$ algebras at self-dual level, using previously unnoticed Lie subalgebras of the mode algebra.
• Prove that the resulting operators generate a graded Lie subalgebra, ensuring the existence and uniqueness of a solution to the constraints.
• Establish equivalence between the higher abstract loop equations and the Bouchard–Eynard topological recursion via residue formulas and symmetric forms.
• Use the spectral curve condition $r \equiv \pm1 \mod s$ to characterize admissible curves for which topological recursion yields symmetric $ω_{g,n}$.
• Derive a Givental-like decomposition of the partition function $Z = \exp(\sum \hbar^{g-1} F_{g,n} x_{\alpha_1} \cdots x_{\alpha_n} / n!)$ from the higher quantum Airy structure.

Experimental results

Research questions

• RQ1 Which higher-order differential operators can form consistent quantum Airy structures beyond the quadratic case?
• RQ2 How can $χ$-algebras at self-dual level be used to construct higher quantum Airy structures for $τ_{N+1}$, $σ_{2N}$, and exceptional types?
• RQ3 Under what conditions on the spectral curve does the Bouchard–Eynard topological recursion produce symmetric $ω_{g,n}$ forms?
• RQ4 Can the higher abstract loop equations be used to derive topological recursion in a way that guarantees symmetry and well-definedness?
• RQ5 What is the Givental-like decomposition structure of the partition function arising from higher quantum Airy structures?

Key findings

• The paper constructs new classes of higher quantum Airy structures from $χ(τ_{N+1})$, $χ(σ_{2N})$, and $χ(ε_N)$ algebras at self-dual level, extending known Virasoro-type constraints.
• It identifies a new class of Lie subalgebras within the mode algebra of $χ(τ_{N+1})$, enabling the construction of higher-order Airy structures.
• For spectral curves with $r \equiv \pm1 \mod s$, the Bouchard–Eynard topological recursion produces symmetric $ω_{g,n}$ forms, and this is equivalent to the $χ(τ_{N+1})$-algebra constraints realized as higher quantum Airy structures.
• The solution to the higher abstract loop equations exists and is unique for admissible spectral curves, providing a direct proof of the well-definedness of topological recursion in this class.
• The partition function associated with these structures admits a Givental-like decomposition, with coefficients $F_{g,n}$ encoded in the generating series $Z = \exp(\sum \hbar^{g-1} F_{g,n} / n! \cdot \prod x_{\alpha_i})$.
• The work provides a new, independent proof (via higher Airy structures) that topological recursion yields symmetric forms for admissible curves, bypassing earlier indirect arguments from [17].