Higher Airy structures, W algebras and topological 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 $\mathfrak{g}={\mathfrak{gl}}_{N+1}$, ${\mathfrak{so}}_{2N}$ 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 ${\omega }_{g,n}$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.

93 pages, v4: references added, many typos corrected

References in corpus (32)

• Vertex Algebras and Algebraic Curves
• Tautological relations and the r-spin Witten conjecture
• Matrix eigenvalue model: Feynman graph technique for all genera
• Towards an enumerative geometry of the moduli space of twisted curves and r-th roots
• W-algebras as coset vertex algebras
• Identification of the Givental formula with the spectral curve topological recursion procedure
• Cosets of affine vertex algebras inside larger structures
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• A new cohomology class on the moduli space of curves
• Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations
• Think globally, compute locally
• Cut-and-join description of generalized Brezin-Gross-Witten model
• CFT and topological recursion
• Reconstructing WKB from topological recursion
• Blobbed topological recursion: properties and applications
• Dubrovin's superpotential as a global spectral curve
• A generalized topological recursion for arbitrary ramification
• Screening operators for W-algebras
• The Witten equation, mirror symmetry and quantum singularity theory
• W-Constraints for the Total Descendant Potential of a Simple Singularity
• Explicit generators in rectangular affine W-algebras of type A
• Matrix models and a proof of the open analog of Witten's conjecture
• The ABCD of topological recursion
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• The combinatorial formula for open gravitational descendents
• Open intersection numbers, matrix models and MKP hierarchy
• Higher Airy structures and topological recursion for singular spectral curves
• Super Quantum Airy Structures
• Uniqueness Theorem of W-Constraints for Simple Singularities
• Introduction to W-algebras and their representation theory
• Open intersection numbers and free fields
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion

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• The ABCD of topological recursion
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• On higher Brézin-Gross-Witten tau-functions
• Taking limits in topological recursion
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions