• Corpus ID: 88516840

Higher Airy structures, W algebras and topological recursion

@article{Borot2018HigherAS,
  title={Higher Airy structures, W algebras and topological recursion},
  author={Gaetan Borot and Vincent Bouchard and Nitin Kumar Chidambaram and Thomas Creutzig and Dmitry Noshchenko},
  journal={arXiv: Mathematical Physics},
  year={2018}
}

• G. Borot, V. Bouchard, +2 authors D. Noshchenko
• Published 20 December 2018
• Mathematics
• arXiv: Mathematical Physics

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 $\omega_{g,n}$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.