Higher Airy structures, W algebras and topological recursion
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arXiv
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| Format: | Preprint |
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2018
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| author | Borot, Gaëtan Bouchard, Vincent Chidambaram, Nitin K. Creutzig, Thomas Noshchenko, Dmitry |
| author_facet | Borot, Gaëtan Bouchard, Vincent Chidambaram, Nitin K. Creutzig, Thomas Noshchenko, Dmitry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_08738 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Higher Airy structures, W algebras and topological recursion Borot, Gaëtan Bouchard, Vincent Chidambaram, Nitin K. Creutzig, Thomas Noshchenko, Dmitry Mathematical Physics High Energy Physics - Theory Algebraic Geometry Representation Theory 81R10, 14N10, 51P05 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. |
| title | Higher Airy structures, W algebras and topological recursion |
| topic | Mathematical Physics High Energy Physics - Theory Algebraic Geometry Representation Theory 81R10, 14N10, 51P05 |
| url | https://arxiv.org/abs/1812.08738 |