Shifted Witten classes and topological recursion
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- by Séverin Charbonnier, Nitin Kumar Chidambaram, Elba Garcia-Failde and Alessandro Giacchetto
- Trans. Amer. Math. Soc. 377 (2024), 1069-1110
- DOI: https://doi.org/10.1090/tran/9046
- Published electronically: October 25, 2023
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Abstract:
The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin–Frobenius manifold using the Givental–Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two $1$-parameter families of spectral curves. By taking the limit as the parameter goes to zero, we prove that the descendant intersection theory of the Witten $r$-spin class can be computed by topological recursion on the $r$-Airy spectral curve. We finally show that this proof suffices to deduce Witten’s $r$-spin conjecture, already proved by Faber, Shadrin and Zvonkine, which claims that the generating series of $r$-spin intersection numbers is the tau function of the $r$-KdV hierarchy that satisfies the string equation.References
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Bibliographic Information
- Séverin Charbonnier
- Affiliation: Université de Paris, UMR 8243 CNRS, Institut de Recherche en Informatique Fondamentale, 75205 Paris Cedex 13, France
- Email: charbonnier@irif.fr
- Nitin Kumar Chidambaram
- Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, United Kingdom
- MR Author ID: 1136134
- ORCID: 0000-0001-8973-7567
- Email: nitin.chidambaram@ed.ac.uk
- Elba Garcia-Failde
- Affiliation: Sorbonne Université, UMR 7586 CNRS, Institut de Mathématiques de Jussieu–Paris Rive Gauche, 75252 Paris, France
- MR Author ID: 1342293
- ORCID: 0000-0001-7901-5819
- Email: egarcia@imj-prg.fr
- Alessandro Giacchetto
- Affiliation: Université Paris-Saclay, UMR 3681 CNRS, CEA, Institut de Physique Théorique, 91191 Gif-sur-Yvette, France
- MR Author ID: 1545702
- ORCID: 0000-0001-8415-5066
- Email: alessandro.giacchetto@ipht.fr
- Received by editor(s): July 19, 2022
- Received by editor(s) in revised form: June 19, 2023
- Published electronically: October 25, 2023
- Additional Notes: The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. ERC-2016-STG 716083 “CombiTop”). The third author was supported by the same grant when this work started. This work was partly a result of the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum) which received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 810573. The fourth author had been supported by the Institut de Physique Théorique Paris (IPhT), CEA, Université de Saclay.
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1069-1110
- MSC (2020): Primary 14H10, 14H70; Secondary 81R10, 34E05
- DOI: https://doi.org/10.1090/tran/9046