A generalization of the Witten conjecture through spectral curve
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909689489391616 |
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| author | Guo, Shuai Ji, Ce Zhang, Qingsheng |
| author_facet | Guo, Shuai Ji, Ce Zhang, Qingsheng |
| contents | We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a correspondence between the geometric descendent potential (apart from ancestors) and the topological recursion of specific spectral curve data $(Σ, x,y)$; Part II (Integrability) claims that the TR descendent potential, defined at the boundary points of the spectral curve (where $dx$ has poles), is a tau-function of a certain reduction of the multi-component KP hierarchy.
In this paper, we show the geometric part of the conjecture for any formal descendent theory by using a generalized Laplace transform. Subsequently, we prove the integrability conjecture for the one-boundary cases. As applications, we generalize and prove the $r$KdV integrability of negative $r$-spin theory conjectured by Chidambaram, Garcia-Failde and Giacchetto. We also show the KdV integrability of the total descendent potential associated with the Hurwitz space $M_{1,1}$, whose Frobenius manifold was initially introduced by Dubrovin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12271 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A generalization of the Witten conjecture through spectral curve Guo, Shuai Ji, Ce Zhang, Qingsheng Mathematical Physics High Energy Physics - Theory Algebraic Geometry 14N35, 37K10, 14J33 We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a correspondence between the geometric descendent potential (apart from ancestors) and the topological recursion of specific spectral curve data $(Σ, x,y)$; Part II (Integrability) claims that the TR descendent potential, defined at the boundary points of the spectral curve (where $dx$ has poles), is a tau-function of a certain reduction of the multi-component KP hierarchy. In this paper, we show the geometric part of the conjecture for any formal descendent theory by using a generalized Laplace transform. Subsequently, we prove the integrability conjecture for the one-boundary cases. As applications, we generalize and prove the $r$KdV integrability of negative $r$-spin theory conjectured by Chidambaram, Garcia-Failde and Giacchetto. We also show the KdV integrability of the total descendent potential associated with the Hurwitz space $M_{1,1}$, whose Frobenius manifold was initially introduced by Dubrovin. |
| title | A generalization of the Witten conjecture through spectral curve |
| topic | Mathematical Physics High Energy Physics - Theory Algebraic Geometry 14N35, 37K10, 14J33 |
| url | https://arxiv.org/abs/2309.12271 |