Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915483314290688 |
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| author | Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro |
| author_facet | Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro |
| contents | We construct and study various properties of a negative spin version of the Witten $ r $-spin class. By taking the top Chern class of a certain vector bundle on the moduli space of twisted spin curves that parametrises $ r $-th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class $ Θ^r $. This CohFT does not have a flat unit and its associated Dubrovin--Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on $ \overline{\mathcal{M}}_{g,n} $. We further consider the descendant potential of the Theta class and prove that it is the unique solution to a set of $ \mathcal{W} $-algebra constraints, which implies a recursive formula for the descendant integrals. Using this result for $ r = 2 $, we prove Norbury's conjecture which states that the descendant potential of $ Θ^2 $ coincides with the Brézin--Gross--Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of $ Θ^r $ is the $ r $-BGW tau function of the $ r $-KdV hierarchy and prove the conjecture for $ r = 3 $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_15621 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro Algebraic Geometry Mathematical Physics 14H10, 14H70 (Primary) 37K20, 81R12 (Secondary) We construct and study various properties of a negative spin version of the Witten $ r $-spin class. By taking the top Chern class of a certain vector bundle on the moduli space of twisted spin curves that parametrises $ r $-th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class $ Θ^r $. This CohFT does not have a flat unit and its associated Dubrovin--Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on $ \overline{\mathcal{M}}_{g,n} $. We further consider the descendant potential of the Theta class and prove that it is the unique solution to a set of $ \mathcal{W} $-algebra constraints, which implies a recursive formula for the descendant integrals. Using this result for $ r = 2 $, we prove Norbury's conjecture which states that the descendant potential of $ Θ^2 $ coincides with the Brézin--Gross--Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of $ Θ^r $ is the $ r $-BGW tau function of the $ r $-KdV hierarchy and prove the conjecture for $ r = 3 $. |
| title | Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture |
| topic | Algebraic Geometry Mathematical Physics 14H10, 14H70 (Primary) 37K20, 81R12 (Secondary) |
| url | https://arxiv.org/abs/2205.15621 |