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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2311.04200 |
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| _version_ | 1866916336524853248 |
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| author | Yang, Di |
| author_facet | Yang, Di |
| contents | Let $M$ be an $n$-dimensional Frobenius manifold. Fix $κ\in\{1,\dots,n\}$. Assuming certain invertibility, Dubrovin introduced the Legendre-type transformation $S_κ$, which transforms $M$ to an $n$-dimensional Frobenius manifold $S_κ(M)$. In this paper, we show that these $S_κ(M)$ share the same monodromy data at the Fuchsian singular point of the Dubrovin connection, and that for the case when $M$ is semisimple they also share the same Stokes matrix and the same central connection matrix. A straightforward application of the monodromy identification is the following: if we know the monodromy data of some semisimple Frobenius manifold $M$, we immediately obtain those of its Legendre-type transformations. Another application gives the identification between the $κ$th partition function of a semisimple Frobenius manifold $M$ and the topological partition function of $S_κ(M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04200 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analytic theory of Legendre-type transformations for a Frobenius manifold Yang, Di Differential Geometry Mathematical Physics Let $M$ be an $n$-dimensional Frobenius manifold. Fix $κ\in\{1,\dots,n\}$. Assuming certain invertibility, Dubrovin introduced the Legendre-type transformation $S_κ$, which transforms $M$ to an $n$-dimensional Frobenius manifold $S_κ(M)$. In this paper, we show that these $S_κ(M)$ share the same monodromy data at the Fuchsian singular point of the Dubrovin connection, and that for the case when $M$ is semisimple they also share the same Stokes matrix and the same central connection matrix. A straightforward application of the monodromy identification is the following: if we know the monodromy data of some semisimple Frobenius manifold $M$, we immediately obtain those of its Legendre-type transformations. Another application gives the identification between the $κ$th partition function of a semisimple Frobenius manifold $M$ and the topological partition function of $S_κ(M)$. |
| title | Analytic theory of Legendre-type transformations for a Frobenius manifold |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2311.04200 |