Smooth Calabi-Yau structures and the noncommutative Legendre transform
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912109700317184 |
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| author | Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis |
| author_facet | Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis |
| contents | We elucidate the relation between smooth Calabi-Yau structures and pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on an $A_\infty$-category $A$, one can produce a pre-Calabi-Yau structure on $A$; as defined in our previous work, this is a shifted noncommutative version of an integrable polyvector field. We explain how this relation is an analogue of the Legendre transform, and how it defines a one-to-one mapping, in a certain homological sense. For concreteness, we apply this formalism to chains on based loop spaces of (possibly non-simply connected) Poincaré duality spaces, and fully calculate the case of the circle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01567 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth Calabi-Yau structures and the noncommutative Legendre transform Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis Algebraic Topology Mathematical Physics Category Theory We elucidate the relation between smooth Calabi-Yau structures and pre-Calabi-Yau structures. We show that, from a smooth Calabi-Yau structure on an $A_\infty$-category $A$, one can produce a pre-Calabi-Yau structure on $A$; as defined in our previous work, this is a shifted noncommutative version of an integrable polyvector field. We explain how this relation is an analogue of the Legendre transform, and how it defines a one-to-one mapping, in a certain homological sense. For concreteness, we apply this formalism to chains on based loop spaces of (possibly non-simply connected) Poincaré duality spaces, and fully calculate the case of the circle. |
| title | Smooth Calabi-Yau structures and the noncommutative Legendre transform |
| topic | Algebraic Topology Mathematical Physics Category Theory |
| url | https://arxiv.org/abs/2301.01567 |