Pre-Calabi-Yau algebras and topological quantum field theories
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913573820694528 |
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| author | Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis |
| author_facet | Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis |
| contents | We introduce a notion generalizing Calabi-Yau structures on A-infinity algebras and categories, which we call pre-Calabi-Yau structures. This notion does not need either one of the finiteness conditions (smoothness or compactness) which are required for Calabi-Yau structures to exist. In terms of noncommutative geometry, a pre-CY structure is as a polyvector field satisfying an integrability condition with respect to a noncommutative analogue of the Schouten-Nijenhuis bracket. We show that a pre-CY structure defines an action of a certain PROP of chains on decorated Riemann surfaces. In the language of the cobordism perspective on TQFTs, this gives a partially defined extended 2-dimensional TQFT, whose 2-dimensional cobordisms are generated only by handles of index one. We present some examples of pre-CY structures appearing naturally in geometric and topological contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14667 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Pre-Calabi-Yau algebras and topological quantum field theories Kontsevich, Maxim Takeda, Alex Vlassopoulos, Yiannis Algebraic Geometry Mathematical Physics Category Theory We introduce a notion generalizing Calabi-Yau structures on A-infinity algebras and categories, which we call pre-Calabi-Yau structures. This notion does not need either one of the finiteness conditions (smoothness or compactness) which are required for Calabi-Yau structures to exist. In terms of noncommutative geometry, a pre-CY structure is as a polyvector field satisfying an integrability condition with respect to a noncommutative analogue of the Schouten-Nijenhuis bracket. We show that a pre-CY structure defines an action of a certain PROP of chains on decorated Riemann surfaces. In the language of the cobordism perspective on TQFTs, this gives a partially defined extended 2-dimensional TQFT, whose 2-dimensional cobordisms are generated only by handles of index one. We present some examples of pre-CY structures appearing naturally in geometric and topological contexts. |
| title | Pre-Calabi-Yau algebras and topological quantum field theories |
| topic | Algebraic Geometry Mathematical Physics Category Theory |
| url | https://arxiv.org/abs/2112.14667 |