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arXiv · 2112.14667

Pre-Calabi-Yau algebras and topological quantum field theories

Maxim Kontsevich·Alex Takeda·Yiannis Vlassopoulos
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Abstract

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.

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Maxim Kontsevich, Alex Takeda, Yiannis Vlassopoulos. 2024-11-08. Pre-Calabi-Yau algebras and topological quantum field theories. https://arxiv.org/abs/2112.14667

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