光滑 Calabi-Yau 结构与非交换 Legendre 变换
Smooth Calabi-Yau structures and the noncommutative Legendre transform
AI总结：
本文阐明光滑 Calabi-Yau 结构与预 Calabi-Yau 结构间的联系，证明前者可产生后者，类比 Legendre 变换并定义同调意义下的一一对应，且应用于环路空间链并完全计算圆的情形。
摘要：
我们阐明了光滑 Calabi-Yau 结构与预 Calabi-Yau 结构之间的关系。我们证明，从 $A_\infty$-范畴 $A$ 上的光滑 Calabi-Yau 结构出发，可以产生 $A$ 上的预 Calabi-Yau 结构；如我们先前工作中所定义的，这是可积多向量场的平移非交换版本。我们解释了这种关系如何类似于 Legendre 变换，以及它如何在某种同调意义上定义一一对应。为具体起见，我们将此形式应用于（可能非单连通的）Poincaré 对偶空间上基于环路空间的链，并完全计算了圆的情形。
英文摘要：
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.
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