Smooth Calabi-Yau structures and the noncommutative Legendre transform

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) Poincar\'e duality spaces, and fully calculate the case of the circle.