Alex Takeda (IHES), Calabi-Yau structures, noncommutative Legendre transform and 'the tube'.

Abstract:

I will start this talk by recalling the definition of Calabi-Yau and pre-Calabi-Yau structure; this latter is a weakening of the former, and should be thought of as a noncommutative version of a Poisson structure. I will then discuss the precise relation between absolute/relative smooth Calabi-Yau structures and pre-Calabi-Yau structures; this turns out to be a noncommutative version of the Legendre transform. This relation can be made explicit by thinking of a certain space of 'tubes': cylinders with certain markings and colorings. Time allowing I will discuss some of the examples where this formalism could be used.