The purpose of this short note is to present two existence results concerning symplectic and lagrangian structures in the derived setting, in situations where the constructions of [Ca] and [PTVV] do not apply. For this we show that symplectic structures can be constructed out of Calabi-Yau structures on sheaves of dg-categories, or out of \emph{ orientations} on sheaves of rigid dg-categories. These results follow from two main theorems: the HKR theorem and the cyclotomic aspect of traces in rigid infty-categories.
Categories: math.AG math.AT