Calabi–Yau structures on dg categories and shifted symplectic structures on moduli

We define a relative notion of Calabi–Yau structure for a morphism of dg categories, which in the absolute case reduces to the standard notion of Calabi–Yau structure on a finite type dg category. We explain how a Calabi–Yau structure on a finite type dg category induces a shifted symplectic structure on its derived moduli stack of objects and how a Calabi–Yau structure on a morphism of dg categories with Calabi–Yau source induces a Lagrangian structure on the corresponding map of moduli stacks. Our construction gives a new construction of many of the known examples of shifted symplectic structures on moduli stacks, as well as providing some new examples not accessible by previous constructions. This is joint work in progress with Tobias Dyckerhoff.