Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects

We show that a Calabi–Yau structure of dimension d on a smooth dg category C C induces a symplectic form of degree $$2-d$$ 2 - d on ‘the moduli space of objects’ M_C M C . We show moreover that a relative Calabi–Yau structure on a dg functor C→ D C → D compatible with the absolute Calabi–Yau structure on C induces a Lagrangian structure on the corresponding map of moduli M_ D → M_C M D → M C .