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

Abstract

AbstractWe 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’ $${{\mathcal {M}}}_{{C}}$$ M C . We show moreover that a relative Calabi–Yau structure on a dg functor $${C}\rightarrow { D}$$ C → D compatible with the absolute Calabi–Yau structure on C induces a Lagrangian structure on the corresponding map of moduli $${{\mathcal {M}}}_{{ D}} \rightarrow {{\mathcal {M}}}_{{C}}$$ M D → M C .