Shifted symplectic rigidification

Abstract

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any $B\mathbb{G}_m$-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.