Shifted symplectic rigidification

TL;DR

This paper develops shifted symplectic structures on rigidified moduli spaces of sheaves on Calabi-Yau varieties and explores Hamiltonian actions of $B\mathbb{G}_m$ on symplectic stacks.

Contribution

It introduces a symplectic rigidification functor and demonstrates Hamiltonian $B\mathbb{G}_m$-actions on non-positively-shifted symplectic derived stacks.

Findings

Constructed shifted symplectic structures on rigidified moduli spaces.

Proved Hamiltonian property of $B\mathbb{G}_m$-actions on certain stacks.

Established a symplectic rigidification functor as a left adjoint.

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.