Shifted symplectic reduction of derived critical loci

TL;DR

This paper demonstrates that derived critical loci of invariant functions possess a shifted moment map, and their symplectic reduction aligns with the critical locus of the reduced function, revealing new structural insights in derived symplectic geometry.

Contribution

It establishes the existence of shifted moment maps on derived critical loci and proves that symplectic reduction commutes with lagrangian intersections in this setting.

Findings

Derived critical loci have shifted moment maps.

Symplectic reduction of derived critical loci equals the critical locus of the reduced function.

Symplectic reduction commutes with lagrangian intersections.

Abstract

We prove that the derived critical locus of a $G$-invariant function $S:X\to\mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red}:X/G\to\mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections.