Shifted symplectic pushforwards

TL;DR

This paper develops a framework for pushing forward shifted symplectic structures along base changes, providing local structure theorems and resolving deformation invariance issues in Donaldson-Thomas theory of Calabi-Yau 4-folds.

Contribution

It introduces methods to pushforward shifted symplectic fibrations, establishes local structure theorems, and addresses deformation invariance in Donaldson-Thomas theory.

Findings

Symplectic pushforwards include symplectic zero loci and quotients.

Equivalence between symplectic fibrations and Lagrangians to critical loci.

Resolution of deformation invariance issues in Donaldson-Thomas theory.

Abstract

We introduce how to pushforward shifted symplectic fibrations along base changes. This is achieved by considering symplectic forms that are closed in a stronger sense. Examples include: symplectic zero loci and symplectic quotients. Observing that twisted cotangent bundles are symplectic pushforwards, we obtain an equivalence between symplectic fibrations and Lagrangians to critical loci. We provide two local structure theorems for symplectic fibrations: a smooth local structure theorem for higher stacks via symplectic zero loci and twisted cotangents, and an {\'e}tale local structure theorem for $1$-stacks with reductive stabilizers via symplectic quotients of the smooth local models. We resolve deformation invariance issue in Donaldson-Thomas theory of Calabi-Yau $4$-folds. Abstractly, we associate virtual Lagrangian cycles for oriented $(-2)$-symplectic fibrations as unique functorial bivariant classes over the exact loci. For moduli of perfect complexes, we show that the exact loci consist of deformations for which the $(0,4)$-Hodge pieces of the second Chern characters remain zero.