Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone

TL;DR

This paper extends shifted symplectic geometry to relative and stack contexts, introducing shifted cotangent bundles, symplectic groupoids, and a deformation to the normal cone, enriching the theoretical framework.

Contribution

It develops the theory of shifted symplectic structures for non-geometric stacks and introduces new constructions like shifted cotangent bundles and a deformation to the normal cone.

Findings

Generalization of shifted symplectic structures to relative and stack contexts

Construction of shifted cotangent bundles and symplectic groupoids

Definition of deformation to the normal cone for shifted Lagrangian morphisms

Abstract

This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.