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

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.

v2: 53 pages. To appear in American Journal of Mathematics. Several typos have been corrected. We also made slightly more significant changes in Section 4 (we had to change the definition of a shifted symplectic groupoid) -- v1: 50 pages