Shifted Symplectic Geometry by Examples
Damien Calaque, Stefano Ronchi
TL;DR
The notes provide a bottom-up introduction to shifted symplectic geometry using derived geometry, starting from linear, two-term and one-term complexes and ending with mapping-stack constructions. They demonstrate how to encode nondegenerate shifted 2-forms via $\,\omega^\flat$ as quasi-isomorphisms, develop the theory of Lagrangian structures and correspondences in both linear and derived settings, and apply PTVV and AKSZ ideas to produce shifted symplectic structures on mapping stacks. Key contributions include explicit linear-algebra descriptions, derived-intersection techniques, and concrete descriptions of non-degeneracy and lagrangian data across affine schemes and stacks, with connections to quasi-symplectic groupoids, moment maps, and symplectic reductions in the derived arena. The framework offers a cohesive, intrinsic approach to constructing and comparing shifted symplectic structures, and it clarifies how classical Poisson/geometric constructions fit into the derived, higher-categorical paradigm. This can impact future work on moduli spaces, character varieties, and topological field theories by providing robust tools for deriving and gluing symplectic structures.
Abstract
These notes are intended to be an introduction to shifted symplectic geometry, targeted to Poisson geometers with a serious background in homological algebra. They are extracted from a mini-course given by the first author at the Poisson 2024 summer school that took place at the Accademia Pontaniana in Napoli.