Lecture notes on the symplectic geometry of graded manifolds and higher Lie groupoids
Miquel Cueca, Antonio Maglio, Fabricio Valencia
TL;DR
These lecture notes survey the symplectic geometry of graded manifolds and higher Lie groupoids, weaving together the infinitesimal (Q-manifolds, Lie algebroids, and their shifted analogues) and global (Lie n-groupoids and Morita invariance) perspectives. The approach emphasizes graded differential geometry, shifted symplectic structures, and Lagrangian subobjects, with explicit ties to Poisson geometry, Courant algebroids, and AKSZ field theories. Key contributions include graded tubular neighborhood results, the formulation of shifted Lagrangian structures and their relation to moment maps and symplectic reduction, and a clear map between infinitesimal data and higher groupoid structures. The notes blend foundational constructions with concrete examples (Poisson, Courant, and higher Courant algebroids) to illuminate applications in mathematics and physics, particularly in topological field theories via AKSZ models and BV formalism.
Abstract
In this work, we study symplectic structures on graded manifolds and their global counterparts, higher Lie groupoids. We begin by introducing the concept of graded manifold, starting with the degree 1 case, and translating key geometric structures into classical differential geometry terms. We then extend our discussion to the degree 2 case, presenting several illustrative examples with a particular emphasis on equivariant cohomology and Lie bialgebroids. Next, we define symplectic Q-manifolds and their Lagrangian Q-submanifolds, introducing a graded analogue of Weinstein's tubular neighborhood theorem and applying it to the study of deformations of these submanifolds. Shifting focus, we turn to higher Lie groupoids and the shifted symplectic structures introduced by Getzler. We examine their Morita invariance and provide several examples drawn from the literature. Finally, we introduce shifted Lagrangian structures and explore their connections to moment maps and symplectic reduction procedures. Throughout these notes, we illustrate the key constructions and results with concrete examples, highlighting their applications in mathematics and physics. These lecture notes are based on two mini-courses delivered by the first author at Geometry in Algebra and Algebra in Geometry VII (2023) in Belo Horizonte, Brazil, and at the INdAM Intensive Period: Poisson Geometry and Mathematical Physics (2024) in Napoli, Italy.