Lecture notes on the symplectic geometry of graded manifolds and higher Lie groupoids (2510.09448v1)

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.

• The paper presents a rigorous extension of symplectic geometry to higher Lie groupoids by defining m-shifted symplectic structures with non-degenerate tangent complexes.
• It employs advanced differential-geometric tools to establish the non-degeneracy and Morita invariance of these structures, linking Poisson geometry with higher categories.
• Applications discussed include the integration of Poisson manifolds, analysis of classifying spaces, and connections to topological field theories via the AKSZ formalism.

Introduction to Shifted Symplectic Structures on Higher Lie Groupoids

The paper of symplectic geometry in the field of Lie groupoids has expanded significantly with the introduction of shifted symplectic structures. Traditionally, symplectic geometry is well-understood for manifolds, where a symplectic form is a closed, non-degenerate 2-form. However, for higher Lie groupoids and stacks, the concept of symplectic forms requires modification to accommodate additional structure, leading to the notion of shifted symplectic structures.

Shifted symplectic structures are particularly motivated by the need to provide both a differential-geometric treatment of structures found in derived algebraic geometry and to extend the reach of symplectic geometry into the domain of higher categorical structures. This framework is crucial for understanding the intersection of Poisson geometry, higher categories, and applications in mathematical physics, including topological field theories.

Tangent Complex of Lie n-Groupoids

For a Lie $n$-groupoid $K_\bullet$, the tangent complex $(T^K_\bullet, \partial)$ provides a cochain complex of vector bundles over $K_0$. It captures the infinitesimal structure associated with the groupoid's vertices and morphisms, extending the concept of tangent bundles from manifolds to the context of higher groupoids. This complex is integral to defining shifted symplectic structures, as it acts as the domain for the induced pairing from the shifted symplectic form.

Definition of Shifted Symplectic Structures

An $m$-shifted symplectic structure on a Lie $n$-groupoid $K_\bullet$ is given by a closed $m$-shifted 2-form $\omega_\bullet$ which is non-degenerate at the level of the tangent complex. The non-degeneracy condition ensures that the induced pairing

$$(\lambda^{\omega_m})^\flat:(T^K_\bullet,\partial)\to (T^{*K}[m],\partial^*)$$

is a quasi-isomorphism. This notion extends symplectic geometry's core ideas to a broader, higher-dimensional category.

Morita Invariance and Symplectic Equivalence

An important feature of shifted symplectic structures is their Morita invariance. Theorems such as the Morita invariance theorem allow one to compare and relate different presentations of a symplectic stack by verifying that their shifted symplectic structures agree up to exact forms correlated by hypercovers. This property asserts the ability to work within different groupoid representatives of a single stack.

Examples and Applications

• Symplectic Groupoids and Beyond: A key example includes the integration of Poisson manifolds to symplectic groupoids, where shifted symplectic structures provide a geometrical framework that accommodates symmetries and deformation quantization.
• AMM Groupoid: Originally constructed to model group-valued moment maps, this groupoid exemplifies the role of shifted symplectic structures in understanding equivalences between moduli of flat connections and quasi-Hamiltonian spaces.
• Double Groupoids: This structure exemplifies a powerful case in which double groupoids, once integrated into a Lie 2-groupoid, provide a natural setting for shifted symplectic structures that respect both vertical and horizontal compositions.
• Classifying Spaces: The 2-shifted symplectic structures found on the classifying spaces of Lie groups link to broader topological features such as Pontryagin classes, emphasizing the deep intersection between symplectic geometry and topological invariants.

Connection with Topological Field Theories

Shifted symplectic structures extend to applications in topological field theories via the AKSZ formalism, where symplectic $Q$-manifolds underpin the classical BV formality. This connection highlights the centrality of shifted symplectic structures in linking geometry with quantum field theoretical constructs, offering a symplectic lens to mirror geometrical actions and quantization procedure.

In summary, the framework of shifted symplectic structures enriches the landscape of symplectic geometry by incorporating higher categorical and algebraic structures, opening avenues for exploration in mathematical physics and beyond. This development not only provides a deeper understanding of symplectic geometry's foundational principles but also aligns with advancements in the paper of derived stacks and higher symplectic categories.