Symplectic Groupoids

• Symplectic groupoids are Lie groupoids with a multiplicative symplectic form that provide a global geometric integration of Poisson manifolds.
• They are constructed via cotangent path equivalences, yielding unique source-simply-connected integrations and local groupoid models for Poisson structures.
• Their structure underpins applications in discrete mechanics, quantization, and gauge theory through Lagrangian bisections and Poisson automorphisms.

A symplectic groupoid is a Lie groupoid equipped with a symplectic form subject to a "multiplicativity" condition, which encodes the integration of Poisson manifolds and provides the global geometric framework for Poisson geometry. Symplectic groupoids are central objects in the study of the symplectic and Poisson categories, with deep connections to integrable systems, quantization, discrete mechanics, and the structure theory of Poisson and cluster varieties.

1. Definitions and Fundamental Properties

A symplectic groupoid consists of a Lie groupoid $G\rightrightarrows M$ with source and target maps $\alpha, \beta: G \to M$, multiplication $m: G^{(2)} \to G$, unit embedding $u: M \hookrightarrow G$, inversion $i: G \to G$, and a symplectic form $\Omega \in \Omega^2(G)$ such that:

• The multiplicativity condition holds:

$$m^*\Omega = \mathrm{pr}_1^*\Omega + \mathrm{pr}_2^*\Omega \quad \text{on}\quad G^{(2)} \subset G \times G$$

Equivalently, the graph of the multiplication is a Lagrangian submanifold of $$(G \times G \times G,\, \Omega \oplus \Omega \oplus -\Omega)$$ (Cosserat, 2022, Marle, 2014).

• The unit manifold $u(M) \subset G$ is a Lagrangian submanifold.
• The inversion $i$ is an anti-symplectomorphism: $i^*\Omega = -\Omega$.

The base $(M, \pi)$ acquires a unique Poisson structure such that $\alpha: (G, \Omega) \to (M, \pi)$ is a Poisson map and $\beta$ is anti-Poisson. The Lie algebroid of $G$ can be identified with $(T^*M, \pi^\sharp)$, the Koszul bracket integrating to the groupoid bracket structure. At the units, the identification

$$t_*\bigl(\Omega^{-1}\bigr) = \pi,\quad s_*\bigl(\Omega^{-1}\bigr) = -\pi$$

holds (Marle, 2014, Cosmo et al., 2023). The symplectic orthogonals of the source and target fibers coincide. These structures lead to the interpretation of symplectic groupoids as global symplectic realizations of Poisson manifolds.

2. Integrability and Constructions

Given any integrable Poisson manifold $(M, \pi)$, there exists a (unique up to isomorphism) source-simply-connected symplectic groupoid integrating it (Marle, 2014). The construction proceeds by considering equivalence classes of cotangent paths in $T^*M$ up to $T^*M$-homotopy, generalizing the framework of Lie group integration (Contreras et al., 2012, Cosmo et al., 2023).

For arbitrary Poisson manifolds, a local symplectic groupoid always exists in a neighborhood of the zero section of $T^*M$, with the canonical symplectic form and appropriately deformed source and target maps (Cosserat, 2022, Kupriyanov et al., 2023). However, global integrability may be obstructed by monodromy phenomena.

There exist parallel infinite-dimensional constructions, such as the relational symplectic groupoid, formulated in the extended symplectic category with objects as (possibly infinite-dimensional) weak symplectic manifolds and morphisms as immersed canonical/Lagrangian relations (Cattaneo et al., 2014, Contreras, 2013). This provides a universal integration of any Poisson manifold, even in non-integrable cases.

3. Lagrangian Bisections, Automorphisms, and Discrete Dynamics

Bisections of a symplectic groupoid are submanifolds $L \subset G$ such that both $\alpha|_L$ and $\beta|_L$ are diffeomorphisms, and $L$ is Lagrangian with respect to $\Omega$. Each Lagrangian bisection induces a Poisson automorphism of the base that preserves symplectic leaves, implemented as $\phi_L = \beta \circ (\alpha|_L)^{-1}$.

Families of Lagrangian bisections correspond bijectively to families of closed 1-forms whose Hamiltonian vector fields are complete. If these forms are exact, the associated functions are called variation functions (Cosserat, 2022, Kupriyanov et al., 2023). This relation is central to Poisson integrators, where formal Taylor expansions lead to recursion relations for generating functions, and the Magnus expansion provides backward-error analysis of geometric integrators (Cosserat, 2022).

Discrete Lagrangian mechanics on symplectic groupoids is formulated via Lagrangian submanifolds, which give rise to implicit discrete dynamical systems. The regularity of such systems, reduction results, and Noether symmetries are expressible in the groupoid formalism, using generating functions in the sense of Śniatycki and Tulczyjew (Marrero et al., 2011).

4. Advanced Constructions: Double Groupoids, Hopfoids, and Deformation Theory

Symplectic double groupoids are symplectic manifolds with two compatible groupoid structures, such that both are symplectic groupoids in the sense above. Their structure is fully classified in terms of "symplectic hopfoids": groupoid-like objects in the symplectic category whose morphisms are canonical (Lagrangian) relations. There is a precise one-to-one correspondence between symplectic double groupoids and symplectic hopfoids (Canez, 2017, Canez, 2011). The core of a symplectic double groupoid is a symplectic quotient, which itself inherits a symplectic groupoid structure.

Deformation theory of symplectic groupoids is controlled by a mapping-cone complex integrating both the groupoid deformation and the deformation of its multiplicative symplectic form, with effective computational models given by the Bott–Shulman–Stasheff double complex. The Moser theorem has a direct symplectic groupoid analogue: first-order triviality classes in deformation cohomology correspond to trivializations by groupoid automorphisms and symplectic isotopies (Cárdenas et al., 2021).

5. Examples and Applications

Symplectic groupoids occur in a broad spectrum of contexts:

• Pair groupoids of symplectic manifolds: $M \times M \rightrightarrows M$ with $$\omega = -\mathrm{pr}_1^*\omega_0 + \mathrm{pr}_2^*\omega_0$$ integrates $(M,\omega_0^{-1})$ (Marle, 2014).
• Cotangent bundles of Lie groups: $T^*G \rightrightarrows \mathfrak{g}^*$ with the canonical symplectic form integrates the dual Poisson Lie group (Marle, 2014, Kupriyanov et al., 2023).
• Cluster varieties: Global and local symplectic groupoids integrating log-canonical Poisson structures on cluster varieties are constructed by gluing local groupoid charts related via Hamiltonian lifts of cluster mutations (Li et al., 2018).
• Log symplectic and Dirac manifolds: Hausdorff symplectic groupoids can be constructed via blow-ups and gluing, with explicit combinatorial classification in terms of the fundamental group data of the base and symplectic leaves (Gualtieri et al., 2012, Mehta et al., 2013).
• Double Bruhat cells: Symplectic leaves in double Bruhat cells of semisimple Lie groups are realized as symplectic groupoids, and possess actions by Poisson and symplectic groupoids associated to Weyl group elements (Lu et al., 2016).
• Poisson sigma models and quantization: Symplectic groupoids arise as reduced phase spaces of Poisson sigma models with boundary, providing a geometric bridge between Poisson geometry and quantum theory (Contreras et al., 2012, Bonechi et al., 2010).

6. Symplectic Groupoids in Poisson Geometry and Gauge Theory

Symplectic groupoids serve as the global objects integrating Poisson manifolds, in analogy with Lie group integration of Lie algebras (Marle, 2014, Kupriyanov et al., 2023). They play a central role in Poisson geometry, quantization, and integrable systems. The construction of a symplectic groupoid provides a symplectic realization of the base Poisson manifold and often governs the properties of associated deformation quantizations and representation theories.

In Poisson electrodynamics and noncommutative geometry, symplectic groupoids provide the classical phase space for point particles in noncommutative spacetime. Bisections of the symplectic groupoid correspond to gauge fields, and Lagrangian bisections correspond to gauge transformations. The action of the groupoid and its dual foliations encode classical duality symmetries and the geometry of physical fields (Cosmo et al., 2023, Kupriyanov et al., 2023).

7. Relational Extensions and Higher Structures

Relational symplectic groupoids generalize the notion of a symplectic groupoid to the extended symplectic category, defining structures via canonical relations instead of maps, enabling integration of every Poisson manifold independent of integrability (Cattaneo et al., 2014, Contreras, 2013). In the integrable case, the reduction of these infinite-dimensional groupoids recovers the usual finite-dimensional symplectic groupoids.

Higher categorical analogues, such as symplectic 2-groupoids integrating exact Courant algebroids, extend these ideas further and yield presymplectic groupoids for Dirac structures as reductions of infinite-dimensional symplectic 2-groupoids (Mehta et al., 2013).

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Key References:

• (Cosserat, 2022) Symplectic Groupoids for Poisson Integrators
• (Marle, 2014) Lie, symplectic and Poisson groupoids and their Lie algebroids
• (Cosmo et al., 2023) Symplectic realizations and Lie groupoids in Poisson Electrodynamics
• (Li et al., 2018) Symplectic groupoids for cluster manifolds
• (Canez, 2017) Double Groupoids and the Symplectic Category
• (Cattaneo et al., 2014) Relational symplectic groupoids
• (Contreras et al., 2012) Groupoids and Poisson sigma models with boundary
• (Marrero et al., 2011) Symplectic groupoids and discrete constrained Lagrangian mechanics
• (Cárdenas et al., 2021) Deformations of symplectic groupoids
• (Lu et al., 2016) Double Bruhat cells and symplectic groupoids
• (Contreras, 2013) Relational symplectic groupoids and Poisson sigma models with boundary
• (Cosmo et al., 2023) Symplectic realizations and Lie groupoids in Poisson Electrodynamics
• (Bonechi et al., 2010) The quantization of the symplectic groupoid of the standard Podles sphere
• (Gualtieri et al., 2012) Symplectic groupoids of log symplectic manifolds
• (Álvarez, 2020) Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds
• (Mehta et al., 2013) Symplectic structures on the integration of exact Courant algebroids