Poisson sigma models and symplectic groupoids

By Alberto S. Cattaneo, Giovanni Felder. 2000

241 citations recorded by the source corpus.

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7fba3f3e5e2d0b742baef55f19a2825042ccb95c

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Citations in the corpus

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model (2003, background): “One may show however ([18] for the closed and [5] for the open case) that C(M) is coisotropic iff π is a Poisson bivector field.”
The momentum map in Poisson geometry (2007, background): “Moreover it is a symplectic groupoid: £(Af) carries a natural symplectic 2-form ß e Q2(L(M)) ([4], [9]) which is multiplicative.”
Calculus on Lie algebroids, Lie groupoids and Poisson manifolds (2008, background): “That groupoid, when the Lie algebroid is the cotangent bundle to a Poisson manifold, is the same as that previously obtained by Cattaneo and Felder by another method.”
n-Groupoids and Stacky Groupoids (2008, background): “Other examples come from a kind of étale stacky groupoid (called a Weinstein groupoid) [36] built upon the very important work of [12, 13].”
Poisson geometry and Morita equivalence (2004, background): “∼ T∗P, where A(G) is the Lie algebroid of G, so the symplectic groupoid Gintegrates T∗P in the sense of Lie algebroids. It follows from (4.6) that dim(G) = 2dim(P). In the work of Cattaneo and Felder [22], symplectic groupoids arise as reduced phase spaces of Poisson sigma models. This means that on”
Linear and Multiplicative 2-Forms (2009, methodology): “[16]), in such a way that ω is obtained by a variation of the infinite dimensional reduction procedure of [5].”
Universal lifting theorem and quasi-Poisson groupoids (2005, methodology): “Such a characterization was first obtained by Cattaneo-Felder motivated by the Poisson sigma model when the Lie algebroid is the cotangent Lie algebroid associated to a Poisson manifold [4].”
Lectures on integrability of Lie brackets (2006, unknown): “A.S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, (ed.”
Integration of twisted Dirac brackets (2003, background): “In general, this quotient space may not be a manifold, but it is the same as the Weinstein groupoid introduced in [13] (see also [7]).”
Symplectic groupoids of log symplectic manifolds (2012, unknown): “Since that time, increasingly powerful general theories have been developed to address the question of existence of integrations of Lie algebroids, culminating in the work of Crainic and Fernandes [2, 3], who described the obstruction theory for the existence of integrations, using a general constru”
Integrability of Poisson Brackets (2002, background): “Weinstein groupoid of a Poisson manifold give the precise relationship between our integrability approach introducedin [10], which isvalid foranyLie algebroid, andthe approachofCattaneo and Felder in [5]. Corollary 2. If Σ(M) is smooth, then it admits a symplectic form which turns Σ(M) into a symple”