Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model (original) (raw)

Geometric quantization and non-perturbative Poisson sigma model

Advances in Theoretical and Mathematical Physics, 2006

In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class [α]. The same condition was considered before by Crainic and Zhu but in a different context. In the case when [α] is in the image of the sharp map we reproduce the Vaisman's condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show, with a different procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end we discuss the case of a generic coisotropic D-brane.

Poisson Sigma Models and Deformation Quantization

Modern Physics Letters A, 2001

This is a review aimed at the physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the noncommutativity of the string endpoint coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.

Classical covariant Poisson structures and Deformation Quantization

2014

Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through the causal Green functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to introduce a generalized version of Wick's theorem. Finally, we include a couple of examples to explicitly test our method: the real scalar field and the bosonic string. For both models we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra in the bosonic string case.

A ] 1 1 Ju n 20 07 Deformation Quantization and Reduction

2007

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞and A∞-algebras, and bimodule structures are recalled. As an application, an “almost” functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson–Lie groups.

Deformation quantization and reduction

Contemporary Mathematics, 2008

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞-and A∞-algebras, and bimodule structures are recalled. As an application, an "almost" functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson-Lie groups.

Deformation Quantization of Poisson manifolds

2005

I shall focus, in this presentation of Deformation Quantization, on the construction of star products on symplectic and Poisson manifolds. The first lectures will be about the general concept of Deformation Quantisation, with examples, with Fedosov’s construction of a star product on a symplectic manifold and with the classification of star products on a symplectic manifold. The next lectures will introduce the notion of formality and its link with star products, give a flavour of Kontsvich’s construction of a formality for Rd and a sketch of the globalisation of a star product on a Poisson manifold following the approach of Cattaneo, Felder and Tomassini. In the last lecture I shall only briefly mention different aspects of the deformation quantisation programme such as action of a Lie group on a deformed product, reduction procedures in deformation quantisation, states and representations in deformed algebras, convergence of deformations, leaving out many interesting and deep aspe...

Classical deformations, Poisson–Lie contractions, and quantization of dual Lie bialgebras

Journal of Mathematical Physics, 1995

A Poisson-Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincare, and Heisenberg groups by using a classical r-matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson-Lie quantization that deforms the standard Moyal-Weyl ah-product. 0 1995 American Institute of Physics.

Deformation quantisation of Poisson manifolds

2011

In the last lectures I only briefly mentioned different aspects of the deformation quantisation programme such as action of a Lie group on a deformed product, reduction procedures in deformation quantisation, states and representations in deformed algebras, convergence of deformations, leaving out many interesting and deep aspects of the theory (such as traces and index theorems, extension to fields theory); these are not included in these notes and I include a bibliography with many references to those topics.

Deformation quantization of Poisson algebras

Proceedings of the Japan Academy. Series A, Mathematical sciences, 1992

O. Introduction. Let M be a C Poisson manifOld, and C(M) the set .of all C-valued C functions on M. In what follows, we put a=C-(M) for simplicity. By definition of Poisson manifolds, there exists a bilinear map. { }'a a-a, called the Poisson bracket, with the following properties" For anyf, g, hea,