Poisson Geometry, Deformation Quantisation and Group Representations (original) (raw)
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...
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.
From local to global deformation quantization of Poisson manifolds
Duke Mathematical Journal, 2002
Dedicated to James Stasheff on the occasion of his 65th birthday ABSTRACT. We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.
SG ] 2 9 A pr 2 01 9 Group Actions in Deformation Quantisation
2019
This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -with intersectionthe introduction to formal deformation quantization and group actions published in [38], corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are ...
Deformation quantization and the action of Poisson vector fields
Lobachevskii Journal of Mathematics, 2017
As one knows, for every Poisson manifold M there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra g act by derivations on the functions on M. The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of g is 1, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of g is greater than 1, the previous methods do not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension.
Group Actions in Deformation Quantisation
arXiv (Cornell University), 2019
This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements-with intersection-the introduction to formal deformation quantization and group actions published in [38], corresponding to a course given in Villa de Leyva in July 2015. After an introduction to the concept of deformation quantization, we briefly recall existence, classification and representation results for formal star products. We come then to results concerning the notion of formal star products with symmetries; one has a Lie group action (or a Lie algebra action) compatible with the Poisson structure, and one wants to consider star products such that the Lie group acts by automorphisms (or the Lie algebra acts by derivations). We recall in particular the link between left invariant star products on Lie groups and Drinfeld twists, and the notion of universal deformation formulas. Classically, symmetries are particularly interesting when they are implemented by a moment map and we give indications to build a corresponding quantum moment map. Reduction is a construction in classical mechanics with symmetries which allows to reduce the dimension of the manifold; we describe one of the various quantum analogues which have been considered in the framework of formal deformation quantization. We end up by some considerations about convergence of star products.
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.
Star products and quantization of poisson-lie groups
Journal of Geometry and Physics, 1992
We prove some theorems by Drinfeld about solutions of the triangular quantum Yang-Baxter equation and corresponding quantum groups. These theorems are to be understood in the natural setting of invariant star products on a Lie group. We also set out and prove another theorem about the invariant Hochschild cohomological meaning of the quantum Yang-Baxter equation, which underlies the others.
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,
This paper extends Kontsevich's ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds.
Deformation Quantisation via Kontsevich Formality Theorem
arXiv (Cornell University), 2022
This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal product as the first example. Then we develop the deformation theory via differential graded Lie algebras and L ∞-algebras, which allows us to reformulate the classification of deformation quantisation as the existence of a L ∞-quasi-isomorphism between two differential graded Lie algebras, known as the formality theorem. Next we present Kontsevich's proof of the formality theorem in R d and his construction of the star product. We conclude with a brief discussion of the globalisation of Kontsevich star product on Poisson manifolds. Acknowledgement I am grateful to my supervisor Prof. Christopher Beem for his guidance in the project. I would like to thank Haiqi Wu, Shuwei Wang, and Dekun Song for the constructive discussions. I would also like to thank Shuwei Wang for his kind assistance in L A T E X related problems, without whom my dissertation would not be presented in such a clean manner. Finally, I would like to express my sincere appreciation to my college tutors, Prof. Balázs Szendrői and Prof. Lionel Mason, who have provided invaluable support for my academic development along my undergraduate life. Declaration of Authorship I hereby declare that the dissertation I am submitting is entirely my own work except where otherwise indicated. It has not been submitted, either wholly or substantially, for another Honour School or degree of this University, or for a degree at any other institution. I have clearly signalled the presence of quoted or paraphrased material and referenced all sources. I have not copied from the work of any other candidate.
Kontsevich's Universal Formula for Deformation Quantization and the Campbell–Baker–Hausdorff Formula
International Journal of Mathematics, 2000
We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.
Deformation Quantisation and Connections
European Women in Mathematics - Proceedings of the 13th General Meeting, 2010
After a brief introduction to the concept of formal Deformation Quantisation, we shall focus on general konwn constructions of star products, enhancing links with linear connections. We first consider the symplectic context: we recall how any natural star product on a symplectic manifold determines a unique symplectic connection and we recall Fedosov's construction which yields a star product, given a symplectic connection. In the more general context, we consider universal star products, which are defined by bidifferential operators expressed by universal formulas for any choice of a linear torsionfree connection and of a Poisson structure. We recall how formality implies the existence (and classification) of star products on a Poisson manifold. We present Kontsevich formality on R d and we recall how Cattaneo-Felder-Tomassini globalisation of this result proves the existence of a universal star product.
Deformation Quantization : an introduction
2005
I shall focus, in this presentation of Deformation Quantization, to some mathematical aspects of the theory of star products. 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. We will then introduce the notion of formality and its link with star products, give a flavour of Kontsvich’s construction of a formality for R and a sketch of the globalisation of a star product on a Poisson manifold following the approach of Cattaneo, Felder and Tomassini. The last lectures will be devoted to the action of a Lie group on a deformed product. The notes here are a brief summary of those lectures; I start with a Further Reading section which includes expository papers with details of what is presented. I shall not mention many very important aspects of the deformation quantisation programme such as reduction p...
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.
Relational symplectic groupoid quantization for constant poisson structures
Letters in Mathematical Physics
As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization constructed in this paper induces Kontsevich's deformation quantization on the underlying Poisson manifold, i.e., the Moyal product, which is known in full details. This allows focussing on the BV-BFV technology and testing it. For the unexperienced reader, this is also a practical and reasonably simple way to learn it.
A Path Integral Approach¶to the Kontsevich Quantization Formula
Communications in Mathematical Physics, 2000
We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin-Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. 1 2. The Kontsevich formula In [K], M. Kontsevich wrote a beautiful explicit solution to the problem of deformation quantization of the algebra of functions on a Poisson manifold M. The problem is to find a deformation of the product on the algebra of smooth functions on a Poisson manifold, which to first order in Planck's constant is given by the Poisson bracket. If M is an open set in R d with a Poisson structure {f, g} given by a skew-symmetric bivector field α, obeying the Jacobi identity α il ∂ l α jk + α jl ∂ l α ki + α kl ∂ l α ij = 0, (1)