Classification of Equivariant Star Products on Symplectic Manifolds

Reichert, T.; Waldmann, Stefan

doi:10.1007/s11005-016-0834-x

Cited by 13 publications

(23 citation statements)

References 21 publications

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“…Next, we assume to have a star product invariant under the action of G which admits a deformation J of J into a quantum momentum map. In the symplectic case such star products always exist since we assume the action of G to be proper, see [29] for a complete classification and further references. In the general Poisson case the situation is less clear.…”

Section: Example I: Brst Reductionmentioning

confidence: 99%

Deformation and Hochschild cohomology of coisotropic algebras

Dippell

Esposito

Waldmann

2021

Annali di Matematica

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Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

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Section: Example I: Brst Reductionmentioning

confidence: 99%

Deformation and Hochschild cohomology of coisotropic algebras

Dippell

Esposito

Waldmann

2021

Annali di Matematica

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“…Both c and c g are bijections. One can even give explicit expressions of both characteristic classes for the case of (equivariant) Fedosov star products (see [13]), which are essentially all (equivariant) star products (by [3], [32]). Strictly speaking, the Fedosov construction maps pairs of a torsion-free, symplectic connection ∇ and a formal series of closed two-forms Ω ∈ νZ 2 (M ) ν to star products.…”

Section: Characteristic Classes Of Reduced Star Productsmentioning

confidence: 99%

“…where H G,2 dR (M ) denotes the second invariant de Rham cohomology of M with respect to the action of G (note that, for noncompact G, this is different from the invariant part of the de Rham cohomology), ev 0 is induced by the evaluation at 0 ∈ g and i is induced by the inclusion of invariant differential forms into differential forms. Both are compatible with taking (equivariant, invariant) characteristic classes of star products, that is the following diagram commutes [32] Star g (M )…”

Section: Introductionmentioning

confidence: 96%

“…hold (where we denoted by X ξ the fundamental vector field of ξ). The pair ( , J) is then called an equivariant star product and the equivalence classes of equivariant star products where recently shown to be characterized by the second equivariant cohomology (in the Cartan model, see [7,18,22]) [32]. Star products on the Marsden-Weinstein reduced symplectic manifold, which we will throughout denote by M red , on the other hand, are classified by the second de Rham cohomology H 2 dR (M red ), see [3,10,14,19,30,34] for the symplectic and [25] for the more general Poisson case.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

Reichert

2016

Lett Math Phys

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In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.

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“…Again, the question is whether one also has a corresponding momentum map on the quantum side. Here one has several positive answers [98,116,117] including a complete classification in the symplectic case [131]. One important construction for classical mechanical systems with symmetry is the Marsden-Weinstein reduction [112] which allows to reduce the dimension by fixing the values of conserved quantities build out of the momentum map.…”

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confidence: 99%

Convergence of star products: From examples to a general framework

Waldmann

2019

EMS Surv. Math. Sci.

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We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincaré disc. * Awarded with the Prix du Concour annuel 2018: On demande une contribution à la construction d'un cadre convergent pour la quantification par déformation of the Académie royale des Sciences, des Lettres et des Beaux-Arts de Belgique.In the by now classical paper [3], Bayen, Frønsdal, Flato, Lichnerowicz, and Sternheimer introduced the notion of a formal star product on a Poisson manifold as a general method to pass from a classical mechanical system encoded by the Poisson manifold to its corresponding quantum system. The main difference compared to other, and more ad-hoc, quantization schemes is the emphasis on the role of the observable algebra. The observable algebra is constructed not as particular operators on a Hilbert space as this is usually done. Instead, one stays with the same vector space of smooth functions on the Poisson manifold and just changes the commutative pointwise product into a new noncommutative product using Planck's constant as a deformation parameter. This way one tries to model the quantum mechanical commutation relations.While in [3] it is shown that this leads to the same results as expected in quantum mechanics for those systems where alternative quantization schemes are available, the proposed scheme of deformation quantization has several conceptual advantages: the first is of course its vast generality concerning the formulation. While other quantization schemes make much more use of specific features of the classical system, deformation quantization can be seen as almost universal in the sense that its requirements are virtually minimal. Only a Poisson algebra of classical observables is needed to formulate the program of deformation quantization. Of course, the hard part of the work consists then in actually proving the existence (and possible classifications) of star products, but in any case, the conceptual framework is fixed from the beginning. A second advantage is that the physical interpretation of the observables is fixed from the beginning: the observables simply stay the same elements of the same underlying vector space. Thus it is immediately clear which quantum observable is the Hamiltonian, the momentum etc. since they are the same as classical. It is only the product law which changes, the correspondence of classical and quantum observables is implemented trivially. A third advantage of this approach to focus on the algebra first is shared also by other formulations of quantum theory, most notably by axiomatic quantum field theory, see e.g. [100]: having the focus on the...

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Classification of Equivariant Star Products on Symplectic Manifolds

Cited by 13 publications

References 21 publications

Deformation and Hochschild cohomology of coisotropic algebras

Deformation and Hochschild cohomology of coisotropic algebras

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

Convergence of star products: From examples to a general framework

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