Deformation quantization of surjective submersions and principal fibre bundles

Bordemann, Martin; Neumaier, Nikolai; Waldmann, Stefan; Weiss, Stefan

doi:10.1515/crelle.2010.009

Cited by 14 publications

(26 citation statements)

References 39 publications

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“…From this theorem and the previous considerations we obtain immediately the following result [5]: In particular, the deformation for the trivial bundle as in Example 4 is the unique one up to equivalence. Here the cohomological method is not sufficient even though in [5] rather explicit homotopies were constructed which allow to determine further properties of •. 2.…”

Section: Deformed Principal Bundlesmentioning

confidence: 57%

“…Now the obstruction lies in the first cohomology HH 1 diff (C ∞ (M ), Diffop(P ) G ). The following (nontrivial) theorem solves the problem of existence and uniqueness of deformation quantizations now in a trivial way [5]: Theorem 4.3. Let pr : P −→ M be a surjective submersion.…”

Section: Deformed Principal Bundlesmentioning

confidence: 99%

“…By a suitable partition of unity one can glue things together to end up with the global statement. For a detailed proof we refer to [5].…”

Section: Deformed Principal Bundlesmentioning

confidence: 99%

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Noncommutative Field Theories from a Deformation Point of View

Waldmann

2009

Quantum Field Theory

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In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fiber bundle and show how the deformation of associated vector bundles can be obtained. (2000). Primary 53D55; Secondary 58B34, 81T75. Mathematics Subject Classification

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Section: Deformed Principal Bundlesmentioning

confidence: 57%

Section: Deformed Principal Bundlesmentioning

confidence: 99%

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Noncommutative Field Theories from a Deformation Point of View

Waldmann

2009

Quantum Field Theory

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“…The third part is clear from general considerations on principal bundles [8]. For the last part we have to show that D is adjointable with respect to ·, · can .…”

Section: Complete Positivitymentioning

confidence: 99%

Involutions and representations for reduced quantum algebras

Gutt

Waldmann

2010

Advances in Mathematics

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In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space M red . We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M , with a moment map J for which zero is a regular value. For the quantization, we follow [6] (with a simplified approach) and build a star product ⋆ red on M red from a strongly invariant star product ⋆ on M . The new questions which are addressed in this paper concern the existence of natural * -involutions on the reduced quantum algebra and the representation theory for such a reduced * -algebra.We assume that ⋆ is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C = J −1 (0), with some equivariance property, defines a * -involution for ⋆ red on the reduced space. Looking into the question whether the corresponding * -involution is the complex conjugation (which is a * -involution in the Marsden-Weinstein context) yields a new notion of quantized unimodular class.We introduce a left (C ∞ (M ) [[λ]], ⋆)-submodule and a right (]-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C ∞ (M red ) [[λ]] and the finite rank operators on C ∞ cf (C) [[λ]]. The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly nondegenerate * -representations of (C ∞ (M red ) [[λ]], ⋆ red ) on pre-Hilbert right D-modules to those of (C ∞ (M ) [[λ]], ⋆) , for any auxiliary coefficient * -algebra D over [[λ]].

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“…where each c k is bidifferential, c 0 (F, Ψ) = F Ψ and the driver of • λ is c 1 (F, Ψ) = Λ # (dF, dΨ). Theorem 1.6 of [5] implies that such a deformation quantization of H exists and is unique up to equivalence. 8 Finally, we suppose that (X, ω), with dim X = 2n, is equipped with a polarization J.…”

Section: The Quantization Constructionmentioning

confidence: 99%