Noncommutative Rigidity

Hawkins, Eli

doi:10.1007/s00220-004-1036-4

Cited by 53 publications

(71 citation statements)

References 16 publications

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“…More generally, in order to formulate a gravity theory associated with a Lie algebroid A, we need the following materials: The first two objects in the list, 1 and 2, are well-known (see for example [23]), and the objects in 3-5 have already been studied in mathematical literature [17][18][19][20][21][22].…”

Section: Riemannian Geometry On Poisson Manifoldmentioning

confidence: 99%

“…In this section we study Riemannian geometry compatible with a Poisson structure, following [17][18][19][20][21]. Let M be a Riemannian manifold as well as a Poisson manifold with local coordinates {x i }, and G be a "Riemannian metric" on the cotangent bundle T * M , i.e.…”

Section: Contravariant Levi-civita Connectionmentioning

confidence: 99%

“…To show this, it is first required to formulate a "general relativity," which is invariant under β-diffeomorphisms on a Poisson manifold, because no well-accepted theory of gravity of such kind has been known at least in the physics literature. Following mathematical literature [17][18][19][20][21][22], we construct such a gravity based on the Lie algebroid (T * M ) θ of a Poisson manifold, by replacing the Levi-Civita connection with its contravariant analogue. We then extend this construction to that in PGG coupled to an R-flux by applying the same strategy as used in GG [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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Gravity theory on Poisson manifold with R‐flux

Asakawa

Muraki

Watamura

2015

Fortschritte der Physik

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A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi-Civita connection, which is based on the Lie algebroid of a Poisson manifold. Then, we show that in Poisson Generalized Geometry the R-fluxes are consistently coupled with such a gravity. An R-flux appears as a torsion of the corresponding connection in a similar way as an H-flux which appears as a torsion of the connection formulated in the standard Generalized Geometry. We give an analogue of the Einstein-Hilbert action coupled with an R-flux, and show that it is invariant under both β-diffeomorphisms and β-gauge transformations.

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Section: Riemannian Geometry On Poisson Manifoldmentioning

confidence: 99%

Section: Contravariant Levi-civita Connectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gravity theory on Poisson manifold with R‐flux

Asakawa

Muraki

Watamura

2015

Fortschritte der Physik

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“…This plays the role of the algebra of differential forms and is like specifying a differential structure on a space. The data for the differential structure at the semiclassical level was analysed in [3,10] by looking at a…”

Section: Introductionmentioning

confidence: 99%

Quantum Riemannian geometry of phase space and nonassociativity

Beggs

Majid

2017

Demonstratio Mathematica

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Noncommutative or 'quantum' differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical 'functorial' way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of CP n where the commutation relations have the canonical form [w i ,w j ] = iλδ ij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.

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“…Geometric quantization comes into this as a potential means of constructing noncommutative spaces (but see [13]). On a given manifold, the geometric quantization construction depends upon a parameterh (Planck's constant).…”

mentioning

confidence: 99%

Quantization of Multiply Connected Manifolds

Hawkins

2005

Commun. Math. Phys.

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ABSTRACT. The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C * -algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form.

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Noncommutative Rigidity

Cited by 53 publications

References 16 publications

Gravity theory on Poisson manifold with R‐flux

Gravity theory on Poisson manifold with R‐flux

Quantum Riemannian geometry of phase space and nonassociativity

Quantization of Multiply Connected Manifolds

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