Examples of Derivation-Based Differential Calculi Related to Noncommutative Gauge Theories

Masson, Thierry

doi:10.1142/s021988780800334x

Cited by 11 publications

(15 citation statements)

References 45 publications

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“…In [4] a general and purely algebraic framework to associate to A a differential calculus was introduced. This approach was named the derivation-based differential calculus and has been further developed in [5,6], see also the reviews [7,8,9].…”

Section: Preliminaries On Derivation-based Differential Calculimentioning

confidence: 99%

“…There are examples showing that there is also a gauge copy problem in noncommutative geometry, see e.g. Proposition 3 in this paper, where we review some results of [10,7,8,9]. This means that considering only observables which are constructed from the curvature, we can in general not extract the complete gauge invariant information of the connection.…”

Section: Introductionmentioning

confidence: 99%

“…This description requires the notion of a differential calculus on A, generalizing differential forms and the exterior differential to the noncommutative setting. Depending on the algebra A, there are different choices for a calculus at hand, for example the twist deformed differential calculus [2] in deformation quantization using Drinfeld twists, the bicovariant differential calculus for quantum groups [3] and the derivation-based differential calculus [4,5,6,7,8,9], being particularly useful for finite matrix algebras [10], such as the fuzzy sphere [11].…”

Section: Introductionmentioning

confidence: 99%

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Module parallel transports in fuzzy gauge theory

Schenkel

2014

Int. J. Geom. Methods Mod. Phys.

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In this article we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

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Section: Preliminaries On Derivation-based Differential Calculimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Module parallel transports in fuzzy gauge theory

Schenkel

2014

Int. J. Geom. Methods Mod. Phys.

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“…The statement and proof require the detailed theory of roots and weights for semisimple Lie groups and their representations, and we prefer to state our main result (Theorem 4.1) after we have established our notation and conventions for this detailed theory. The particular case in which G = SU (n) and π is the defining representation of G on C n was treated earlier in [11,10,18].…”

Section: Introductionmentioning

confidence: 99%

Cotangent bundles for “matrix algebras converge to the sphere”

Rieffel

2020

Expositiones Mathematicae

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In the high-energy quantum-physics literature one finds statements such as "matrix algebras converge to the sphere". Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. More recently I have dealt with corresponding statements in the literature about vector bundles on spheres and matrix algebras. But physicists want, even more, to treat structures on spheres (and other spaces) such as Dirac operators, Yang-Mills functionals, etc., and they want to approximate these by corresponding structures on matrix algebras. In preparation for understanding what the Dirac operators should be we determine here what the corresponding "cotangent bundles" should be for the matrix algebras, since it is on them that a "Riemannian metric" must be defined, which is then the information needed to determine a Dirac operator. (In the physics literature there are at least 3 inequivalent suggestions for the Dirac operators.)2000 Mathematics Subject Classification. Primary 53C30; Secondary 46L87, 58J60, 53C05.

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“…where the derivations are defined by (1.16) (see [5,36,46]). We leave the remaining cases to a further work.…”

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confidence: 99%

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

Marmo

Vitale

Zampini

2019

Journal of Geometry and Physics

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Examples of Derivation-Based Differential Calculi Related to Noncommutative Gauge Theories

Abstract: Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.

Cited by 11 publications

References 45 publications

Module parallel transports in fuzzy gauge theory

Module parallel transports in fuzzy gauge theory

Cotangent bundles for “matrix algebras converge to the sphere”

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

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