Elements of Noncommutative Geometry

Gracia‐Bondía, José M.; Várilly, Joseph C.; Figueroa, Héctor

doi:10.1007/978-1-4612-0005-5

Cited by 651 publications

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“…while (6) By (15) we get e 11 − e 2 11 = e 22 − e 2 22 , so that (16) shows that e 12 e 21 = e 21 e 12 . We also see that e 12 and e 21 both commute with e 11 .…”

Section: Components Of the Chern Character And The Instanton Algebramentioning

confidence: 92%

“…Moreover, the real structure is given by the twisted involution J defined in (16). One checks using the results of [24] and [8] that Poincaré duality continues to hold for the deformed spectral triple.…”

Section: Isospectral Deformationsmentioning

confidence: 99%

“…The corner stone of the general theory is an operator theoretic index formula [6], [12], [16] which expresses the above index pairing (4) by explicit local cyclic cocycles on the algebra A. These local formulas become extremely simple in the special case where only the top component of the Chern character Ch(e) in cyclic homology fails to vanish.…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Connes

Landi

2001

Commun. Math. Phys.

291

523

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We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of R n . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.The new examples include the instanton algebra and the NC-4-spheres S 4 θ . We construct the noncommutative algebras A = C ∞ (S 4 θ ) of functions on NCspheres as solutions to the vanishing, ch j (e) = 0 , j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M 4 (A) , e 2 = e , e = e * . We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere intimately related to quantum group deformations SU q (2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of q-analogues, q = exp(2πiθ).We then construct the noncommutative geometry of S 4 θ as given by a spectral triple (A, H, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g µν on S 4 whose volume form √ g d 4 x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,where is the projection on the commutant of 4 × 4 matrices. We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S 4 θ so that the previous equation continues to hold without any change. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to noncommutative geometries.

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“…while (6) By (15) we get e 11 − e 2 11 = e 22 − e 2 22 , so that (16) shows that e 12 e 21 = e 21 e 12 . We also see that e 12 and e 21 both commute with e 11 .…”

Section: Components Of the Chern Character And The Instanton Algebramentioning

confidence: 92%

Section: Isospectral Deformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Connes

Landi

2001

Commun. Math. Phys.

291

523

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“…It was shown in [15] that the datum (L θ (C ∞ (M)), H, D) satisfies all properties of a noncommutative spin geometry [13] (see also [22]); there is also a grading γ (for the even case) and a real structure J. In particular, boundedness of the commutators [D,…”

Section: Deforming a Torus Actionmentioning

confidence: 99%

“…, by using the induced dual representation ρ ′ on the dual vector space W ′ given by 22) we have that (4.24) and write T = T ij ⊗ e ij with respect to the basis {e ij } of L(W ) induced from the basis…”

Section: Associated Bundlesmentioning

confidence: 99%

Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere S 4 θ and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on S 4 θ . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the "tangent space" to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.

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Noncommutative geometry and fundamental interactions: the first ten years

Gracia‐Bondía

2002

Ann. Phys.

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This is the full text of a survey talk for nonspecialists, delivered at the 66th Annual Meeting of the German Physical Society in Leipzig, March 2002. We have not taken pains to suppress the colloquial style. References are given only insofar as they help to underline the points made; this is not a full-blooded survey. The connection between noncommutative field theory and string theory is mentioned, but deemphasized. Contributions to noncommutative geometry made in Germany are emphasized. their primitive form in the early eighties. Also pioneer work by Rieffel [6] contributed to shape noncommutative geometry in its early stages.Noncommutative geometry, in essence, is an operator algebraic, variational reformulation of the foundations of geometry, extending to noncommutative spaces. NCG allows consideration of "singular spaces", erasing the distinction between the continuous and the discrete. Its main specific tools are Dirac operators, Fredholm modules, the noncommutative integral, Hochschild and cyclic cohomology of algebras, C * -modules and Hopf algebras. NCG has many affinities with quantum field theory, and it is not unusual for noncommutative homological constructs to crop up in that context [7].On the mathematical side, NCG has had a vigorous development. Current topics of interest include index theory and groupoids, mathematical quantization, the Novikov and Baum-Connes conjectures and the relation of the latter to the Langlands program, locally compact quantum groups, the dressing of fermion propagators in the framework of spectral triples, and the famous Riemann hypothesis. In the mainstream of mathematics, the noncommutative program needs no tribute. It is here to stay.

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Elements of Noncommutative Geometry

Cited by 651 publications

References 0 publications

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Noncommutative Instantons from Twisted Conformal Symmetries

Noncommutative geometry and fundamental interactions: the first ten years

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