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Noncommutative manifolds, the instanton algebra and isospectral deformations
| Venue: | Comm. Math. Phys |
| Citations: | 162 - 29 self |
BibTeX
@ARTICLE{Connes_noncommutativemanifolds,,
author = {Alain Connes and Giovanni Landi},
title = {Noncommutative manifolds, the instanton algebra and isospectral deformations},
journal = {Comm. Math. Phys},
year = {},
pages = {141--159}
}
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Abstract
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 S4 θ. We construct the noncommutative algebras A = C ∞ (S4 θ) of functions on NCspheres as solutions to the vanishing, chj(e) = 0,j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M4(A), e2 = 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 SUq(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 S4 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 gµν on S4 whose volume form √ g d4x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation, e − 1
Keyphrases
noncommutative manifold instanton algebra isospectral deformation new example nc-4-spheres s4 idempotent m4 noncommutative algebra corresponding dirac operator group deformation suq space gr universal noncommutative space following quartic equation riemannian metric noncommutative geometry basic consideration previous paper admissible morphisms moyal deformation complex value non-trivial global feature chern character noncommutative grassmanian spectral triple unusual value volume form d4x cyclic homology corresponding notion noncommutative differential topology
