Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Abstract:

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝⁿ. 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 ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2).

We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, 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 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 _θ ⁴ 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.