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Abstract

We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations S 3 u of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4 . For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. It follows that different S 3 u can span the same R 4 u . This equivalence generates a foliation of the parameter space Σ. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two 1-parameter families C± ⊂ Σ. Up to the simple operation of taking the fixed algebra by an involution, these two families are identical and we concentrate here on C+. For u ∈ C+ the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θ-deformations, a notion which we generalize in any dimension and various contexts, and study in some details. Here, and this point is crucial, the dimension is not an artifact, i.e. the dimension of the classical model, but is the Hochschild dimension of the corresponding algebra which remains constant during the deformation. Besides the standard noncommutative tori, examples of θ-deformations include the recently defined noncommutative 4-sphere S 4 θ as well as m-dimensional generalizations, noncommutative versions of spaces R m and quantum groups which are deformations of various classical groups. We develop general tools such as the twisting of the Clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative 2n-dimensional spherical manifolds S 2n θ . A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.

Families of Noncommutative Instantons

“…There are corresponding generators E r together with two mutually commuting generators H 1 , H 2 of the Cartan subalgebra. The universal enveloping algebra U(so (5)) is the algebra generated by elements {H j , E r } modulo relations given by the Lie brackets of so (5). The twisted universal enveloping algebra U θ (so (5)) is generated as above (i.e.…”

“…we take the relations of U(so(5, 1)), as we did for U θ (so(5)). We thus define U θ (so(5, 1)) as the algebra U θ (so (5)) with five extra generators adjoined, H 0 , G r , r = (±1, 0), (0, ±1), subject to the relations coming from the Lie brackets of so (5,1). Although the algebra structure is unchanged, again the Hopf algebra structure of U θ (so(5, 1)) gets twisted.…”

A FIELD-THEORETIC APPROACH TO CONNES' GAUGE THEORY ON M4 × Z2

“…In §3 we review two possible extensions 9), 10) of the ordinary differential geometry and show that they precisely lead to the two different field strengths. We shall derive in §4 Asquish's representation 11) of Connes' color-flavor algebra 3) of the standard model using the double sum prescription 12) and discuss its consequence regarding the electric charge quantization in the presence or absence of ν R . The final section is devoted to discussion.…”

Noncommutative instantons come in families

“…At the smooth level this is not problematic. The algebra C ∞ ( S 4 θ ) is defined as a fixed point algebra [7] and one finds that the spectrum of ρ 2 is positive and does not contain the point 0.…”

“…Toric noncommutative manifolds -introduced in [8] (with further elaborations in [7]) and called isospectral deformations -are deformations of classical Riemannian manifolds M satisfying all the properties of noncommutative spin geometries. The main idea consists in deforming the spectral triple of the Riemannian geometry of M along a torus embedded in the isometry group.…”