Michel Dubois-Violette

Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples

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 S3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations of the standard 3-sphere S3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space ℝ4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed ℝ4u 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 can span the same . 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 . 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 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 detail. 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 as well as m-dimensional generalizations, noncommutative versions of spaces 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 -dimensional spherical manifolds . A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.

Noncommutative differential geometry of matrix algebras

The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C). A canonical symplectic structure is exhibited for Mn (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for Mn (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.

Noncommutative Differential Geometry and New Models of Gauge Theory

The noncommutative differential geometry of the algebra C∞(V)⊗Mn(C) of smooth Mn(C)‐valued functions on a manifold V is investigated. For n≥2, the analog of Maxwell’s theory is constructed and interpreted as a field theory on V. It describes a U(n)–Yang–Mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential. The Euclidean action, which is positive, vanishes on exactly two distinct gauge orbits, which are interpreted as two vacua of the theory. In one of the corresponding vacuum sectors, the SU(n) part of the Yang–Mills field is massive. For the case n=2, analogies with the standard model of electroweak theory are pointed out. Finally, a brief description is provided of what happens if one starts from the analog of a general Yang–Mills theory instead of Maxwell’s theory, which is a particular case.

Gauge bosons in a noncommutative geometry

A noncommutative extension of abelian gauge theory is proposed and is compared with standard nonabelian gauge theory.

Lectures on graded differential algebras and noncommutative geometry

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

Tensor fields of mixed Young symmetry type and N complexes

Abstract: We construct N-complexes of non-completely antisymmetric irreducible tensor fields on ℝD which generalize the usual complex (N=2) of differential forms. Although, for N≥ 3, the generalized cohomology of these N-complexes is nontrivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10].

Classical bosons in a non-commutative geometry

The classical theory of the scalar field is developed within the context of a noncommutative geometry. A non-commutative extension of Abelian gauge theory is proposed and is compared with standard non-Abelian gauge theory.

Generalized cohomology for irreducible tensor fields of mixed Young symmetry type

We construct N-complexes of noncompletely antisymmetric irreducible tensor fields on ℝD, thereby generalizing the usual complex (N=2) of differential forms. These complexes arise naturally in the description of higher spin gauge fields. Although, for N ⩾ 3, the generalized cohomology of these N-complexes is nontrivial, we give a generalization of the Poincaré lemma. Several results which have appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma.

The quantum group of a non-degenerate bilinear form

Abstract In complete analogy with the definition of the classical groups we introduce the quantum group which preserves a non-degenerate bilinear form. This is a Hopf algebra defined by a multiplicative matrix (i.e. a matrix quantum group) with an R matrix which we explicitly compute. If one restricts attention to the two-dimensional case, one recovers the known quantum deformations of SL(2).

General solution of the consistency equation

Abstract We produce the general solution of the Wess-Zumino consistency condition for gauge theories of the Yang-Mills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology of the BRS operator, modulo the differential on space-time. This in particular includes the presence of external fields and non-trivial topologies of space-time.