A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl-Wigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative Yang-Mills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an in-depth study of the gauge group of noncommutative Yang-Mills theory. Some of the more mathematical ideas and

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×— × = D −1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita’s involution J. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.

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We exhibit large classes of examples of noncommutative finitedimensional 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 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 are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θ-deformations, which we generalize in any dimension and various contexts, and study in some details. Here, and

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Non-commutative heat kernel

Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the Connes-Lott action [15] previously computed by Gayral [23] for symplectic Θ.

Abstract. Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 (G). Following Connes, Mℓ can be used as a “Dirac ” operator for C ∗ r (G). It defines a Lipschitz seminorm on C ∗ r (G), which defines a metric on the state space of C ∗ r (G). We investigate whether the topology from this metric coincides with the weak- ∗ topology (our definition of a “compact quantum metric space”). We give an affirmative answer for G = Zd when ℓ is a word-length, or the restriction to Zd of a norm on Rd. This works for C ∗ r (G) twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes ’ cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays. The group C ∗-algebras of discrete groups provide a much-studied class of “compact non-commutative spaces ” (that is, unital C ∗-algebras). In [11] Connes showed that the “Dirac ” operator of an unbounded

Abstract. We show that allowing the metric dimension of a space to be independent of its KO-dimension and turning the finite noncommutative geometry F – whose product with classical 4-dimensional space-time gives the standard model coupled with gravity–into a space of KO-dimension 6 by changing the grading on the antiparticle sector into its opposite, allows to solve three problems of the previous noncommutative geometry interpretation of the standard model of particle physics: The finite geometry F is no longer put in “by hand ” but a conceptual understanding of its structure and a classification of its metrics is given. The fermion doubling problem in the fermionic part of the action is resolved. The spectral action of our joint work with Chamseddine now automatically generates the full standard model coupled with gravity with neutrino mixing and see-saw mechanism for neutrino masses. The predictions of the Weinberg angle and the Higgs scattering parameter at unification scale are the same as in our joint work but we also find a mass relation (to be imposed at unification scale). 1.