Renormalizability conditions for almost-commutative geometries

We formulate conditions for almost-commutative (spacetime) manifolds under which the asymptotically expanded spectral action is renormalizable. These conditions are of a graph-theoretical nature, involving the Krajewski diagrams that classify such geometries. This applies in particular to the Standard Model of particle physics, giving a graph-theoretical argument for its renormalizability. A promising potential application is in the selection of physical (renormalizable) field theories described by almost-commutative geometries, thereby going beyond the Standard Model.Comment: 7 pages. To appear in Phys. Lett.

Renormalization of the spectral action for the Yang-Mills system

We establish renormalizability of the full spectral action for the Yang-Mills system on a flat 4-dimensional background manifold. Interpreting the spectral action as a higher-derivative gauge theory, we find that it behaves unexpectedly well as far as renormalization is concerned. Namely, a power counting argument implies that the spectral action is superrenormalizable. From BRST-invariance of the one-loop effective action, we conclude that it is actually renormalizable as a gauge theory.Comment: 6 pages; 4 figures; minor correction

Renormalization of gauge fields: A Hopf algebra approach

We study the Connes-Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Ward identities and the Slavnov-Taylor identities (in the abelian and non-abelian case respectively) are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a purely combinatorial and rigorous proof of compatibility of the Slavnov-Taylor identities with renormalization.Comment: 24 pages; uses feynm

Noncommutative Bundles and Instantons in Tehran

We present an introduction to the use of noncommutative geometry for gauge theories with emphasis on a construction of instantons for a class of four dimensional toric noncommutative manifolds. These instantons are solutions of self-duality equations and are critical points of an action functional. We explain the crucial role of twisted symmetries as well as methods from noncommutative index theorems.Comment: 64 pages; v2: major changes, parts rewritten, references added. Based on lectures delivered by GL at the ``International Workshop on Noncommutative Geometry NCG2005'', Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, September 200

Principal fibrations from noncommutative spheres

We construct noncommutative principal fibrations S_\theta^7 \to S_\theta^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion A(S_\theta^4) \into A(S_\theta^7) is an example of a not trivial quantum principal bundle.Comment: 23 pages. Latex. v3: Additional minor corrections, version published in CM