Inner fluctuations of the spectral action

We prove in the general framework of noncommutative geometry that the inner fluctuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less or equal to four the obtained terms add up to a sum of a Yang-Mills action with a Chern-Simons action.Comment: 18 pages, 4 figures Equation 1.6 correcte

On the Topological Interpretation of Gravitational Anomalies

We consider the mixed gravitational-Yang-Mills anomaly as the coupling between the $K$-theory and $K$-homology of a $C^*$-algebra crossed product. The index theorem of Connes-Moscovici allows to compute the Chern character of the $K$-cycle by local formulae involving connections and curvatures. It gives a topological interpretation to the anomaly, in the sense of noncommutative algebras.Comment: 16 pages, LaTex, no figure

Unique factorization in perturbative QFT

We discuss factorization of the Dyson--Schwinger equations using the Lie- and Hopf algebra of graphs. The structure of those equations allows to introduce a commutative associative product on 1PI graphs. In scalar field theories, this product vanishes if and only if one of the factors vanishes. Gauge theories are more subtle: integrality relates to gauge symmetries.Comment: 5pages, Talk given at "RadCor 2002 - Loops and Legs 2002", Kloster Banz, Germany, Sep 8-13, 200

Hopf Algebra Primitives in Perturbation Quantum Field Theory

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.Comment: 21 pages, accepted for publication in J.Geom.and Phy

Geometry of Quantum Spheres

Spectral triples on the q-deformed spheres of dimension two and three are reviewed.Comment: 23 pages, revie

Curved noncommutative torus and Gauss--Bonnet

We study perturbations of the flat geometry of the noncommutative two-dimensional torus T^2_\theta (with irrational \theta). They are described by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A_\theta of T_\theta. We show, up to the second order in perturbation, that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.Comment: 13 pages, LaTe

BPS states on noncommutative tori and duality

We study gauge theories on noncommutative tori. It was proved in [5] that Morita equivalence of noncommutative tori leads to a physical equivalence (SO(d,d| Z)-duality) of the corresponding gauge theories. We calculate the energy spectrum of maximally supersymmetric BPS states in these theories and show that this spectrum agrees with the SO(d,d| Z)-duality. The relation of our results with those of recent calculations is discussed.Comment: Misprints corrected, appendices added, minor changes in the main body of the paper; Latex, 32 page

Noncommutative Geometry and The Ising Model

The main aim of this work is to present the interpretation of the Ising type models as a kind of field theory in the framework of noncommutative geometry. We present the method and construct sample models of field theory on discrete spaces using the introduced tools of discrete geometry. We write the action for few models, then we compare them with various models of statistical physics. We construct also the gauge theory with a discrete gauge group.Comment: 12 pages, LaTeX, TPJU - 18/92, December 199

BFV-BRST analysis of equivalence between noncommutative and ordinary gauge theories

Constrained hamiltonian structure of noncommutative gauge theory for the gauge group U(1) is discussed. Constraints are shown to be first class, although, they do not give an Abelian algebra in terms of Poisson brackets. The related BFV-BRST charge gives a vanishing generalized Poisson bracket by itself due to the associativity of *-product. Equivalence of noncommutative and ordinary gauge theories is formulated in generalized phase space by using BFV-BRST charge and a solution is obtained. Gauge fixing is discussed.Comment: Minor changes, ref. added, to be pub. PL

Diffeomorphisms and orthonormal frames

There is a natural homomorphism of Lie pseudoalgebras from local vector fields to local rotations on a Riemannian manifold. We address the question whether this homomorphism is unique and give a positive answer in the perturbative regime around the flat metric.Comment: 11 pages LaTeX, title and abstract changed, published versio