4,434 research outputs found

### 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

### The Standard Model a la Connes-Lott

The relations among coupling constants and masses in the standard model \`a
la Connes-Lott with general scalar product are computed in detail. We find a
relation between the top and the Higgs masses. For $m_t=174\pm22\ GeV$ it
yields $m_H=277\pm40\ GeV$. The Connes-Lott theory privileges the masses
$m_t=160.4\ GeV$ and $m_H=251.8\ GeV$.Comment: 20 pages, LaTe

### 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

### Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

We examine the hypothesis that space-time is a product of a continuous
four-dimensional manifold times a finite space. A new tensorial notation is
developed to present the various constructs of noncommutative geometry. In
particular, this notation is used to determine the spectral data of the
standard model. The particle spectrum with all of its symmetries is derived,
almost uniquely, under the assumption of irreducibility and of dimension 6
modulo 8 for the finite space. The reduction from the natural symmetry group
SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that
the two layers of space-time are finite distance apart but is non-dynamical.
The square of the Dirac operator, and all geometrical invariants that appear in
the calculation of the heat kernel expansion are evaluated. We re-derive the
leading order terms in the spectral action. The geometrical action yields
unification of all fundamental interactions including gravity at very high
energies. We make the following predictions: (i) The number of fermions per
family is 16. (ii) The symmetry group is U(1)xSU(2)xSU(3). (iii) There are
quarks and leptons in the correct representations. (iv) There is a doublet
Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of
170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism.
Moreover, the zeroth order spectral action obtained with a cut-off function is
consistent with experimental data up to few percent. We discuss a number of
open issues. We prepare the ground for computing higher order corrections since
the predicted mass of the Higgs field is quite sensitive to the higher order
corrections. We speculate on the nature of the noncommutative space at
Planckian energies and the possible role of the fundamental group for the
problem of generations.Comment: 56 page

### 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

### 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

### 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

### Geometry of Quantum Spheres

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

### The Structure of the Ladder Insertion-Elimination Lie algebra

We continue our investigation into the insertion-elimination Lie algebra of
Feynman graphs in the ladder case, emphasizing the structure of this Lie
algebra relevant for future applications in the study of Dyson-Schwinger
equations. We work out the relation of this Lie algebra to some classical
infinite dimensional Lie algebra and we determine its cohomology.Comment: 24 pages, LaTex, typos correcte

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