Gauge Theory of the Star Product

The choice of a star product realization for noncommutative field theory can be regarded as a gauge choice in the space of all equivalent star products. With the goal of having a gauge invariant treatment, we develop tools, such as integration measures and covariant derivatives on this space. The covariant derivative can be expressed in terms of connections in the usual way giving rise to new degrees of freedom for noncommutative theories.Comment: 16 page

Elementary Derivation of the Chiral Anomaly

An elementary derivation of the chiral gauge anomaly in all even dimensions is given in terms of noncommutative traces of pseudo-differential operators.Comment: Minor errors and misprints corrected, a reference added. AmsTex file, 12 output pages. If you do not have preloaded AmsTex you have to \input amstex.te

String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy

We consider scalar-Gauss-Bonnet and modified Gauss-Bonnet gravities and reconstruct these theories from the universe expansion history. In particular, we are able to construct versions of those theories (with and without ordinary matter), in which the matter dominated era makes a transition to the cosmic acceleration epoch. It is remarkable that, in several of the cases under consideration, matter dominance and the deceleration-acceleration transition occur in the presence of matter only. The late-time acceleration epoch is described asymptotically by de Sitter space but may also correspond to an exact $\Lambda$CDM cosmology, having in both cases an effective equation of state parameter $w$ close to -1. The one-loop effective action of modified Gauss-Bonnet gravity on the de Sitter background is evaluated and it is used to derive stability criteria for the ensuing de Sitter universe.Comment: LaTeX20 pages, 4 figures, version to apear in PR

Noncommutative geometry and lower dimensional volumes in Riemannian geometry

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all.Comment: 12 page

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.Comment: 76 pages, 8 figures, final version, one section on open problems added, and references expanded. Appears in "Advances in Noncommutative Geometry - on the occasion of Alain Connes' 70th birthday

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.Comment: 14 page

Muon capture on light nuclei

This work investigates the muon capture reactions 2H(\mu^-,\nu_\mu)nn and 3He(\mu^-,\nu_\mu)3H and the contribution to their total capture rates arising from the axial two-body currents obtained imposing the partially-conserved-axial-current (PCAC) hypothesis. The initial and final A=2 and 3 nuclear wave functions are obtained from the Argonne v_{18} two-nucleon potential, in combination with the Urbana IX three-nucleon potential in the case of A=3. The weak current consists of vector and axial components derived in chiral effective field theory. The low-energy constant entering the vector (axial) component is determined by reproducting the isovector combination of the trinucleon magnetic moment (Gamow-Teller matrix element of tritium beta-decay). The total capture rates are 393.1(8) s^{-1} for A=2 and 1488(9) s^{-1} for A=3, where the uncertainties arise from the adopted fitting procedure.Comment: 6 pages, submitted to Few-Body Sys

Interaction of Low - Energy Induced Gravity with Quantized Matter -- II. Temperature effects

At the very early Universe the matter fields are described by the GUT models in curved space-time. At high energies these fields are asymptotically free and conformally coupled to external metric. The only possible quantum effect is the appearance of the conformal anomaly, which leads to the propagation of the new degree of freedom - conformal factor. Simultaneously with the expansion of the Universe, the scale of energies decreases and the propagating conformal factor starts to interact with the Higgs field due to the violation of conformal invariance in the matter fields sector. In a previous paper \cite{foo} we have shown that this interaction can lead to special physical effects like the renormalization group flow, which ends in some fixed point. Furthermore in the vicinity of this fixed point there occur the first order phase transitions. In the present paper we consider the same theory of conformal factor coupled to Higgs field and incorporate the temperature effects. We reduce the complicated higher-derivative operator to several ones of the standard second-derivative form and calculate an exact effective potential with temperature on the anti de Sitter (AdS) background.Comment: 12 pages, LaTex - 2 Figure

Noncommutative Induced Gauge Theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative $\phi^4$-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.Comment: 20 pages, 6 figure

On the scalar curvature for the noncommutative four torus

The canonical flat metric of the noncommutative four torus is conformally perturbed and the term corresponding to the scalar curvature in the heat kernel expansion of perturbed Laplacian is calculated by using an analog of the Wodziski residue. This allows the calculation to be done without using the so called rearrangement lemma. Thus, because of the simplicity of the method, the structure of the one and two variable of functions of a modular automorphism that appear in the formula for the curvature can be understood and functional relations among them are discovered. Also the gradient of the analog of the Einstein-Hilbert action is calculated explicitly, which prepares the ground for defining certain geometric flows (such as the Yamabe flow) on the noncommutative four torus