242 research outputs found
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
CDM cosmology, having in both cases an effective equation of state
parameter 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
Gravity, Non-Commutative Geometry and the Wodzicki Residue
We derive an action for gravity in the framework of non-commutative geometry
by using the Wodzicki residue. We prove that for a Dirac operator on an
dimensional compact Riemannian manifold with , even, the Wodzicki
residue Res is the integral of the second coefficient of the heat
kernel expansion of . We use this result to derive a gravity action for
commutative geometry which is the usual Einstein Hilbert action and we also
apply our results to a non-commutative extension which, is given by the tensor
product of the algebra of smooth functions on a manifold and a finite
dimensional matrix algebra. In this case we obtain gravity with a cosmological
constant.Comment: 17p., MZ-TH/93-3
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
Poisson-Lie group of pseudodifferential symbols
We introduce a Lie bialgebra structure on the central extension of the Lie
algebra of differential operators on the line and the circle (with scalar or
matrix coefficients). This defines a Poisson--Lie structure on the dual group
of pseudodifferential symbols of an arbitrary real (or complex) order. We show
that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP
Poisson structures are naturally realized as restrictions of this Poisson
structure to submanifolds of this ``universal'' Poisson--Lie group.
Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in
physical terminology) can be viewed as subspaces of the quotient (or Poisson
reduction) of this Poisson--Lie group by the dressing action of the group of
functions.
Finally, we define an infinite set of functions in involution on the
Poisson--Lie group that give the standard families of Hamiltonians when
restricted to the submanifolds mentioned above. The Poisson structure and
Hamiltonians on the whole group interpolate between the Poisson structures and
Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical
meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes
to 0.Comment: 64 pages, no figure
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
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
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
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
Periodicity and the determinant bundle
The infinite matrix `Schwartz' group is a classifying group for
odd K-theory and carries Chern classes in each odd dimension, generating the
cohomology. These classes are closely related to the Fredholm determinant on
We show that while the higher (even, Schwartz) loop groups of
again classifying for odd K-theory, do \emph{not} carry
multiplicative determinants generating the first Chern class, `dressed'
extensions, corresponding to a star product, do carry such functions. We use
these to discuss Bott periodicity for the determinant bundle and the eta
invariant. In so doing we relate two distinct extensions of the eta invariant,
to self-adjoint elliptic operators and to elliptic invertible suspended
families and show that the corresponding invariant is a determinant in
this sense
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