117 research outputs found
Exact Partition Functions for Gauge Theories on
The noncommutative space , a deformation of
, supports a -parameter family of gauge theory models with
gauge-invariant harmonic term, stable vacuum and which are perturbatively
finite to all orders. Properties of this family are discussed. The partition
function factorizes as an infinite product of reduced partition functions, each
one corresponding to the reduced gauge theory on one of the fuzzy spheres
entering the decomposition of . For a particular
sub-family of gauge theories, each reduced partition function is exactly
expressible as a ratio of determinants. A relation with integrable 2-D Toda
lattice hierarchy is indicated.Comment: 20 pages. Title modified. Typos corrected. Version to appear in
Nucl.Phys.
Derivations of the Moyal Algebra and Noncommutative Gauge Theories
The differential calculus based on the derivations of an associative algebra
underlies most of the noncommutative field theories considered so far. We
review the essential properties of this framework and the main features of
noncommutative connections in the case of non graded associative unital
algebras with involution. We extend this framework to the case of
-graded unital involutive algebras. We show, in the case of the
Moyal algebra or some related -graded version of it, that the
derivation based differential calculus is a suitable framework to construct
Yang-Mills-Higgs type models on Moyal (or related) algebras, the covariant
coordinates having in particular a natural interpretation as Higgs fields. We
also exhibit, in one situation, a link between the renormalisable NC
-model with harmonic term and a gauge theory model. Some possible
consequences of this are briefly discussed.Comment: 25 pages, 1 figure. Based on a talk given at the XVIIth International
Colloquium on Integrable Systems and Quantum Symmetries, June 19-22, 2008,
Pragu
Noncommutative field theories on : Towards UV/IR mixing freedom
We consider the noncommutative space , a deformation of
the algebra of functions on which yields a "foliation" of
into fuzzy spheres. We first construct a natural matrix base
adapted to . We then apply this general framework to the
one-loop study of a two-parameter family of real-valued scalar noncommutative
field theories with quartic polynomial interaction, which becomes a non-local
matrix model when expressed in the above matrix base. The kinetic operator
involves a part related to dynamics on the fuzzy sphere supplemented by a term
reproducing radial dynamics. We then compute the planar and non-planar 1-loop
contributions to the 2-point correlation function. We find that these diagrams
are both finite in the matrix base. We find no singularity of IR type, which
signals very likely the absence of UV/IR mixing. We also consider the case of a
kinetic operator with only the radial part. We find that the resulting theory
is finite to all orders in perturbation expansion.Comment: 31 pages, 4 figures. Improved version. Sections 5.1 and 5.2 have been
clarified. A minor error corrected. References adde
A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model
In this paper we consider the Spontaneous Symmetry Breaking Mechanism (SSBM)
in the Standard Model of particles in the unitary gauge. We show that the
computation usually presented of this mechanism can be conveniently performed
in a slightly different manner. As an outcome, the computation we present can
change the interpretation of the SSBM in the Standard Model, in that it
decouples the SU(2)-gauge symmetry in the final Lagrangian instead of breaking
it.Comment: 16 page
Metrics and causality on Moyal planes
Metrics structures stemming from the Connes distance promote Moyal planes to
the status of quantum metric spaces. We discuss this aspect in the light of
recent developments, emphasizing the role of Moyal planes as representative
examples of a recently introduced notion of quantum (noncommutative) locally
compact space. We move then to the framework of Lorentzian noncommutative
geometry and we examine the possibility of defining a notion of causality on
Moyal plane, which is somewhat controversial in the area of mathematical
physics. We show the actual existence of causal relations between the elements
of a particular class of pure (coherent) states on Moyal plane with related
causal structure similar to the one of the usual Minkowski space, up to the
notion of locality.Comment: 33 pages. Improved version; a summary added at the end of the
introduction, misprints corrected. Version to appear in Contemporary
Mathematic
Spectral Distances: Results for Moyal Plane and Noncommutative Torus
The spectral distance for noncommutative Moyal planes is considered in the
framework of a non compact spectral triple recently proposed as a possible
noncommutative analog of non compact Riemannian spin manifold. An explicit
formula for the distance between any two elements of a particular class of pure
states can be determined. The corresponding result is discussed. The existence
of some pure states at infinite distance signals that the topology of the
spectral distance on the space of states is not the weak * topology. The case
of the noncommutative torus is also considered and a formula for the spectral
distance between some states is also obtained
Involutive representations of coordinate algebras and quantum spaces
We show that Lie algebras of coordinate operators related to
quantum spaces with noncommutativity can be conveniently
represented by -covariant poly-differential involutive representations.
We show that the quantized plane waves obtained from the quantization map
action on the usual exponential functions are determined by polar decomposition
of operators combined with constraint stemming from the Wigner theorem for
. Selecting a subfamily of -representations, we show that the
resulting star-product is equivalent to the Kontsevich product for the Poisson
manifold dual to the finite dimensional Lie algebra . We
discuss the results, indicating a way to extend the construction to any
semi-simple non simply connected Lie group and present noncommutative scalar
field theories which are free from perturbative UV/IR mixing.Comment: 29 pages, several paragraphs added, published in JHE
Vacuum energy and the cosmological constant problem in -Poincar\'e invariant field theories
We investigate the vacuum energy in -Poincar\'e invariant field
theories. It is shown that for the equivariant Dirac operator one obtains an
improvement in UV behavior of the vacuum energy and therefore the cosmological
constant problem has to be revised.Comment: improved version, 15 page
Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus
Derivations of a noncommutative algebra can be used to construct differential
calculi, the so-called derivation-based differential calculi. We apply this
framework to a version of the Moyal algebra . We show that the
differential calculus, generated by the maximal subalgebra of the derivation
algebra of that can be related to infinitesimal symplectomorphisms,
gives rise to a natural construction of Yang-Mills-Higgs models on
and a natural interpretation of the covariant coordinates as Higgs fields. We
also compare in detail the main mathematical properties characterizing the
present situation to those specific of two other noncommutative geometries,
namely the finite dimensional matrix algebra and the
algebra of matrix valued functions . The
UV/IR mixing problem of the resulting Yang-Mills-Higgs models is also
discussed.Comment: 23 pages, 2 figures. Improved and enlarged version. Some references
have been added and updated. Two subsections and a discussion on the
appearence of Higgs fiels in noncommutative gauge theories have been adde
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