126 research outputs found

### Equivariant Fields in an $SU({\cal N})$ Gauge Theory with new Spontaneously Generated Fuzzy Extra Dimensions

We find new spontaneously generated fuzzy extra dimensions emerging from a
certain deformation of $N=4$ supersymmetric Yang-Mills (SYM) theory with cubic
soft supersymmetry breaking and mass deformation terms. First, we determine a
particular four dimensional fuzzy vacuum that may be expressed in terms of a
direct sum of product of two fuzzy spheres, and denote it in short as $S_F^{2\,
Int}\times S_F^{2\, Int}$. The direct sum structure of the vacuum is revealed
by a suitable splitting of the scalar fields in the model in a manner that
generalizes our approach in \cite{Seckinson}. Fluctuations around this vacuum
have the structure of gauge fields over $S_F^{2\, Int}\times S_F^{2\, Int}$,
and this enables us to conjecture the spontaneous broken model as an effective
$U(n)$ $(n < {\cal N})$ gauge theory on the product manifold $M^4 \times
S_F^{2\, Int} \times S_F^{2\, Int}$. We support this interpretation by
examining the $U(4)$ theory and determining all of the $SU(2)\times SU(2)$
equivariant fields in the model, characterizing its low energy degrees of
freedom. Monopole sectors with winding numbers $(\pm
1,0),\,(0,\pm1),\,(\pm1,\pm 1)$ are accessed from $S_F^{2\, Int}\times S_F^{2\,
Int}$ after suitable projections and subsequently equivariant fields in these
sectors are obtained. We indicate how Abelian Higgs type models with vortex
solutions emerge after dimensionally reducing over the fuzzy monopole sectors
as well. A family of fuzzy vacua is determined by giving a systematic treatment
for the splitting of the scalar fields and it is made manifest that suitable
projections of these vacuum solutions yield all higher winding number fuzzy
monopole sectors. We observe that the vacuum configuration $S_F^{2\, Int}\times
S_F^{2\, Int}$ identifies with the bosonic part of the product of two fuzzy
superspheres with $OSP(2,2)\times OSP(2,2)$ supersymmetry and elaborate on this
feature.Comment: 38+1 pages, published versio

### Magnetic Field and Curvature Effects on Pair Production II: Vectors and Implications for Chromodynamics

We calculate the pair production rates for spin-$1$ or vector particles on
spaces of the form $M \times {\mathbb R}^{1,1}$ with $M$ corresponding to
${\mathbb R}^2$ (flat), $S^2$ (positive curvature) and $H^2$ (negative
curvature), with and without a background (chromo)magnetic field on $M$. Beyond
highlighting the effects of curvature and background magnetic field, this is
particularly interesting since vector particles are known to suffer from the
Nielsen-Olesen instability, which can dramatically increase pair production
rates. The form of this instability for $S^2$ and $H^2$ is obtained. We also
give a brief discussion of how our results relate to ideas about confinement in
nonabelian theories.Comment: 24 pages, 9 figure

### Waves on Noncommutative Spacetimes

Waves on ``commutative'' spacetimes like R^d are elements of the commutative
algebra C^0(R^d) of functions on R^d. When C^0(R^d) is deformed to a
noncommutative algebra {\cal A}_\theta (R^d) with deformation parameter \theta
({\cal A}_0 (R^d) = C^0(R^d)), waves being its elements, are no longer
complex-valued functions on R^d. Rules for their interpretation, such as
measurement of their intensity, and energy, thus need to be stated. We address
this task here. We then apply the rules to interference and diffraction for d
\leq 4 and with time-space noncommutativity. Novel phenomena are encountered.
Thus when the time of observation T is so brief that T \leq 2 \theta w, where w
is the frequency of incident waves, no interference can be observed. For larger
times, the interference pattern is deformed and depends on \frac{\theta w}{T}.
It approaches the commutative pattern only when \frac{\theta w}{T} goes to 0.
As an application, we discuss interference of star light due to cosmic strings.Comment: 19 pages, 5 figures, LaTeX, added references, corrected typo

### Magnetic Field and Curvature Effects on Pair Production I: Scalars and Spinors

The pair production rates for spin-zero and spin-$\frac{1}{2}$ particles are
calculated on spaces of the form $M \times {\mathbb R}^{1,1}$ with $M$
corresponding to ${\mathbb R}^2$ (flat), $T^2$ (flat, compactified), $S^2$
(positive curvature) and $H^2$ (negative curvature), with and without a
background magnetic field on $M$. The motivation is to elucidate the effects of
curvature and background magnetic field. Contrasting effects for positive and
negative curvature on the two cases of spin are obtained. For positive
curvature, we find enhancement for spin-zero and suppression for
spin-$\frac{1}{2}$, with the opposite effect for negative curvature.Comment: 28 pages, 10 figure

### Interacting Quantum Topologies and the Quantum Hall Effect

The algebra of observables of planar electrons subject to a constant
background magnetic field B is given by A_theta(R^2) x A_theta(R^2) the product
of two mutually commuting Moyal algebras. It describes the free Hamiltonian and
the guiding centre coordinates. We argue that A_theta(R^2) itself furnishes a
representation space for the actions of these two Moyal algebras, and suggest
physical arguments for this choice of the representation space. We give the
proper setup to couple the matter fields based on A_theta(R^2) to
electromagnetic fields which are described by the abelian commutative gauge
group G_c(U(1)), i.e. gauge fields based on A_0(R^2). This enables us to give a
manifestly gauge covariant formulation of integer quantum Hall effect (IQHE).
Thus, we can view IQHE as an elementary example of interacting quantum
topologies, where matter and gauge fields based on algebras A_theta^prime with
different theta^prime appear. Two-particle wave functions in this approach are
based on A_theta(R^2) x A_theta(R^2). We find that the full symmetry group in
IQHE, which is the semi-direct product SO(2) \ltimes G_c(U(1)) acts on this
tensor product using the twisted coproduct Delta_theta. Consequently, as we
show, many particle sectors of each Landau level have twisted statistics. As an
example, we find the twisted two particle Laughlin wave functions.Comment: 10 pages, LaTeX, Corrected typos, Published versio

### Spontaneous Lorentz Violation: The Case of Infrared QED

It is by now clear that infrared sector of QED has an intriguingly complex
structure. Based on earlier pioneering works on this subject, two of us
recently proposed a simple modification of QED by constructing a generalization
of the $U(1)$ charge group of QED to the "Sky" group incorporating the known
spontaneous Lorentz violation due to infrared photons, but still compatible in
particular with locality. There it was shown that the "Sky" group is generated
by the algebra of angle dependent charges and a study of its superselection
sectors has revealed a manifest description of spontaneous breaking of Lorentz
symmetry. We further elaborate this approach here and investigate in some
detail the properties of charged particles dressed by the infrared photons. We
find that Lorentz violation due to soft photons may be manifestly codified in
an angle dependent fermion mass modifying therefore the fermion dispersion
relations. The fact that the masses of the charged particles are not Lorentz
invariant affects their spin content too.Time dilation formulae for decays
should also get corrections. We speculate that these effects could be measured
possibly in muon decay experiments.Comment: 18+1 pages, revised version, expanded discussion in section 5

### Quantum Hall Effect on the Grassmannians $\mathbf{Gr}_2(\mathbb{C}^N)$

Quantum Hall Effects (QHEs) on the complex Grassmann manifolds
$\mathbf{Gr}_2(\mathbb{C}^N)$ are formulated. We set up the Landau problem in
$\mathbf{Gr}_2(\mathbb{C}^N)$ and solve it using group theoretical techniques
and provide the energy spectrum and the eigenstates in terms of the $SU(N)$
Wigner ${\cal D}$-functions for charged particles on
$\mathbf{Gr}_2(\mathbb{C}^N)$ under the influence of abelian and non-abelian
background magnetic monopoles or a combination of these thereof. In particular,
for the simplest case of $\mathbf{Gr}_2(\mathbb{C}^4)$ we explicitly write down
the $U(1)$ background gauge field as well as the single and many-particle
eigenstates by introducing the Pl\"{u}cker coordinates and show by calculating
the two-point correlation function that the Lowest Landau Level (LLL) at
filling factor $\nu =1$ forms an incompressible fluid. Our results are in
agreement with the previous results in the literature for QHE on ${\mathbb
C}P^N$ and generalize them to all $\mathbf{Gr}_2(\mathbb{C}^N)$ in a suitable
manner. Finally, we heuristically identify a relation between the $U(1)$ Hall
effect on $\mathbf{Gr}_2(\mathbb{C}^4)$ and the Hall effect on the odd sphere
$S^5$, which is yet to be investigated in detail, by appealing to the already
known analogous relations between the Hall effects on ${\mathbb C}P^3$ and
${\mathbb C}P^7$ and those on the spheres $S^4$ and $S^8$, respectively.Comment: 34 pages, revtex 4-1, Minor Correction

### Quantum aspects of a noncommutative supersymmetric kink

We consider quantum corrections to a kink of noncommutative supersymmetric
phi^4 theory in 1+1 dimensions. Despite the presence of an infinite number of
time derivatives in the action, we are able to define supercharges and a
Hamiltonian by using an unconventional canonical formalism. We calculate the
quantum energy E of the kink (defined as a half-sum of the eigenfrequencies of
fluctuations) which coincides with its' value in corresponding commutative
theory independently of the noncommutativity parameter. The renormalization
also proceeds precisely as in the commutative case. The vacuum expectation
value of the new Hamiltonian is also calculated and appears to be consistent
with the value of the quantum energy E of the kink.Comment: 20 pages, v2: a reference adde

### Star Product and Invariant Integration for Lie type Noncommutative Spacetimes

We present a star product for noncommutative spaces of Lie type, including
the so called ``canonical'' case by introducing a central generator, which is
compatible with translations and admits a simple, manageable definition of an
invariant integral. A quasi-cyclicity property for the latter is shown to hold,
which reduces to exact cyclicity when the adjoint representation of the
underlying Lie algebra is traceless. Several explicit examples illuminate the
formalism, dealing with kappa-Minkowski spacetime and the Heisenberg algebra
(``canonical'' noncommutative 2-plane).Comment: 21 page

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