New Fuzzy Extra Dimensions from SU(N)SU({\cal N}) Gauge Theories

We start with an $SU(\cal {N})$ Yang-Mills theory on a manifold ${\cal M}$, suitably coupled to two distinct set of scalar fields in the adjoint representation of $SU({\cal N})$, which are forming a doublet and a triplet, respectively under a global $SU(2)$ symmetry. We show that a direct sum of fuzzy spheres $S_F^{2 \, Int} := S_F^2(\ell) \oplus S_F^2 (\ell) \oplus S_F^2 \left ( \ell + \frac{1}{2} \right ) \oplus S_F^2 \left ( \ell - \frac{1}{2} \right )$ emerges as the vacuum solution after the spontaneous breaking of the gauge symmetry and lay the way for us to interpret the spontaneously broken model as a $U(n)$ gauge theory over ${\cal M} \times S_F^{2 \, Int}$. Focusing on a $U(2)$ gauge theory we present complete parameterizations of the $SU(2)$-equivariant, scalar, spinor and vector fields characterizing the effective low energy features of this model. Next, we direct our attention to the monopole bundles $S_F^{2 \, \pm} := S_F^2 (\ell) \oplus S_F^2 \left ( \ell \pm \frac{1}{2} \right )$ over $S_F^2 (\ell)$ with winding numbers $\pm 1$, which naturally come forth through certain projections of $S_F^{2 \, Int}$, and discuss the low energy behaviour of the $U(2)$ gauge theory over ${\cal M} \times S_F^{2 \, \pm}$. We study models with $k$-component multiplet of the global $SU(2)$, give their vacuum solutions and obtain a class of winding number $\pm (k-1)$ monopole bundles $S_F^{2 \,, \pm (k-1)}$ as certain projections of these vacuum solutions. We make the observation that $S_F^{2 \, Int}$ is indeed the bosonic part of the $N=2$ fuzzy supersphere with $OSP(2,2)$ supersymmetry and construct the generators of the $osp(2,2)$ Lie superalgebra in two of its irreducible representations using the matrix content of the vacuum solution $S_F^{2 \, Int}$. Finally, we show that our vacuum solutions are stable by demonstrating that they form mixed states with non-zero von Neumann entropy.Comment: 27+1 pages, revised version, added references and corrected typo

Equivariant Fields in an SU(N)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

A U(3)U(3) Gauge Theory on Fuzzy Extra Dimensions

In this article, we explore the low energy structure of a $U(3)$ gauge theory over spaces with fuzzy sphere(s) as extra dimensions. In particular, we determine the equivariant parametrization of the gauge fields, which transform either invariantly or as vectors under the combined action of $SU(2)$ rotations of the fuzzy spheres and those $U(3)$ gauge transformations generated by $SU(2) \subset U(3)$ carrying the spin $1$ irreducible representation of $SU(2)$. The cases of a single fuzzy sphere $S_F^2$ and a particular direct sum of concentric fuzzy spheres, $S_F^{2 \, Int}$, covering the monopole bundle sectors with windings $\pm 1$ are treated in full and the low energy degrees of freedom for the gauge fields are obtained. Employing the parametrizations of the fields in the former case, we determine a low energy action by tracing over the fuzzy sphere and show that the emerging model is abelian Higgs type with $U(1) \times U(1)$ gauge symmetry and possess vortex solutions on ${\mathbb R}^2$, which we discuss in some detail. Generalization of our formulation to the equivariant parametrization of gauge fields in $U(n)$ theories is also briefly addressed.Comment: 27+1 page

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

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

Mesencephalon; Midbrain

The mesencephalon is the most rostral part of the brainstem and sits above the pons and is adjoined rostrally to the thalamus. It comprises two lateral halves, called the cerebral peduncles; which is again divided into an anterior part, the crus cerebri, and a posterior part, tegmentum. The tectum is lay dorsal to an oblique coronal plane which includes the aquaduct, and consist of pretectal area and the corpora quadrigemina. In transvers section, the cerebral peduncles are seen to be composed of dorsal and ventral regions separated by the substantia nigra. Tegmentum mesencephali contains red nucleus, oculomotor nucleus, thochlear nucleus, reticular nuclei, medial lemnisci, lateral lemnisci and medial longitudinal fasciculus. In tectum, the inferior colliculus and superior colliculus have main nucleus, which are continuous with the periaqueductal grey matter. The mesencephalon serves important functions in motor movement, particularly movements of the eye, and in auditory and visual processing. The mesencephalic syndrome cause tremor, spastic paresis or paralysis, opisthotonos, nystagmus and depression or coma. In addition cranial trauma, brain tumors, thiamin deficiency and inflammatory or degenerative disorders of the mesencephalon have also been associated with the midbrain syndrome

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

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