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

How to decompose arbitrary continuous-variable quantum operations

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multi-mode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g. in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.Comment: Ver. 3: published version with supplementary materia

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

THE EFFECTS OF GLOBAL DYNAMICS TO URBAN HIERARCHY: COMPROMISING CITIES INSTEAD OF COMPETITIVE CITIES

Especially after 1990ââ¬â¢s, important results have emerged with the rapid transformation process in the world by the concept of globalization. It is emphasized that cities are the center of spatial transformation where the globalization mostly effected. The economic and technological developments have caused the fluidity of capital in this manner cities have to bring out their potentials more than ever to attract the capital and disparities reveals in the pattern of urban hierarchy. The presence of cities depends on the renewal of existing potentials according to the changing conditions of today. In this process, many cities around the world, especially with economic, spatial, environmental advantages come forward in the international arena. While the competition between cities significantly increased, the cities that cannot keep up with varying conditions are destined to lose. The context of this paper is about the effects of the globalization and changes in the economic structure to urban hierarchy which is examined in the Aydın Providence of Turkey. Aydın is a developed province of Aegean Region which is situated in the seaside with its natural amenities that has to be protected. There are sub-regions which have different identities stands out in the province. The settlements in the inner sides and the coast sides differentiate in social, economic, demographic characteristics. While the population of settlements in the coast side of the province are increased as a result of investments and tourism potentials, the population of inner settlements which are based on agricultural economies, are decreased. Consequently, a dual structure is present in the very same region; we can see developing and an under developed sub-regions. The material example of this structure can be observed in two different districts of Aydın; Kuşadası which is situated in the seaside; Söke which is situated in the inner side. At this point, the basic problematic of this paper is about the positive-negative effects of rapid development of certain areas. As a result, in the context of emerging problems, the two settlements which are located in close proximity are examined and the concept of compromising cities suggested instead of competitive cities.

Spontaneous Breaking of Lorentz Symmetry and Vertex Operators for Vortices

We first review the spontaneous Lorentz symmetry breaking in the presence of massless gauge fields and infraparticles. This result was obtained long time ago in the context of rigorious quantum field theory by Frohlich et. al. and reformulated by Balachandran and Vaidya using the notion of superselection sectors and direction-dependent test functions at spatial infinity for the non-local observables. Inspired by these developments and under the assumption that the spectrum of the electric charge is quantized, (in units of a fundamental charge e) we construct a family of vertex operators which create winding number k, electrically charged Abelian vortices from the vacuum (zero winding number sector) and/or shift the winding number by k units. In particular, we find that for rotating vortices the vertex operator at level k shifts the angular momentum of the vortex by k \frac{{\tilde q}}{q}, where \tilde q is the electric charge of the quantum state of the vortex and q is the charge of the vortex scalar field under the U(1) gauge field. We also show that, for charged-particle-vortex composites angular momentum eigenvalues shift by k \frac{{\tilde q}}{q}, {\tilde q} being the electric charge of the charged-particle-vortex composite. This leads to the result that for \frac{{\tilde q}}{q} half-odd integral and for odd k our vertex operators flip the statistics of charged-particle-vortex composites from bosons to fermions and vice versa. For fractional values of \frac{{\tilde q}}{q}, application of vertex operator on charged-particle-vortex composite leads in general to composites with anyonic statistics.Comment: Published version, 15+1 pages, 1 figur

Consumption with Habit Formation

In a representative-agent model of intertemporal consumption-saving with stochastic income and habit formation, we have shown that precautionary savings observed in the data cannot be attributed only to income uncertainty, but also to the time-non-separability of preferences. We have found that, with habit-forming preferences, consumption depends not only on permanent income but also on past consumption and the stronger the habits the lower the effect of income uncertainty on consumption. For a given constant coefficient of risk aversion, habit-forming consumer will have smaller precautionary savings per unit of income risk faced than the one with time-separable preferences. By allowing habit forming preferences in consumption, a closed form solution, explaining excess-sensitivity, excess-smoothness and excess-growth puzzles of consumption, and thus, providing a better framework for empirically testing the behavior of consumption over the business cycle, is found with only i.i.d. income innovations. Dans un modèle intertemporel de consommation et d'épargne avec revenu stochastique et formation d'habitudes, nous avons démontré que l'épargne précautionnelle observée dans les données peut être attribuée non seulement à l'incertitude de revenu0501s aussi à l'inséparabilité des préférences. Nous avons trouvé que, avec les préférences qui forment des habitudes, la consommation dépend non seulement du revenu permanent0501s aussi de niveaux de consommations passés. De plus, plus les habitudes sont résistantes, moins grand sera l'effet de l'incertitude de revenu sur la consommation. Pour un coéfficient constant fixé de l'aversion au risque, le consommateur avec habitudes va avoir une épargne précautionnelle plus basse par unité de risque de revenu par rapport à celle avec des préférences temps-séparables. En introduisant la formation d'habitudes dans la consommation, et en supposant seulement des innovations i.i.d., nous avons trouvé une solution fermée qui explique les trois énigmes de consommation, l'excès de sensitivité, l'excès de lisseté et l'excès de croissance anticipée, et qui propose un meilleur modèle pour tester le trajet de la consommation pendant le cycle économique.Intertemporal consumption-saving, habit formation, dynamic programming, Consommation/épargne intertemporelles, formation d'habitudes, programmation dynamique