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

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

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 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

Non-Linear Sigma Model on the Fuzzy Supersphere

In this note we develop fuzzy versions of the supersymmetric non-linear sigma model on the supersphere S^(2,2). In hep-th/0212133 Bott projectors have been used to obtain the fuzzy CP^1 model. Our approach utilizes the use of supersymmetric extensions of these projectors. Here we obtain these (super) -projectors and quantize them in a fashion similar to the one given in hep-th/0212133. We discuss the interpretation of the resulting model as a finite dimensional matrix model.Comment: 11 pages, LaTeX, corrected typo

Explorations in fuzzy physics and non-commutative geometry

Fuzzy spaces arise as discrete approximations to continuum manifolds. They are usually obtained through quantizing coadjoint orbits of compact Lie groups and they can be described in terms of finite-dimensional matrix algebras, which for large matrix sizes approximate the algebra of functions of the limiting continuum manifold. Their ability to exactly preserve the symmetries of their parent manifolds is especially appealing for physical applications. Quantum Field Theories are built over them as finite-dimensional matrix models preserving almost all the symmetries of their respective continuum models. In this dissertation, we first focus our attention to the study of fuzzy supersymmetric spaces. In this regard, we obtain the fuzzy supersphere [Special characters omitted.] through quantizing the supersphere, and demonstrate that it has exact supersymmetry. We derive a finite series formula for the [low *]-product of functions over [Special characters omitted.] and analyze the differential geometric information encoded in this formula. Subsequently, we show that quantum field theories on [Special characters omitted.] are realized as finite-dimensional supermatrix models, and in particular we obtain the non-linear sigma model over the fuzzy supersphere by constructing the fuzzy supersymmetric extensions of a certain class of projectors. We show that this model too, is realized as a finite-dimensional supermatrix model with exact supersymmetry. Next, we show that fuzzy spaces have a generalized Hopf algebra structure. By focusing on the fuzzy sphere, we establish that there is a [low *]-homomorphism from the group algebra SU (2)* of SU (2) to the fuzzy sphere. Using this and the canonical Hopf algebra structure of SU (2)* we show that both the fuzzy sphere and their direct sum are Hopf algebras. Using these results, we discuss processes in which a fuzzy sphere with angular momenta J splits into fuzzy spheres with angular momenta K and L . Finally, we study the formulation of Chern-Simons (CS) theory on an infinite strip of the non-commutative plane. We develop a finite-dimensional matrix model, whose large size limit approximates the CS theory on the infinite strip, and show that there are edge observables in this model obeying a finite-dimensional Lie algebra, that resembles the Kac-Moody algebra