Extended dynamical symmetries of Landau levels in higher dimensions

Continuum models for time-reversal (TR) invariant topological insulators (Tis) in d >= 3 dimensions are provided by harmonic oscillators coupled to certain SO(d) gauge fields. These models are equivalent to the presence of spin-orbit (SO) interaction in the oscillator Hamiltonians at a critical coupling strength (equivalent to the harmonic oscillator frequency) and leads to flat Landau Level (LL) spectra and therefore to infinite degeneracy of either the positive or the negative helicity states depending on the sign of the SO coupling. Generalizing the results of [1] to d >= 4, we construct vector operators commuting with these Hamiltonians and show that SO(d, 2) emerges as the non-compact extended dynamical symmetry. Focusing on the model in four dimensions, we demonstrate that the infinite degeneracy of the flat spectra can be fully explained in terms of the discrete unitary representations of SO(4,2), i.e. the doubletons. The degeneracy in the opposite helicity branch is finite, but can still be explained exploiting the complex conjugate doubleton representations. Subsequently, the analysis is generalized to d-dimensions, distinguishing the cases of odd and even d. We also determine the spectrum generating algebra in these models and briefly comment on the algebraic organization of the LL states w.r.t. an underlying "deformed" AdS geometry as well as on the organization of the surface states under open boundary conditions in view of our results

Constructing the quantum Hall system on the Grassmannians Gr(2)(C-N)

In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(CN). We set up the Landau problem in Gr2(CN), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D-functions for charged particles on Gr2(CN) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C4) we provide explicit constructions of the single and many- particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor v = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr2(C4) and the Hall effect on the odd sphere S5, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S4 and S8, respectively. The talk is given by S. Kürkçüoğlu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S. Kürkçüoğlu, G.Ünal [1]

Quantum Hall effect on odd spheres

We solve the Landau problem for charged particles on odd dimensional spheres S2k-1 in the background of constant SO(2k - 1) gauge fields carrying the irreducible representation (I/2,I/2, . . . , I/2). We determine the spectrum of the Hamiltonian, the degeneracy of the Landau levels and give the eigenstates in terms of the Wigner D-functions, and for odd values of I, the explicit local form of the wave functions in the lowest Landau level (LLL). The spectrum of the Dirac operator on S2k-1 in the same gauge field background together with its degeneracies is also determined, and in particular, its number of zero modes is found. We show how the essential differential geometric structure of the Landau problem on the equatorial S2k-2 is captured by constructing the relevant projective modules. For the Landau problem on S-5, we demonstrate an exact correspondence between the union of Hilbert spaces of LLLs, with I ranging from 0 to I-max = 2K or I-max = 2K or I-max = 2K + 1 to the Hilbert spaces of the fuzzy CP3 or that of winding number +/- 1 line bundles over CP3 at level K, respectively

Gauge Theories with Fuzzy Extra Dimensions and Noncommutative Vortices and Fluxons

A U(2) Yang-Mills theory on the space M × S 2 F is considered, where it is assumed that M is an arbitrary noncommutative space and S 2 F is a fuzzy sphere spontaneously generated from a noncommutative U(N ) Yang-Mills theory on M coupled to a triplet of scalars in the adjoint of U(N ). SU(2)-equivariant reduction of this theory leads to a noncommutative U(1) gauge theory coupled adjointly to a set of scalar fields. The emergent model is studied on the Groenewald-Moyal plane R 2 θ and it is found that, in certain limits, it admits noncommutative, non-BPS vortex as well as fluxon solutions

Equivariant Reduction of Gauge Theories over Fuzzy Extra Dimensions

In SU(N) Yang-Mills theories on a manifold M, which are suitably coupled to a set of scalars, fuzzy spheres may be generated as extra dimensions by spontaneous symmetry breaking. This process results in gauge theories over the product space of the manifold M and the fuzzy spheres with smaller gauge groups. Here we present the SU(2)- and SU(2) x SU(2)-equivariant parametrization of U(2) and U(4) gauge fields on S-F(2), and S-F(2), x S-F(2), respectively and outline the dimensional reduction of these theories over the fuzzy extra dimensions. The emerging dimensionally reduced theories are Higgs type models. Some vortex type solutions of these theories are briefly discussed

Edge Currents in Non commutative Chern Simons Theory

This paper discusses the formulation of the non-commutativ e Chern-Simons (CS) theory where the spatial slice, an infinite strip, is a manifold with boundaries. As standard ∗-products are not correct for suc h manifolds, the standard non-commutativ e CS theory is not also appropriate here. Instead w e formulate a new finite-dimensional matrix CS model as an approximation to the CS theory on the strip. A work whic h has points of contact with ours is due to Lizzi, Vitale and Zampini where the authors obtain a description for the fuzzy disc. The gauge fields in our approac h are operators supported on a subspace of finite dimension N + η of the Hilbert space of eigenstates of a simple harmonic oscillator with N , η ∈ Z + and N 6= 0. This oscillator is associated with the underlying Mo yal plane. The resultan t matrix CS model has a fuzzy edge. It becomes the required sharp edge when N and η → ∞ in a suitable sense. The non-commutativ e CS theory on the strip is defined b y this limiting procedure. After performing the canonical constrain t analysis of the matrix theory , w e find that there are edge observables in the theory generating a Lie algebra with properties similar to that of a non-abelian Kac-Mo ody algebra. Our study shows that there are ( η + 1) 2 abelian charges(observables) given b y the matrix elements ( Aˆ i ) N − 1 N − 1 and ( Aˆ i )nm (where n or m ≥ N ) of the gauge fields, that obey certain standard canonical commutation relations. In addition, the theory contains three unique non-abelian charges, localized near the Nth level. We observ e that all non-abelian edge observables except these three can b e constructed from the ( η +1) 2 abelian charges ab o ve. Using some of the results of this analysis w e discuss in detail the limit where this matrix model approximates the CS theory on the infinite strip

Noncommutative Q-lumps

Q-lumps associated with the noncommutative CP(N) model in 2+1 dimensions are constructed. These are solitonic configurations which are time dependent and rotate with constant angular frequency. Energy of the Q-lumps is E=2 pi k+alpha|Q|, and we find that in a regime in which the noncommutativity parameter theta is related to the moduli determining the size of the lumps, it can be viewed to depend on theta via the Noether charge Q. We present a collective coordinate-type analysis signalling that CP(1) Q-lumps remain stable under small radiative perturbations

Equivariant reduction of U(4) gauge theory over S-F(2) x S-F(2) and the emergent vortices

We consider a U(4) Yang-Mills theory on M x S-F(2) x S-F(2) where M is an arbitrary Riemannian manifold and S-F(2) x S-F(2) is the product of two fuzzy spheres spontaneously generated from a SU(N) Yang-Mills theory on M which is suitably coupled to six scalars in the adjoint of U(N). We determine the SU(2) x SU(2)-equivariant U(4) gauge fields and perform the dimensional reduction of the theory over S-F(2) x S-F(2). The emergent model is a U(1)(4) gauge theory coupled to four complex and eight real scalar fields. We study this theory on R-2 and find that, in certain limits, it admits vortex type solutions with U(1)(3) gauge symmetry and discuss some of their properties

A fuzzy supersymmetric non-linear sigma model

We present the ideas leading to the construction of non-linear sigma model on the fuzzy supersphere S-F((2,2)). In a recent article(1) Bott projectors have been used to obtain the fuzzy CP1 model. Here we obtain the supersymmetric extensions of these projectors and their fuzzy versions. The latter are then used W construct the non-linear sigma model on S-F((2,2)). We discuss the interpretation of the resulting model as a finite-dimensional matrix model