non-Abelian Systems Research Articles

Generalizing the Coleman–Hill Theorem

Following the work of Khare et al., we show that the generalization to systems with spontaneous symmetry breaking of the Coleman–Hill theorem to one-loop order can be extended to the case including fermions with the most general interactions. Although the correction to the parity-odd part of the vacuum polarization looks complicated in the Higgs phase, it turns out that the correction to the Chern–Simons term is identical to that in the symmetric phase, with the difference coming only from the contribution of the would be Chern–Simons term. We also discuss the implication of our result to non-Abelian systems.

THE FIVE-DIMENSIONAL KEPLER PROBLEM AS AN SU(2) GAUGE SYSTEM: ALGEBRAIC CONSTRAINT QUANTIZATION

Starting from the structural similarity between the quantum theory of gauge systems and that of the Kepler problem, an SU(2) gauge description of the five-dimensional Kepler problem is given. This non-Abelian gauge system is used as a testing ground for the application of an algebraic constraint quantization scheme which can be formulated entirely in terms of observable quantities. For the quantum-mechanical reduction, only the quadratic Casimir of the constraint algebra, interpreted as an observable, is needed.

Higher grading generalisations of the Toda systems

In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher Z-grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one.

Screening in two-dimensional non-abelian vortex systems

We study charge screening in a system of two-dimensional non-abelian vortices, at finite temperature. Such vortices are generated after an SO(3) global symmetry group is spontaneously broken to a discrete subgroup Q 8 , where the latter is isomorphic to the non-abelian group of quatemions. Poisson-Boltzmann like equations are derived for the various inter-vortex potentials, and the solutions to lowest order in a small fugacity expansion, are shown to behave much like those in abelian or Coulomb charge systems. The consequences for the phase structure of the system are discussed, where the fugacities associated to the thermal production of the various types of non-abelian vortices, are shown to play a key role.

The Chern-Simons coefficient in the Higgs phase

We study one-loop corrections to the Chern-Simons coefficient ϰ in abelian self-dual Chern-Simons Higgs systems and their N=2 and N=3 supersymmetric generalizations in both symmetric and asymmetric phases. One-loop corrections to the Chern-Simons coefficient of these systems turn out to be integer multiples of 1 4 π in both phases. Especially in the maximally supersymmetric N=3 case, the correction in the symmetric phase vanishes and that in the asymmetric phase is ϰ/(2 π| ϰ|). Our results suggest that nonabelian self-dual system might enjoy similar features. We also discuss various issues arising from our results.

Chiral operator product algebra and edge excitations of a fractional quantum Hall droplet

In this paper we study the spectrum of low-energy edge excitations of a fractional quantum Hall (FQH) droplet. We show how to generate, by conformal field theory (CFT) techniques, the many-electron wave functions for the edge states. And we propose to classify the spectrum of the edge states by the same chiral operator product algebra (OPA) that appears in the CFT description of the ground state in the bulk. This bulk-edge correspondence is suggested particularly for FQH systems that support quasiparticles obeying non-abelian braid statistics, including the v = 5 2 Haldane-Rezayi state. Numerical diagonalization to count the low-lying edge states has been done for several non-abelian FQH systems, showing good agreement in all cases with the chiral OPA predictions. The specific heat of the edge excitations in those non-abelian states is also calculated.

Supersymmetry and non-Abelian Chern-Simons systems.

We study $\mathrm{SU}(n)$ $N=1$ supersymmetric Chern-Simons systems in 2+1 dimensions with and without a Maxwell term. Fixing the potential in a specific way, the supersymmetry is extended to $N=2$ which leads to a system with nontopological soliton solutions. The central charge of the extended supersymmetry is calculated and self-dual equations for the background fields are derived.

EDGE STATES IN 4D AND THEIR 3D GROUPS AND FIELDS

It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a non-chiral field theory. We study the edge states of the Abelian BF system in four dimensions (4D) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2D generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a "self-dual" or "anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field (the phrase "self-dual" here being used essentially in its sense in monopole theory). There are similar results for the non-Abelian BF system as well. These shared features of edge states in 3D and 4D suggest that the edge Lagrangians for BF systems are certain natural generalizations of field theory Lagrangians related to KM groups.

Kosterlitz-Thouless-type transition in two-dimensional non-Abelian field theory.

A renormalization-group recursion formula in a two-dimensional O(N)-invariant model is described by an integral kernel ${\mathit{K}}_{\mathit{N}}$, a probability density exhibiting perturbative asymptotic freedom for all N>1. The ${\mathit{K}}_{\mathit{N}}$'s are qualitatively the same for all N>1 and thus there may be Kosterlitz-Thouless--type transitions in some non-Abelian systems. In fact, we establish that the correlation functions of the O(4) \ensuremath{\sigma} model are bounded from below by ones of a random-coupling XY model in which the strong-coupling regions percolate over ${\mathit{Z}}^{2}$ as the original coupling constant becomes large.

Algebraic set-up for the gauge-fixing of BF and super BF systems

We construct the general gauge-fixed action for non-abelian BF systems on arbitrary dimensional manifolds. We exhibit an algebraic structure underlying the Batalin-Vilkovisky framework which permits us to obtain straighforwardly the action solution of the master equation, thus avoiding its usual iterative construction. The final gauge-fixed action is found to involve at most trilinear couplings among ghosts and antighosts. We exhibit another algebraic structure underlying the super BF systems allowing to construct automatically the corresponding off-shell nilpotent BRST operation. The gauge-fixing for these latter systems is also presented as well as possible topological invariants. Finally, we discuss the metric independence of the partition function for the non-abelian BF systems.

Massive fermions in the non-Abelian vortex field.

We obtain the electric charge of the non-Abelian vortex-fermion system using a dimensional reduction technique from (2+1)- to (1+1)-dimensional space-time. This reduction is possible if we restrict ourselves to a J=0 (s-wave) component of the fermionic wave function. We also quantize the fermion-vortex system in the J=0 channel, considering quantum fluctuation around the classical (background) vortex field. We show that in this channel, the quantum Higgs fields do not interact with ${A}_{\ensuremath{\mu}}^{q}$ (nor do the fermion fields), so this sector can be decoupled and a two-dimensional fermion-Abelian gauge-field system is obtained.

Algebraic approach to the geometry of the (super-) string model

We discuss the extension of constraint algebras to include subsidiary constraints within a larger algebra. The interplay between various mathematical aspects of this procedure is described. Tools from Lie algebra cohomology and differential geometry are used to gain new insights into BRS techniques for nonabelian constrained systems. We show that cohomology considerations restrict our formalism to non-semisimple constraint algebras, such as the (super-) string model; we illustrate the ideas by presenting concrete results for this case.

Bose-Einstein condensation in many-particle gauge theories and external charge

The realization of the charged Bose-Einstein condensation in gauge theories with finite particle density is investigated. Examples are found for a situation in which two charged Bose-Einstein condensates coexist in equilibrium in systems described by Abelian gauge theories. Also Abelian and non-Abelian systems are constructed for which a charged Bose-Einstein condensate is balanced by the charge of a fermionic subsystem. A limiting transition to infinitely heavy fermions with infinite chemical potential is pointed out, which turns the fermionic subsystem into an external charge in an Abelian theory. On the contrary, the non-Abelian gauge theory with external charge cannot be obtained by the same dynamical limiting process since it requires a special quantization scheme. It is realized for the Yang-Mills-Higgs model with a constant external charge. Eigenfrequencies of the system are found, which include an additional quantum degree of freedom induced by the external charge. The contribution of the eigenfrequencies to the thermodynamical potential is found.

Stochastic Quantization of System under Constraints through Phase Space Formulation

The Parisi-Wu stochastic quantization method is extended to a new method in the phase space. We first show that this method applied to a regular dynamical system yields the equilibrium distribution, in the stochastic process with respect to a fictitious time, which is the same formula as given by the conventional path-integral method. Next we present an extension of this phase space formulation applicable to non-Abelian gauge systems. The generalized Fokker-Planck equation gives naturally the equilibrium distribution which coincides with the Faddeev-Popov formula. The relation between this method and stochastic gauge fixing schemes is briefly discussed.

Origins of global anomalies in quantum mechanics

Simple and tractable examples of abelian and non-abelian gauge systems with global anomalies are presented in quantum mechanics. Explicit calculations are done both in the path-integral and hamiltonian formalism. Algebraic criteria are given for the existence of global gauge anomalies. These criteria are applied for every gauge group and many representations. The inconsistency of theories with a global gauge anomaly is discussed.

Renormalization Group Methods in Non-Abelian Systems

Renormalization Group Methods in Non-Abelian Systems

Absence of the Kosterlitz-Thouless fixed points in the Migdal-Kadanoff recursion formulas.

It is shown that the Migdal approximate-renormalization recursion formulas always bring the initial state into the high-temperature region for the U(1)-invariant models at the critical dimensions. This means that the Migdal recursion formulas fail to exhibit the Kosterlitz-Thouless transitions. This is also the case for the recursion formulas of Kadanoff type. The method developed here is extended to non-Abelian systems by some additional tricks.

Fractional charges for the non-Abelian fermion-monopole system.

The Dirac equation is studied in the background of a non-Abelian magnetic monopole. The inclusion of chirally rotated scalar and isotriplet masses induces a dependence of the charge and fermion number densities on the chiral angles. These densities, and the associated total charges, are computed in the limit of infinite monopole mass, and compared to the results for the Abelian monopole. Choosing an appropriate self-adjoint extension, non-Abelian fermion-monopole systems can always be modeled by Abelian ones.

Field-strength formulation of classical dynamics for spherically symmetric non-Abelian systems

We employ the conditions of spherical symmetry to study the field equations for a classical S${\mathrm{U}}_{2}$ gauge theory coupled to specified currents. To emphasize the overlap with classical electrodynamics we create a field-strength formulation employing electric and magnetic non-Abelian fields as dynamic variables. We find that the covariant field equations which are the non-Abelian generalizations of Maxwell's equations can be simplified by using gauge-invariant projections. These projections allow the residual gauge variables to be removed from the formalism and replaced by the topological current. We eliminate transverse degrees of freedom and find that the topological charge ${E}_{i}^{a}{B}_{i}^{a}$, along with the Euclidean density ${E}_{i}^{a}{E}_{i}^{a}+{B}_{i}^{a}{B}_{i}^{a}$, constitute nonlinear source terms in wave equations for radial projections of ${G}_{\ensuremath{\mu}\ensuremath{\nu}}^{a}$. The ready comparison with electrodynamics allowed by the field-strength approach clarifies several dynamical issues.

Finite element hamiltonian Monte Carlo of SU(2)

Following Bender et al., we apply the finite element method to a compact non-abelian system. We do a hamiltonian Monte Carlo on a 0+1 dimensional lattice and compare the results with those obtained by a finite difference method. For the kinetic energy, the Feynman-Hibbs prescription is followed and the finite element method is shown to be distinctly superior.