Abelian Gauge Group Research Articles

Unraveling Spontaneously Broken Abelian Chern-Simons Theories

In this talk I describe recent work (hep-th/9606029) in which I classified all conceivable 2+1 dimensional Chern-Simons (CS) theories with continuous compact abelian gauge group or finite gauge group. The CS theories with finite abelian gauge group that can be obtained from the spontaneous breakdown of a CS theory with gauge group the direct product of various compact U(1) gauge groups were also identified. Those that can not be reached in this was are actually the most interesting since they lead to nonabelian pheomena such as nonabelian braid statistics, Alice fluxes and Cheshire charges and quite generally lead to dualities with 2+1 dimensional theories with a nonabelian finite gauge group.

(Spontaneously broken) Abelian Chern-Simons theories

A detailed analysis of Chem-Simons (CS) theories in which a compact Abelian direct product gauge group U(1) k is spontaneously broken down to a direct product of cyclic groups H ⋍ Z N (1) × … x Z N (k) is presented. The spectrum features global H charges, vortices carrying magnetic flux labeled by the elements of H and dyonic combinations. Due to the Aharonov-Bohm effect these particles exhibit topological interactions. The remnant of the U(1) k CS term in the discrete H gauge theory describing the effective long distance physics of such a model is shown to be a 3-cocycle for H governing the non-trivial topological interactions for the magnetic fluxes implied by the U(1) k CS term. It is noted that there are in general three types of 3-cocycles for a finite Abelian gauge group H: one type describes topological interactions between vortices carrying flux with respect to the same cyclic group in the direct product H, another type gives rise to topological interactions among vortices carrying flux with respect to two different cyclic factors of H and a third type leading to topological interactions between vortices carrying flux with respect to three different cyclic factors. Among other things, it is demonstrated that only the first two types can be obtained from a spontaneously broken U(1) k CS theory. The 3-cocycles that cannot be reached in this way turn out to be the most interesting. They render the theory non-Abelian and in general lead to dualities with planar theories with a non-Abelian finite gauge group. In particular, the CS theory with finite gauge group H ⋍ Z 2 × Z 2 × Z 2 defined by such a 3-cocycle is shown to be dual to the planar discrete D 4 gauge theory with D 4 the dihedral group of order 8.

Mixed (open/closed) N = (2, 2) string theory as an integrable deformation of self-duality

The exact effective field equations of motion, corresponding to the perturbative mixed theory of open and closed (2,2) world-sheet supersymmetric strings, are investigated. It is shown that they are only integrable in the case of an abelian gauge group. The gravitational equations are then stationary with respect to the Born-Infeld-type effective action.

Dual Higgs model with dual Dirac strings

We investigate a dual Higgs model with dual Dirac strings invariant under U(1) gauge transformations, where U(1) is the Abelian gauge group of dual-vector potential transformations spontaneously broken by the Higgs mechanism. The action of a dual Dirac string calculated in leading order in the large Higgs field mass expansion coincides with a dual version of that obtained by Nambu. For the example of the string tension at the classical tree-level approach we develop a procedure for the computation of contributions of string-string interactions. Non-perturbative phenomena of compact QED are discussed.

Generalized free field-like U(1)-Gauge theories within the wightman framework

Generalized free field-like U(1)-gauge theories are considered as an application of the Wightman theory to some special gauge field theories within the Borchers-Uhlmann approach to general QFT (without linear program). Along these lines, conditions upon the pseudo-Wightman functional are given such that the reconstructed indefinite-metric space D of state vectors decomposes into the direct sum of charge sectors labelled by the elements of the Pontrjagin dual group of a compact and Abelian gauge group G . All these gauge groups G are characterized in such a way that the positive definiteness of states of the vacuum sector holds. Furthermore, observable fields not carrying any charges as well as charge carrying fields are given. Finally, an example covering charged scalar free fields of mass m > 0 is discussed within that context.

Supersymmetry and gauge invariance constraints in a U(1) x U(1)'-Higgs superconducting cosmic string model.

A supersymmetric extension of the $U(1)\times U(1)^{\prime }$-Higgs bosonic superconducting cosmic string model is considered,and the constraints imposed upon such a model due to renormalizability, supersymmetry, and gauge invariance are examined. For a simple model with a single $U(1)$ chiral superfield and a single $% U(1)^{\prime }$ chiral superfield, the Witten mechanism for bosonic superconductivity (giving rise to long range gauge fields outside of the string) does not exist. The simplest model that can accommodate the requisite interactions requires five chiral supermultiplets. This superconducting cosmic string solution is investigated.

Magnetic translation groups as group extensions

Extensions of a direct product T of two cyclic groups Z N 1 and Z N 2 by an Abelian (gauge) group G with the trivial action of T on G are considered. All possible (inequivalent) factor systems are determined using the Mac Lane method. Some of the resulting groups describe magnetic translation groups. Extensions with G = U(1) and G = Z N are considered and discussed as examples.

Monopoles and four-manifolds

Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.

Color confinement, abelian gauge and renormalization group flow

Under the assumption that the color charge can be written in a BRST exact form, the color confinement mechanism proposed by Kugo and Ojima (KO) explains the confinement of any colored particles including dynamical quarks and gluons. This mechanism, however, is known to break down in the abelian gauge which treats the maximal abelian subgroup of the gauge group in a special manner. In order to study whether the failure of the KO mechanism is particular only to the abelian gauge or whether this failure occurs in a wide class of gauges including the ordinary Lorentz type guage, we carry out a renormalization group study of the SU(2) gauge theory in the gauge fixing space. Our gauge fixing space consists of four distinct regions that are not connected with each other by renormalization group flows, and we find that the abelian gauge is infrared unstable in three regions which include the Lorentz type gauge. This suggests that the failure of the KO mechanism is a phenomenon which occurs only in the abelian gauge. We also find that the Lorentz gauge is infrared stable.

Parafermions from coset models

Conformally invariant coset models in two dimensions can be described by the gauged Wess-Zumino-Witten lagrangian. We present a semi-classical treatment of these theories in a unitary gauge. In particular, we give an explicit construction of the conserved parafermion fields in the case of an abelian gauge group, and also show that the Poisson brackets between these field close in a simple fashion. We make use of these results to find a free field realization of this Poisson structure.

THE ADLER-BARDEEN THEOREM FOR THE AXIAL U(1) ANOMALY IN A GENERAL NON-ABELIAN GAUGE THEORY

A general, regularization-scheme-independent proof of the nonrenormalization theorem for the anomaly of a U(1) axial current in a renormalizable gauge theory is presented. The gauge group may be an arbitrary compact Lie group. The validity of the theorem is traced back to some finiteness properties allowing for a well defined but particular choice of the anomaly operators. Whereas in the case of a purely Abelian gauge group this choice amounts to a physically reasonable normalization at zero energy, the general non-Abelian case awaits a deeper understanding.

The gravitational field at spatial infinity

This paper treats the formulation of the gravitational field variables and the equations obeyed by them at spatial infinity. The variables consist of a three-dimensional tensor and a scalar, which satisfy separate field equations, which in turn can be obtained from two distinct Lagrangians. Aside from Lorentz rotations, the symmetry operations include an Abelian gauge group and an Abelian Lie group, leading to a number of conservation laws and to differential identities between the field equations.

Cluster properties and bound states of the Yang-Mills model with compact gauge group. I

The authors obtain the cluster structure of the gauge-invariant part of the transfer matrix and the higher branches of its spectrum are investigated for a lattice Yang-Mills model with compact Abelian gauge group.

Decoupling inCP n−1 models with fermions

We investigate fermions coupled in various ways to the bosons of two-dimensionalCPn−1 models with Abelian or non-Abelian gauge groups. There always exist chiral transformations which effectively decouple the fermion and boson sectors of these theories and which allow the fermions to be treated exactly. The remaining effective bosonic theory, which in the non-Abelian case can only be displayed somewhat implicitly, may be treated by using some approximation scheme like the 1/n expansion.

Confinement of static quarks in two dimensional lattice gauge theories

In two dimensional Higgs models on a lattice with Abelian or Nonabelian compact gauge group, fractional static charges are always confined.

Hamiltonian treatment of the spherically symmetric einstein-Yang-Mills system

A canonical formalism of the dynamics of interacting spherically symmetric Yang-Mills and gravitational fields is presented. The work is based on Dirac's technique for constrained hamiltonian systems. The gauge freedom of the Yang-Mills field is treated in the same footing with the coordinate transformation freedom of the gravitational field. In particular, the fixation of coordinates and the fixation of the internal gauge are achieved by totally similar techniques. Two classes of spherically symmetric motions are considered: (i) the class for which the Yang-Mills potentials themselves are spherically symmetric (“manifest spherical symmetry”). In this case the results are valid for an arbitrary gauge group; and (ii) the class for which, in the SO(3) gauge group, a rotation in physical space is compensated by a rotation of equal magnitude but opposite direction in isospin space (“spherical symmetry up to a gauge transformation”). For manifest spherical symmetry the problem amounts to effectively dealing with an abelian gauge group and the most general solution of the field equations turns out to be the Reissner-Nordström metric with a Coulomb field. For spherical symmetry up to a gauge transformation the problem is more interesting. the formalism contains then, besides the gravitational variables, three pairs of functions of the radial coordinate that describe the degrees of freedom of the Yang-Mills field. Two pairs of these functions can be combined into a complex field ψ and its conjugate. The hamiltonian is then invariant under r-dependent rotations in the complex ψ-plane. The third degree of freedom plays the role of a compensating field associated with this invariance under localized U( l) rotations. The compensating field can always be brought to zero by a gauge transformation. After this is done the gauge is completely fixed but the problem remains invariant under position independent rotations in the ψ plane. Static solutions of the field equations in this gauge are of the form ψ( r) = ϱ( r) exp ( iΘ) with Θ independent of position. The particular case Θ = 0 corresponds to the Wu-Yang ansatz. A nontrivial static solution can be found in closed form. The Yang-Mills field is of the generalized Wu-Yang type with an extra electric term, and the metric is the Reissner-Nordström one. It is pointed out that a Higgs field can be easily introduced in the formalism. The addition of the Higgs field does not destroy the invariance of the Hamiltonian under r-dependent rotations in the ψ-plane. The conserved quantity associated with the invariance under ψ → exp ( i(const)) ψ coincides with the electric charge as defined by 't Hooft in a more general context.

Spontaneous breaking of supersymmetry in non-abelian gauge theories

The Higgs mechanism gives at present the only reliable method of spontaneous symmetry breaking in quantum field theory. However, straightforward application of this mechanism to supersymmetric theories meets certain difficulties due to specific form of the supersymmetric potential. Fayet and Illipoulosl succeeded to solve the problem for the case of the supersymmetric theory invariant with respect to Abelian gauge group (see also2). Unfortunately this method is not directly applicable to the case of semi-simple gauge group which is the most interesting from the practical point of view. In particular it does not allow one to construct a model which is simultaneously infra-red convergent and asymptotically free, for which the supersymmetric Yang-Mills theory is a natural candidate.

Compact gauge fields and the infrared catastrophe

It is shown that infrared phenomena in the gauge theories are guided by certain classical solutions of the Yang-Mills equations. The existence of such solutions can lead to a finite correlation length which stops infrared catastrophe. In the present paper we deal only with theories with a compact but abelian gauge group. In this case the problems of correlation length and charge confinement are completely solved.

Spontaneous symmetry breakdown and Regge intercepts

By Regge slope methods we exhibit a Nambu-Goldstone spontaneous symmetry breakdown and a Higgs mechanism inherent in a unit-intercept dual meson amplitude. Introducing an Abelian gauge group corresponding to conservation of a pseudocharge, Regge intercept shifts are calculated; to lowest order the leading Regge intercept of the Neveu-Schwarz model is lowered from α(0) = 1 to α(0) = 1 2 . Our method has the advantage over Bardakci's spurion method that massless scalar are not required at any stage.

A Theory of SpontaneousTViolation

A theory of spontaneous $T$ violation is presented. The total Lagrangian is assumed to be invariant under the time reversal $T$ and a gauge transformation (e.g., the hypercharge gauge), but the physical solutions are not. In addition to the spin-1 gauge field and the known matter fields, in its simplest form the theory consists of two complex spin-0 fields. Through the spontaneous symmetry-breaking mechanism of Goldstone and Higgs, the vacuum expectation values of these two spin-0 fields can be characterized by the shape of a triangle and their quantum fluctuations by its vibrational modes, just like a triangular molecule. $T$ violations can be produced among the known particles through virtual excitations of the vibrational modes of the triangle which has a built-in $T$-violating phase angle. Examples of both Abelian and non-Abelian gauge groups are discussed. For renormalizable theories, all spontaneously $T$-violating effects are finite. It is found that at low energy, below the threshold of producing these vibrational quanta, $T$ violation is always quite small.