Abelian Theory Research Articles

Non-Abelian Chern-Simons coefficient in the Higgs phase

We calculate the one loop corrections to the Chern-Simons coefficient $\kappa$ in the Higgs phase of Yang-Mills Chern-Simons Higgs theories. When the gauge group is SU(N), we show, by taking into account the effect of the would be Chern-Simons term, that the corrections are always integer multiples of ${1\over 4\pi}$, as they should for the theories to be quantum-mechanically consistent. In particular, the correction is vanishing for SU(2). The same method can also be applied to the case that the gauge group is SO(N). The result for SO(2) agrees with that found in the abelian Chern-Simons theories. Therefore, the calculation provides with us a unified understanding of the quantum correction to the Chern-Simons coefficient.

Reducible connections in massless topological QCD and 4-manifolds

The role of reducible connections in non-Abelian Seiberg-Witten invariants is analyzed with massless topological QCD where the monopole is extended to the non-Abelian group version. By giving small external fields, we found that the vacuum expectation value can be separated into a part from Donaldson theory, a part from Abelian monopole theory and a part from non-Abelian monopole theory. As a by-product, we find identities of U(1) topological invariants. In our proof, the duality relation and the Higgs mechanism are not necessary.

Ward identity and dynamical mass generation in Abelian gauge theory

We develop Ward-Takahashi identities to include composite fields and utilize them to study gauge symmetry dynamical breaking in Abelian theory. In terms of the equivalence between elementary field description and the composite field description, we obtain a matching condition. The mechanism of mass generation and the mass spectra are investigated in Cornwall-Norton, Jackiw-Johnson, and Schwinger models. The gauge boson masses are in agreement with previous works; meanwhile the composite Higgs boson and decay constant ${F}_{\ensuremath{\pi}},$ which have not been obtained in previous works and other methods, can be easily obtained. It turns out that this approach can be easily applied to study not only the mass generation of fermion and gauge bosons, but also that of composite fields.

Soft dynamics and gauge theories

Infrared divergences obscure the underlying soft dynamics in gauge theories. They remove the pole structures associated with particle propagation in the various Green's functions of gauge theories. Here we present a solution to this problem. We give two equations which describe how charged particles must be dressed by gauge degrees of freedom. One follows from gauge invariance, the other, which is new, from velocity superselection rules familiar from the heavy quark effective theory. The solution to these equations in the Abelian theory is proven to lead to on-shell Green's functions that are free of soft divergences at all orders in perturbation theory.

Topology and duality in abelian lattice theories

We show how to obtain the dual of any lattice model, with inhomogeneous local interactions, based on an arbitrary Abelian group in any dimension and on lattices with arbitrary topology. It is shown that in general the dual theory contains disorder loops on the generators of the cohomology group of a particular dimension. An explicit construction for altering the statistical sum to obtain a self-dual theory, when these obstructions exist, is also given. We discuss some applications of these results, particularly the existence of non-trivial self-dual 2-dimensional Z N theories on the torus. In addition, we explicitly construct the n -point functions of plaquette variables for the U (1) gauge theory on the 2-dimensional g -tori.

The Brauer Group of a Braided Monoidal Category

Representation theory of finite groups prompted the introduction of the Brauer group of a field and also of its Schur subgroups. The theory of quadratic forms and spaces introduced the Zr2Z-graded variant of the Brauer group together with the typical Clifford algebras and related w x representations; cf. the work of Wall 29 . The extension of the Zr2Zgraded theory to gradings by other cyclic groups and consequently to abelian groups, replacing the usual Clifford algebras by the so-called generalized Clifford algebras, is then natural from the algebraic point of view. However, extension to nonabelian groups did not seem to be possible, at least not by simple modifications of the abelian theory. At the same time, the generalization of the Brauer group of graded algebras obtained w x Ž . by Long 12, 13 made use of Hopf algebras but again cocommutativity conditions were present. The recent interest in quantum groups motivated the authors to introduce the Brauer group of a quantum group as the Ž Brauer group of crossed module algebras also called quantum Yang] . w x Baxter module algebras in 4, 5 . The use of the category of crossed w x modules originated in the case of group algebras; cf. Whitehead 30 . Ž . Module algebras or coalgebras of this crossed type were introduced by w x Radford in 24 and used by Majid in the description of modules over Drinfel’d quantum doubles. Ž . The noncocommutativity aspects of the theory concerning the Brauer group in terms of crossed module algebras do not have a counterpart in classical theory. Note that a first unifying categorical theory has been w x obtained by Pareigis in 23 ; this theory deals with the Brauer group of a

Ward identities for Yang-Mills theory in Abelian gauges: Abelian dominance at high energies

We consider Yang-Mills theory in a general class of Abelian gauges. Exploiting the residual Abelian symmetry on a quantum level, we derive a set of Ward identities in functional form, valid to all orders in perturbation theory. As a consequence, the coupling constant is only renormalised through the Abelian two-point function. This implies that asymptotic freedom in all Abelian gauges can be understood from an effective Abelian theory alone, which can be interpreted as Abelian dominance in the high energy regime.

Monopole percolation in the compact Abelian Higgs model

We have studied the monopole-percolation phenomenon in the four dimensional Abelian theory that contains compact U(1) gauge fields coupled to unitary norm Higgs fields. We have determined the location of the percolation transition line in the plane $({\ensuremath{\beta}}_{g},{\ensuremath{\beta}}_{H}).$ This line overlaps the confined-Coulomb and the confined-Higgs phase transition lines, originated by a monopole-condensation mechanism, but continues away from the end point where this phase transition line stops. In addition, we have determined the critical exponents of the monopole percolation transition away from the phase transition lines. We have performed the finite size scaling in terms of the monopole density instead of the coupling, because the density seems to be the natural parameter when dealing with percolation phenomena.

Asymptotic Padé approximant predictions: Up to five loops in QCD and SQCD

We use Asymptotic Pade Approximants (APAP's) to predict the four- and five-loop \beta-functions in QCD and N=1 supersymmetric QCD (SQCD), as well as the quark mass anomalous dimensions in Abelian and non-Abelian gauge theories. We show how the accuracy of our previous \beta-function predictions at the four-loop level may be further improved by using estimators weighted over negative numbers of flavours (WAPAP's). The accuracy of the improved four-loop results encourages confidence in the new five-loop \beta-function predictions that we present. However, the WAPAP approach does not provide improved results for the anomalous mass dimension, or for Abelian theories.

Non-abelian gravity and antisymmetric tensor gauge theory

A non-abelian generalisation of a theory of gravity coupled to a 2-form gauge field and a dilaton is found, in which the metric and 3-form field strength are Lie algebra-valued. In the abelian limit, the curvature with torsion is self-dual in four dimensions, or has SU( n) holonomy in 2 n dimensions. The coupling to self-dual Yang-Mills fields in 4 dimensions, or their higher dimensional generalisation, is discussed. The abelian theory is the effective action for (2,1) strings, and the non-abelian generalisation is relevant to the study of coincident branes in the (2,1) string approach to M-theory. The theory is local when expressed in terms of a vector prepotential.

Infrared Stable Supersymmetry in Chern-Simons Theories with Matter and Quenched Disorder

We study Abelian Chern-Simons field theories with matter fields and global $\mathrm{SU}(N)$ symmetry in the presence of random weak quenched disorder. In the absence of disorder, these theories possess $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ supersymmetric fixed points and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ supersymmetric fixed lines in the infrared limit. We show that, although the presence of disorder forbids any supersymmetry of the bare action, infrared stable supersymmetric fixed points and fixed lines are realized in the disorder-averaged effective theories.

Constraining differential renormalization in abelian gauge theories

We present a procedure of differential renormalization at the one loop level which avoids introducing unnecessary renormalization constants and automatically preserves abelian gauge invariance. The amplitudes are expressed in terms of a basis of singular functions. The local terms appearing in the renormalization of these functions are determined by requiring consistency with the propagator equation. Previous results in abelian theories, with and without supersymmetry, are discussed in this context.

Abelian Chern-Simons theory and quantum symmetry

Abstract Quantized abelian Chern-Simons theory on a Riemann surface of arbitrary nonzero genus g carries a quantum symmetry; the underlying quantum algebra is the deformation U q ( sl g+1 ), with q a root of unity.

Coexistence of Vortices and Antivortices in an Abelian Gauge Theory

We present an Abelian field theory model that allows the coexistence of vortices and antivortices. Some of the interesting properties are that the magnetic flux is proportional to the difference of the numbers of vortices and antivortices but the energy is proportional to the sum of these numbers. In the context of cosmology, these vortices represent cosmic strings and antistrings with opposite magnetic charges. In addition to the above properties, we show that the total quantized curvature is proportional to the sum of the numbers of strings and antistrings which also determines the completeness of the gravitational metric.

Moving frames hierarchy and BF theory

We show that the one-dimensional projection of Chern–Simons gauged nonlinear Schrödinger model is equivalent to an Abelian gauge field theory of a continuum Heisenberg spin chain. In such a theory, the matter field has geometrical meaning of coordinates in tangent plane to the spin phase space, while the U(1) gauge symmetry relates to rotation in the plane. This allows us to construct the infinite hierarchy of integrable gauge theories and related magnetic models. To each of them a U(1) invariant gauge fixing constraint of non-Abelian BF theory is derived. The corresponding moving frames hierarchy is obtained and the spectral parameter is interpreted as a constant-valued statistical gauge potential constrained by the 1-cocycle condition.

Abelian Chern–Simons theory. II. A functional integral approach

Following Witten, [Commun. Math. Phys. 21, 351–399 (1989)] we approach the Abelian quantum Chern–Simons (CS) gauge theory from a Feynman functional integral point of view. We show that for 3-manifolds with and without a boundary the formal functional integral definitions lead to mathematically proper expressions that agree with the results from the rigorous construction [J. Math. Phys. 39, 170–206 (1998)] of the Abelian CS topological quantum field theory via geometric quantization.

Confinement in partially broken abelian Chern-Simons theories

Planar Chern-Simons (CS) theories in which a compact abelian gauge group U(1) x U(1) is spontaneously broken to U(1) x Z_N are investigated. Among other things, it is noted that the theories just featuring the mixed CS term coupling the broken to the unbroken U(1) gauge fields in general exhibits an interesting form of confinement: only particles carrying certain multiples of the fundamental vortex flux unit and certain multiples of the fundamental charge of the unbroken U(1) gauge field can appear as free particles. Adding the usual CS term for the broken U(1) gauge fields does not change much. It merely leads to additional Aharonov-Bohm interactions among these particles. Upon introducing the CS term for the unbroken U(1) gauge fields, in contrast, the confinement phenomenon completely disappears.

Massive Quantum Vortex Excitations in a Pure Gauge Abelian Theory in 2 + 1D

We introduce and study a pure gauge Abelian theory in 2 + 1D in which massive quantum vortex states do exist in the spectrum of excitations. This theory can be mapped in a three-dimensional gas of point particles with a logarithmic interaction, in the grand-canonical ensemble. We claim that this theory is the 2 + 1D analog of the sine–Gordon, the massive vortices being the counterparts of sine–Gordon solitons. We show that a symmetry breaking, order parameter, similar to the vacuum expectation value of a Higgs field does exist.

Fermions in an external SU( 2) magnetic field

We consider fermions in a constant and uniform external SU(2) magnetic field. We find that the results for the energy levels depend on the choice of gauge. Choosing a Landau-type gauge yields results analagous to those of Abelian theories. On the other hand, using another gauge potential one obtains a different continuous eigenvalue spectrum.

Scaling and confinement aspects of tadpole-improved SU(2) lattice gauge theory and its Abelian projection

Using a tadpole improved SU(2) gluodynamics action, the nonabelian potential and the abelian potential after the abelian projection are computed. Rotational invariance is found restored at coarse lattices both in the nonabelian theory and in the effective abelian theory resulting from maximal abelian projection. Asymptotic scaling is tested for the SU(2) string tension. Deviation of the order of $6%$ is found, for lattice spacings between 0.27 and 0.06 fm. Evidence for asymptotic scaling and scaling of the monopole density in maximal abelian projection is also seen, but not at coarse lattices. The scaling behavior is compared with analyses of Wilson action results, using bare and renormalized coupling schemes. Using extended monopoles, evidence is found that the gauge dependence of the abelian projection reflects short distance fluctuations, and may thus disappear at large scales.