Non-Abelian Gauge Theories on Non-Commutative Spaces

The construction of a non-abelian gauge theory on non-commutative spaces is based on enveloping algebra-valued gauge fields. The number of independent field components is reduced to the number of gauge fields in a usual gauge theory. This is done with the help of the Seiberg–Witten map. The dynamics is formulated with a Lagrangian where additional couplings appear. Received: 9 August 2000 / Accepted: 12 August 2000     

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.     

Quaternion-Octonion Analyticity for Abelian and Non-Abelian Gauge Theories of Dyons

Einstein-Schrödinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill’s fields leading to two SU(2) gauge theories of internal spaces due to the presence of electric and magnetic charges on dyons.     

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) _e ×U(1) _m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) _e ×SU(2) _m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) _e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) _m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.     

Vacuum-Induced Abelian and Non-Abelian Gauge Potentials in CavityQuantum Electrodynamics

Gauge potential plays an important role in exploring exotic phenomena in the single- and many-body quantum systems. In this paper, we propose a scheme to create both new Abelian and non-Abelian gauge potentials by adiabatically controlling the degenerate Dicke model in cavity quantum electrodynamics. It is shown that a non-Abelian gauge potential is achieved only for a single atom, whereas an Abelianizen diagonal gauge potential is realized for the atomic ensemble. More importantly, two interesting quantum phenomena such as the geometric phase and the magnetic monopole induced by our created gauge potentials are also predicted. The possible physical realization is presented in the macroscopic circuit quantum electrodynamics with the Cooper pair boxes, which act as the artificial two-level atoms controlled by the gate voltage and the external magnetic flux.     

Non-Abelian Gauge Theory in the Lorentz Violating Background

In this paper, we will discuss a simple non-Abelian gauge theory in the broken Lorentz spacetime background. We will study the partial breaking of Lorentz symmetry down to its sub-group. We will use the formalism of very special relativity for analysing this non-Abelian gauge theory. Moreover, we will discuss the quantisation of this theory using the BRST symmetry. Also, we will analyse this theory in the maximal Abelian gauge.     

Unification of Non-Abelian SU(N) Gauge Theory and Gravitational Gauge Theory

In this paper, a general theory on unification of non-Abelian SU(N) gauge interactions and gravitationalinteractions is discussed. SU(N) gauge interactions and gravitational interactions are formulated on the similar basisand are unified in a semi-direct product group GSU(N). Based on this model, we can discuss unification of fundamentalinteractions of Nature.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD (Quantum Chromodynamics) leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of thevertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Enhancing Gauge Symmetries of Non-Abelian Supersymmetric Chern-Simons Model

In this article, we study gauge symmetries of the Non-Abelian Supersymmetric Chern-Simons model (SCS) of SU(2) group at (2+1)-dimensions in the framework of the formalism of constrained systems. Since, broken gauge symmetries in this physical system lead to the presence of nonphysical degrees of freedom, the Non-Abelian SCS model is strictly constrained to second-class constraints. Hence, by introducing some auxiliary fields and using finite order BFT method, we obtain a gauge symmetric model by converting second-class constraint to first-class ones. Ultimately, the partition function of the model is obtained in the extended phase space.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD （Quantum Chromodynamics） leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of the vertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Twisted Gauge Theories

Gauge theories on a space-time that is deformed by the Moyal–Weyl product are constructed by twisting the coproduct for gauge transformations. This way a deformed Leibniz rule is obtained, which is used to construct gauge invariant quantities. The connection will be enveloping algebra valued in a particular representation of the Lie algebra. This gives rise to additional fields, which couple only weakly via the deformation parameter θ and reduce in the commutative limit to free fields. Consistent field equations that lead to conservation laws are derived and some properties of such theories are discussed.     

Radiative Corrections to Non-Abelian Gauge Theory in Homogeneous Self-Dual Backgrounds

We study the Euclidean version of non-Abelian gauge theories in homogeneous background fields with due regard of negative and zero modes of the gauge field kernel. For general gauge group and general background but without external currents we prove a relation between the propagators belonging to different gauge fixings. Specializing to a self-dual homogeneous background we derive nonperturbatively the propagators. Next we calculate the matter field contribution to the renormalized polarization tensor using a modified dimensional regularization scheme. We prove the validity of a modified Ward identity.     

Non-Abelian Vortices in Supersymmetric Gauge Field Theory via Direct Methods

Vortices in supersymmetric gauge field theory are important constructs in a basic conceptual phenomenon commonly referred to as the dual Meissner effect which is responsible for color confinement. Based on a direct minimization approach, we present a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations governing color-charged multiply distributed flux tubes, which provide an essential mechanism for linear confinement. Over a doubly periodic domain, existence results are obtained under explicitly stated necessary and sufficient conditions that relate the size of the domain, the vortex numbers, and the underlying physical coupling parameters of the models. Over the full plane, existence results are valid for arbitrary vortex numbers and coupling parameters. In all cases, solutions are unique.     

Topological Charges in Gauge Theories

Topological and geometric aspects of gauge theories are examined. The geometry of the fiber-bundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non-singular soliton configurations is carried out.     

Quadratic Lagrangians and Topology in Gauge Theory Gravity

We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang–Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler than those found in the literature.     

Spin-2 Quantum Gauge Theories and Perturbative Gauge Invariance

In the framework of causal perturbation theory we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the S-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator Q is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein–Hilbert action plus divergences and coboundaries.     

Action Principle and Algebraic Approach to Gauge Transformations in Gauge Theories

The action principle is used to derive, by an entirely algebraic approach, gauge transformations of the full vacuum-to-vacuum transition amplitude (generating functional) from the Coulomb gauge to arbitrary covariant gauges and in turn to the celebrated Fock–Schwinger (FS) gauge for the Abelian (QED) gauge theory without recourse to path integrals or to commutation rules and without making use of delta functionals. The interest in the FS gauge, in particular, is that it leads to Faddeev–Popov ghosts-free non-Abelian gauge theories. This method is expected to be applicable to non-Abelian gauge theories including supersymmetric ones.