Superalgebras in N = 1 gauge theories

N = 1 supersymmetric gauge theories with global flavor symmetries contain a gauge invariant W-superalgebra which acts on its moduli space of gauge invariants. With adjoint matter, this superalgebra reduces to a graded Lie algebra. When the gauge group is SO(n_c), with vector matter, it is a W-algebra, and the primary invariants form one of its representation. The same superalgebra exists in the dual theory, but its construction in terms of the dual fields suggests that duality may be understood in terms of a charge conjugation within the algebra. We extend the analysis to the gauge group E₆.     

Generalized coupling constants for broken gauge symmetries in geometrical terms

A geometrical treatment of the gauge coupling constant is proposed in terms of a generalized connection form using fibre-bundle language. This extends the notion of the coupling constant to a notion of a field. The reduction of a curvature form for the generalized connection form is described in the case of a reduction of a structure group G to a subgroup H (broken gauge symmetry), and a coupling constant for the gauge group H is constructed from the corresponding one for the gauge group G.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Some issues in a gauge model of unparticles

We address in a recent gauge model of unparticles the issues that are important for consistency of a gauge theory, i.e., unitarity and the Ward identity of the physical amplitudes. We find that non-integrable singularities arise in physical quantities like the cross section and the decay rate from the gauge interactions of unparticles. We also show that the Ward identity is violated due to the lack of a dispersion relation for charged unparticles although the Ward–Takahashi identity for general Green functions is incorporated in the model. A previous observation that the contribution of the unparticle (with scaling dimension d) to the gauge boson self-energy is a factor (2−d) of the particle’s self-energy has been extended to the Green function of triple gauge bosons. This (2−d) rule may be generally true for Green functions for any number of points of the gauge bosons. This implies that the model would be trivial even as one that mimics certain dynamical effects on gauge bosons in which unparticles serve as an interpolating field.     

Chan–Paton soliton gauge states of the compactified open string

We study the mechanism of the enhanced gauge symmetry of the bosonic open string compactified on a torus by analyzing the zero-norm soliton (non-zero winding of the Wilson line) gauge states in the spectrum. Unlike the closed string case, we find that the soliton gauge state exists only at massive levels. These soliton gauge states correspond to the existence of enhanced massive gauge symmetries with transformation parameters containing both Einstein and Yang–Mills indices. In the T-dual picture, these symmetries exist only at some discrete values of compactified radii when N D-branes are coincident. Received: 14 May 1999 / Published online: 17 March 2000     

Conformal symmetry breaking and mass generation

We consider an extension of the supersymmetry formalism in order to include gauge fields. We construct a fiber bundle P(M ₄×{θ}, G) over the superspace with the gauge group as the structural group. We obtain the equations of interacting pure Yang-Mills and massless Higgs fields, considering these fields as the components of the same gauge field. Moreover, by fixing a gauge we generate a mass as a result of the supersymmetry breaking. Supported by Instituto Nacional de Investigacao Cientifica (Lisboa).     

Boundaries forSU(3) c xU(1)el lattice gauge theory with a chemical potential

It is shown forSU(N) andU(1) gauge groups that periodic spatial boundary conditions, as commonly used in lattice simulations, are not possible in the charged sectors of a local gauge theory. For charge-conjugate (C-)periodic boundary conditions the effective gauge action of fermions is derived. For nonzero chemical potential, the breakdown of translational invariance induced by the breakdown ofC symmetry is discussed. If translational invariance is abandoned, (anti)periodic spatial b.c. for fermions and for theSU(3) gauge field andC-periodic b.c. for theU(1) gauge field can be used.     

Chaotic Quantization of Classical Gauge Fields

We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum field theory emerges, and calculate Planck's constant from quantities defined in the underlying classical gauge theory.     

Local BRST Cohomology and Covariance

The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. Received: 25 July 1996 / Accepted: 23 April 1997     

On the Atiyah-Drinfeld-Hitchin-Manin construction for self-dual gauge fields

The vector spacesA, B, C, in terms of which the general construction due to Atiyah, Drinfeld, Hitchin and Manin for self-dual gauge fields defined over some region of Euclidean space is phrased, are shown to be expressible in terms of the spaces spanned by the solutions of certain linear covariant differential equations depending on the gauge field. The corresponding linear maps betweenA andB, B andC are given with the properties required by ADHM and the results then necessary to verify the construction informally proved. The local problems associated with assuming the gauge field to obey the self-duality equations are separated from the global problems of assuring the required boundary conditions for a particular solution. With suitable global conditionsC is shown to be the dual ofA and a natural scalar product defined onB so as to reconstruct the gauge field in the standard form given by the construction. A discussion is given of the requirements entailed by the condition of a symmetry on the gauge field and the relation to the usual cohomological treatment is outlined in an appendix.On leave of absence from DAMTP, Silver Street, Cambridge, England     

Gauge invariance properties of transition amplitudes in gauge theories. I

The functional approach developed earlier for scattering theory in quantum field theory makes it possible to make an explicit and complete study of the gauge invariance properties oftransition amplitudes (not just of the gauge transformations of Green's functions) in covariant and noncovariant gauges. This paper is devoted to the Abelian gauge theory of quantum electrodynamics. Using the powerful technique of functional differentiation and starting from the Coulomb gauge, the gauge invariance property of transition amplitudes,up to gauge-dependent scaling factors, isexplicitly established in arbitrary gauges. The key ingredients in the analysis are the derived exact expression for the vacuum-to-vacuum transition amplitude, introducing in the process arbitrary gauges, and the idea of stimulated emissions by external sources studied earlier.     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Geometrical approach to the reduction of gauge theories with spontaneously broken symmetry

The reduction of a theory with gauge group G to a theory which is gauge invariant with respect to a subgroup H of G is formulated in a geometrical language. It is assumed that among the physical fields considered as cross-sections of fibre bundles with structure group G there exists a section of the fibre bundle with fibre isomorphic to G/H — a Higgs field. The investigation of the broken gauge symmetry is based on the reduction theorem for structure groups of principal fibre bundles. The reduction of fields and their covariant derivatives is studied.     

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.     

Towards a quantization of gauge fields on de Sitter group by functional integral method

A formulation of the de Sitter symmetry as a purely inner symmetry defined on a fixed Minkowski space-time is presented. We define the generators of the de Sitter group and write the structure equations using a constant deformation parameter λ. The conserved gauge currents are calculated, and their physical meaning is given. Local gauge transformations and the corresponding covariant derivative depending on the gauge fields are also obtained. We study the behavior of gauge fields, the torsion and curvature tensors and give a regularization technique in terms of the ζ function.     

An alternative to the horizontality condition in the superfield approach to BRST symmetries

We provide an alternative to the gauge covariant horizontality condition, which is responsible for the derivation of the nilpotent (anti-) BRST symmetry transformations for the gauge and (anti-) ghost fields of a (3+1)-dimensional (4D) interacting 1-form non-Abelian gauge theory in the framework of the usual superfield approach to the Becchi–Rouet–Stora–Tyutin (BRST) formalism. The above covariant horizontality condition is replaced by a gauge invariant restriction on the (4,2)-dimensional supermanifold, parameterised by a set of four spacetime coordinates, x^μ(μ=0,1,2,3), and a pair of Grassmannian variables, θ and θ̄. The latter condition enables us to derive the nilpotent (anti-) BRST symmetry transformations for all the fields of an interacting 1-form 4D non-Abelian gauge theory in which there is an explicit coupling between the gauge field and the Dirac fields. The key differences and the striking similarities between the above two conditions are pointed out clearly. PACS 11.15.-q; 12.20.-m; 03.70.+k     

Reducibility and Gribov Problem in Topological Quantum Field Theory

In spite of its simplicity and beauty, the Mathai–Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: i) the existence of reducible field configurations on which the action of the gauge group is not free and ii) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai–Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt within a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson–Witten model. Received: 22 July 1996 / Accepted: 21 October 1996     

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.     

Linear bosonic quantum field theories arising from causal variational principles

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.