Lagrangian interactions within a special class of covariant mixed-symmetry type tensor gauge fields

Consistent non-trivial interactions within a special class of covariant mixed-symmetry type tensor gauge fields of degree three are constructed from the deformation of the solution to the master equation combined with specific cohomological techniques. In spacetime dimensions strictly greater than four, the only consistent interaction terms are those gauge invariant under the original symmetry. Only in four spacetime dimensions the gauge symmetry is found to be deformed. Received: 21 November 2002 / Revised version: 9 December 2002 / Published online: 14 February 2003     

Non-self-dual gauge fields

The Ward construction is generalized to non-self-dual gauge fields. Reality and currentless conditions are specified.     

On symmetric gauge fields

The subgroups of the symmetry group of the gauge invariant Lagrangian are studied. For given subgroupG theG-invariant gauge fields are listed.     

Geometry of gauge fields in a multidimensional universe

Let S be a group of automorphisms of a principal fibre bundle (U, , E, R), both groups S and R being compact. Let I (resp. H) be the isotropy group of S (resp. S x R) acting on E (resp. U), and let N(I) (resp. N(H)) be the normalizer of I (resp. H) in S (resp. S x R). We construct two principal bundles P(M, N(I)|I) E and Q(M, N(H)|H) U, where M=E/S is the space of orbits of S in E, and we prove that, given a connection A in P, there is a one-to-one correspondence between S-invariant connections in U and triples (B, , ), where B is a connection in Q, part of which is a pullback ^* A of A, and , are scalars which are crosssections of certain vector bundles associated with Q. The resulting final gauge group N(H)|H is shown to contain as a normal subgroup the centralizer of I in R, known from earlier works of other authors. A dimensional reduction of the Einstein-Yang-Mills system on E is briefly discussed.Partially supported by the Polish Ministry of Science, Higher Education and Technology under the Project MRI-7.     

Topology of lattice gauge fields

Non-Abelian gauge fields on a four-dimensional hypercubic lattice with small action density [Tr{U( )} for SU(2) gauge fields] are shown to carry an integer topological chargeQ, which is invariant under continuous deformations of the field. A concrete expression forQ is given and it is verified thatQ reduces to the familiar Chern number in the classical continuum limit.Work supported in part by Schweizerischer Nationalfonds     

The geometry of gauge fields

Principal fibre bundles with connections provide geometrical models of gauge theories. Bundles allow for a global formulation of gauge theories: the potentials used in physics are pull-backs, by means of local sections, of the connection form defined on the total spaceP of the bundle. Given a representationP of the structure (gauge) groupG in a vector spaceV, one defines a (generalized) Higgs field as a map fromP toV, equivariant under the action ofG inP. If the image of is an orbitW V ofG, then a breaks (spontaneously) the symmetry: the isotropy (little) group ofw ₀ W is the unbroken groupH. The principal bundleP is then reduced to a subbundleQ with structure groupH. Gravitation corresponds to a linear connection, i.e. to a connection on the bundle of frames. This bundle has more structure than an abstract principal bundle: it is soldered to the base. Soldering results in the occurrence of torsion. The metric tensor is a Higgs field breaking the symmetry fromGL (4,R) to the Lorentz group.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.Work on this paper was supported in part by the Polish Research Programme MR. I. 7.This paper is based in part on the research done in 1976–77 when I was Visiting Professor at the State University of New York at Stony Brook. I thankChen Ning Yang for encouragement, discussions and hospitality at the Institute for Theoretical Physics, SUSB. I have also learned much from conversations with D. Z.Freedman, A. S.Goldhaber, P.van Nieuwenhuizen, J.Smith, P. K.Townsend, W. I.Weisberger, and D.Wilkinson.     

Analytic structure of gauge fields

The analytic structure of gauge fields in the presence of fermions is studied in arbitrary symmetry. A Hamiltonian formalism is developed which relates Cauchy-Riemann equations to the symmetry. The formalism is applied to three problems in (2+1)-dimensional Euclidean space: (1) a free fermion, (2) a fermion interacting with a massless scalar field, and (3) a fermion interacting with a vector field. We find that the Hamiltonian for the free fermion is analytic and single-valued in a finite region of momentum space. With the addition of an auxiliary field, the Hamiltonian can be analytic in the entire momentum space. The scalar field then acquires spin-dependent coordinates by interaction with the fermion; the interactions break the Abelian symmetry of so that ₁ 1/(x₁ –-im ₁ ^–1 (x₁ –-im ₁ ^–1 ), wherem ₁ are spin-dependent and multivalued. There are four solutions for each chirality eigenvalue of the fermion. For spinless fermions gives the Jackiw-Nohl-Rebbi solution and is separable into Coulomb-like 1/x analytic functions on the first and fourth quadrants. For a vector field the results are similar except that the coordinates are not spindependent or multivalued; interactions break the initial symmetry andA (x )A ¹ (x ) and theA ¹ have a non-Abelian algebra. Thel indices represent directions fixed by spin matrices in a spin-dependent color space.     

Linear perturbations of gauge fields

It is shown that under certain weak conditions (the vanishing of the field strength along a family of self-dual or anti-self-dual geodesic two-surfaces), in a curved or flat space-time, the linear perturbations of a given gauge field configuration can be expressed in terms of the solutions of a single second-order linear partial differential equation for a matrix potential. The particular case of the self-dual gauge fields is treated in some detail.     

The radiation gauge static potential for non-abelian gauge fields

The static potential between a fermion and an anti-fermion in a group singlet state is calculated, through two loops, in the radiation gauge first order formalism. The results of this calculation imply that the Coulomb propagator is not sufficient to determine the static potential: a new function of the coupling constant α_s(?t) is also required.     

Analytic loops and gauge fields

In this paper the linear representations of analytic Moufang loops are investigated. We prove that every representation of semisimple analytic Moufang loop is completely reducible and find all nonassociative irreducible representations. We show that such representations are closely associated with the (anti-)self-dual Yang–Mills equations in R⁸.     

Factorizations for self-dual gauge fields

For a particular class of patching matrices onP ₃(?), including those for the complex instanton bundles with structure group Sp(k,?) orO(2k,?), we show that the associated Riemann-Hilbert problemG(x, λ)=G?(x, λ)·G ₊ ^?1 (x, λ) can be generically solved in the factored formG _?=φ ₁ φ ₂.....φ _n . IfГ=Г _n is the potential generated in the usual way fromG _?, and we setψ _i =φ ₁.....,φ _i withψ _n =G _?, then eachψ _i also generates a selfdual gauge potentialΓ _i . The potentials are connected via the “dressing transformations” $$\Gamma _\iota = \phi _i^{ - 1} \cdot \Gamma _{\iota - 1} \cdot \phi _i + \phi _i ^{ - 1} D\phi _i$$ of Zakharov-Shabat. The factorization is not unique; it depends on the (arbitrary) ordering of the poles of the patching matrix.     

From points to gauge fields

We show that several classes of free field theories with local gauge invariance (e.g. the Yang-Mills, Einstein and p-forms linearized actions) can be constructed from classical actions for a finite number of points by applying the BRST quantization initiated by Siegel. We briefly outline the generalization of this construction for strings.     

Equivalence principles and gauge fields

We investigate the implication of the week equivalence principles and Eötvös-Dicke experiments for gauge fields in a general framework. In particular, we show that the Galileo weak equivalence principle (WEP[I]) implies the Einstein equivalence principle (EEP) with one exception; however, the second weak equivalence statement (WEP[II]) implies EEP. For the exceptional case, there are anomalous torques on polarized test bodies. As an example, we apply our results to quantum chromodynamics.     

Topology and lattice gauge fields

A general discussion of the topology of continuum gauge fields and the problems involved in defining and computing the topology of a lattice gauge field configuration is given. Two definitions of the topological charge for 4-dimensional SU(n) lattice gauge theory are presented. The first of these is geometrically the most straightforward, the second the most useful for efficient numerical calculations.     

Charge conjugation of non-Abelian gauge fields

Questions associated with the determination of the charge conjugation in non-Abelian gauge fields are investigated. It is shown that the properties of gauge groups allow the charge conjugation to be simply determined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 62–64, April, 1982.     

Non-standard gauge-boson self-interactions within a gauge invariant model

We examine dimension-six extensions of the standard electroweak Lagrangian which are invariant under localSU(2) _L ×U(1) _Y -transformations. The dimensionfour trilinear and quadrilinear effective interactions of the vector bosons with one another are found to coincide with the vector boson interactions previously derived from globalSU(2) weak isospin symmetry broken by electromagnetism. Supplementing the model by a well-known dimension-six single-parameter quadrupole interaction leads to the most general vector boson self-couplings that can be obtained by addition of dimension-six terms to the standard Lagrangian. We examine in some detail anotherSU(2) _L ×U(1) _Y -symmetric interaction which containsW ₃ B mixing and modifies both vector boson self-couplings and fermionic interactions. Independently of being strongly constrained by the LEP 1 data, the addition of this interaction to the above-mentioned non-standard ones does not change the form of the trilinear and quadrilinear non-standard self-couplings of the vector bosons. Therefore, while being interesting in itself with respect to LEP 1 physics, this term is irrelevant with respect to the phenomenology of the vector-boson self-interactions.