Lattice gauge fields and noncommutative geometry

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified space-time or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD θ-term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.     

Two questions on the geometry of gauge fields

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.     

Unified geometry of antisymmetric tensor gauge fields and gravity

Vector gauge fields are known to be related to connections on principal fibre bundles P(M,G) over space-time M. Here we relate abelian antisymmetric tensor gauge fields of rank r to constrained connections on P(Ω_r?1M,U(1)), the U(1) bundles over the space of all (r?1)-dimensional closed submanifolds of M. In particular, the Kalb-Ramond field (r=2) is seen as such a connection on the bundle over the space of all loops in space-time. Moreover, a Kaluza-Jordan construction on P(Ω₁M,U(1)) leads to the bosonic sector of the N = 1 supergravity action in ten dimensions. This result is highly reminiscent of the α′→ 0 limit of the closed string dual model.     

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.     

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.     

Classification of gauge fields: The Eigenspinor-Eigenvalue equation

The eigenspinor-eigenvalue equation for SU(2) gauge fields is presented. A classification of gauge fields is consequently obtained. The whole classification is described in terms of a diagram.     

The riemannian geometry of the configuration space of gauge theories

We state some new results about the configuration space of pure Yang-Mills theory. These results come from the study of the kinetic energy term of the Lagrangian of the theory. This term defines a riemannian metric on the space of non-equivalent gauge potentials. We develop a riemannian calculus on the configuration space, compute the riemannian connection, the curvature tensor, and solve for the geodesics, etc. We show that the Gribov ambiguity is more than an artefact of the choice of a gauge condition, and is related to the existence of conjugate points on the geodesics, and is thus an intrinsic feature of the theory.Laboratoire associé au CNRS     

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     

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 computation of characteristic classes of lattice gauge fields

AGL(p,C)-valued lattice gauge fieldu on a simplicial complex determines a principalGL(p,C)-bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters. A representative cocyclec _q for theq ^th Chern class of can be computed on each 2q-simplex by takingc _q() to be the intersection number of a certain singular 2q-cubeM with a Schubert-type variety _q in the space of allp×p matrices. This reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling-type procedures. An application of this method is suggested for the computation of the topological charge of anSU(3)-valued lattice gauge field on a 4-complex.Partially supported by NSF grant DMS 8607168Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

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.     

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.     

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.     

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.