Double field formulation of Yang-Mills theory

Based on our previous work on the differential geometry for the closed string double field theory, we construct a Yang-Mills action which is covariant under O(D,D) T-duality rotation and invariant under three-types of gauge transformations: non-Abelian Yang-Mills, diffeomorphism and one-form gauge symmetries. In double field formulation, in a manifestly covariant manner our action couples a single O(D,D) vector potential to the closed string double field theory. In terms of undoubled component fields, it couples a usual Yang-Mills gauge field to an additional one-form field and also to the closed string background fields which consist of a dilaton, graviton and a two-form gauge field. Our resulting action resembles a twisted Yang-Mills action.     

Quantum scattering from classical field theory

We show that scattering amplitudes between initial wave packet states and certain coherent final states can be computed in a systematic weak coupling expansion about classical solutions satisfying initial-value conditions. The initial-value conditions are such as to make the solution of the classical field equations amenable to numerical methods. We propose a practical procedure for computing classical solutions which contribute to high energy two-particle scattering amplitudes. We consider in this regard the implications of a recent numerical simulation in classical SU(2) Yang-Mills theory for multiparticle scattering in quantum gauge theories and speculate on its generalization to electroweak theory. We also generalize our results to the case of complex trajectories and discuss the prospects for finding a solution to the resulting complex boundary value problem, which would allow the application of our method to any wave packet to coherent state transition. Finally, we discuss the relevance of these results to the issues of baryon number violation and multiparticle scattering at high energies.     

The theory of overdetermined linear systems and its applications to non-linear field equations

We review some aspects of the theory of overdetermined linear systems of partial differential equations and use it to interpret some non-linear equations of classical field theory as integrability conditions of linear one. In particular, it is shown that the Einstein and the Yang-Mills equations are equivalent to the existence of flat connections in affine subspaces of connections on some vector bundles, i.e. they may be written as zero-curvature conditions.     

Witten's gauge field equations and an infinite-dimensional Grassmann manifold

Witten's gauge fields are interpreted as motions on an infinite-dimensional Grassmann manifold. Unlike the case of self-dual Yang-Mills equations in Takasaki's work, the initial data must satisfy a system of differential equations since Witten's equations comprise a pair of spectral parameters. Solutions corresponding to (anti-) self-dual Yang-Mills fields are characterized in the space of initial data and in application, some Yang-Mills fields which are not self-dual, anti-self-dual nor abelian can be constructed.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Classical solutions to topologically massive Yang-Mills theory

Classical solutions to (2 + 1)-dimensional Yang-Mills theory in the presence of the Chern-Simons invariant are considered. The SO(3)-invariant solutions to the Euclidean field equations are complex, whereas the equations in Minkowski space-time possess real SO(2, 1)-invariant solutions. The field equations for time independent axially symmetric vector potentials are derived and some solutions are obtained. The behavior of general Euclidean spacetime solutions is discussed. It is also shown that, because of the gauge dependence of the Chern-Simons invariant, the reduced field equations cannot be uniquely obtained from the reduced action. Applications of the results to the infrared structure of finite temperature QCD are discussed; in particular, it is argued that the Chern-Simons invariant cannot be consistently incorporated as a gauge-invariant magnetic mass term in a three-dimensional effective long distance theory at high temperatures.     

The source galerkin method for scalar field theory

This is part I of a two-part series on the Source Galerkin method. This approach is based on the differential formulation of quantum field theory. On a finite lattice, the functional differential equations for a theory in the presence of an external source becomes a set of coupled differential equations for the generating functional Z. Systematic approximations to these equations are found using the Galerkin method. Calculations are straightforward to perform, and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. In addition, bosons and fermions are treated in a symmetric manner. In this paper, we consider power series solutions for scalar field theory in D = 2, 3,4. Propagators and mass gaps are calculated for a number of systems. The calculations in this paper were made on a work station of modest power using a fourth order polynomial expansion for lattices of size 8², 4³, 2⁴ in 2D 3D, and 4D. In part II we consider the fermionic formulation.     

De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

A quantization of field theory based on the De Donder-Weyl (DW) covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets on differential forms which were put forward for the DW theory earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DW Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DW canonical field equations are satisfied for expectation values of properly chosen operators.     

A geometric foundation for a unified field theory

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists an N-dimensional linear vector space with N5. This space is decomposed into a four-dimensional tangent space and an (N - 4)-dimensional internal space. On the basis of geometric considerations, one arrives at a number of fields, the field equations being derived from a variational principle. Among the fields obtained there are the electromagnetic field, Yang-Mills gauge fields, and fields that can be interpreted as describing matter. As a simple example, the case N=6 is considered.     

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities ² and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions E. The functions E are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. ³ These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.     

The parallel propagator as basic variable for Yang-Mills theory

The parallel propagator (associated with a Yang-Mills connection) taken along all null geodesics from a field point to null infinity is introduced as a basic variable in Yang-Mills theory. It is shown that the Yang-Mills connection can be reconstructed from this parallel propagator.The Yang-Mills equations are expressed as an equation for the parallel propagator. This equation can be given as a sum of two parts. The first of these, when set equal to zero on its own, satisfies the Huygens property and is soluble. When the second part is included, the Huygens property is destroyed. This leads to an approximation scheme which at first order is soluble yet already captures much of the non-linearity of Yang-Mills theory.     

Interpretation of Einstein's theory of gravitation including the cosmological term as a de Sitter-invariant field theory on the de Sitter space

In order to remove the singular nature of Einstein's theory of gravitation including theΛ term, within the domain of field theories, it is shown that this theory can be consistently interpreted as a field theory on the fixed de Sitter space, invariant with respect to the de Sitter groupO(4,1). The corresponding field equations as well as the equations of motion are derived, their structural properties are discussed, and they are integrated for a spherical mass source of the field.     

Dyons in topological field theories

We examine a class of topological field theories defined by Lagrangians that under certain conditions can be written as the sum of two characteristic numbers or winding numbers. Therefore, the action or the energy is a topological invariant and stable under perturbations. The sufficient conditions required for stability take the form of first-order field equations, analogous to the self-duality and Bogomol'nyi equations in Yang-Mills(-Higgs) theory. Solutions to the first-order equations automatically satisfy the full field equations. We show the existence of nontrivial, nonsingular, minimum energy spherically symmetric dyon solutions and that they are stable. We also discuss evidence for a dual field theory to Yang-Mills-Higgs in topological field theory. The existence of dual field theories and electric monopoles is predicted by Montonen and Olive.     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.     

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL _n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space.A summary of the results of the second part of this work appears in the Introduction.Most of this work was done while Darby Fellow of Mathematics at Lincoln College, Oxford     

Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

High-temperature limit of Landau-gauge Yang-Mills theory

The infrared properties of the high-temperature limit of Landau-gauge Yang-Mills theory are investigated. In a first step the high-temperature limit of the Dyson-Schwinger equations is taken. The resulting equations are identical to the Dyson-Schwinger equations of the dimensionally reduced theory, a three-dimensional Yang-Mills theory coupled to an effective adjoint Higgs field. These equations are solved analytically in the infrared and ultraviolet, and numerically for all Euclidean momenta. We find infrared enhancement for the Faddeev-Popov ghosts, infrared suppression for transverse gluons and a mass for the Higgs. These results imply long-range interactions and over-screening in the chromomagnetic sector of high-temperature Yang-Mills theory while in the chromoelectric sector only screening is observed.Received: 5 August 2004, Published online: 21 September 2004