D.B. Fairlie

First-order equations for gauge fields in spaces of dimension greater than four

Abstract We consider the possibility of satisfying the gauge field equations in dimensions greater than four by imposing linear relations amongst the components of the field strength tensor, F μν , generalising the idea of self-duality in four dimensions.

Trigonometric structure constants for new infinite-dimensional algebras

Abstract Novel infinite-dimensional algebras of the Virasoro/Kac-Moody/Floratos-Iliopoulos type are introduced, which involve trigonometric functions in their structure constants. They are then supersymmetrized, and relevant features of them are explored. An associated “lazy tongs” formulation of the SU (2) Kac-Moody algebra is also given.

Features of Time-independent Wigner Functions

The Wigner phase-space distribution function provides the basis for Moyal{close_quote}s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional ({open_quotes}star{close_quotes}) eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ({open_quotes}supersymmetric{close_quotes}) isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P{umlt o}schl-Teller potential, and the Liouville potential. {copyright} {ital 1998} {ital The American Physical Society}

The construction of self-dual solutions to SU(2) gauge theory

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansätze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansätze. There is a six-parameter family of these Bäcklund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.

Infinite-dimensional algebras, sine brackets, and SU(∞)

Abstract We investigate features of the infinite dimensional algebras we have previously introduced, which involve trigonometric functions in their structure constants. We find a realization for them which leads to a basis-independent formulation. A special family of them, the cyclotomic ones, contain SU ( N ) as invariant subalgebras. In this basis, it is evident by inspection that the algebra of SU(∞) is equivalent to the centerless algebra of SDiff 0 on two-dimensional manifolds. Gauge theories of SU(∞) are thus simply reformulated in terms of surface coordinates.

Infinite‐dimensional algebras and a trigonometric basis for the classical Lie algebras

This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

A green function for the general self-dual gauge field

Abstract The recent general solution by Atiyah, Hitchin, Drinfeld and Manin of the self-duality equations for an arbitrary compact classical group is discussed. The Green function for a scalar field transforming as a vector under the group is shown to take an elegant form in the background field of the general self-dual solution. The massless solutions of the Dirac equation, for this representation of the group, are also explicitly exhibited.

Off-Shell States in Dual Resonance Theory

Abstract We discuss ooff-shell states guided by an analogue model approach. This leads us to a more complete understanding of a model proposed recently by Schwarz with critical dimension 16. We are led, by algebraic considerations, to off-shell states in the Neveu- Schwarz-Ramond model, which obey the gauge conditions in the same critical dimension as the on-shell theory, the amplitudes factorizing on the usual positive definite states in 10 dimensions. Brief calculations reveal that some of the divergences present in the orbital model disappear in the fermion theory.

Scalar field theory and exact solutions to a classical SU (2) gauge theory

We demonstrate a relationship between the solutions of a φ4 scalar field theory and a class of solutions to an SU (2) gauge theory. Most known exact solutions belong to this class.

Universal field equations with covariant solutions

Abstract Metric-independent σ-models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the independent variables, one has invariance of solutions of the field equations under arbitrary functional redefinitions of the field quantities. Among the many interesting properties of these new models is the existence of a hierarchical structure which is illustrated by the following result. Given an arbitrary lagrangian, dependent only upon first derivatives of the field, and homogeneous of weight one, an iterative procedure for calculating a sequence of equations of motion is discovered, which ends with a universal, possibly integrable equation, which is independent of the starting lagrangian. A generalisation to more than one field is given.