H. Osborn

Topological charges for N = 4 supersymmetric gauge theories and monopoles of spin 1

The central charges in the supersymmetry algebra of the N = 4 supersymmetry gauge theory are obtained. When spontaneous symmetry breaking is imposed it is shown that the spins of the topological monopole states should be identical to those of the massive elementary particles.

General Background Field Calculations With Fermion Fields

Background field calculations including fermion background fields are undertaken to determine one- and two-loop counterterms, and β-functions, for a general vector, scalar, spinor field gauge theory. Dimensional regularisation, and also the dimensional reduction variant, with a non-perturbative treatment of the background fields are used. In the dimensional reduction case two alternative forms are given for the results depending on the precise treatment of e-scalars.

Two-loop background field calculations for arbitrary background fields

Abstract The ultraviolet divergences of two-loop vacuum graphs in the presence of an arbitrary background field are determined for four-dimensional φ 4 and gauge theories on flat space. Dimensional regularisation is employed and the heat kernel is used to analyse the short-distance singularities in the products of propagators, in the presence of the background field, that occur for two-loop graphs. The single and double poles in ϵ = 4 − d are determined in a concise fashion, giving known results for the β function. A procedure for determining the remaining finite parts in terms of explicit convergent integrals in four dimensions is described.

Background field calculations in curved spacetime: (I). General formalism and application to scalar fields

Abstract The ultraviolet divergences for vacuum graphs in the presence of arbitrary background fields and euclidean metric in curved space are determined for all two- and some three-loop graphs. The method is applicable to general field theories and can be extended to higher orders but is here applied to a general scalar theory to calculate the corresponding counterterms and associated β functions. The heat kernel is used to derive an expansion for the propagator that explicitly isolates the singular terms in the short-distance limit. With dimensional regularisation the poles in ϵ = 4− d are determined from explicit distribution formulae for the products of the singular functions that arise. The results maintain general covariance and generalise previous results for flat space. A justification for the continuation of integral expressions, in the presence of the arbitrary background, to general dimension d is given.

Zeta-function regularization and multi-instanton determinants☆

Abstract The determinant of the covariant laplacian is discussed using the ζ-function regularization procedure. For a self-dual solution to a general euclidean Yang-Mills gauge theory, an integral expression for its variation, with respect to the parameters of the solution, is obtained in terms of the general construction of Atiyah et al. The conformal properties are analyzed and an explicit expression is obtained for a part of the determinant containing all the non-trivial behavior. The remaining conformally invariant part has not been determined. The removal of the infrared divergences by considering the problem on a sphere is discussed in some detail.

Renormalisation and composite operators in non-linear σ models

Abstract The renormalisation of two-dimensional non-linear σ models, without torsion, is considered when the metric g depends explicitly on both coordinates for the σ model manifold φ and the spacetime coordinates x . This allows finite local composite operators to be defined by functional differentiation of the renormalised action with respect to g . A well defined expression for the trace of the energy-momentum tensor in terms of normal products is obtained and the results of Tseytlin and also of Curci and Paffuti recovered. Additional counterterms for x -dependent g are necessary but these are calculated, for up to two loops, using background field methods and dimensional regularisation and a technique which does not depend on any expansion about flat space. A confirmatory three-loop calculation for the dependence on the two-dimensional scalar curvature is presented.

Gauging the general σ-model with a Wess-Zumino term

Abstract We show how to gauge the general two-dimensional σ-model with a Wess-Zumino term. We discuss quantisation with the gauge field as background and show that renormalisability requires additional terms involving the gauge field strength. We also illustrate the subtleties involved in quantising the gauge field by considering explicit examples.

Solutions of the Dirac equation for general instanton solutions

Abstract The zero mode normalisable solutions of the Dirac equation in the presence of a general self-dual gauge field of instanton number k for the groups SU( n ) or Sp( r ) are found. The method uses the explicit Green functions constructed by Corrigan et al. and gives the full complement of k -independent solutions and also determines their normalisations.

String theory effective actions from bosonic σ models

Abstract An expression for the string effective action for the metric g ij and dilation Φ fields corresponding to a bosonic non-linear σ model, without torsion is found in terms of the corresponding β functions, modifying an earlier formula of Tseytlin. Although the analysis is based on dimensional regularization the result allows for a certain class of perturbative redifinitions of the metric and dilaton fields corresponding to non-minimal subtraction. Two-point functions of scalar fields and the energy momentum tensor ale also discussed, relating the present results to the general discussions of Zamolodchikov, and providing an alternative derivation of the essential relation on which the construction of the action is based. The expressions obtained include a quadratic form on tensor fields h ij which can be calculated in the loop expansion and acts as a metric on the space of couplings and is here explicitly given to two loops. The relation between the dilaton β function and the central charge of the Virasoro algebra is also obtained for any conformally flat two-dimensional space.

Relativistic extension of the electromagnetic current for composite systems

Abstract We formulate a general method for evaluating relativistic and binding energy corrections to matrix elements of the electromagnetic current for multiparticle composite systems. Application is made to the calculation of the g-factor for bound electrons and compared with recent experiments.