Abstract.It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine KacMoody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.

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An account is given of the new insight into the theory of magnetic monopoles originating from the work of 't Hooft and Polyakov. Their magnetic monopole, associated with the conventional electromagnetic gauge group U( l), occurs as a finite-energy smooth soliton solution to an SU(2) gauge theory. A precise picture of its internal structure, the values of its magnetic charge and its mass are obtained. These new developments bring together previously unrelated fields of study, namely the Dirac monopole (with point structure) and the Sine-Gordon soliton in twodimensional space-time.Properties of more general monopoles, associated with large gauge groups now thought to be relevant in physics, are discussed. Particular attention is paid to topological properties. Based on this new viewpoint, conjectures can be made about a future quantum theory of monopoles.

A proof is given of the formula, recently proposed by Cachazo, He and Yuan (CHY) for gluon tree amplitudes in pure Yang-Mills theory in arbitrary dimension. The approach is to first establish the corresponding result for massless φ 3 theory using the BCFW recurrence relation and then to extend this to the gauge theory case. Additionally, it is shown that the scattering equations introduced by CHY can be generalized to massive particles, enabling the description of tree amplitudes for massive φ 3 theory.

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