We present a conjecture for the n-gluon one-loop amplitudes with maximal helicity violation. The conjecture emerges from the powerful requirement that the amplitudes have the correct behavior in the collinear limits of external momenta. One implication is that the corresponding amplitudes where three or more gluon legs are replaced by photons vanish for n > 4.

Factorization properties of one-loop gauge theory amplitudes have been used as checks on explicitly computed amplitudes and in the construction of ansätze for higher-point ones. In massless theories, such as QCD at high energies, infrared divergences complicate factorization. Here we prove that factorization in such theories is described by a set of universal functions. In particular, a proof of the universality of one-loop splitting functions as the momenta of two particles become collinear is presented. Factorization in multi-particle channels is also given. The discontinuity functions that appear in the splitting functions may also be used to obtain infrared divergent box integral functions from finite ones.

We provide an action for self-dual Yang-Mills theory which is a simple truncation of the usual Yang-Mills action. Only vertices that violate helicity conservation maximally are included. One-loop amplitudes in the self-dual theory then follow as a subset of the Yang-Mills ones. In light-cone gauges this action is almost identical to previously proposed actions, but in this formulation the vanishing of all higher-loop amplitudes is obvious; the explicit perturbative S-matrix is known. Similar results apply to gravity.

The coulomb branch of N = 4 supersymmetric Yang-Mills gauge theories in d = 2 + 1 is studied. A direct connection between gauge theories and monopole moduli spaces is presented. It is proposed that the hyper-Kähler metric of supersymmetric N = 4 SU(N) Yang-Mills theory is given by the charge N centered moduli space of BPS monopoles in SU(2). The theory is compared to N = 2 supersymmetric Yang-Mills theory in four dimensions through compactification on a circle of the latter. It is found that rational maps are appropriate to this comparison. A BPS mass formula is also written for particles in three dimensions and strings in four dimensions.