Scattering amplitudes and AdS/CFT

Scattering amplitudes in the maximally supersymmetric N = 4 gauge theory are known to possess some remarkable features. The reasons for this are not completely understood, but some possible candidates are: the integrability properties of the theory, the existence of a gravitational dual and the existence of a twistor space description. I will not try to give a detailed description of the N = 4 supersymmetric gauge theory, but only mention some of its important features. The particle spectrum of the theory can be described by a gauge field, four Weyl fermions and six real scalar fields, all transforming in the adjoint representation of a gauge group which I take to be SU(Nc). The theory has a SU(4)R symmetry group under which the four Weyl fermions transform in a 4 representation and the six scalars transform in a 6 representation. In fact, this theory has an even larger symmetry, described by the supergroup PSU(2, 2|4), which contains the conformal group SU(2, 2) as a subgroup. However, the implications of this symmetry for the scattering amplitudes are not easy to work out. In part, this is due to the fact the asymptotic states used in the scattering theory do not transform in irreducible representations of the global symmetry supergroup PSU(2, 2|4). There is a natural action of the (super-)conformal group on the transform of the amplitudes to twistor space. At this time, however, the transform to twistor space is best understood for tree-level amplitudes. In the following, I will restrict the discussion to planar, on-shell scattering amplitudes. The data needed to describe the external states is given by the on-shell momenta, the helicities of the particles, the color indices and by the transformations under the R-symmetry group SU(4). This data can be neatly encoded in a super-wavefunction Ψ(p, η), where p is the momentum and η , with I = 1, . . . , 4 are four Grassmann