Aspects of N = 4 Super Yang-Mills: Amplitudes, Operators and Invariants

Abstract

In this thesis, we study aspects of scattering amplitudes, half-BPS operators and Yangian Invariants in N = 4 super Yang Mills. We begin by exploring the geometry of Wilson loop diagrams. The Wilson loop in supertwistor space gives an explicit description of perturbative superamplitude integrands in N = 4 super Yang-Mills as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that the geometrical images of the diagrams would give a tessellation of the amplituhedron. This turns out to be true for NMHV amplitudes, however we prove that for N^2MHV and beyond this is not the case. Specifically, we show that there is no choice of geometric image of the Wilson loop Feynman diagrams that gives a geometric object with no spurious boundaries.

We then move to investigate a set of half-BPS operators in N = 4 super Yang-Mills which are appropriate for describing single particle states of superstring theory on AdS5×S5; we refer to these as single particle operators. They are defined to have vanishing two-point function with all multi-trace operators, and so correspond to admixtures of single- and multi-traces. We find explicit formulae for these operators and their two-point function normalisation. We prove that single particle operators in the U(N) gauge theory are single particle operators in the SU(N) theory, and show that at large N these operators interpolate between the single trace operator and the sphere giant graviton. A multipoint orthogonality theorem is presented and proved, which as a consequence enforces all near-extremal correlators to vanish. We compute all maximally and next-to-maximally extremal free correlators, and provide some explicit results for subsets of two- and three-point functions for multi-particle operators.

Finally, we calculate the N^2MHV Yangian invariants for N = 4 SYM in amplituhedron coordinates, and see that some have suggestively simple forms.