On-shell methods for off-shell quantities in N=4 Super Yang-Mills: from scattering amplitudes to form factors and the dilatation operator

TL;DR

This paper explores how on-shell methods in N=4 Super Yang-Mills theory can be extended beyond scattering amplitudes to compute off-shell quantities like form factors and the dilatation operator, including non-planar corrections.

Contribution

It introduces new techniques for calculating form factors and studying the dilatation operator using on-shell methods, and generalizes on-shell diagrams to non-planar regimes.

Findings

Computed form factors of half-BPS operators up to two loops

Studied the dilatation operator in specific sectors using amplitude techniques

Generalized on-shell diagrams to include non-planar corrections

Abstract

Planar maximally supersymmetric Yang-Mills theory (N=4 SYM) is a special quantum field theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory, the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N=4 SYM in depth, one can learn about other, more realistic quantum field theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2|3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. to take into account non-planar corrections.