On the singularities of scattering amplitudes: from cluster adjacency tropical geometry

Abstract

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a unique quantum field theory. Its interesting properties include conformal symmetry at the quantum level, expected integrability, and being a prime example of the AdS/CFT correspondence. Witten’s twistor string theory [1] reignited an interest in this theory, leading to enormous progress in utilising its special features to uncover new mathematical structures, such as the Grassmannian [2], and to better understand gauge theory in general. One such feature is the hidden dual conformal symmetry in SYM [3, 4] which led to the introduction of momentum twistors [5] as useful variables when describing scattering amplitudes in SYM. It was then shown in [6] that the kinematic space parametrised by these momentum twistors, itself a Grassmannian, comes equipped with a particular mathematical structure called a cluster algebra. It was shown that scattering amplitudes in SYM depend on cluster coordinates and, at least for six and seven-points, the cluster algebra provided all the singularities, or alphabet, for all known amplitudes in SYM. The ultimate hope is that by understanding all the analytic structure of the functions scattering amplitudes consist of, one could simply write down the form of an amplitude without the need for an explicit calculation. The overarching theme of this thesis is to better understand the analytic properties of scattering amplitudes, mainly (but not exclusively) in SYM. Although this thesis has two parts, the main focus is on cluster algebras; extracting the information they contain about amplitudes and exploring their relationship with other areas of mathematics, as well as what those relationships imply for the singularity properties of amplitudes. In the first part, we introduce and develop the notion of cluster adjacency and how it controls the poles and branch cuts an amplitude is allowed to have. We also provide an example of how cluster adjacency can aid in computations in calculating the seven-point, four-loop, NMHV scattering amplitude. The second part expands on the discussion started in [7] linking tropical geometry to scattering amplitudes in the biadjoint φ 3 theory. We demonstrate how the connection between tropical Grassmannians and cluster algebras allows for straightforward calculation of amplitudes in this theory. We go on to utilise this connection to generalise the notion of cluster adjacency and conjecture how different helicity configurations (MHV, NMHV, etc.) may each have their own cluster adjacency rules. Finally, tropical geometry allows us to approach certain issues with eight-point scattering in SYM. Namely, the infinite set of cluster coordinates provided by cluster algebra associated with eight-point scattering, as well as generating the algebraic singularities known to appear even at one-loop for N2MHV amplitudes. This thesis is based on [8–13] with considerable overlap with these papers.

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