Singularities of Scattering Amplitudes: Symbol Letters, Cluster Adjacency, and Plabic Graphs

Mago Trejo, Jorge Leonardo

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Year:: 2022
Contributor:: Mago Trejo, Jorge Leonardo (creator); Volovich, Anastasia (Advisor); Jevicki, Antal (Reader); Gates, Sylvester James (Reader); Brown University. Department of Physics (sponsor)
Genre:: theses
Subject:: Physics; scattering amplitudes; Mathematical physics; Cluster algebras; Supersymmetry; Yang-Mills Theory
Extent:: , None p.

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Abstract:: In this thesis, we present a study of the singularity structure of scattering amplitudes in maximally supersymmetric Yang-Mills (N=4 SYM). We first consider the cluster adjacency properties of the singularities, and later we discuss a method to generate symbol letters from plabic graphs. We start our discussion by making use of the Sklyanin Poisson bracket on Gr(4,n) to investigate pair-wise cluster adjacency of singularities of rational Yangian invariants in N=4 SYM, and to establish that the n-point one-loop NMHV ratio function satisfies Steinmann cluster adjacency. We also present a series of conjectures about cluster adjacency. We continue our study of Yangian invariants in N=4 SYM by classifying all positive n-particle N^kMHV Yangian invariants in the theory with n=5k. We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons and use this to enumerate the cyclic classes of these invariants. We provide an alternative (but equivalent) classification by showing that a product of k five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated. Finally, we suggest an algorithm for computing symbol alphabets from plabic graphs by solving matrix equations of the form C Z = 0 to associate functions on Gr(m,n) to parameterizations of certain cells of Gr(k,n) indexed by plabic graphs. For m=4 and n=8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs. We also show that it is possible to obtain all rational symbol letters by solving matrix equations of the form C Z = 0 if one allows C to be an arbitrary cluster parameterization of the top cell of Gr_+(n-4,n). We finish our discussion by identifying sets of parameterizations of the top cell of Gr_+(5,9) for which the solutions produce all of (and only) the cluster variable letters of the two-loop nine-particle NMHV amplitude, and identify plabic graphs from which all of its algebraic letters originate.
Notes:: Thesis (Ph. D.)--Brown University, 2022

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Mago Trejo, Jorge Leonardo, "Singularities of Scattering Amplitudes: Symbol Letters, Cluster Adjacency, and Plabic Graphs" (2022). Physics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:h8aqjqke/

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Physics Theses and Dissertations

Theses and Dissertations for the Physics department.
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