Symbol alphabets from plabic graphs
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Springer
Received: August 3, 2020 Accepted: September 15, 2020 Published: October 20, 2020
Jorge Mago,a Anders Schreiber,a Marcus Spradlina,b and Anastasia Volovicha a
Department of Physics, Brown University, Providence, RI 02912, U.S.A. b Brown Theoretical Physics Center, Brown University, Providence, RI 02912, U.S.A.
E-mail: jorge [email protected], anders [email protected], marcus [email protected], anastasia [email protected] Abstract: Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these 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. Keywords: Scattering Amplitudes, Supersymmetric Gauge Theory, Differential and Algebraic Geometry ArXiv ePrint: 2007.00646
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)128
JHEP10(2020)128
Symbol alphabets from plabic graphs
Contents 1
2 A motivational example
2
3 Six-particle cluster variables
4
4 Towards non-cluster variables
7
5 Algebraic eight-particle symbol letters
9
6 Discussion
11
A Some six-particle details
13
B Notation for algebraic eight-particle symbol letters
13
1
Introduction
A central problem in studying the scattering amplitudes of planar N = 4 super-Yang-Mills (SYM) theory is to understand their analytic structure. Certain amplitudes are known or expected to be expressible in terms of generalized polylogarithm functions. The branch points of any such amplitude are encoded in its symbol alphabet—a finite collection of multiplicatively independent functions on kinematic space called symbol letters [1]. In [2] it was observed that for n = 6, 7, the symbol alphabet of all (then-known) n-particle amplitudes is the set of cluster variables [3, 4] of the Gr(4, n) Grassmannian cluster algebra [5]. The hypothesis that this remains true to arbitrary loop order provides the bedrock underlying a bootstrap program that has enabled the computation of these amplitudes to impressively high loop order and remains supported by all available evidence (see [6] for a recent review). For n > 7 the Gr(4, n) cluster algebra has infinitely many cluster variables [4, 5]. While it has long been known that the symbol alphabets of some n > 7 amplitudes (such as the two-loop MHV amplitudes [7]) are given by finite subsets of cluster variables, there was no candidate guess for a “theory” to explain why amplitudes would select the subsets that they do. At the same time, it was expected [8, 9] that the symbol alphabets of even MHV amplitudes for n > 7 would generically require letters that are not cluster varia
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