Cluster adjacency for m = 2 Yangian invariants

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Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ 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.

Section: Jhep10(2020)128mentioning

confidence: 98%

Symbol alphabets from plabic graphs

Mago

Schreiber

Spradlin

et al. 2020

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“…n,k , see [17] for details. In particular, the partial triangulations which we construct in the definition of the positive region ∆ n (β) correspond to the graphical labels for generalized triangles described in [17].…”

Section: Jhep07(2021)111mentioning

confidence: 99%

Kleiss-Kuijf relations from momentum amplituhedron geometry

Damgaard

Ferro

Łukowski

et al. 2021

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In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.

“…Namely they provide not only the entire symbol alphabet, but also which letters thereof are allowed to appear in adjacent entries of the symbol [28]. This property of cluster adjacency has also been extended to the rational factors that may be present in the amplitudes [29], for more recent applications see [30][31][32][33]. Very interestingly, while the direct physical origin of cluster adjacency remains obscure, for the space of functions with physical branch cuts containing six-and seven-particle amplitudes, cluster adjacency is equivalent to the physically more transparent extended Steinmann relations 2 [35][36][37].…”

Section: Introductionmentioning

confidence: 99%

How tropical are seven- and eight-particle amplitudes?

Henke¹,

Παπαθανασίου²

2020

164

We study tropical Grassmanians Tr(k, n) in relation to cluster algebras, and assess their applicability to n-particle amplitudes for n = 7, 8. In N = 4 super Yang-Mills theory, we first show that while the totally positive part of Tr(4, 7) may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration. Then, using Tr(4, 8) we propose a finite set of 356 cluster A-coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences. Finally, we use a triangulation of the totally positive part of Tr(3, 8) to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.