Polylogarithm Identities, Cluster Algebras and the $$\mathcal {N} = 4$$  Supersymmetric Theory

Vergu, Cristian

doi:10.1007/978-3-030-37031-2_7

Cited by 6 publications

(6 citation statements)

References 63 publications

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“…There already exist many excellent introductions to cluster algebras [85,86,87,88], as well as articles with detailed review sections on their relation to scattering amplitudes [89,35], so we will not attempt to repeat this here. Instead, we will briefly highlight their main features, and outline a simple example of a Graßmannian cluster algebra, which is relevant for our discussion.…”

Section: Symbol Alphabets and Cluster Algebrasmentioning

confidence: 99%

The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes

Caron-Huot¹,

Dixon²,

Drummond³

et al. 2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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We review the bootstrap method for constructing six-and seven-particle amplitudes in planar N = 4 super Yang-Mills theory, by exploiting their analytic structure. We focus on two recently discovered properties which greatly simplify this construction at symbol and function level, respectively: the extended Steinmann relations, or equivalently cluster adjacency, and the coaction principle. We then demonstrate their power in determining the six-particle amplitude through six and seven loops in the NMHV and MHV sectors respectively, as well as the symbol of the NMHV seven-particle amplitude to four loops.

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Section: Symbol Alphabets and Cluster Algebrasmentioning

confidence: 99%

The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes

Caron-Huot¹,

Dixon²,

Drummond³

et al. 2020

Proceedings of Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2019)

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“…There are several excellent reviews on cluster algebras and their appearance in planar N = 4 sYM (see for example [1,2,8,35,36]); here we review only the bare minimum needed for our purposes. The two main takeaways from this section are (i) how to compute a Poisson bracket between two cluster A-coordinates, and (ii) that this provides an efficient test for determining whether there exists a cluster containing two given cluster coordinates a 1 and a 2 ; namely, this is the case if and only if {log a 1 , log a 2 } ∈ 1 2 Z.…”

Section: Cluster Algebras and Poisson Bracketsmentioning

confidence: 99%

The Sklyanin bracket and cluster adjacency at all multiplicity

Golden

McLeod²,

Spradlin

et al. 2019

J. High Energ. Phys.

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We argue that the Sklyanin Poisson bracket on Gr(4, n) can be used to efficiently test whether an amplitude in planar N = 4 supersymmetric Yang-Mills theory satisfies cluster adjacency. We use this test to show that cluster adjacency is satisfied by all one-and two-loop MHV amplitudes in this theory, once suitably regulated. Using this technique we also demonstrate that cluster adjacency implies the extended Steinmann relations at all particle multiplicities. 1 We in particular thank M. Gekhtman for correspondence and discussions on this topic.

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“…This structure respects mutation, implying that the entry b ij (which counts the number of arrows from x i to x j in a given cluster's quiver) will be the same in all clusters containing both x i and x j . The Poisson bracket (and associated Sklyanin bracket) will play a larger role in forthcoming work [39], so we defer further discussion of this structure (for existing discussions in the literature, see [40,41]). We therefore refer to the collection of clusters that involve this pentagon as an A 2 subalgebra of Gr (2,6).…”

Section: Cluster X -Coordinatesmentioning

confidence: 99%

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

Golden

McLeod

2019

J. High Energ. Phys.

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We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N = 4 supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory's two-loop MHV amplitudes-considered as functions, symbols, and at the level of their Lie cobracket-and recount how the 'nonclassical' part of these amplitudes can be decomposed into specific functions evaluated on the A 2 or A 3 subalgebras of Gr(4, n). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the D 5 and A 5 subalgebras of Gr(4, 7), and that these decompositions are themselves decomposable in terms of the same A 4 function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra. arXiv:1810.12181v3 [hep-th] 5 Sep 2019 (2) 7 52 6 Conclusion 54 A Counting Subalgebras of Finite Cluster Algebras 56 B Cobracket Spaces in Finite Cluster Algebras 59 -1 --6 -

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Polylogarithm Identities, Cluster Algebras and the $$\mathcal {N} = 4$$ Supersymmetric Theory

Cited by 6 publications

References 63 publications

The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes

The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes

The Sklyanin bracket and cluster adjacency at all multiplicity

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

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