Motivic amplitudes and cluster coordinates

  • J. K. Golden
  • A. B. Goncharov
  • M. SpradlinEmail author
  • C. Vergu
  • A. Volovich
Open Access


In this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf n (ℙ3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude \( \mathcal{A}_{7,2}^{\mathcal{M}} \) and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster \( \mathcal{X} \)-coordinates on Conf n (ℙ3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to \( \mathcal{A}_{7,2}^{\mathcal{M}} \) being expressible in terms of classical

polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster \( \mathcal{X} \)-coordinates of a single algebra. In this respect it is similar to Abel’s 5-term dilogarithm identity.


Supersymmetric gauge theory Scattering Amplitudes 


Open Access

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Copyright information

© The Author(s) 2014

Authors and Affiliations

  • J. K. Golden
    • 1
  • A. B. Goncharov
    • 2
  • M. Spradlin
    • 1
    • 3
    Email author
  • C. Vergu
    • 4
  • A. Volovich
    • 1
    • 3
  1. 1.Department of PhysicsBrown UniversityProvidenceU.S.A.
  2. 2.Department of MathematicsYale UniversityNew HavenU.S.A.
  3. 3.Theory Division, Physics Department, CERNGeneva 23Switzerland
  4. 4.ETH ZürichInstitut für Theoretische PhysikZürichSwitzerland

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