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

  • Cristian VerguEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


Scattering amplitudes in \(\mathcal {N} = 4\) super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in \(\mathbb {CP}^3\) and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a \(40\)-term trilogarithm identity which was discovered by accident while studying the physical results.


Scattering amplitudes Polylogarithms Cluster algebras Twistors 



First, I would like to thank the organizers of the Opening Workshop of the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory: José I. Burgos Gil, Kurusch Ebrahimi-Fard, D. Ellwood, Ulf Kühn, Dominique Manchon and P. Tempesta.

I would also like to thank the participants and particularly Frédéric Chapoton and Herbert Gangl for discussions during the opening workshop Numbers and Physics (NAP2014). Finally, I am grateful to my coauthors in Refs. [41, 46] for collaboration.


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Authors and Affiliations

  1. 1.Department of MathematicsKing’s College London The StrandLondonUK

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