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Polylogarithm identities, cluster algebras and the \(\mathcal{N} = 4\) supersymmetric theory. (English) Zbl 1444.81037

Burgos Gil, José Ignacio (ed.) et al., Periods in quantum field theory and arithmetic. Outcome of the “Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory”, ICMAT 2014, Madrid, Spain, September 15–19, 2014. Cham: Springer. Springer Proc. Math. Stat. 314, 145-172 (2020).
Summary: 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.
For the entire collection see [Zbl 1446.81002].

MSC:

81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
81R60 Noncommutative geometry in quantum theory
13F60 Cluster algebras
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