Abstract
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.
Keywords
Expanded version of a talk given at the Opening Workshop of the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory, organized by José I. Burgos Gil, Kurusch Ebrahimi-Fard, D. Ellwood, Ulf Kühn, Dominique Manchon and P. Tempesta.
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Notes
- 1.
Analyticity survives after adding quantum corrections, but factorization becomes more subtle in case there are infrared divergences (see Ref. [13]). Since scattering amplitudes in gauge theories are infrared divergent, exploiting factorization at loop level seems to be much harder.
- 2.
The sum over particle histories is not well-defined mathematically. Nevertheless, we can use it formally to compute the perturbative expansion. A similar statement holds for a string theory, where we sum over string histories also called worldsheets.
- 3.
A similar construction can be done for Minkowski space \(\mathbb {M}\) instead, in which case we obtain the Penrose’s twistor space (see Ref. [58]).
- 4.
- 5.
This holds in many explicit examples, but I have not found a proof in the literature.
- 6.
Any conic is determined by five points. Given four points there is an infinity of conics which contain them.
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Acknowledgements
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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Vergu, C. (2020). Polylogarithm Identities, Cluster Algebras and the \(\mathcal {N} = 4\) Supersymmetric Theory. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_7
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