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

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Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

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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.

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. 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. 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. 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. 4.

    Here we are presented a flipped version of the quiver and with the arrows reversed with respect to the quivers of Refs. [40, 51].

  5. 5.

    This holds in many explicit examples, but I have not found a proof in the literature.

  6. 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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Authors and Affiliations

  1. Department of Mathematics, King’s College London The Strand, London, WC2R 2LS, UK

    Cristian Vergu

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  1. Cristian Vergu
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Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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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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