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Workshop Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory

ICMAT-MZV 2014: Periods in Quantum Field Theory and Arithmetic pp 77–103Cite as

The Number Theory of Superstring Amplitudes

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

Abstract

The following article is intended as a survey of recent results at the interface of number theory and superstring theory. We review the expansion of scattering amplitudes—central observables in field and string theory—in the inverse string tension where elegant patterns of multiple zeta values occur. More specifically, the Drinfeld associator and the Hopf algebra structure of motivic multiple zeta values are shown to govern tree-level amplitudes of the open superstring. Partial results on the quantum corrections are discussed where elliptic analogues of multiple zeta values play a central r\(\hat{\text {o}}\)le.

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Notes

  1. 1.

    The terminology here and in later places relies on the commonly trusted conjectures on the transcendentality of MZVs.

  2. 2.

    The sign convention for \(e_1\) varies in the literature.

  3. 3.

    In the original disk integrals (2), rearranging the curly bracket of the integrand as

    $$ \prod ^{n-2}_{l=2} \sum ^{l-1}_{m=1} {s_{ml} \over z_{ml}} \rightarrow \prod ^{\nu }_{l=2} \sum ^{l-1}_{m=1} {s_{ml} \over z_{ml}} \prod ^{n-2}_{p=\nu +1} \sum ^{n-1}_{q=p+1} {s_{pq} \over z_{pq}} $$

    amounts to adding total derivatives w.r.t. \(z_2,\ldots ,z_{n-2}\) which vanish in presence of the Koba-Nielsen factor \(\prod _{i<j}^{n-1} |z_{ij}|^{s_{ij}}\). Tentative boundary contributions at \(z_j=z_{j\pm 1}\) are manifestly suppressed by \(|z_j - z_{j\pm 1}|^{s_{j,j\pm 1}}\) for positive real part of \(s_{j,j\pm 1}\) which propagates to generic complex values by analytic continuation.

  4. 4.

    The derivative w.r.t. \(z_0\) directly acts at the level of the integrand since the boundary contribution from the \(z_0\)-dependence in the upper limit is suppressed as \(\lim _{z_{n-2} \rightarrow z_0} (z_0 - z_{n-2})^{s_{0,n-2}} = 0\). As before, the limit is obvious if \(s_{0,n-2}\) has a positive real part and otherwise follows from analytic continuation.

  5. 5.

    Also, none of the odd \(\zeta \)-values has been proven to be transcendental so far: the only known facts are the irrationality of \(\zeta _3\) as well as the existence of an infinite number of odd irrational \(\zeta \)’s [36, 37].

  6. 6.

    In the pure spinor framework [19], kinematic building blocks suitable to describe the anomaly sector have been constructed in [56], see [57] for their appearance in the integrand of ten-dimensional field-theory amplitudes.

  7. 7.

    A set of indecomposable eMZVs of weight w and length r is a minimal set of eMZVs such that any other eMZV of the same weight and length can be expressed as a linear combination of elements from this set as well as products of eMZVs with strictly positive weights and eMZVs of lengths smaller than r or weight lower than w. The coefficients are understood to comprise MZVs (including rational numbers) and integer powers of \(2\pi i\).

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Acknowledgements

I am very grateful to Johannes Broedel, Carlos Mafra, Nils Matthes, Stephan Stieberger and Tomohide Terasoma for collaboration on the projects on which this article is based. Moreover, I would like to thank Johannes Broedel, Nils Matthes and Federico Zerbini for valuable comments on the draft. I am indebted to the organizers of the conference “Numbers and Physics” in Madrid in September 2014 which strongly shaped the research directions leading to [16, 53] and possibly further results. I also acknowledge financial support by the European Research Council Advanced Grant No. 247252 of Michael Green.

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Authors and Affiliations

  1. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476, Potsdam, Germany

    Oliver Schlotterer

  2. Department of Physics and Astronomy, Uppsala University, 75108, Uppsala, Sweden

    Oliver Schlotterer

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  1. Oliver Schlotterer
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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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Schlotterer, O. (2020). The Number Theory of Superstring Amplitudes. 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_4

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