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Single-valued integration and double copy

  • Francis Brown EMAIL logo and Clément Dupont

Abstract

In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves 0,n of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

Funding statement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 724638). Both authors thank the IHES for hospitality. The second author was partially supported by ANR grant ANR-18-CE40-0017.

Acknowledgements

This paper was initiated during the trimester “Periods in number theory, algebraic geometry and physics” which took place at the HIM Bonn in 2018, to which both authors offer their thanks. Many thanks to Andrey Levin, whose talk during this programme on the dilogarithm inspired this project, to Federico Zerbini for discussions, and to the anonymous reviewer for helpful comments.

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Received: 2020-09-30
Published Online: 2020-12-15
Published in Print: 2021-06-01

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