Single-valued integration and double copy

Francis Brown (1, 2) , Clément Dupont (3)

1 MI - Mathematical Institute [Oxford]
2 IHES - Institut des Hautes Études Scientifiques
3 IMAG - Institut Montpelliérain Alexander Grothendieck

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 formulas 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. In a sequel to this paper \cite{BD2} we apply this formalism to the moduli space of curves of genus zero with marked points, to deduce a recent conjecture due to Stieberger in string perturbation theory, which states that closed string amplitudes are the single-valued projections of open string amplitudes.

Domaines: Théorie des nombres [math.NT]; Physique des Hautes Energies - Théorie [hep-th]; Géométrie algébrique [math.AG]

Francis Brown, Clément Dupont. Single-valued integration and double copy. Journal für die reine und angewandte Mathematik, 2021, 775, pp.145-196. ⟨10.1515/crelle-2020-0042⟩. ⟨hal-01899456⟩