Single-valued integration and double copy

Abstract

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 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.

arXiv commentThe results are unchanged. We made some minor corrections and added several remarks and clarifications

References in corpus (7)

• Closed String Amplitudes as Single-Valued Open String Amplitudes
• Single-valued harmonic polylogarithms and the multi-Regge limit
• Closed Superstring Amplitudes, Single-Valued Multiple Zeta Values and Deligne Associator
• On manifolds with corners
• Single-valued integration and superstring amplitudes in genus zero
• A spectral sequence for stratified spaces and configuration spaces of points
• Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions

Cited by in corpus (9)

• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Single-valued integration and superstring amplitudes in genus zero
• Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions
• A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• On the Bloch-Kato conjecture for GSp(4)
• Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values
• Genus-zero and genus-one string amplitudes and special multiple zeta values
• On coefficients of Poincaré series and single-valued periods of modular forms