Single-valued integration and double copy

TL;DR

This paper introduces a single-valued integration pairing that generalizes classical constructions, providing a framework to express complex volume integrals as quadratic forms of period integrals, with applications in mathematics and string theory.

Contribution

It develops a general theory of single-valued integration, including formulas and examples, and connects it to the double copy structure and string amplitude conjectures.

Findings

Canonical single-valued versions of period integrals are constructed.

Double copy formula expresses complex integrals as quadratic forms of periods.

Applications include modular forms, heights on curves, multiple zeta values, and string amplitudes.

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 N\'{e}ron-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.