One-loop Double Copy Relation from Twisted (Co)homology

TL;DR

This paper establishes a geometric one-loop double copy relation between closed and open string amplitudes using twisted (co)homology and intersection theory, revealing new mathematical structures in string theory.

Contribution

It introduces a novel geometric framework connecting closed and open string amplitudes at one-loop via twisted intersection theory and homological splitting.

Findings

Derived a one-loop double copy relation using twisted intersection theory.

Linked inner products on differential forms to twisted homology and cohomology groups.

Provided a new geometric perspective on string amplitude relations.

Abstract

We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of allowed differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.