A double copy from twisted (co)homology at genus one

Authors: Rishabh Bhardwaj, Andrzej Pokraka, Lecheng Ren, Carlos Rodriguez

Preprint number: UUITP–37/23

We study the twisted (co)homology of a family of genus-one integrals — the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated.

While not true one-loop string integrals, they share many similarities and are simple enough that the associated twisted (co)homologies have been completely characterized (Goto, 2022). Using intersection numbers — an inner product on the vector space of allowed differential forms — we derive Gauss-Manin connections for a basis of the twisted cohomology.

In particular, we provide an independent check of the Gauss-Manin connection derived in (Mano & Watanabe, 2012). We also use the intersection index — an inner product on the vector space of allowed contours — to define a double copy (or KLT) kernel. We numerically verify that this kernel correctly double copies Riemann-Wirtinger integrals into single-valued Riemann-Wirtinger integrals: the analogues of closed string integrals.