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

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 actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [Goto2022,arXiv:2206.03177]. Using intersection numbers – an inner product on the vector space of allowed differential forms – we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [Mano&Watanabe2012]. We also use the intersection index – an inner product on the vector space of allowed contours – to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.