  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.
