  We express one-loop closed string amplitudes as weighted sums over squares of
open string one-loop subamplitudes. These findings generalize - subject to
final complex structure modulus integration - the celebrated tree-level
relationships known as Kawai-Lewellen-Tye (KLT) relations to higher loops and
can be applied for both the massless and massive case - with or without
supersymmetry. As a consequence in the field-theory limit our relations
capitalize solid one-loop gauge-gravity relations including loop-level color
kinematics duality. In particular, in gravitational one-loop amplitudes a
graviton is traded for two gluons just like at tree-level. Our results are
derived on the underlying string world-sheet torus by splitting each complex
string coordinate into a pair of two real cylinder coordinates by means of an
analytic continuation on the elliptic curve. This manipulation involves the
loop momentum, which mediates between the two one-loop open string cylinder
amplitudes.
