  The monodromy relations in string theory provide a powerful and elegant
formalism to understand some of the deepest properties of tree-level field
theory amplitudes, like the color-kinematics duality. This duality has been
instrumental in tremendous progress on the computations of loop amplitudes in
quantum field theory, but a higher-loop generalisation of the monodromy
construction was lacking.
  In this letter, we extend the monodromy relations to higher loops in open
string theory. Our construction, based on a contour deformation argument inside
open string diagrams, leads to new identities that relate planar and non-planar
topologies in string theory. We write one and two-loop monodromy formulae
explicitly at any multiplicity. In the field theory limit, at one-loop we
obtain identities that reproduce known results. At two loops, we check our
formulae by unitarity in the case of the four-point N=4 super-Yang-Mills
amplitude.
