Higher-loop amplitude monodromy relations in string and gauge theory

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.