Higher-loop amplitude monodromy relations in string and gauge theory

TL;DR

This paper extends monodromy relations from string theory to higher loops, providing new identities that relate different topologies and reproduce known field theory results, with explicit formulae at one and two loops.

Contribution

The authors develop a higher-loop generalization of monodromy relations in open string theory using contour deformation, linking planar and non-planar topologies.

Findings

One-loop identities reproduce known field theory results.

Two-loop identities verified via unitarity in N=4 super-Yang-Mills.

Explicit monodromy formulae provided for any multiplicity.

Abstract

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.