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Higher-loop amplitude monodromy relations in string and gauge theory

Piotr Tourkine, Pierre Vanhove

TL;DR

The construction of a higher-loop generalization of the monodromy construction, based on a contour deformation argument of the open string diagram integrands, leads to new identities that relate planar and nonplanar topologies in string theory.

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.

Higher-loop amplitude monodromy relations in string and gauge theory

TL;DR

The construction of a higher-loop generalization of the monodromy construction, based on a contour deformation argument of the open string diagram integrands, leads to new identities that relate planar and nonplanar topologies in string theory.

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.

Paper Structure

This paper contains 7 sections, 30 equations, 2 figures.

Table of Contents

  1. Monodromies on the annulus
  2. Local and global monodromies
  3. Open string relations
  4. Field theory relations
  5. Toward Higher-loop relations
  6. A two-loop example.
  7. Planar and non-planar Green function

Figures (2)

  • Figure 1: The $\nu_1$ contour integral (red) vanishes. The two boundaries (black) have opposite orientation.
  • Figure 2: Two-loop integrand monodromy. Integration over the red contour vanishes. Given the definition of the loop momentum in eq. \ref{['eq:higher-g-ell']}, parallel integrations along $a_1,a_2$ cancel only up to a shift in the loop momentum.