Holomorphic Linking, Loop Equations and Scattering Amplitudes in Twistor Space

TL;DR

This paper explores a complex analogue of Wilson Loops in holomorphic Chern-Simons theory, establishing a duality with N=4 super Yang-Mills scattering amplitudes through loop equations and holomorphic linking in twistor space.

Contribution

It introduces a holomorphic loop equation framework that connects Wilson Loops with scattering amplitudes in twistor space, revealing new geometric and algebraic structures.

Findings

Duality between twistor Wilson Loop and N=4 SYM S-matrix

Loop equations reduce to BCFW recursion relations

Scattering amplitudes linked to holomorphic linking in twistor space

Abstract

We study a complex analogue of a Wilson Loop, defined over a complex curve, in non-Abelian holomorphic Chern-Simons theory. We obtain a version of the Makeenko-Migdal loop equation describing how the expectation value of these Wilson Loops varies as one moves around in a holomorphic family of curves. We use this to prove (at the level of the integrand) the duality between the twistor Wilson Loop and the all-loop planar S-matrix of N=4 super Yang-Mills by showing that, for a particular family of curves corresponding to piecewise null polygons in space-time, the loop equation reduce to the all-loop extension of the BCFW recursion relations. The scattering amplitude may be interpreted in terms of holomorphic linking of the curve in twistor space, while the BCFW relations themselves are revealed as a holomorphic analogue of skein relations.