Holomorphic Linking, Loop Equations and Scattering Amplitudes in Twistor Space

TL;DR

The paper embeds planar N=4 SYM amplitudes into a holomorphic setting on twistor space by defining a complex Wilson loop along nodal curves. It derives holomorphic Makeenko–Migdal-type loop equations and shows that, in the planar limit, these reduce to BCFW-like recursions for the all-loop integrand, with an extended equation in the full theory that includes forward-limit contributions. A key concept is holomorphic linking, which encodes scattering data as a holomorphic invariant of curves in twistor space. Together, these results provide a field-theoretic justification for the amplitude/Wilson loop duality and connect skein-type relations to all-loop recursion in a holomorphic framework.

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.

Figures (2)

• Figure 1: Depending on the location of the vertices, the twistor lines may form a polygonal knot. The plane projection of the nodal curve can thus be far more involved than figure 1 suggests.
• Figure 2: Going beyond the application to scattering amplitudes, we may allow the component curves to have arbitrary degrees $\geq1$.