A Direct Proof of BCFW Recursion for Twistor-Strings

TL;DR

The paper provides a direct, worldsheet-level proof that the leading trace sector of the genus-$0$ twistor-string path integral obeys the BCFW recursion, thereby establishing that the twistor-string computes all tree amplitudes in $\mathcal{N}=4$ SYM. By performing a contour deformation in the path integral and localising on boundary divisors of the moduli space of stable maps to $\mathbb{CP}^{3}$, the authors derive a twistor-space version of BCFW that decomposes amplitudes into products of lower-degree maps glued at nodal points. The geometric picture reveals how BCFW channels correspond to intersections of twistor-space curves of degrees $(d_1,d_2)$, offering a natural link between tree-level recursion and leading singularities of higher-loop amplitudes. The results also illuminate the relation to Grassmannian formulations and suggest avenues for generalisation to higher genus and a refined worldsheet gravity contour.

Abstract

This paper gives a direct proof that the leading trace part of the genus zero twistor-string path integral obeys the BCFW recursion relation. This is the first complete proof that the twistor-string correctly computes all tree amplitudes in maximally supersymmetric Yang-Mills theory. The recursion has a beautiful geometric interpretation in twistor space that closely reflects the structure of BCFW recursion in momentum space, both on the one hand as a relation purely among tree amplitudes with shifted external momenta, and on the other as a relation between tree amplitudes and leading singularities of higher loop amplitudes. The proof works purely at the level of the string path integral and is intimately related to the recursive structure of boundary divisors in the moduli space of stable maps to CP^3.