The S-Matrix in Twistor Space

TL;DR

The S-Matrix in Twistor Space advocates a (2,2) signature dual to ordinary spacetime that makes scattering-amplitude structures manifest via twistor-space formulations. By recasting BCFW recursions into twistor space, Arkani-Hamed, Cachazo, Cheung, and Kaplan develop Hodges diagrams and a link representation that render tree-level amplitudes highly transparent and reveal a holographic, S-matrix–based description of N=4 SYM and N=8 SUGRA. They extend the framework to loops, showing that split-signature, PV-regularized integrals yield well-defined, often remarkably simple results, with deep connections between twistor-space structures and momentum-space amplitudes. The work suggests a sharp, nonlocal dual theory living in twistor space, offering new computational tools and a fresh conceptual viewpoint on the S-matrix of gauge and gravity theories.

Abstract

The simplicity and hidden symmetries of (Super) Yang-Mills and (Super)Gravity scattering amplitudes suggest the existence of a "weak-weak" dual formulation in which these structures are made manifest at the expense of manifest locality. We suggest that this dual description lives in (2,2) signature and is naturally formulated in twistor space. We recast the BCFW recursion relations in an on-shell form that begs to be transformed into twistor space. Our twistor transformation is inspired by Witten's, but differs in treating twistor and dual twistor variables more equally. In these variables the three and four-point amplitudes are amazingly simple; the BCFW relations are represented by diagrammatic rules that precisely define the "twistor diagrams" of Andrew Hodges. The "Hodges diagrams" for Yang-Mills theory are disks and not trees; they reveal striking connections between amplitudes and suggest a new form for them in momentum space. We also obtain a twistorial formulation of gravity. All tree amplitudes can be combined into an "S-Matrix" functional which is the natural holographic observable in asymptotically flat space; the BCFW formula turns into a quadratic equation for this "S-Matrix", providing a holographic description of N=4 SYM and N=8 Supergravity at tree level. We explore loop amplitudes in (2,2) signature and twistor space, beginning with a discussion of IR behavior. We find that the natural pole prescription renders the amplitudes well-defined and free of IR divergences. Loop amplitudes vanish for generic momenta, and in twistor space are even simpler than their tree-level counterparts! This further supports the idea that there exists a sharply defined object corresponding to the S-Matrix in (2,2) signature, computed by a dual theory naturally living in twistor space.