Lectures on Twistor Strings and Perturbative Yang-Mills Theory

TL;DR

The notes survey Witten's twistor-string proposal and its implications for perturbative Yang-Mills theory, emphasizing how twistor space and holomorphic curves organize gauge-theory amplitudes. They develop the open B-model on super-twistor space, the role of D-instantons, and how tree-level amplitudes emerge from instanton sectors and MHV-building blocks; they also discuss Berkovits's open twistor string as an alternative framework. The text connects these geometric ideas to practical computational tools, including BCFW recursion and unitarity-based methods with quadruple cuts, and it addresses loop-level issues such as conformal supergravity in the closed-string sector. Overall, it presents a cohesive picture where twistor geometry provides both a conceptual framework and concrete calculational techniques for perturbative Yang-Mills amplitudes, while also identifying open questions about removing conformal gravity contributions at loop level and realizing pure YM dual descriptions.

Abstract

Recently, Witten proposed a topological string theory in twistor space that is dual to a weakly coupled gauge theory. In this lectures we will discuss aspects of the twistor string theory. Along the way we will learn new things about Yang-Mills scattering amplitudes. The string theory sheds light on Yang-Mills perturbation theory and leads to new methods for computing Yang-Mills scattering amplitudes.

Figures (14)

• Figure 1: A scattering amplitude of $n$ gluons in Yang-Mills theory. Each gluon comes with the color factor $T_i,$ spinors $\lambda_i,\tilde{\lambda}_i$ and helicity label $h_i=\pm1.$
• Figure 2: (a) In complex twistor space $\Bbb{CP}^3$, the MHV amplitude localizes to a $\Bbb{CP}^1.$ (b) In the real case, the amplitude is associated to a real line in $\Bbb R^3.$
• Figure 3: An amplitude with tree negative helicity gluons has contribution from two configurations: (a) Connected $d=2$ instanton. (b) Two disjoint $d=1$ instantons. The dashed line represents an open string connecting the instantons.
• Figure 4: Localization of the connected instanton contribution to next to MHV amplitude; (a) the integral over the moduli space of connected degree two curves, localizes to an integral over the degenerate curves of (b), that is intersecting complex lines. In the figure, we draw the real section of the curves.
• Figure 5: Two representations of a degree three MHV diagram. (a) In Minkowski space, the MHV vertices are represented by points. (b) In twistor space, each MHV vertex corresponds to a line. The three lines pairwise intersect.