The SAGEX Review on Scattering Amplitudes, Chapter 6: Ambitwistor Strings and Amplitudes from the Worldsheet

TL;DR

Ambitwistor-string theories provide a direct worldsheet route to massless scattering amplitudes, producing compact CHY-type formulæ localized on the scattering equations and revealing a natural double-copy structure. The chapter develops two 4d formulations—the vector/current-algebra model and the twistorial model—then extends tree-level results to loop level via higher-genus correlators, finally reformulating loop integrands on nodal spheres and clarifying their forward-limit structure. It also discusses extensions to curved backgrounds and connections to conventional strings and celestial holography, illustrating a broad, unified framework linking CHY, ambitwistor geometry, and loop-level amplitudes across dimensions. The work highlights how BRST quantization, vertex-operator constructions, and polarization data encode YM, gravity, and beyond, while preserving a tractable, factorized structure amenable to double-copy realizations and forward-limit techniques.

Abstract

Starting with Witten's twistor string, chiral string theories have emerged that describe field theory amplitudes without the towers of massive states of conventional strings. These models are known as ambitwistor strings due to their target space; the space of complexified null geodesics, also called ambitwistor space. Correlators in these string theories directly yield compact formulae for tree-level amplitudes and loop integrands, in the form of worldsheet integrals fully localized on solutions to constraints known as the scattering equations. In this chapter, we discuss two incarnations of the ambitwistor string: a 'vector representation' starting in space-time and structurally resembling the RNS superstring, and a four-dimensional twistorial version closely related to, but distinct from Witten's original model. The RNS-like models exist for several theories, with 'heterotic' and type II models describing super-Yang-Mills and 10d supergravities respectively, and they manifest the double copy relations directly at the level of the worldsheet models. In the second half of the chapter, we explain how the underlying models lead to diverse applications, ranging from extensions to new sectors of theories, loop amplitudes and to scattering on curved backgrounds. We conclude with a brief discussion of connections to conventional strings and celestial holography.

Figures (7)

• Figure 1: Boundary divisor $\mathfrak{M}_{0,n_I+1}\times\mathfrak{M}_{0,n_{\bar{I}}+1}\subset \partial\widehat{\mathfrak{M}}_{0,n}$ of the moduli space, corresponding to a pair of marked spheres $\Sigma_I$ and $\Sigma_{\bar{I}}$. We parametrize the spheres by $x\in \Sigma_I$, and $\sigma\in \Sigma_{\bar{I}}$, with a nodal point $\sigma_I\in \Sigma_{\bar{I}}$ and $x_I=\infty\in\Sigma_I$, and subject to $\sigma=\sigma_I+\varepsilon x$. Depicted is the case for $n=6$, with $I=\{1,2,3\}$ and $\bar{I} = \{4,5,6\}$.
• Figure 2: Illustration of ambitwistor space. A fixed twistor $Z\in \mathbb{P}\mathbb{T}$ corresponds to a totally null 2-plane on space-time, known as an $\alpha$-plane; similarly for $\tilde{Z}\in \mathbb{P}\mathbb{T}^*$ and $\beta$-planes. When $Z\cdot\tilde{Z}=0$, these planes intersect in a light-ray $L$.
• Figure 3: Schematic loop expansion of the amplitude. In worldsheet models such as the ambitwistor string, $g$-loop amplitudes correspond to correlators on genus-$g$ Riemann surfaces.
• Figure 4: The homology basis at genus two.
• Figure 5: A graphic depiction of the residue theorem on the fundamental domain for $g=1$. Illustrated on the left is the localisation of the integrand on solutions to $u=0$ on the support of the remaining scattering equations. This is equal to the integrand, now localised on the boundary $\tau=i\infty$ of the moduli space, with the two expressions related by a residue theorem on the fundamental domain.