# Twistor theory at fifty: from contour integrals to twistor strings

## TL;DR

This paper reviews fifty years of twistor theory, highlighting its mathematical structures, physical applications in gauge and gravity solutions, integrability, and recent advances in twistor string theories for quantum scattering amplitudes.

## Contribution

It provides a comprehensive overview of twistor theory's development, including new insights into twistor strings and their role in quantum physics.

## Key findings

- Explicit constructions of solutions to ASD equations
- New formulae for quantum scattering amplitudes from twistor strings
- Discussion of the Newtonian limit and quantum gravity implications

## Abstract

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills, and gravitational instantons which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of ASD Yang--Mills equations, and Einstein--Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally we discuss the Newtonian limit of twistor theory, and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.