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Twistor actions for non-self-dual fields; a new foundation for twistor-string theory

L. J. Mason

TL;DR

The paper builds a principled bridge from a twistor-space action with non-integrable almost complex structures to tree-level twistor-string generating functionals for full Yang-Mills theory and conformal gravity. By formulating an ASD core action and a nonlocal correction I, it shows how full dynamics can be recovered, and how the classical path-integral expansion reproduces space-time generating functionals. A supersymmetric D-instanton reformulation and a supertwistor-space extension are developed to express the generating functionals and connect them to twistor-string constructions, with parallel treatment for conformal gravity. The framework disentangles YM from conformal gravity modes, clarifies the gauge structure, and provides a foundation for loop-level extensions and SUSY generalizations in future work.

Abstract

Twistor space constructions and actions are given for full Yang-Mills and conformal gravity using almost complex structures that are not, in general, integrable. These are used as the basis of a derivation of the twistor-string generating functionals for tree level perturbative scattering amplitudes of Yang-Mills and conformal gravity. The derivation follows by expanding and resumming the classical approximation to the path integral obtained from the twistor action. It provides a basis for exploring whether the equivalence can be made to extend beyond tree level and allows one to disentangle conformal supergravity modes from the Yang-Mills modes.

Twistor actions for non-self-dual fields; a new foundation for twistor-string theory

TL;DR

The paper builds a principled bridge from a twistor-space action with non-integrable almost complex structures to tree-level twistor-string generating functionals for full Yang-Mills theory and conformal gravity. By formulating an ASD core action and a nonlocal correction I, it shows how full dynamics can be recovered, and how the classical path-integral expansion reproduces space-time generating functionals. A supersymmetric D-instanton reformulation and a supertwistor-space extension are developed to express the generating functionals and connect them to twistor-string constructions, with parallel treatment for conformal gravity. The framework disentangles YM from conformal gravity modes, clarifies the gauge structure, and provides a foundation for loop-level extensions and SUSY generalizations in future work.

Abstract

Twistor space constructions and actions are given for full Yang-Mills and conformal gravity using almost complex structures that are not, in general, integrable. These are used as the basis of a derivation of the twistor-string generating functionals for tree level perturbative scattering amplitudes of Yang-Mills and conformal gravity. The derivation follows by expanding and resumming the classical approximation to the path integral obtained from the twistor action. It provides a basis for exploring whether the equivalence can be made to extend beyond tree level and allows one to disentangle conformal supergravity modes from the Yang-Mills modes.

Paper Structure

This paper contains 14 sections, 3 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The Yang-Mills actions
  3. The action on space-time
  4. The actions on twistor space
  5. The twistor action, field equations and solutions
  6. Twistor-string Yang-Mills generating functionals from the twistor action
  7. Supersymmetric D-instanton reformulation of twistor action
  8. A digression concerning generating functions for scattering amplitudes
  9. The twistor string generating functionals
  10. Extension to the full quantum field theory
  11. Conformal gravity
  12. The extension to non anti-self-dual fields
  13. Reformulation on supertwistor space
  14. Twistor-string theory for conformal gravity

Key Result

Proposition 2.1

The action $S_T=S_{\mathrm{asd}}[a,g]- \frac{\epsilon}{2} I[a,g]$ is equivalent at the classical level to $S_{\mathrm{YM}}$. This is true both in the sense that gauge equivalence classes of solutions to the Euler Lagrange equations on twistor space are in $1:1$ correspondence with gauge equivalence

Theorems & Definitions (3)

  • Proposition 2.1
  • Lemma 4.1
  • Proposition 4.1