Higher Spin Theories in Twistor Space

Abstract

In this thesis we formulate an action principle for conformal higher spin theory on twistor space. For this theory, and for a unitary sub-sector that we identify, we construct an MHV amplitude expansion by considering anti-self-dual fluctuations moving in a self-dual background. We begin by reviewing some elements of gauge theory, including gravity as a gauge theory and its higher spin generalisations. We start with Yang-Mills theory, introducing the Chalmers-Siegel action and the spinor-helicity formalism, before turning to the structure of Einstein-Cartan theory, conformal gravity, and its classical truncation to Einstein gravity. We continue with an outline of linearised massless higher spin theory and the higher-derivative conformal theory. Finally, we give a short introduction to twistor theory, which leads to a discussion of twistor space actions for Yang-Mills theory and gravity. In the second half of the thesis we consider the twistor description of conformal higher spin theories and give twistor space actions for the self-dual sector of theories with spin greater than two that produce the correct flat space-time spectrum. Upon identifying a ghost-free, unitary sub-sector, analogous to the embedding of Einstein gravity with cosmological constant in Weyl gravity, we can generate the unique three- point anti-MHV amplitude for arbitrary spin consistent with Poincaré invariance and helicity constraints. By including interactions between the infinite tower of higher-spin fields, we give a geometric interpretation to the twistor equations of motion as the integrability condition for a holomorphic structure on an infinite jet bundle, and introduce anti-self-dual interaction terms to define a twistor action for the full conformal higher spin theory. This leads to finding expressions for all three-point anti-MHV amplitudes and all MHV amplitudes, involving positive helicity conformal gravity particles and two negative helicity higher spins, which provides the on-shell analogue for the covariant coupling of conformal higher-spin fields to a conformal gravity background. We study the flat-space limit and show that the restricted amplitudes vanish, supporting the conjecture that in the unitary sector the S-matrix of conformal higher spin theories is trivial. However, by appropriately rescaling the amplitudes we find non-vanishing results which we compare with chiral flat-space higher spin theories.