Twistor theory and its applications in asymptotically flat spacetimes

Abstract

This thesis provides an overview of the recent progress in understanding dynamics in asymptotically flat spacetimes inspired by the use of twistor theory. We begin by introducing scattering amplitudes of QFTs in 4d asymptotically flat spaces, which after Mellin transforms, can be recast as 2d conformal correlation functions. It is interesting to ask whether there is a 2d celestial conformal field theory (CCFT) underlying such correlation functions on the celestial sphere, thereby giving an alternative approach to studying the asymptotic dynamics of Minkowski space using powerful CFT techniques. We also include the models used to generate such amplitudes, the (ambi-)twistor string worldsheet theories and twistor actions.

The first part (chapter 4 to 6) of the thesis is concerned with infrared behaviours of scattering amplitudes, in particular, collinear limits, which can be reinterpreted as the CCFT operator product expansion (OPE) limits on the celestial sphere. We focus on deriving these OPEs from the worldsheet theories of (ambi-)twistor strings and twistor sigma models. Interestingly, the worldsheet OPEs in (ambi-)twistor strings localize directly on the celestial OPE limit, producing the singular coefficients for gauge theory and gravity, as well as their supersymmetric extensions. As a byproduct, mode expansions of the OPEs in the soft regime straightforwardly give infinite dimensional symmetry algebras. By allowing spacetime non-commutativity, deformations of such soft algebras can be obtained, in the case of Einstein gravity, a W-infinity algebra is found to be the unique deformation. In the MHV sector, the twistor string worldsheet is holomorphically identified with the celestial sphere and the first expression for the full, non-singular gluon celestial OPE is obtained. Using semi-classical twistor sigma models, the results above are extended to self-dual radiative spacetimes.

In the second part (chapter 7) of the thesis, we explore the conjecture on the holographic nature of the putative 2d CCFT from a geometric point of view. Action functionals for theories which admit perturbative expansions around integrable subsectors have been written down on twistor space, where holomorphic QFT techniques can be applied to study the dynamics. On the level of an action functional, using ideas of holographic reduction and mechanisms of twistor theory, we derive an explicit 2d CFT living on the celestial sphere by performing large gauge transformations which gives residual degrees of freedom living at the asymptotic boundary. This CFT produces dynamics of Yang-Mills theory such as an infinite dimensional symmetry algebra and form factors. By lifting local operators to twistor space using holomorphic frames, conformal blocks in 4 dimensional N=4 Yang-Mills are phrased in terms of 2d correlation functions on the celestial sphere. This gives a concrete realisation of the weak-weak 4d-2d Koszul duality envisioned by Costello and Paquette.