Celestial Chiral Algebras and Self-Dual Gravity

Abstract

Celestial holography suggests, among other things, that collinear singularities of graviton scattering amplitudes are described by the OPEs of some putative dual CFT. One of the great successes has been the insight that this duality is true at tree-level which led to the discovery of new infinite dimensional symmetry algebras of tree-level amplitudes in flat space closely related to $w_{1+\infty}$. This thesis studies these celestial chiral algebras in the light of twistor theory and derives tree-level deformations thereof induced by non-trivial background geometries that solve some form of the self-dual Einstein equations. After an elaborate introduction, we begin by reviewing how holomorphic collinear singularities of gravity and gauge theory amplitudes in a certain basis are reminiscent of OPEs in a $2$-dimensional CFT. Then, we discuss how a non-commutative $\mathbb{R}^4$-background deforms these celestial OPEs in an interesting way. The following chapter reviews some basic twistor theory and various actions on twistor space and spacetime that describe self-dual gravity and self-dual Yang-Mills theory at the classical level. In the following chapter, we give a detailed analysis of celestial symmetries in an asymptotically (locally) Euclidean space, Eguchi-Hanson space, that solves the equations of self-dual Einstein gravity. This deformation arises naturally from a backreaction on twistor space analogous to parts of Burns holography, the top-down construction of Costello, Paquette and Sharma and we will highlight similarities and differences. We explain how the deformed celestial OPEs are closely related to certain chiral algebras, from now on referred to as celestial chiral algebras, supported on twistor lines. In the final two chapters, we discuss the presence of a non-zero cosmological constant, which has many subtleties. Twistor theory allows us to also include a cosmological constant in the self-dual Einstein equations, and after reviewing the relevant background material, we discuss how the cosmological constant deforms the gravitational celestial chiral algebra. This gives an independent derivation of a deformed algebra previously found by Taylor and Zhu. Repeating the twistorial backrection in the presence of a cosmological constant leads us to self-dual limits of Pleba'{n}ski-Demia'{n}ski black hole metrics. From their twistor perspective we derive a two-parameter deformation which generalises both the Eguchi-Hanson and cosmological constant deformations we previously discussed and in a sense interpolates between them.