Lectures on Celestial Holography

TL;DR

The notes develop celestial holography for four-dimensional asymptotically flat spacetimes, connecting subleading soft graviton theorems to Virasoro symmetry and reformulating scattering as celestial amplitudes on the 2D celestial sphere. They construct conformal primary wavefunctions for massive and massless states, build a Milne/AdS3-inspired framework, and map momentum-space amplitudes to 2D CFT correlators, illustrating with a tree-level 2 massless/1 massive 3-point example. Poincaré invariance and soft theorems impose strong constraints on celestial correlators, including a 2D stress tensor Ward identity and a rich tower of conformally soft symmetries that constrain OPEs and correlator data. The discussions culminate in the emergence of holographic symmetry algebras from soft currents, suggesting a highly structured CCFT describing 4D gravity in the celestial setting.

Abstract

These notes consist of 3 lectures on celestial holography given at the Pre-Strings school 2021. We start by reviewing how semiclassically, the subleading soft graviton theorem implies an enhancement of the Lorentz symmetry of scattering in four-dimensional asymptotically flat gravity to Virasoro. This leads to the construction of celestial amplitudes as $\mathcal{S}$-matrices computed in a basis of boost eigenstates. Both massless and massive asymptotic states are recast as insertions on the celestial sphere transforming as global conformal primaries under the Lorentz SL$(2, \mathbb{C})$. We conclude with an overview of celestial symmetries and the constraints they impose on celestial scattering.

Figures (5)

• Figure 1: The soft limit relates an amplitude with a low-energy massless particle to the same amplitude without the massless particle.
• Figure 2: In the soft limit, the amplitude will include contributions from Feynman diagrams where the soft particle attaches to external and internal lines. Diagrams where the soft particle attaches to an internal line are subleading in the soft limit.
• Figure 3: Penrose diagram of Minkowski space.
• Figure 4: Penrose diagram of Minkowski space where each pair of points represents a two-sphere. Massive particles come in from $i^-$ and go out at $i^+$, while massless particles enter and exit spacetime at $\mathcal{I}^{\pm}$.
• Figure 5: Minkowski space split into four regions: the past and future lightcones are covered by $\mathbb{H}_3$ slices while the causally disconnected Rindler regions are covered by $dS_3$ slices.