Lectures on Celestial Amplitudes

TL;DR

Celestial Holography proposes a duality between 4D scattering in asymptotically flat spacetimes and a 2D CFT on the celestial sphere. The notes develop a dictionary that maps S-matrix elements to celestial CFT correlators using conformal primary wavefunctions and Mellin transforms, and show how soft theorems become 2D currents. They connect asymptotic symmetries (BMS) to Ward identities and derive celestial OPEs from collinear limits, revealing a rich operator algebra with a proposed stress-tensor current. The lectures also survey active research directions and provide exercises and literature guidance for CCFT.

Abstract

Lecture notes prepared for the 2021 SAGEX PhD School in Amplitudes hosted by the University of Copenhagen August 10th through 13th. Topics covered include: the manifestation of asymptotic symmetries via soft theorems, their organization into currents in a celestial CFT, aspects of the holographic dictionary, a literature guide, and accompanying exercises.

Figures (3)

• Figure 1: Penrose diagram for Minkowski space. Massive particles enter and exit at past ($i^-$) and future ($i^+$) timelike infinity. Massless excitations enter and exit at past ($\cal{I}^-$) and future ($\cal{I}^+$) null infinity. Meanwhile, the boundary of a Cauchy slice will limit to spatial infinity ($i^0$).
• Figure 2: In the limit where the energy of a graviton or gauge boson goes 'soft,' the leading contribution will come from attaching to each of the 'hard' external legs.
• Figure 3: Celestial Diamond illustrating the nested submodule structure of $SL(2,\mathbb{C})$ primary descendants that exist for special values of the conformal weights.