Infinite Symmetry Algebras in Four-Dimensional Conformal Field Theories

Abstract

In generic interacting four-dimensional Lorentzian conformal field theories, an infinite set of universal light-ray operators constructed from the stress tensor is shown to generate the wedge subalgebra of the loop algebra of ${\rm w}_{1+\infty}$. This algebra was recently identified among the asymptotic symmetries of asymptotically flat spacetimes. The one-point functions of the ${\rm w}_{1+\infty}$ generators in scalar states (also known as one-point event shapes) are explicitly demonstrated to be finite and are precisely related to universal soft factors in the infinite tower of soft graviton theorems. A second universal class of light-ray operators that generates the ''$S$ algebra,'' the gauge-theoretic analog of ${\rm w}_{1+\infty}$, is also constructed and shown to have finite one-point functions in four-dimensional conformal field theories with a spin-one conserved current. Along with the details of these results, this paper presents a general classification of stress-tensor and conserved current light-ray operators by scaling dimension and Lorentz ${\rm SL}(2,\mathbb{C})$ weights, a general technique for computing commutators by Poincar\'e recursion, results for other light-ray operator algebras including a local version of the four-dimensional conformal symmetry algebra, and examples for free scalar fields.