On the stress tensor light-ray operator algebra

Abstract

We study correlation functions involving generalized ANEC operators of the form $$ \int {dx}^{-}{\left({x}^{-}\right)}^{n+2}{T}_{--}\left(\overrightarrow{x}\right) $$ ∫ dx − x − n + 2 T − − x → in four dimensions. We compute two, three, and four-point functions involving external scalar states in both free and holographic Conformal Field Theories. From this information, we extract the algebra of these light-ray operators. We find a global subalgebra spanned by n = {−2, −1, 0, 1, 2} which annihilate the conformally invariant vacuum and transform among themselves under the action of the collinear conformal group that preserves the light-ray. Operators outside this range give rise to an infinite central term, in agreement with previous suggestions in the literature. In free theories, even some of the operators inside the global subalgebra fail to commute when placed at spacelike separation on the same null-plane. This lack of commutativity is not integrable, presenting an obstruction to the construction of a well defined light-ray algebra at coincident $$ \overrightarrow{x} $$ x → coordinates. For holographic CFTs the behavior worsens and operators with n ≠ −2 fail to commute at spacelike separation. We reproduce this result in the bulk of AdS where we present new exact shockwave solutions dual to the insertions of these (exponentiated) operators on the boundary.

On local and integrated stress-tensor commutators

“…The obvious 4d generalization of these operators arises by taking f (u) ∼ (iR + u) 2−n (iR − u) 2+n . These operators close under the subgroup of the conformal group which preserves the null ray [25,42], and also have the property that they annihilate the vacuum on the left and/or on the right, but it is not yet clear whether these operators give rise to interesting Ward identities or sum rules. This paper is organized as follows.…”

“…We first discuss the action of the collinear subalgebra of the conformal algebra on these operators following [25,42]. The collinear subalgebra is defined as the subalgebra of the full conformal algebra that leaves the light-ray v = 0, x ⊥ = 0 invariant, and is generated by 13…”

“…Convergence of the Lorentzian OPE is a subtle issue and it is not a priori clear whether one is actually allowed to use the OPE when computing the commutator inside general correlation functions. In fact there is a disagreement between a holographic computation and the field theory computation of the BMS commutator [K, E] [25], which could be a large N artifact but also a signal that there is an issue with the precise definition of these operators.…”

“…If light-ray operators are strictly speaking ill-defined, one could try to modify their definition to make them well-defined, but it is not clear what such a modification could be. One suggestion is to add an explicit regulator [25] so that the large u behavior is improved, and which has the effect that one can generically close contours in the u-plane as long as the u-behavior is analytic and there are no branch cuts. Alternatively, one could think more seriously about the conformal embedding of Minkowski space in the Lorentzian cylinder and only consider states on Minkowski spacetime which admit a proper extension to the full cylinder.…”

“…The main motivation for this work is to compute, as much as possible from first principles, commutators of various types of light-ray operators built out of components of the stress-tensor. Such operators are very interesting, as one can use them to build a BMS algebra [10,11], to construct the Averaged Null Energy Operator (ANEC) [12,13] (see also [14][15][16][17][18][19][20][21] for example of other uses of the ANEC operator JHEP07(2021)148 in CFT), and to possibly uncover new symmetry-like structures generated by operators of the form duf (u)T uu with u a lightcone coordinate [10,[22][23][24][25][26]. For a general discussion of light-ray operators in CFT, see e.g.…”