Abstract
We study correlation functions involving generalized ANEC operators of the form int {dx}^{-}{left({x}^{-}right)}^{n+2}{T}_{--}left(overrightarrow{x}right) 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} 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.
Highlights
Introduction and summary of resultsTwo-dimensional conformal field theories occupy a special place in the landscape of all Conformal Field Theories (CFTs)
We study correlation functions involving generalized averaged null energy condition (ANEC) operators of the form dx− (x−)n+2 T−−(x) in four dimensions
The easiest way to compute the contribution of individual conformal blocks is to map the null plane x+ = 0 to the celestial sphere. After briefly reviewing this map, we demonstrate that O exchange gives rise to a nonzero commutator at finite separation, even for the simplest case of [L−1, L−2], which suggests the existence of a non-trivial sum rule, where subleading conformal blocks must cancel this finite-separation commutator in physical CFTs
Summary
Two-dimensional conformal field theories occupy a special place in the landscape of all Conformal Field Theories (CFTs). A intuitive way to think about the Virasoro symmetry is that in two dimensions, only the stress-tensor T and its composites can appear in the T × T OPE, with coefficients uniquely determined by the central charge This immediately presents serious challenges for higher-dimensional CFTs since all (neutral) operators of the theory can in principle appear in the stress-tensor OPE, making any form of universality seem hopeless. One may hope that this universality is controlled by a Virasoro symmetry, emergent at large N and large ∆gap, which can recast gravitational dynamics of Einstein gravity in terms of a symmetry It is with this overarching goal in mind that we will study the algebra of light-ray operators (1.1). The algebra (1.3) as advocated for in [15], does not seem to hold, neither in free field theory nor in holographic CFTs where one would expect the most universality.
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