Abstract
We analyze the commutation relations of light-ray operators in conformal field theories. We first establish the algebra of light-ray operators built out of higher spin currents in free CFTs and find explicit expressions for the corresponding structure constants. The resulting algebras are remarkably similar to the generalized Zamolodchikov’s W∞ algebra in a two-dimensional conformal field theory. We then compute the commutator of generalized energy flow operators in a generic, interacting CFTs in d > 2. We show that it receives contribution from the energy flow operator itself, as well as from the light-ray operators built out of scalar primary operators of dimension ∆ ≤ d − 2, that are present in the OPE of two stress-energy tensors. Commutators of light-ray operators considered in the present paper lead to CFT sum rules which generalize the superconvergence relations and naturally connect to the dispersive sum rules, both of which have been studied recently.
Highlights
We show that it receives contribution from the energy flow operator itself, as well as from the light-ray operators built out of scalar primary operators of dimension ∆ ≤ d − 2, that are present in the OPE of two stress-energy tensors
Where a local conformal operator Oμ1...μS with scaling dimension ∆ and Lorentz spin S is sent to null infinity in the direction specified by the null vector n = (1, n), with n being a unit vector on the sphere Sd−2, in a d-dimensional CFT
In this paper we explored commutation relations of the light-ray operators (1.1) built out of local operators in a d-dimensional CFT
Summary
These operators are built out of two fundamental fields (scalars, fermions and gauge field strength tensor) and their derivatives They carry the leading twist τ = ∆ − S = d − 2 and have the following schematic form. Notice that for d = 2 and jφ = 0, the normalization factor looks as N ∼ Γ( − 1) and it diverges for = 0 and = 1 The reason for this is that the corresponding scalar operators, O0(φ) = φφ and O1(φ) = −iφ(∂→− − ∂←−)φ, are not conformal primary operators in d = 2. For S = 2, the operators (2.12) are proportional to the stress-energy tensor T−− = Tμνnμnν in the corresponding free CFTs. In a free theory of a single complex scalar, Dirac fermion, and gauge field we have [16, 29].
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