Abstract
We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator ∂μ, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator mathcal{D} ijA used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents 〈JJJJ〉 and four energy-momentum tensors 〈TTTT〉, we mostly focus on conservation conditions for leftlangle JJmathcal{O}rightrangle and leftlangle TTmathcal{O}rightrangle . Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.
Highlights
Quantum field theories are often classified in terms of their symmetry group and their matter content, with the global symmetry group fixing the allowed charges of the matter fields
The final results may be divided into three distinct parts: one contribution that can be made conformally covariant and that annihilates two-point correlation functions; one contribution which can be discarded since it annihilates all correlation functions; and one contribution which is not conformally covariant and that vanishes only when the unitarity bound is saturated
Once we enforce the conservation of two-point correlation functions by saturating the unitarity bound — with “twists” τ ∗ = (d − 2)/2 for scalars and spinors and τ = τ ∗ = d − 1 − n for quasi-primary operators in general irreducible representations N — we may derive the conformally-covariant conserved current differential operators in embedding space directly from their position space counterparts as
Summary
Quantum field theories are often classified in terms of their symmetry group and their matter content, with the global symmetry group fixing the allowed charges of the matter fields. We use the embedding space formalism of [3, 4] to analyze conservation conditions of quasi-primary operators in arbitrary irreducible representations of the Lorentz group. Three-point correlation functions of conserved vector currents and energy-momentum tensors were discussed extensively and in all generality in [14, 15], while conserved currents in the O(N ) model were investigated in [16]. The relevant result from the point of view of unitarity, where conformally-covariant equations contract the position space differential operator with a vector index of the smallest row of the Young tableau associated to a conserved current in a mixed irreducible representation, were used in [19, 20] to analyze conserved vector currents and the energy-momentum tensor. Apart from developing the techniques necessary for the study of conservation conditions in the embedding space formalism, this paper can be seen as a complement to [19], where the conservation conditions for JJO and T T O for the case of a symmetric-traceless operator O were first computed
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