Conformal Conserved Currents in Embedding Space

TL;DR

This paper develops a formalism using embedding space techniques to analyze conformal conserved currents across various representations, providing explicit differential operators for conservation conditions and implications for conformal bootstrap.

Contribution

It introduces a covariant differential operator in embedding space for conservation, extending the analysis of conformal currents beyond symmetric traceless cases.

Findings

Derived explicit conservation operators in embedding space.

Analyzed conservation conditions for diverse current representations.

Connected three-point coefficients to charges via conformal Ward identities.

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 $\partial_\mu$, 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 $\langle JJJJ\rangle$ and four energy-momentum tensors $\langle TTTT\rangle$, we mostly focus on conservation conditions for $\langle JJ\mathcal{O}\rangle$ and $\langle TT\mathcal{O}\rangle$. Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.