Currents in Celestial CFT

In this review we discuss currents in celestial CFT and the consistency of their naive symmetry algebras. In particular we study in detail the Jacobi identity and the double residue condition for soft insertions, hard momentum space insertions, and hard celestial insertions. In the latter case we introduce the notion of a "hard current" in CFT and work through examples in the 2D critical Ising model. The current algebra of hard insertions in pure Einstein gravity is a slight conceptual generalization of the familiar $w_{1+\infty}$-wedge current algebra. We also review branch cut terms in the celestial OPE, which indicate new primary content and were previously missed until recently. We work through an explicit toy example illustrating the mechanism by which such branch cut terms can arise. These branch cut terms prevent a symmetry interpretation but are fully compatible with a consistent OPE.