Dimensional uplift in conformal field theories

The n-point functions of any conformal field theory (CFT) in d dimensions can always be interpreted as spatial restrictions of corresponding functions in a higher-dimensional CFT with dimension d′  > d. In particular, when a four-point function in d dimensions has a known conformal block expansion, this expansion can be easily extended to d′ = d + 2 due to a remarkable identity among conformal blocks, discovered by Kaviraj, Rychkov and Trevisani (KRT) as a consequence of Parisi–Sourlas supersymmetry and confirmed to hold in any CFT with d  > 1. In this note, we provide an elementary proof of this identity using simple algebraic properties of the Casimir operators. Additionally, we construct five differential operators, Λi, which promote a conformal block in d dimensions to five conformal blocks in d + 2 dimensions. These operators can be normalized such that ∑iΛi=1, from which the KRT identity immediately follows. Similar, simpler identities have been proposed, all of which can be reformulated in the same way.