Dimensional Uplift in Conformal Field Theories

Abstract

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, $\Lambda_i$, which promote a conformal block in $d$ dimensions to five conformal blocks in $d+2$ dimensions. These operators can be normalized such that $\sum_i \Lambda_i = 1$, from which the KRT identity immediately follows. Similar, simpler identities have been proposed, all of which can be reformulated in the same way.