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.

Questo articolo esplora i giri e le loro implicazioni.

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