Dimensional Uplift in Conformal Field Theories

TL;DR

The paper addresses whether the dimensional uplift of conformal blocks from $(d-2)$ to $d$ dimensions can be derived purely from conformal invariance. It presents an elementary algebraic proof of the Kaviraj–Rychkov–Trevisani (KRT) identity by constructing five differential operators $Y_i$ that yield a closed algebra with the quadratic Casimir, and then defines normalized operators $\Lambda_i=V_i^jY_j$ whose action maps a $(d-2)$-dimensional block $g^{(d-2)}_{\Delta,\ell}$ to a linear combination of five $d$-dimensional blocks $g^{(d)}_{\Delta+\alpha,\ell-\beta}$ with $(\alpha,\beta)\in\{0,1,2\}$ and even $\alpha+\beta$. The key result is a normalization $\,\sum_i\Lambda_i=1$, which immediately yields the KRT uplift $g^{(d-2)}_{\Delta,\ell}=\sum_{\alpha,\beta} k_{\alpha\beta}\, g^{(d)}_{\Delta+\alpha,\ell-\beta}$; scalar blocks further simplify to fewer terms. The method clarifies how dimensional reduction and uplift are encoded in conformal symmetry and extends to related uplift identities, including boundary CFT contexts. Overall, the work provides a concrete, algebraic mechanism to relate spectra across dimensions and reconstruct higher-dimensional conformal data from lower-dimensional information.

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