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 ′ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>&gt;</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>&gt;</mml:mo> </mml:mrow> </mml:math> 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> , which promote a conformal block in d dimensions to five conformal blocks in d + 2 dimensions. These operators can be normalized such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:munder> <mml:mo>∑</mml:mo> <mml:mi>i</mml:mi> </mml:munder> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , from which the KRT identity immediately follows. Similar, simpler identities have been proposed, all of which can be reformulated in the same way.