Conformal Blocks in Two and Four Dimensions from Oscillator Representations

The explicit computation of higher-point conformal blocks in any dimension is usually a challenging task. For two-dimensional conformal field theories in Euclidean signature, the oscillator formalism proves to be very efficient. We demonstrate this by reproducing the general $n$-point global conformal block in the comb channel in an elegant and direct manner. Exploiting similarities to the representation theory of two-dimensional CFTs, we extend the oscillator formalism to the computation of higher-point conformal blocks in four Euclidean dimensions. As a proof of concept, we explicitly compute the scalar four-point block with scalar exchange within this framework and discuss the extension to the higher-point case.