Thermal $n$-Point Conformal Blocks in Four Dimensions from Oscillator Representations

We define and compute the four-dimensional thermal $n$-point conformal block in the projection channel using oscillator representations on ${\mathbb{S}}_{\beta }^1\times {\mathbb{S}}^3$. This is done by evaluating a class of integrals over the homogeneous space ${\mathbb{D}}_4$ of the four-dimensional conformal group. We restrict ourselves to scalar external operators and scalar exchange. In the low-temperature limit, our result reduces correctly to the vacuum $(n+2)$-point block in the comb channel. The corresponding expressions can be written as a series of terminating hypergeometric functions or equivalently, a series of weighted SU(2) spin-networks. Alternatively, functions adapted to the SU(2,2) representation are introduced and some properties are discussed.