We define and compute the four-dimensional thermal n-point conformal block in the projection channel using oscillator representations on \( {\mathbbm{S}}_{\beta}^1\times {\mathbbm{S}}^3 \) . This is done by evaluating a class of integrals over the homogeneous space \( {\mathbbm{D}}_4 \) 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.
