Thermal n-point conformal blocks in four dimensions from oscillator representations

A bstract We define and compute the four-dimensional thermal n -point conformal block in the projection channel using oscillator representations on $$ {\mathbbm{S}}_{\beta}^1\times {\mathbbm{S}}^3 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . This is done by evaluating a class of integrals over the homogeneous space $$ {\mathbbm{D}}_4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> 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.