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

TL;DR

This paper introduces a method to compute four-dimensional thermal n-point conformal blocks using oscillator representations, providing explicit formulas and analyzing their properties in the low-temperature limit.

Contribution

It presents a novel approach to calculating thermal conformal blocks in four dimensions via oscillator representations and integral evaluations on the conformal group.

Findings

Reduces to vacuum (n+2)-point block at low temperature

Expresses blocks as hypergeometric series or SU(2) spin-networks

Introduces functions adapted to SU(2,2) representations

Abstract

We define and compute the four-dimensional thermal $n$-point conformal block in the projection channel using oscillator representations on $\mathbb{S}^1_\beta \times \mathbb{S}^3$. 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.