Intertwining Operator in Thermal CFT$_{d}$

Abstract: It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to fulfill some operator identities---the intertwining relations---in the representation space of conformal algebra. Meanwhile, it has been known that the S-matrix operator in scattering theory is nothing but the intertwining operator between the Hilbert spaces of in- and out-particles. Inspired by this algebraic resemblance, in this paper we develop a simple Lie-algebraic approach to momentum-space two-point functions of thermal CFT living on the hyperbolic spacetime $\mathbb{H}^{1}\times\mathbb{H}^{d-1}$ by exploiting the idea of Kerimov's intertwining operator approach to exact S-matrix. We show that in thermal CFT on $\mathbb{H}^{1}\times\mathbb{H}^{d-1}$ the intertwining relations reduce to certain linear recurrence relations for two-point functions in the complex momentum space. By solving these recurrence relations, we obtain the momentum-space representations of advanced and retarded two-point functions as well as positive- and negative-frequency two-point Wightman functions for a scalar primary operator in arbitrary spacetime dimension $d\geq3$.