Finite Temperature Correlation Functions and Modular Forms in a Globally Conformal Invariant QFT

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space–time dimensions implies the Huygens’ principle, and hence, rationality of correlation functions of observable fields [16]. The conformal hamiltonian H has discrete spectrum and we assume that the partition function trD ( q ) , q = e2πi τ , Im τ > 0 (|q| < 1) as well as the finite temperature expectation values of the field products are well defined in the finite energy space D (an assumption that is verified for free fields). We then demonstrate that the finite temperature expectation values are expressed by (doubly periodic) elliptic functions in appropriate coordinates. We compute examples of 2-point functions of free fields and study the modular transformation properties of the mean value of the energy in an equilibrium state with respect to the (complex) inverse temperature parameter τ .