Temperature states on loop groups, Theta functions and the Luttinger model

Abstract

We consider projective representations of the loop group of U( N) suggested by statistical mechanics. These representations are defined by certain positive definite functions (called temperature states) on the universal central extension of the loop group. We find that the Boson-Fermion correspondence, known to exist in a mathematically precise form at zero temperature, persists at non-zero temperature. By expressing the temperature state for U(1) in two different ways we obtain identities between elliptic functions. We apply these ideas to a simple model in statistical physics (the Luttinger model) establishing the existence of a projective representation of the loop group of U(1) × U(1) in the presence of interaction and at all temperatures. A rigorous derivation of the correlation functions for this model is given using the Boson-Fermion correspondence.