A note on the boson-fermion correspondence and infinite dimensional groups

We show how to construct irreducible projective representations of the infinite dimensional Lie group Map (S¹,), by embedding it into the group of Bogoliubov automorphisms of the CAR. Using techniques of G. Segal for extending certain representations of Map (S¹, SU(2)) we show that our representations extend to give representations of a certain infinite dimensional superalgebra. We relate our work to the well known boson-fermion correspondence which exists in 1+1 dimensions.