Projective representations of the loop group and the Boson-Fermion correspondence

We consider some particular projective representations of the restricted orthogonal and symplectic groups. These representations are related to the second quantization. We derive very simple and natural cocycles associated with these representations, as we get explicit and simple expressions for the corresponding vacuum functionals. We apply our results to the loop group LS1 and the diffeomorphism group Diff+(S1). Finally, we derive the bosonfermion correspondence in a particular case in a quite transparent fashion.