Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory

Abstract

We establish an isomorphism between the vertex and spinor representations of affine Lie algebras for types D l (1) and D l + 1 (2) . We also study decomposition of spinor representations using the infinite family of Casimir operators and prove that they are either irreducible or have two irreducible components. We show that the vertex and spinor constructions of the representations can be reformulated in the language of two-dimensional quantum field theory. In this physical context, the two constructions yield the generalized sine-Gordon and Thirring models, respectively, already in renormalized form. The isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson-fermion correspondence