1206.4026 (Iana I. Anguelova)

Twisted vertex algebras, bicharacter construction and boson-fermion
correspondences    [PDF]

Iana I. Anguelova

The boson-fermion correspondences are a fascinating phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. The boson-fermion correspondence of type A is an isomorphism between two super vertex algebras (and so has singularities in the operator product expansions only at $z = w$). But most boson-fermion correspondences (the correspondence of type B and others) cannot be described by the concept of a super vertex algebra, as they have more general singularities in their operator product expansions. Thus the answer to the question "what is a boson-fermion correspondence" needs a more general concept than that of a super vertex algebra. In this paper we present such a notion: the concept of a twisted vertex algebra, which generalizes the concept of a super vertex algebra. We also present the bicharacter construction which allows us to describe many and varied examples of twisted vertex algebras, with the boson fermion correspondence of type B among them. We construct another, new, example of a boson-fermion correspondence: the boson-fermion correspondence of type D-C. The correspondence of type D-C is also an example of isomorphism of twisted vertex algebras. The bicharacter construction is the main working set of tools in this paper, as it is uniquely suited for calculating operator product expansions (OPEs), analytic continuations and vacuum expectation values using the underlying Hopf algebra structure. We present also general bicharacter formulas for the vacuum expectation values for three important groups of examples.

View original: http://arxiv.org/abs/1206.4026