Abstract
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). The boson-fermion correspondence of type B plays similarly important role in many areas, including representation theory, integrable systems, random matrix theory and random processes. But the vertex operators describing it have singularities in their operator product expansions at both z = w and \(z = -w\), and thus need a more general notion 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 super vertex algebra. The two sides of the correspondence of type B constitute two examples of twisted vertex algebras. The boson-fermion correspondence of type B is thus an isomorphism between two twisted vertex algebras.
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Notes
- 1.
- 2.
The reason for this notation is that the resulting vertex algebra is a lattice vertex algebra.
- 3.
The reason for the name is that there is a free neutral fermion of type D, which is commonly referred to as just “the free neutral fermion”. In fact, there is a boson-fermion correspondence of type D-C, see [1].
- 4.
For details on normal ordered products in this more general case see [1], the construction uses an additional Hopf algebra structure, similar to Laplace pairing.
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Anguelova, I.I. (2013). Boson-Fermion Correspondence of Type B and Twisted Vertex Algebras. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_28
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DOI: https://doi.org/10.1007/978-4-431-54270-4_28
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