Boson-Fermion Correspondence of Type B and Twisted Vertex Algebras

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.

Multilocal Fermionization

“…This is possible by a "change of localization", and gives rise to new multilocal symmetries generated by the corresponding multilocal current and stress-energy tensor. The result gives a common underlying explanation of several remarkable recent results on the representation of the free Bose field in terms of free Fermi fields [1,2], and on the modular theory of the free Fermi algebra in disjoint intervals [7,15].…”

“…The fact that the current (3.1) j(x) = i : ψ (1) (x)ψ (2) (x) : satisfies the CCR (3.2) is purely algebraic, and therefore independent of the representation. Taking ψ (1) in the Ramond representation and ψ (2) in the vacuum representation, the isomorphism (5.1) embeds the resulting current into the Ramond algebra π R CAR(S 1 ) . The result is (in the compact picture)…”

“…It is well known that there is an algebraic isomorphism (the "split isomorphism") A(x)B(y) → A(x) ⊗ t B(y), x ∈ O 1 , y ∈ O 2 between the fields of a given QFT localized in two spacetime regions O i , and two independent copies of the same fields localized in the same regions, as long as the regions are sufficiently well spacelike separated from each other. 1 Naively, this is true because A (anti)commute with B by graded locality, while A ⊗ t 1 (anti)commute with 1 ⊗ t B by construction, so that all algebraic relations are preserved. However, this isomorphism does not preserve the vacuum state, because it eliminates all correlations between fields in O 1 and fields in O 2 : ω(A(x) ⊗ t B(y)) = ω(A(x)) · ω(B(y)) = ω(A(x)B(y)).…”

N-point locality for vertex operators: Normal ordered products, operator product expansions, twisted vertex algebras

“…Then the answer to our "generating descendants" problem is: the fields a i (z) will generate a twisted vertex algebra. Twisted vertex algebras of general order N were defined in [Ang12] (of order 2 in [Ang13]) in order to answer the question: what are the boson-fermion correspondences of type B, C and D-A? These correspondences must be isomorphisms between some type of structures, and we wanted to understand those structures.…”

Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space $\mathit {F^{\otimes \frac{1}{2}}}$F⊗12

“…The correspondence of type A is an isomorphism of super vertex algebras, but most boson-fermion correspondences cannot be described by the concept of a super vertex algebra due to the more general singularities in their operator product expansions. In order to describe the more general cases, including the correspondences of types B, C and D-A, in [Ang13a] and [Ang13b] we defined the concept of a twisted vertex algebra which generalizes super vertex algebra. By utilizing the bicharacter construction in [Ang13b] we showed that the correspondences of types B, C and D-A are isomorphisms of twisted vertex algebras.…”