Twisted vertex algebras, bicharacter construction and boson-fermion correspondences

Abstract

The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new bosonfermion correspondence, of type D-A. Further, we define a new concept of twisted vertex algebra of order N , which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions (OPEs), analytic continuations and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for three important groups of examples. We show that the correspondences of type B, C and D-A are isomorphisms of twisted vertex algebras. Contents

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].…”

“…turns into the new bilocal fermionization formula [2] embedding the current into the real CAR algebra…”

“…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)…”

“…The first is the fact [2] that besides the standard fermionization formula j(x) = : φ * (x)φ(x) : of the free chiral Bose current in terms of a complex Fermi field, one can obtain the same current as a bilocal Wick product of a real Fermi field at two different points: j(x) ∼ : ψ(x 1 )ψ(x 2 ) : , where x 2 = −1/x 1 are the two solutions of q(x i ) ≡ 2x i /(1 −x 2 i ) = x. (The differential algebra underlying this relation was described much earlier in [8] in the context of hierarchies of integrable systems.…”

A universe field theory for JT gravity

“…We treat it as the indefinite integral of J (z). For an account of the boson-fermion correspondence for twisted fields, see[64,65].…”