Iana I. Anguelova

Twisted vertex algebras, bicharacter construction and boson-fermion correspondences

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 boson-fermion 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, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.

Boson-Fermion Correspondence of Type B and Twisted Vertex Algebras

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.

Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space F⊗12

We construct the bosonization of the Fock space F⊗12 of a single neutral fermion by using a 2-point local Heisenberg field. We decompose F⊗12 as a direct sum of irreducible highest weight modules for the Heisenberg algebra HZ, and thus we show that under the Heisenberg HZ action the Fock space F⊗12 of the single neutral fermion is isomorphic to the Fock space F⊗1 of a pair of charged free fermions, thereby constructing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on F⊗12. We construct a family of 2-point-local Virasoro fields with central charge −2+12λ−12λ2,λ∈C, on F⊗12. We construct a W1 + ∞ representation on F⊗12 and show that under this W1 + ∞ action F⊗12 is again isomorphic to F⊗1.

The second bosonization of the CKP hierarchy

In this paper we discuss the second bosonization of the Hirota bilinear equation for the CKP hierarchy introduced by Date, Jimbo, Kashiwara and Miwa. We show that there is a second, untwisted, Heisenberg action on the Fock space, in addition to the twisted Heisenberg action suggested by Date, Jimbo, Kashiwara and Miwa and studied by van de Leur, Orlov and Shiota. We derive the decomposition of the Fock space into irreducible Heisenberg modules under this action. We show that the space spanned by the highest weight vectors of the irreducible Heisenberg modules has a structure of a super vertex algebra, specifically the symplectic fermions vertex algebra. We complete the second bosonization of the CKP Hirota equation by expressing the generating field via exponentiated boson vertex operators acting on a polynomial algebra with two infinite sets of variables.

Virasoro Structures in the Twisted Vertex Algebra of the Particle Correspondence of Type C

In this paper we study the existence of Virasoro structures in the twisted vertex algebra describing the particle correspondence of type C. We show that this twisted vertex algebra has at least two distinct Virasoro structures: one with central charge 1, and a second with central charge − 1.

Representations of a∞ and d∞ with central charge 1 on the single neutral fermion Fock space

We construct a new representation of the infinite rank Lie algebra a∞ with central charge c = 1 on the Fock space of a single neutral fermion. We show that is a direct sum of irreducible integrable highest weight modules for a∞ with central charge c = 1. We prove that as a∞ modules is isomorphic to the Fock space F⊗1 of the charged free fermions. As a corollary we obtain the decompositions of certain irreducible highest weight modules for d∞ with central charge into irreducible highest weight modules for d∞ with central charge c = 1.

Quadratic Differential Operators, Bicharacters and • Products

For a commutative cocommutative Hopf algebra, we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator, and a • product obtained via twisting by a bicharacter. This new relationship between • products and exponentials of quadratic differential operators was inspired by studying the exponential of a quadratic differential operator introduced in [7] and used in the theory of twisted modules of lattice vertex algebras.