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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 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.

© 2013 AIP Publishing LLC

Received 08 July 2013
Accepted 22 November 2013
Published online 12 December 2013

Article outline:

I. INTRODUCTION
A. Motivation
B. Overview of the paper
II. NOTATION AND BACKGROUND
III. TWISTED VERTEX ALGEBRAS: DEFINITION, OVERVIEW, EXAMPLES
A. Twisted vertex algebras: Definition and overview
B. Examples of twisted vertex algebras: The boson-fermion correspondence of type B
C. Examples of twisted vertex algebras: The boson-fermion correspondence of type D-A
D. Examples of twisted vertex algebras: The boson-fermion correspondence of type D-A of order *N*
IV. BICHARACTER CONSTRUCTION: CONSTRUCTING EXAMPLES OF TWISTED VERTEX ALGEBRAS
A. Super bicharacters and free Leibnitz modules
B. Examples of free Leibnitz modules
C. Exponential map and its properties: Nonsingular twisted vertex algebras
D. Vertex operators, analytic continuations, OPEs, and normal ordered products from a bicharacter
V. EXAMPLES OF TWISTED VERTEX ALGEBRAS BASED ON A BICHARACTER
A. Twisted vertex algebras based on and a choice of a bicharacter
B. Twisted vertex algebras based on : Pfaffian vacuum expectation values
C. Twisted vertex algebras based on : The neutral free fermion of type B
D. Twisted vertex algebras based on : The neutral free fermion of type D-A
E. Twisted vertex algebras based on and a choice of a bicharacter
F. Twisted vertex algebras based on : Product vacuum expectation values
G. Twisted vertex algebra based on : The free boson of type B
H. Twisted vertex algebra based on : The free boson of type D-A
I. Boson-fermion correspondence of type D-A and order *N*
J. Other examples of boson-fermion correspondences and twisted vertex algebras
VI. SUMMARY

/content/aip/journal/jmp/54/12/10.1063/1.4842075

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/content/aip/journal/jmp/54/12/10.1063/1.4842075

2013-12-12

2016-05-26

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