Algebraic approach to parafermionic conformal field theories

Abstract

Parafermionic conformal field theories are considered on a purely algebraic basis. The generalized Jacobi type identity is presented. Systems of free fermions coupled to each other by nontrivial parafermionic type relations are studied in detail. A new parafermionic conformal algebra is introduced, it describes the sl(2|1) 2 /u(1) 2 coset system.

Classification of symmetric polynomials of infinite variables: Construction of Abelian and non-Abelian quantum Hall states

“…50 The CFT formed by the simple-current operators ψ a is a special case of a generalized vertex algebra.…”

$\Bbb {Z}_2$ x $\Bbb {Z}_2$ graded superconformal algebra of parafermionic type

“…(37) For the derivation see Section 4 in [9]. A m and B n are the modes of the fields A(z) and B(z), and C (l) (z) are the terms in the operator product expansion of A(z)B(w):…”

“…We will follow here Ref. [9] (Sections 2 and 3). This algebraic approach in fact goes back to 1993 [7].…”

“…In our previous work [9] we applied the algebraic approach to calculate the structure constants of the sl(n) 2 /u(1) 2 and the sl(2|1) 2 /u(1) 2 coset parafermions. We called the generators of the former theory the sl(n) fermions.…”

“…The current paper is the direct continuation of [9], we use the same setting and the same tools to derive the Z 2 × Z 2 graded N = 1 superconformal algebra. This algebra resembles in many aspects the sl(3) fermion algebra from [9]. But the new algebra is more complicated: it has more generating fields and has one free continuous parameter.…”