Parafermionic Algebra Research Articles

The third parafermionic chiral algebra with the symmetry [formula omitted

We have constructed the parafermionic chiral algebra with the principal parafermionic fields Ψ,Ψ+ having the conformal dimension ΔΨ=8/3 and realizing the symmetry Z3.

D-branes and extended characters in [formula omitted

We present a detailed study of D-branes in the axially gauged SL ( 2 , R ) k / U ( 1 ) coset conformal field theory for integer level k. Our analysis is based on the modular bootstrap approach and utilizes the extended SL ( 2 , R ) / U ( 1 ) characters and the embedding of the parafermionic coset algebra in the N = 2 superconformal algebra. We propose three basic classes of boundary states corresponding to D0-, D1- and D2-branes. We verify that these boundary states satisfy the Cardy consistency conditions and discuss their physical properties. The D0- and D1-branes agree with those found in earlier work by Ribault and Schomerus using different methods (descent from the Euclidean AdS 3 model). The D2-branes are new. They are not, in general, space-filling but extend from the asymptotic circle at infinity up to a minimum distance ρ min ⩾ 0 from the tip of the cigar.

Twisting of homogeneous algebras

Twisting process for homogeneous algebras is defined. It consists of a particular kind of actual deformations controlled by a multiplicative cosimplicial quasicomplex C • related to each homogeneous algebra. The elements implementing twist transformations are a subclass of the counital 2-cocycles of C • . Such transformations are studied by means of their co-homological properties. It is shown they define an equivalence relation between homogeneous algebras, in such a way that cubic Artin–Schelter regular algebras of type S 1 (including parafermionic and plactic algebras with two generators) belong to the same class.

Conformal field theories with ZN and Lie algebra symmetries

We construct two-dimensional conformal field theories with a ZN symmetry, based on the second solution of Fateev–Zamolodchikov for the parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group (ZN×Z2, where Z2 stands for the ZN charge conjugation), as singlets, ⌊(N−1)/2⌋ different doublets, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra B(N−1)/2 when N is odd, and DN/2 when N is even. The unitary theories are representations of the coset SOn(N)×SO2(N)/SOn+2(N), with n=1,2,…. We suggest that physically they realize the series of multicritical points in statistical systems having a ZN symmetry.

Parafermionic theory with the symmetry ZN for N even

Following our previous papers [Nucl. Phys. B 656 (2003) 259, Nucl. Phys. B 664 (2003) 477] we complete the construction of the parafermionic theory with the symmetry Z N based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. In the present paper we construct the Z N parafermionic theory for N even. Primary operators are classified according to their transformation properties under the dihedral group ( Z N × Z 2, where Z 2 stands for the Z N charge conjugation), as two singlets, doublet 1,2,…, N/2−1, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra D N/2 . The unitary theories are representations of the coset SO n ( N)× SO 2( N)/ SO n+2 ( N), with n=1,2,…. We suggest that physically they realise the series of multicritical points in statistical systems having a Z N symmetry.

Decomposition of the lattice vertex operator algebra [formula omitted

Motivated by the work of Dong et al. [Associative subalgebras of Griess algebra and related topics, in: J. Ferrar, K. Harada (Eds.), Proc. Conf. Monster and Lie Algebras, de Gruyter, Berlin, 1998], we study a decomposition of the lattice vertex operator algebra V 2 A l as a direct sum of irreducible modules of a certain tensor product of Virasoro vertex operator algebras and a parafermion algebra W l+1 (2 l/( l+3)). We find that the vertex operator algebra V 2 A l contains a subalgebra isomorphic to a parafermion algebra W l+1 (2 l/( l+3)) of central charge 2 l/( l+3). A complete decomposition of the vertex operator algebra V 2 A l as a direct sum of irreducible modules of W=L(c 1,0)⊗L(c 2,0)⊗⋯⊗L(c l,0)⊗W l+1(2l/(l+3)) , where c i , i=1,…, l, is given by the discrete series c i =1−6/( i+2)( i+3), is also obtained.

Parafermionic theory with the symmetry ZN, for N odd

We construct a parafermionic conformal theory with the symmetry Z N , for N odd, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group D N , as singlet, doublet 1,2,…, N−1 2 , and disorder operators. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the weight lattice of the Lie algebra B ( N−1)/2 . The unitary theories are representations of the coset SO n ( N)× SO 2( N)/ SO n+2 ( N), with n=1,2,… . Physically, they realise the series of multicritical points in statistical theories having a D N symmetry.

Parafermionic theory with the symmetry Z5

A parafermionic conformal theory with the symmetry Z 5 is constructed, based on the second solution of Fateev–Zamolodchikov for the corresponding parafermionic chiral algebra. The primary operators of the theory, which are the singlet, doublet 1, doublet 2, and disorder operators, are found to be accommodated by the weight lattice of the classical Lie algebra B 2. The finite Kac tables for unitary theories are defined and the formula for the conformal dimensions of primary operators is given.

A(2)2 parafermions: a new conformal field theory

A new parafermionic algebra associated with the homogeneous space A (2) 2/ U(1) and its corresponding Z-algebra have been recently proposed. In this paper, we give a free boson representation of the A (2) 2 parafermion algebra in terms of seven free fields. Free field realizations of the parafermionic energy–momentum tensor and screening currents are also obtained. A new algebraic structure is discovered, which contains a W-algebra type primary field with spin two.

Homogeneous Algebras, Statistics and Combinatorics

After some generalities on homogeneous algebras, we give a formula connecting the Poincare series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one, called the parafermionic (parabosonic) algebra, is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra, i.e., the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D=2, we describe the relations with the cubic Artin–Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group GLp,q(2) on the generic cubic Artin–Schelter regular algebra of type S1; p and q being related to the Artin–Schelter parameters. It is claimed that this has a counterpart for any integer D⩾2.

Quantum parafermions in the SL(2, R)/ U(1) WZNW black hole model

Starting with its classical parafermion algebra, we consider the quantisation of the SL(2, R)/ U(1) WZNW black hole model.

Integration of the SL(2,R) / U(1) gauged WZNW theory by reduction and quantum parafermions

Using a gauge-invariant reduction, we directly integrate the SL(2,ℝ/U(1) WZNW theory. We prove that the conserved parafermions in this theory are coset currents. The parafermion algebra, the energy–momentum tensor, and “auxiliary” parafermions are quantized in a self-consistent manner.

Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan–Schwinger map

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well-known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan–Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.

Parafermionic representation of the affine [formula omitted] algebra at fractional level

The four fermionic currents of the affine superalgebra sl ̂ (2|1; C) at fractional level k= 1 u −1,u∈ N are shown to be realised in terms of a free scalar field, an sl ̂ (2; C) doublet field and primary fields of the parafermionic algebra Z u−1 .

Interpretation and extension of Green's ansatz for paraparticles

The anomalous bilinear commutation relations satisfied by the components of the Green's ansatz for paraparticles are shown to derive from the comultiplication of the paraboson or parafermion algebra. The same provides a generalization of the ansatz, wherein paraparticles of order p= ∑ α=1 r p α are constructed from r paraparticles of order p α , α=1, 2, …, r.

The Rogers-Ramanujan identities, the finite general linear groups, and the Hall-Littlewood polynomials

We connect Gordon's generalization of the Rogers-Ramanujan identities with the Hall-Littlewood polynomials and with generating functions which arise in a probabilistic setting in the finite general linear groups. This yields a Rogers-Ramanujan type product formula for the n -+ oo probability that an element of GL(n, q) or Mat(n, q) is semisimple. 1. BACKGROUND AND NOTATION The Rogers-Ramanujan identities are among the most remarkable partition identities in number theory and combinatorics. This paper will be concerned with the following generalization of the Rogers-Ramanujan identities, due to Gordon. Let (x)n denote (1 x)(1 x2)... (1 xn). Theorem 1 ([A, page 111]). For 1 2, and complex x with Ixl O )*'' r0,?i(mod 2k+l) where Nj = nj + + nk-1. Gordon's generalization of the Rogers-Ramanujan identities has been widely studied and appears in many places in mathematics and physics. Andrews [A] discusses combinatorial aspects of these identities. In an important series of papers, Lepowsky and Wilson [LW1], [LW2], [LW3] connect the Gordon identities with affine Lie algebras and structures that they called Z-algebras (later interpreted as parafermion algebras in conformal field theory). Meurman and Primc [MP] solve a problem left open in [LW3], proving the independence of a Z-algebra basis and obtaining a Z-algebra proof of the Gordon identities. Feigin and Frenkel [FF] interpret the Gordon identities as a character formula for the Virasoro algebra. Andrews, Baxter, and Forrester [ABF] and Warnaar [W] relate the Gordon identities with statistical mechanics. For some number theoretic connections see the conference proceedings [AABRR]. We use the following standard notation from the theory of partitions. Call A = (A1,A2,--.) a partition of n = AIX if A1 > A2 > > 0 where the Ai are Received by the editors March 6, 1998. 1991 Mathematics Subject Classification. Primary 20P05, 05E05. ()1999 American Mathematical Society 17 This content downloaded from 207.46.13.142 on Sat, 28 May 2016 06:40:23 UTC All use subject to http://about.jstor.org/terms

Graded parafermions

A graded generalization of the Z k parafermionic current algebra is constructed. This symmetry is realized in the osp(1|2)/ U(1) coset conformal field theory. The structure of the parafermionic highest-weight modules is analyzed and the dimensions of the fields of the theory are determined. A free field realization of the graded parafermionic system is obtained and the structure constants of the current algebra are found. Although the theory is not unitary, it presents good reducibility properties.

Deformed Heisenberg algebra with reflection

A universality of the deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of parafermionic nature. We demonstrate that finite-dimensional representations are representations of the deformed parafermionic algebra with internal Z 2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supplies us with irreducible representations of the osp(1|2) superalgebra. We show that the normalized form of the deformed Heisenberg algebra with reflection has the structure of a guon algebra related to the generalized statistics.

Majorana parafermions

A Majorana analog to the parafermion algebra is constructed and shown to be isomorphic to so(n). Ansatz realizations of this algebra are constructed and used to establish the properties of particular representations. The construction of specific so(n) representations is discussed for odd and even n, and for even n the procedure is illustrated explicitly in the case of so(4). By contracting the Majorana parafermion algebra, the para-Grassman algebra and its representation theory are obtained.

Tree scattering amplitudes of the spin-4/3 fractional superstring. I. The untwisted sectors.

Scattering amplitudes of the spin-4/3 fractional superstring are shown to satisfy spurious state decoupling and cyclic symmetry (duality) at tree-level in the string perturbation expansion. This fractional superstring is characterized by the spin-4/3 fractional superconformal algebra---a parafermionic algebra studied by Zamolodchikov and Fateev involving chiral spin-4/3 currents on the world-sheet in addition to the stress-energy tensor. Examples of tree scattering amplitudes are calculated in an explicit c=5 representation of this fractional superconformal algebra realized in terms of free bosons on the string world-sheet. The target space of this model is three-dimensional flat Minkowski space-time with a level-2 Kac-Moody so(2,1) internal symmetry, and has bosons and fermions in its spectrum. Its closed string version contains a graviton in its spectrum. Tree-level unitarity (i.e., the no-ghost theorem for space-time bosonic physical states) can be shown for this model. Since the critical central charge of the spin-4/3 fractional superstring theory is 10, this c=5 representation cannot be consistent at the string loop level. The existence of a critical fractional superstring containing a four-dimensional space-time remains an open question.