Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞ (original) (raw)
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Affine Orbifolds and Rational Conformal Field Theory Extensions of W1+8
Communications in Mathematical Physics, 1997
Chiral orbifold models are defined as gauge field theories with a finite gauge group Gamma\GammaGamma. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Gamma\GammaGamma of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra AGammasubsetAA^{\Gamma}\subset AAGammasubsetA of local observables invariant under Gamma\GammaGamma. A set of positive energy AGammaA^{\Gamma}AGamma modules is constructed whose characters span, under some assumptions on Gamma\GammaGamma, a finite dimensional unitary representation of SL(2,Z)SL(2,Z)SL(2,Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W1+inftyW_{1+\infty}W1+infty that appear to provide a bridge between two approaches to the quantum Hall effect.
Rational conformal field theory extensions of W[sub 1+∞] in terms of bilocal fields
Journal of Mathematical Physics, 1998
The rational conformal field theory (RCFT) extensions of W 1+∞ at c = 1 are in one-to-one correspondence with 1-dimensional integral lattices L(m). Each extension is associated with a pair of oppositely charged "vertex operators" of charge square m ∈ N. Their product defines a bilocal field V m (z 1 , z 2) whose expansion in powers of z 12 = z 1 − z 2 gives rise to a series of (neutral) local quasiprimary fields V l (z, m) (of dimension l + 1). The associated bilocal exponential of a normalized current generates the W 1+∞ algebra spanned by the V l (z, 1) (and the unit operator). The extension of this construction to higher (integer) values of the central charge c is also considered. Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter τ , but also on a chemical potential ζ. We compute such a ζ dependence of orbifold characters, thus extending the range of applications of a recent study of affine orbifolds.
The operator algebra of orbifold models
Communications in Mathematical Physics, 1989
We analyze the chiral properties of (orbifold) conformal field theories which are obtained from a given conformal field theory by modding out by a finite symmetry group. For a class of orbifolds, we derive the fusion rules by studying the modular transformation properties of the one-loop characters. The results are illustrated with explicit calculations of toroidal and c = 1 models.
Nuclear Physics B, 2001
We present a pure Chern-Simons formulation of families of interesting Conformal Field Theories describing edge states of non-Abelian Quantum Hall states. These theories contain two Abelian Chern-Simons fields describing the electromagnetically charged and neutral sectors of these models, respectively. The charged sector is the usual Abelian Chern-Simons theory that successfully describes Laughlin-type incompressible fluids. The neutral sector is a 2 + 1-dimensional theory analogous to the 1 + 1-dimensional orbifold conformal field theories. It is based on the gauge group O(2) which contains a Z 2 disconnected group manifold, which is the salient feature of this theory. At level q, the Abelian theory of the neutral sector gives rise to a Z 2q symmetry, which is further reduced by imposing the Z 2 symmetry of charge-conjugation invariance. The remaining Z q symmetry of the neutral sector is the origin of the non-Abelian statistics of the (fermionic) q-Pfaffian states.
Modular invariance of trace functions in orbifold theory
The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise numbers of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlations functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representations of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic...
Quantum Matrix Models for Simple Current Orbifolds
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An algebraic formulation of the stringy geometry on simple current orbifolds of the WZW models of type A_N is developed within the framework of Reflection Equation Algebras, REA_q(A_N). It is demonstrated that REA_q(A_N) has the same set of outer automorphisms as the corresponding current algebra A^{(1)}_N which is crucial for the orbifold construction. The CFT monodromy charge is naturally identified within the algebraic framework. The ensuing orbifold matrix models are shown to yield results on brane tensions and the algebra of functions in agreement with the exact BCFT data.
The quantum symmetry of rational conformal field theories
Nuclear Physics B, 1991
The quantum group symmetry of the c < 1 Rational Conformal Field Theory. in its Coulomb gas version, is formulated in terms of a new type of screened vertex operators, which define the representation spaces of a quantum group Q. The conformal properties of these operators show a deep interplay between the quantum group Q and the Virasoro algebra.
W 1+∞ Field Theories for the Edge Extensions in the Quantum Hall Effect
International Journal of Modern Physics A, 1997
We briefly review these low-energy effective theories for the quantum Hall effect, with emphasis and language familiar to field theorists. Two models have been proposed for describing the most stable Hall plateaus (the Jain series): the multi-component Abelian theories and the minimal W1+∞ models. They both lead to a-priori classifications of quantum Hall universality classes. Some experiments already confirmed the basic predictions common to both effective theories, while other experiments will soon pin down their detailed properties and differences. Based on the study of partition functions, we show that the Abelian theories are rational conformal field theories while the minimal W1+∞ models are not.
A Unified Conformal Field Theory Description¶of Paired Quantum Hall States
Communications in Mathematical Physics, 1999
The wave functions of the Haldane-Rezayi paired Hall state have been previously described by a non-unitary conformal field theory with central charge c = −2. Moreover, a relation with the c = 1 unitary Weyl fermion has been suggested. We construct the complete unitary theory and show that it consistently describes the edge excitations of the Haldane-Rezayi state. Actually, we show that the unitary (c = 1) and non-unitary (c = −2) theories are related by a local map between the two sets of fields and by a suitable change of conjugation. The unitary theory of the Haldane-Rezayi state is found to be the same as that of the 331 paired Hall state. Furthermore, the analysis of modular invariant partition functions shows that no alternative unitary descriptions are possible for the Haldane-Rezayi state within the class of rational conformal field theories with abelian current algebra. Finally, the known c = 3/2 conformal theory of the Pfaffian state is also obtained from the 331 theory by a reduction of degrees of freedom which can be physically realized in the double-layer Hall systems.