Affine orbifolds and rational conformal field theory extensions of W_{1+infinity}

Abstract

Chiral orbifold models are defined as gauge field theories with a finite gauge group $Γ$. 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 $Γ$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^Γ\subset A$ of local observables invariant under $Γ$. A set of positive energy $A^Γ$ modules is constructed whose characters span, under some assumptions on $Γ$, a finite dimensional unitary representation of $SL(2,Z)$. 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 $W_{1+\infty}$ that appear to provide a bridge between two approaches to the quantum Hall effect.