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

TL;DR

This paper develops a framework for chiral orbifold models in conformal field theory, computes their modular properties, and constructs RCFT extensions of W_{1+infinity} that connect different quantum Hall effect approaches.

Contribution

It introduces a method to analyze gauge theories with finite automorphism groups, computes their modular data, and constructs new RCFT extensions of W_{1+infinity}.

Findings

Characters of orbifold modules form finite-dimensional representations of SL(2,Z).

Explicit formulas for modular transformations and fusion rules are derived.

Constructs RCFT extensions linking different quantum Hall effect models.

Abstract

Chiral orbifold models are defined as gauge field theories with a finite gauge group $\Gamma$. 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 $\Gamma$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^{\Gamma}\subset A$ of local observables invariant under $\Gamma$. A set of positive energy $A^{\Gamma}$ modules is constructed whose characters span, under some assumptions on $\Gamma$, 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.