Victor G. Kac, Ivan T. Todorov
Affine Orbifolds and Rational Conformal Field Theory Extensions of $W_{1+\infty}$
The paper is published: Comm. Math. Phys. 190, No. 1 (1997) 57-111

MSC:

81R10 Representations of infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody and other current algebras, See also {17B65, 17B67, 22E65, 22E67, 22E70}

11F22 Relationship to Lie algebras and finite simple groups

17B67 Kac-Moody algebras (structure and representation theory)

81T40 Two-dimensional field theories, conformal field theories, etc.

Abstract: Chiral orbifold models are defined as gauge field theories with a
finite gauge group $\Ga$. We start with a conformal current
algebra $\goth A$ associated with a connected compact Lie group
$G$ and a negative definite integral invariant bilinear form on
its Lie algebra. Any finite group $\Ga$ of inner automorphisms or
$\goth A$ (in particular, any finite subgroup of $G$) gives rise
to a gauge theory with a chiral subalgebra $\goth
A^{\Ga}\subset\goth A$ of {\it local observables} invariant under
$\Ga$. A set of positive energy $\goth A^{\Ga}$ modules is
constructed whose characters span, under some assumptions on
$\Gamma$, a finite dimensional unitary representation of
$SL(2,\Bbb Z)$. We compute their {\it asymptotic dimensions}
(thus singling out the nontrivial orbifold modules) and find
explicit formulae for the modular transformations and hence, for
the {\it fusion rules}.

As an application we construct a family of {\it rational conformal
field theory} (RCFT) extensions of $W_{1+\infty}$ that appear to
provide a bridge between two approaches to the quantum Hall effect.