Decompositions of representations of exceptional affine algebras with respect to conformal subalgebras.

Abstract

We use modular-invariance and conformal-invariance constraints on the highest-weight representations of affine algebras as well as asymptotic behavior of their characters to deduce an algorithm for the decomposition of these representations with respect to conformal subalgebras. We use the algorithm to find explicit decompositions of level-1 representations of all affine exceptional algebras with respect to all their conformal subalgebras. The crucial point of this work is a connection of the decomposition problem with the Frobenius theory of positive matrices, which hints to a possible connection with Markov chains. The problem of computing branching coefficients for conformal subalgebras arises in the string compactifications. We show that a solution to this problem allows one to construct modular-invariant partition functions on a group manifold. An important new general theoretical result of the paper is the uniqueness of the vacuum state in the basic representation of an affine algebra for its conformal subalgebra.