$U(1)^m$ modular invariants, N=2 minimal models, and the quantum Hall effect

TL;DR

This paper classifies all modular invariant partition functions for $u(1)^m$ and $su(2) imes u(1)^m$, providing a comprehensive understanding relevant to quantum Hall effect theories and N=2 superconformal models.

Contribution

It offers a complete classification of modular invariants for $u(1)^m$ and extends to $su(2) imes u(1)^m$, refining the A-D-E classification for N=2 minimal models.

Findings

Classified all modular invariants for $u(1)^m$ using self-dual lattices.

Enumerated the exact number of partition functions for $E_6, E_7, E_8$ types.

Found that invariance under $ au o au+1$ is less restrictive than other conditions.

Abstract

The problem of finding all possible effective field theories for the quantum Hall effect is closely related to the problem of classifying all possible modular invariant partition functions for the algebra $u(1)^m$, as was argued recently by Cappelli and Zemba. This latter problem is also a natural one from the perspective of conformal field theory. In this paper we completely solve this problem, expressing the answer in terms of self-dual lattices, or equivalently, rational points on the dual Grassmannian $G_{m,m}(R)^*$. We also find all modular invariant partition functions for $su(2)\oplus u(1)^m$, from which we obtain the classification of all N=2 superconformal minimal models. The `A-D-E classification' of these, though often quoted in the literature, turns out to be a very coarse-grained one: e.g. associated with the names $E_6,E_7,E_8$, respectively, are precisely 20,30,24 different partition functions. As a by-product of our analysis, we find that the list of modular invariants for su(2) lengthens surprisingly little when commutation with T -- i.e. invariance under $\tau \mapsto \tau+1$ -- is ignored: the other conditions are far more essential.