  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.
