U(1) ^{m} modular invariants, N = 2 minimal models, and the quantum Hall effect
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 overlineu(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 selfdual lattices, or equivalently, rational points on the dual Grassmannian G _{m,m}(R) ^{∗}. We also find all modular invariant partition functions for affine su(2) ⊕ u(1) ^{⊕ m}, from which we obtain the classification of all N = 2 superconformal minimal models. The 'ADE classification' of these, though often quoted in the literature, turns out to be a very coarsegrained one: e.g. associated with the names E_{6}, E_{7}, E_{8}, respectively, are precisely 20, 30, 24 different partition functions. As a byproduct of our analysis, we find that the list of modular invariants for overlinesu(2) lengthens surprisingly little when commutation with T — i.e. invariance under τ → τ + 1 — is ignored: the other conditions are far more essential.
 Publication:

Nuclear Physics B
 Pub Date:
 February 1997
 DOI:
 10.1016/S05503213(97)000321
 arXiv:
 arXiv:hepth/9608063
 Bibcode:
 1997NuPhB.491..659G
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter
 EPrint:
 plain tex, 28 pages