$U(1{)}^m$ 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 $u(1{)}^m$, 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 $G_{m,m}(R{)}^{\ast }$. 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.

plain tex, 28 pages

References in corpus (2)

• Modular Invariant Partition Functions in the Quantum Hall Effect
• Heterotic Coset Models and (0,2) String Vacua

Cited by in corpus (16)

• Finite Group Modular Data
• An Introduction to Conformal Field Theory
• Orbifold equivalent potentials
• $\mathcal{N}=2$ Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack
• D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories
• On the complete classification of the unitary N=2 minimal superconformal field theories
• Free-field realisation of boundary states and boundary correlation functions of minimal models
• Partition Functions and Stability Criteria of Topological Insulators
• N=2 superconformal nets
• Modular Invariants in the Fractional Quantum Hall Effect
• Boundary states, matrix factorisations and correlation functions for the E-models
• Conformal field theory approach to gapless 1D fermion systems and application to the edge excitations of nu = 1/(2p+1) quantum Hall sequences
• Rational conformal field theory with matrix level and strings on a torus
• New conformal field theory from $\mathcal{N}=(0,2)$ Landau-Ginzburg model
• Two-dimensional conformal field theories with matrix-valued level
• Explicit construction of states in orbifolds of products of $N=2$ Superconformal ADE Minimal models