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

Terry J. Gannon

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.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.

Jan 1, 199630 citations

Modular anomalies in (2+1) - and (3+1) -dimensional edge theories

Moon Jip Park, Chen Fang, Bogdan Andrei Bernevig +1 more

The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a $D$ -dimensional system under modular transformation, or modular anomaly, signals the presence of a $(D+1)$ -D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a $(2+1)$ -D and a $(3+1)$ -D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in $(2+1)$ -D and $(3+1)$ -D are modular invariant. However, we show in $(3+1)$ -D that when the edge theory is coupled to a background $U(1)$ gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a $(4+1)$ -D bulk. Using the modular anomaly, we find that the edge theory of a $(4+1)$ -D insulator with space-time inversion symmetry $(PT)$ and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.The classification of topological phases of matter in the presence of interactions is an area of intense interest. One possible means of classification is via studying the partition function under modular transforms, as the presence of an anomalous phase arising in the edge theory of a $D$-dimensional system under modular transformation, or modular anomaly, signals the presence of a $(D+1)$-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a $(2+1)$-D and a $(3+1)$-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in $(2+1)$-D and $(3+1)$-D are modular invariant. However, we show in $(3+1)$-D that when the edge theory is coupled to a background $U(1)$ gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a $(4+1)$-D bulk. Using the modular anomaly, we find that the edge theory of a $(4+1)$-D insulator with space-time inversion symmetry $(PT)$ and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.

arXiv:1604.00407Physical review. B./Physical review. BJun 16, 20171 citation

U(1)^m

modular invariants, N=2 minimal models, and the quantum Hall effect

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.

D

-dimensional system under modular transformation, or modular anomaly, signals the presence of a

(D+1)

-D nontrivial bulk. In this work, we discuss the modular transformations of conformal field theories along a

(2+1)

-D and a

(3+1)

-D edge. Using both analytical and numerical methods, we show that chiral complex free fermions in

(2+1)

-D and

(3+1)

-D are modular invariant. However, we show in

(3+1)

-D that when the edge theory is coupled to a background

U(1)

gauge field, this results in the presence of a modular anomaly that is the manifestation of a quantum Hall effect in a

(4+1)

-D bulk. Using the modular anomaly, we find that the edge theory of a

(4+1)

-D insulator with space-time inversion symmetry

(PT)

and fermion number parity symmetry for each spin becomes modular invariant when eight copies of the edges exist.