Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants

Braided subfactors of von Neumann algebras provide a framework for studying two dimensional conformal field theories and their modular invariants. We review this in the context of SU(3) conformal field theories through corresponding SU(3) braided subfactors and various subfactor invariants including spectral measures for the nimrep graphs, A_2-planar algebras and almost Calabi-Yau algebras.Comment: 45 pages, 25 figures. v3: minor correction to Figure 14; v2: figures of 0-1 parts of graphs included, some minor correction

Spectral Measures for G2G_2 II: finite subgroups

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been studied. In this paper we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of $G_2$. Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of $G_2$ and present their McKay graphs, some of which are new.Comment: 33 pages, 20 figures; minor improvements to exposition. Accepted for publication in Reviews in Mathematical Physic

Spectral Measures for Sp(2)Sp(2)

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.Comment: 41 pages, 45 figures. Title changed and notation corrected. arXiv admin note: substantial text overlap with arXiv:1404.186

Spectral Measures for G2G_2

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the rank two Lie group $G_2$, including the McKay graphs for the irreducible representations of $G_2$ and its maximal torus, and fusion modules associated to all known $G_2$ modular invariants.Comment: 36 pages, 40 figures; correction to Sections 5.4 and 5.5, minor improvements to expositio

Classification of Module Categories for SO(3)2mSO(3)_{2m}

The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$ modular invariants. There are module categories of type $\mathcal{A}$, $\mathcal{E}$ and their conjugates, but there are no orbifold (or type $\mathcal{D}$) module categories. We present a construction of a subfactor with principal graph given by the fusion rules of the fundamental generator of the $SO(3)_{2m}$ modular category. We also introduce a Frobenius algebra $A$ which is an $SO(3)$ generalisation of (higher) preprojective algebras, and derive a finite resolution of $A$ as a left $A$-module along with its Hilbert series.Comment: 56 pages, many figures; corrected error at the end of Section 4 about $\mathcal{E}_8$ nimrep, and corrected computational error in Theorem 5.10 about $\mathcal{E}_8^c$. The main theorem, Theorem 5.12, has been modified to reflect these correction

The Ising model and beyond

We study the SU(3) AVE graphs, which appear in the classification of modular in variant partition functions from numerous viewpoints, including determination of their Boltzmann weights, representations of Hecke algebras, a new notion of A2 planar algebras and their modules, various Hilbert series of dimensions and spectral measures, and the K-theory of associated Cuntz-Krieger algebras. We compute the K-theory of the of the Cuntz-Krieger algebras associated to the SU(3) AVE graphs. We compute the numerical values of the Ocneanu cells, and consequently representations of the Hecke algebra, for the AVE graphs. Some such representations have appeared in the literature and we compare our results. We use these cells to define an SU(3) analogue of the Goodman-de la Harpe-Jones construction of a subfactor, where we embed the j42-Temperley-Lieb algebra in an AF path-algebra of the SU(3) AVE graphs. Using this construction, we realize all SU(3) modular invariants by subfactors previously announced by Ocneanu. We give a diagrammatic representation of the i42-Temperley-Lieb algebra, and show that it is isomorphic to Wenzl's representation of a Hecke algebra. Generalizing Jones's notion of a planar algebra, we construct an 42-planar algebra which captures the structure contained in the SU(3) AVE subfactors. We show that the subfactor for an AVE graph with a flat connection has a description as a flat >12-planar algebra. We introduce the notion of modules over an 42-planar algebra, and describe certain irreducible Hilbert A2- Temperley-Lieb-modules. A partial decomposition of the ,42-planar algebras for the AVE graphs is achieved. We compare various Hilbert series of dimensions associated to ADE models for SU(2), and the Hilbert series of certain Calabi-Yau algebras of dimension 3. We also consider spectral measures for the ADE graphs and generalize to SU(3), and in particular obtain spectral measures for the infinite SU(3) graphs

Spectral measures for Sp(2)

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group B2 and its double cover the compact connected, simply-connected rank two Lie group C2, including the McKay graphs for the irreducible representations of C2 and B2 and their maximal tori, and fusion modules associated to the C2 modular invariants

Spectral measures for Sp(2)

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group SO(5) and its double cover the compact connected, simply-connected rank two Lie group Sp(2), including the McKay graphs for the irreducible representations of Sp(2) and SO(5) and their maximal tori, and fusion modules associated to the Sp(2) modular invariants

Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z)

Spectral measures for fundamental representations of the rank two Lie groups A2, C2 and G2 have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus T2 and are invariant under an action of the corresponding Weyl group, which are all subgroups of GL(2,Z). Here we consider spectral measures invariant under an action of the other finite subgroups of GL(2,Z), which are all associated fundamental representations of other rank two Lie groups