Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus

After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and co-semisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction - restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure).Comment: 25 pages, 5 tables, 9 figures. Version 2: updated references. Typos corrected. Several proofs added. Examples of ADE and generalized ADE trigonometric identities have been removed to shorten the pape

Semi-regular masas of transfinite length

In 1965 Tauer produced a countably infinite family of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each $n\in\mathbb N$, finding masas for which this procedure terminates at the $n$-th stage. Such masas are said to have length $n$. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalising tower.Comment: 14 page

Branching rules of semi-simple Lie algebras using affine extensions

We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW models with chiral algebra g_k. In fact, it turns out that for these models each embedding p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps can be be extended to finite level, we obtain a Verlinde-like formula for branching coefficients.Comment: 11 pages, LaTeX, v2: one reference added, v3: Clarified proof of Theorem 2, completely rewrote and extended Section 5 (relation to CFT), added various references. Accepted for publication in J. Phys.

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected. su(2) example has been removed to shorten the paper. Dual annular matrices for the rejected exceptional su(3) diagram are determine

Exceptional quantum subgroups for the rank two Lie algebras B2 and G2

Exceptional modular invariants for the Lie algebras B2 (at levels 2,3,7,12) and G2 (at levels 3,4) can be obtained from conformal embeddings. We determine the associated alge bras of quantum symmetries and discover or recover, as a by-product, the graphs describing exceptional quantum subgroups of type B2 or G2 which encode their module structure over the associated fusion category. Global dimensions are given.Comment: 33 pages, 27 color figure

Rigid C^*-tensor categories of bimodules over interpolated free group factors

Given a countably generated rigid C^*-tensor category C, we construct a planar algebra P whose category of projections Pro is equivalent to C. From P, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C^*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F_infty).Comment: 50 pages, many figure

Non-critical string pentagon equations and their solutions

We derive pentagon type relations for the 3-point boundary tachyon correlation functions in the non-critical open string theory with generic c_{matter} < 1 and study their solutions in the case of FZZ branes. A new general formula for the Liouville 3-point factor is derived.Comment: 18 pages, harvmac; misprints corrected, section 3.2 extended, a new general formula for the Liouville 3-point factor adde

Spectral measures of small index principal graphs

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $\Delta$ is the adjacency matrix of $X$ we consider the equation $\Delta=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $\epsilon$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $\epsilon$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $\epsilon$ is closely related to Jones' $\Theta$-series.Comment: 23 page

Subfactors of index less than 5, part 1: the principal graph odometer

In this series of papers we show that there are exactly ten subfactors, other than $A_\infty$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete