On Examples of Intermediate Subfactors from Conformal Field Theory

Motivated by our subfactor generalization of Wall’s conjecture, in this paper we determine all intermediate subfactors for conformal subnets corresponding to four infinite series of conformal inclusions, and as a consequence we verify that these series of subfactors verify our conjecture. Our results can be stated in the framework of Vertex Operator Algebras. We also verify our conjecture for Jones-Wassermann subfactors from representations of Loop groups extending our earlier results.     

Generalized Longo–Rehren Subfactors and α-Induction

We study the recent construction of subfactors by Rehren which generalizes the Longo–Rehren subfactors. We prove that if we apply this construction to a non-degenerately braided subfactor N⊂M and α^±-induction, then the resulting subfactor is dual to the Longo–Rehren subfactor M⊗M ^opp⊂R arising from the entire system of irreducible endomorphisms of M resulting from α^plusmn;-induction. As a corollary, we solve a problem on existence of braiding raised by Rehren negatively. Furthermore, we generalize our previous study with Longo and Müger on multi-interval subfactors arising from a completely rational conformal net of factors on S ¹ to a net of subfactors and show that the (generalized) Longo–Rehren subfactors and α-induction naturally appear in this context. Received: 11 September 2001 / Accepted: 7 October 2001     

Cyclotomic Integers,Fusion Categories,and Subfactors

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A _n or D _n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.     

Conformal Subnets and Intermediate Subfactors

 Given an irreducible local conformal net 𝒜 of von Neumann algebras on S ¹ and a finite-index conformal subnet ℬ⊂𝒜, we show that 𝒜 is completely rational iff ℬ is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [27] hold. Our proofs are based on an analysis of the net inclusion ℬ⊂𝒜; among other things we show that, for a fixed interval I, every von Neumann algebra  intermediate between ℬ(I) and 𝒜(I) comes from an intermediate conformal net ℒ between ℬ and 𝒜 with ℒ(I)=. We make use of a theorem of Watatani (type II case) and Teruya and Watatani (type III case) on the finiteness of the set ℑ(𝒩,ℳ) of intermediate subfactors in an irreducible inclusion of factors 𝒩⊂ℳ with finite Jones index [ℳ:𝒩]. We provide a unified proof of this result that gives in particular an explicit bound for the cardinality of ℑ(𝒩,ℳ) which depends only on [ℳ:𝒩]. Received: 21 December 2001 / Accepted: 28 February 2002 Published online: 14 March 2003 RID="⋆" ID="⋆" Supported in part by MIUR and INDAM-GNAMPA. Communicated by K. Fredenhagen     

Modular Invariants from Subfactors:¶Type I Coupling Matrices and Intermediate Subfactors

A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent” coupling matrices Z ^± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z ⁺=Z ⁻, then Z is related to Z ⁺ by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z ⁺≠Z ⁻, and that Z need not be related to a type I invariant by such an automorphism. Received: 8 December 1999 / Accepted: 15 February 2000     

Generalized Goodman–Harpe–Jones Construction of Subfactors, II

We build some exceptional representations of Birman–Wenzl algebras from the data of certain conformal embeddings. As a result we construct new finite depth subfactors whose principal graphs are completely determined. Received: 16 February 1996 / Accepted: 25 July 1996     

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 _∞ 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.     

Chiral Structure of Modular Invariants for Subfactors

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their "ambichiral" intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of f-induced sectors. We show that modular invariants come along naturally with several non-negative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU(2)_k modular invariants.     

Canonical Tensor Product Subfactors

Canonical tensor product subfactors (CTPS's) describe, among other things, the embedding of chiral observables in two-dimensional conformal quantum field theories. A new class of CTPS's is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding two-dimensional theories. Received: 24 November 1999 / Accepted: 6 December 1999     

A Planar Calculus for Infinite Index Subfactors

We develop an analog of Jones’ planar calculus for II ₁-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns’ results on rotations and extremality for infinite index subfactors. These results are obtained without Jones’ basic construction and the resulting Jones projections.     

Generalized Goodman–Harpe–Jones Construction of Subfactors, I

We construct some exceptional finite depth subfactors and determine their principal graphs from some exceptional integrable lattice models. Some of these subfactors are conjectured to be the same as those coming from certain conformal embeddings (, , and ) for which the principal graphs are previously unknown. Received: 24 November 1995 / Accepted: 14 May 1996     

On Haagerup’s List of Potential Principal Graphs of Subfactors

We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author. The first author was sponsored in part by NSF grant #DMS-0504199.     

Valid for Much of “Realistic” Fields: A Non-Generational Conjecture for Deriving All First-Class Constraints at Once

We propose a single-step non-generational conjecture for derivation of all first class constraints, (involving, only, variables compatible with canonical Poisson brackets), of a realistic gauge (singular) field theory. We verify our conjecture for the free electromagnetic field, the Yang-Mills fields in interaction with spinor and scalar fields, and we also verify our conjecture in the case of gravitational field. We show that the first class constraints, which were reached at using the standard Dirac’s multi-generational algorithm, will be reproduced using the proposed conjecture. We make no claim that this conjecture is valid for all “mathematically” plausible Lagrangians; but, nevertheless, the examples we consider here show that this conjecture is valid for a “wide” range or much of realistic fields of Physical interest that are known to exist and be manifested in nature.     

Quantum Subgroups of the Haagerup Fusion Categories

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the “quantum subgroups” in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Subfactors of Index Less Than 5, Part 3: Quadruple Points

One major obstacle in extending the classification of small index subfactors beyond ${3 +\sqrt{3}}$ is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the ??weeds?? ${\mathcal{Q}}$ and ${\mathcal{Q}'}$ from Part 1 (Morrison and Snyder in Commun. Math. Phys., doi:10.1007/s00220-012-1426-y, 2012). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed ${\mathcal{Q}'}$ immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds ${\mathcal{Q}}$ and ${\mathcal{Q}'}$ . Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor using a combination of planar algebra techniques and connections.     

Orbifold subfactors from Hecke algebras

We apply the notion of orbifold models ofSU(N) solvable lattice models to the Hecke algebra subfactors of Wenzl and get a new series of subfactors. In order to distinguish our subfactors from those of Wenzl, we compute the principal graphs for both series of subfactors. An obstruction for flatness of connections arises in this orbifold procedure in the caseN=2 and this eliminates the possibility of the Dynkin diagramsD _2n+1 , but we show that no such obstructions arise in the caseN=3. Our tools are the paragroups of Ocneanu and solutions of Jimbo-Miwa-Okado to the Yang-Baxter equation.     

The Order Parameter of the Chiral Potts Model

An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ‘‘broken rapidity line’’ approach pioneered by Jimbo et al. (J. Phys. A 26:2199--2210 (1993).).     

On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.