The representation theory of a conformal net is a unitary modular tensor category. It is captured by the bimodule category of the Jones-Wassermann subfactor. In this paper, we construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are self-dual. It generalizes and categorifies the self-duality of finite abelian groups and we call it modular self-duality.
Braiding structures on categorical multi-Interval Jones-Wassermann subfactor
Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University（清华大学丘成桐数学科学中心及数学系）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京怀柔数学院）
In this paper we present a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. Motivated by this conjecture, we determine all intermediate subfactors of Goodman-Harpe-Jones subfactors, and as a result we verify that Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a negative answer to a question motivated by a conjecture of Aschbacher-Guralnick.
Surprisingly Richard Thompson's groups have recently appeared in Jones' subfactor theory. Vaughan Jones is famous for linking theories that are a priori completely disconnected; for instance, his celebrated polynomial for links emanating from subfactor theory. This note is about a new beautiful story in mathematics which results from a fortunate accident in the land of quantum field theory.
We complete the realisation by braided subfactors, announced by Ocneanu, of all SU(3)-modular invariant partition functions previously classified by Gannon.
Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein gro…
Given a subfactor planar algebra P, Guionnet, Jones and Shlyakhtenko give a diagrammatic construction of a II_{1} subfactor whose planar algebra is P. They showed if P is finite-depth, then the factors are interpolated free group factors, and they identified the parameters. We prove if P is infinite-depth, then the factors are isomorphic to L(F\infty}$.
By using Ocneanu's result on the classification of all irreducible connections on the Dynkin diagrams, we show that the dual principal graphs as well as the fusion rules of bimodules arising from any Goodman-de la Harpe-Jones subfactors are obtained by a purely combinatorial method. In particular we obtain the dual principal graph and the fusion rule of bimodules arising from the Goodman-de la Harpe-Jones subfactor corresponding to the Dynkin diagram $E_8$. As an application, we also show some subequivalence among $A$-$D$-$E$ paragroups.
We analyse the Guionnet-Jones-Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.
We prove existence of subfactors of finite depth of the hyperfinite II_1 factor with indices (5+sqrt{13})/2= 4.302... and (5+sqrt{17})/2=4.561.... The existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then. These are the only known subfactors with finite depth which do not arise from classical groups, quantum groups or rational conformal field theory.
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.
Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haage…
If $G$ is a countable, discrete group generated by two finite subgroups $H$ and $K$ and $P$ is a II$_1$ factor with an outer G-action, one can construct the group-type subfactor $P^H \subset P \rtimes K$ introduced in \cite{BH}. This construction was used in \cite{BH} to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra (in the sense of Jones \cite{J2}) of this subfactor and prove that any subfactor with an abstract planar algebra of "group type" arises from such a subfactor. The action of Jones' planar op…
Bisch and Jones proposed the classification of planar algebras by simple generators and relations. In this paper, we study the generating problem for a family of group-subgroup subfactors associated with the Kneser graphs, namely, to determine the generators with minimal size. In particular, we prove that this family of subfactors are generated by $2$-boxes and this provides an affirmative answer to a question of Vaughan Jones. This generator problem is also related to the theory of quantum permutation groups, and the main theorem also provides an infinite family of strongly regular graphs wit…
A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete on 2-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theo…
An inclusion of II$_1$ factors $N \subset M$ with finite Jones index gives rise to a powerful set of invariants that can be approached successfully in a number of different ways. We describe Jones' pictorial description of the standard invariant of a subfactor as a so-called planar algebra and show how this point of view leads to new structure results for subfactors.
We introduce a new notion of angle between intermediate subfactors and prove various interesting properties of the angle and relate it with the Jones' index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo pu…
Using Jones' quadratic tangles formulas, we automate the construction of the 4442, 3333, 3311, and 2221 spoke subfactors by finding sets of 1-strand jellyfish generators. The 4442 spoke subfactor is new, and the 3333, 3311, and 2221 spoke subfactors were previously known.
We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of vector spaces which will capture the double complex structure pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system, including both the periodicity three coming from the A_2-Temperley-Lieb algebra as well as the periodicity two coming from the subfactor basic construction. We use an A_2-planar algebra to obtain a description of the …
We construct a certain `cobordism category' ${\cal D}$ whose morphisms are suitably decorated cobordism classes between similarly decorated closed oriented 1-manifolds, and show that there is essentially a bijection between (1+1-dimensional) unitary topological quantum field theories (TQFTs) defined on ${\cal D}$, on the one hand, and Jones' subfactor planar algebras, on the other.
A {\it W$^*$-representation} of a II$_1$ subfactor $N\subset M$ with finite Jones index, $[M:N]<\infty$, is a non-degenerate commuting square embedding of $N\subset M$ into an inclusion of atomic von Neumann algebras $\oplus_{i\in I} \Cal B(\Cal K_i)=\Cal N \subset^{\Cal E} \Cal M=\oplus_{j\in J} \Cal B(\Cal H_j)$. We undertake here a systematic study of this notion, first introduced in [P92], giving examples and considering invariants such as the (bipartite) {\it inclusion graph} $Λ_{\Cal N \subset \Cal M}$, the {\it coupling vector} $(\text{\rm dim}(_M\Cal H_j))_j$ and the {\it RC-algebra} (…
Jones and Penneys showed that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph, via a Markov towers of algebras approach. We relate several equivalent perspectives on the notion of module over a subfactor planar algebra, and show that a Markov tower is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a corollary, we obtain a classification of semisimple pivotal C* modules over Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex weighting. We then generalize the Markov towers of algebras…
To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special sym- metries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra P. This talk is based on joint work w…
The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square
iTHEMS Research Group, RIKEN（理化学研究所 iTHEMS 研究组）；Kavli IPMU (WPI), the University of Tokyo（东京大学 Kavli 宇宙物理数学统一中心（世界顶尖研究中心））
We develop a representation theory for $λ$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $λ$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with propert…
We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When $G$ is compactly generated, we prove that $C$ is essentially equivalent to the planar algebra associated by Jones and Burstein to a group acting on a locally finite bipartite graph. We then concretely realize $C$ as the category of bimodules generated by a hyperfinite subfactor.
We present more planar algebraic construction of subfactors than those of Guionet-Jones-Shlyakhtenko-Walker and Kodiyalam-Sunder which start from a subfactor planar algebra and give in a direct way a subfactor of the same standard invariant with the planar algebra. Our construction is based on using the ordinary concepts in planar algebras such as involution, inclusion and conditional expectation mappings as it is
To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_μU(2). The deformation parameter is determined by -1<μ<0 and [M:N]=|μ+μ^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_μU(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. μis negative as M is a real bimodule.
The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $Δ$ is the adjacency matrix of $X$ we consider the equation $Δ=U+U^{-1}$. 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 $ε$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $ε$ 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 o…
Given a II$_1$-subfactor $A\subset B$ of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as $A-A$ bimodule maps among the bimodules $\bigotimes_A^n L^2(B)$. As a corollary, we get a lower bound on the dimension of the centralizer algebras $A_0'\cap A_{2n}$ for infinite index subfactors, and we also get that $A_0'\cap A_{2n}$ is nonabelian for $n\geq 2$, where $(A_n)_{n\geq 0}$ is the Jones tower for $A_0=A\subset B=A_1$. We also show that the annular GICAR/planar rook category acts as maps amongst the $A$-central…
The article contains a detailed description of the connection between finite depth inclusions of $II_1$-subfactors and finite $C^*$-tensor categories (i.e. $C^*$-tensor categories with dimension function for which the number of equivalence classes of irreducible objects is finite). The $(N,N)$-bimodules belonging to a $II_1$-subfactor $N\subset M$ with finite Jones index form a $C^*$-tensor category with dimension function. Conversely, taking an object of a finite $C^*$-tensor category C we construct a subfactor $A\subset R$ of the hyperfinite $II_1$-factor R with finite index and finite depth…
Jones introduced unitary representations of Thompson group $F$ starting from a given subfactor planar algebra, and all unoriented links arise as matrix coefficients of these representations. Moreover, all oriented links arise as matrix coefficients of a subgroup $\vec{F}$ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup $\vec{F}$ and showed many interesting properties. In this paper, we investigate into a large class of groups which arises as subgroups of Thompson group $F$ and reveal the relation between the skein theory of the subfactor planar algebr…
In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi quadratic categories. One key technique which has previously revealed hidden structure in fusion categories is to study all other fusion categories in the Morita equivalence class, and hope that one of the others is easier to understand. In this paper…
A near-group fusion category is a fusion category C where all but 1 simple objects are invertible. Examples of these include the Tambara-Yamagami categories and the even sectors of the E6 and affine-D5 subfactors, though there are infinitely many others. We classify the near-group fusion categories, and compute their doubles and the modular data relevant to conformal field theory. Among other things, we explicitly construct over 40 new finite depth subfactors, with Jones index ranging from around 6.85 to around 14.93. We expect all of these doubles to be realised by rational conformal field th…
Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\cal O}(p) =\{u p u^*: u\in U_M\} $$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p) $ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ en…
We define Jones's planar algebra as a map of multicategories and constuct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We prove finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus.
Q-systems are unitary versions of Frobenius algebra objects which appeared in the theory of subfactors. In recent joint work with R. Hernández Palomares and C. Jones, the authors defined a notion of Q-system completion for C*/W* 2-categories, which is a unitary version of a higher idempotent completion in the spirit of Douglas--Reutter and Gaiotto--Johnson-Freyd. In this article, we prove that Q-system completion is a dagger 3-functor on the dagger 3-category of C*/W* 2-categories. We also prove that Q-system completion satisfies a universal property analogous to the universal property satisfi…
Popa introduced the tensor category $\tildeχ(M)$ of approximately inner, centrally trivial bimodules of a $\rm{II}_{1}$ factor $M$, generalizing Connes' $χ(M)$. We extend Popa's notions to define the $\rm W^*$-tensor category $\operatorname{End}_{\rm loc}(\mathcal{C})$ of local endofunctors on a $\rm W^*$-category $\mathcal{C}$. We construct a unitary braiding on $\operatorname{End}_{\rm loc}(\mathcal{C})$, giving a new construction of a braided tensor category associated to an arbitrary $\rm W^*$-category. For the $\rm W^*$-category of finite modules over a $\rm{II}_{1}$ factor, this yields a…
In this article, we construct a 2-shaded rigid ${\rm C}^*$ multitensor category with canonical unitary dual functor directly from a standard $λ$-lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard $λ$-lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2-shaded rigid ${\rm C}^*$ multitensor category. As an example, we compute the modules and bimodules for Temperley-Lieb-Jones standard $λ$-lattices in terms of traceless Markov towers and lattices. Translating…
