Subfactors and mathematical physics

Abstract

This paper surveys the long-standing connections and impact between Vaughan Jones’s theory of subfactors and various topics in mathematical physics, namely statistical mechanics, quantum field theory, quantum information, and two-dimensional conformal field theory.

1. Subfactors and mathematical physics

Subfactor theory was initiated by Vaughan Jones 67. This led him to the study of a new type of quantum symmetry. This notion of quantum symmetries led to a diverse range of applications, including the Jones polynomial, a completely new invariant in knot theory which led to the new field of quantum topology. His novel theory has deep connections to various topics in mathematical physics. This renewed interest in known connections between mathematical physics and operator algebras, and it opened up totally novel frontiers. We present a survey on these interconnecting topics with emphasis on statistical mechanics and quantum field theory, particularly two-dimensional conformal field theory.

2. Subfactors and statistical mechanics

Let $N\subset M$ be a subfactor of type II$_1$, and let $[M:N]$ be its Jones index, which is a positive real number or infinity. That is, $N$ and $M$ are infinite-dimensional simple von Neumann algebras with a trace $\mathrm{tr}$. We only consider the case that $[M:N]$ is finite. Vaughan 67 constructed a sequence of projections $e_j$, $j=1,2,3,\dots$, called the Jones projections, and discovered the following relations:

$$\begin{equation} \left\{ \begin{aligned} e_j&=e_j^2=e_j^*,\\ e_j e_k&=e_k e_j,\quad j\neq k,\\ e_j e_{j\pm 1} e_j&=[M:N]^{-1}e_j. \end{aligned} \right. {\tag{2.1}} \end{equation}$$

Using these relations and a trace, Vaughan showed that the set of possible values of the Jones indices is exactly equal to

$$\begin{equation*} \left\{4\cos ^2\frac{\pi }{n}\mid n=3,4,5,\dots \right\}\cup [4,\infty ]. \end{equation*}$$

Vaughan made the substitution

$$\begin{equation} \sigma _j = t e_j-(1-e_j), {\tag{2.2}} \end{equation}$$

where $[M:N] = \lambda ^{-1} = 2 +t + t^{-1}$ to yield the Artin relations of the braid group, where $\sigma _j$ is the braid which interchanges the $j$ and $j+1$ strands. Vaughan’s representation came equipped with a trace $\mathrm{tr}$ satisfying the Markov trace property in the probabilistic sense $\mathrm{tr}(xe_j)=[M:N]^{-1}\mathrm{tr}(x)$, where $x$ belongs to the algebra generated by $e_1,e_2,\dots ,e_{j-1}$. Any link arises as a closure of a braid by a theorem of Alexander, and two braids give the same link if and only if they are related by a series of two types of moves, known as the Markov moves, by a theorem of Markov. The trace property $\mathrm{tr}(xy)=\mathrm{tr}(yx)$ and the above Markov trace property give invariance of a certain adjusted trace value of a braid under the two Markov moves. This is the Jones polynomial 6869 in the variable $t$, a polynomial invariant of a link.

Evans pointed out in 1983 that these relations 2.1 appear in similar formalism to one studied by Temperley and Lieb 114 in solvable statistical mechanics. The Yang–Baxter equation plays an important role in subfactor theory and quantum groups. The two-dimensional Ising model assigns two possible spin values $\pm$ at the vertices of a lattice. Important generalisations include the Potts model, with $Q$ states at each vertex, and vertex or IRF (interaction round a face) models, where the degrees of freedom are assigned to the edges of the lattice. The transfer matrix method, originated by Kramers and Wannier, assigns a matrix of Boltzmann weights to a one-dimensional row lattice. The partition function of a rectangular lattice in general is then obtained by gluing together matrix products of the transfer matrix. Baxter 6 showed how to construct commuting families of transfer matrices via Boltzmann weights satisfying the Yang–Baxter equation (YBE). The YBE is an enhancement of the braid relations in 2.2, as it reduces to them in a certain limit. This commutativity permits simultaneous diagonalisation, with the largest eigenvalue being crucial for computing the free energy. The transfer matrix method transforms the classical statistical mechanical model to a one-dimensional quantum model. A conformal field theory can arise from the scaling limit of a statistical mechanical lattice model at criticality. Temperley and Lieb 114 found that the transfer matrices of the Potts model and an ice-type vertex model could both be described through generators obeying the same relations as in Vaughan’s work 2.1 and in this way demonstrated equivalence of the models. Whilst the relations for the Potts model only occur when $\lambda ^{-1} =Q$ is integral, the partition function is a Tutte–Whitney dichromatic polynomial. One variable is $Q$ which can be extrapolated, and the partition function is then related to the Jones polynomial on certain links associated to the lattice 68, page 108. The ice-type representation though has a continuous parameter $\lambda$. In both cases, the Markov trace did not manifest itself.

Pimsner and Popa 100 discovered that the inverse of the index, namely $[M:N]^{-1}$, is the best constant $c\geq 0$ for which $E_N(x^*x) \geq c x^*x,$ for all $x\in M$, which they called the probabilistic index. Here $E_N$ is the conditional expectation of $M$ onto $N$ which gives rise to the first Jones projection in the tower. This was key to creating the link, with the theory of Doplicher, Haag, and Roberts, by Longo 89 identifying statistical dimension with the Jones index, and by Fredenhagen, Rehren, and Schroer 4445, in the late 1980s, and also key to calculating all of the subsequent entropy quantities/invariants related to subfactors, including the calculation of the entropy of the shift on the Jones projections and the calculation of the Connes–Stormer entropy 25, $H(M|N)=\ln ([M:N])$, for irreducible subfactors.

For a subfactor $N\subset M$ with finite Jones index, we have the Jones tower construction

$$\begin{equation*} N\subset M\subset M_1\subset M_2\subset \cdots , \end{equation*}$$

where $M_k$ is generated by $M$ and $e_1,e_2,\dots ,e_k$. The basic construction from $N \subset M$ to $M \subset M_1$ and its iteration to give the Jones tower of II$_1$ factors has a fundamental role in subfactor theory and applications in mathematical physics. The higher relative commutants $M'_j\cap M_k$, $j\le k$, give a system of commuting squares of inclusions of finite-dimensional $C^*$-algebras with a trace, an object denoted by $\mathcal{G}_{N\subset M}$ and called the standard invariant of $N\subset M$. This exceptionally rich mathematical structure encodes algebraic and combinatorial information about the subfactor, a key component of which is a connected, possibly infinite bipartite graph $\Gamma _{N\subset M}$, of Cayley type, called the principal graph of $N\subset M$, with a canonical weight vector $\vec{v}$, whose entries are square roots of indices of irreducible inclusions in the Jones tower. The weighted graph $(\Gamma , \vec{v})$ satisfies the Perron–Frobenius type condition $\Gamma ^t\Gamma (\vec{v})=[M:N]\vec{v}$, and also $\|\Gamma \|^2\leq [M:N]$.

Of particular relevance to mathematical physics is when $N\subset M$ has finite depth, corresponding to the graph $\Gamma$ being finite, in which case the weights $\vec{v}$ give the (unique) Perron–Frobenius eigenvector, entailing $\|\Gamma \|^2=[M:N]$. Finite depth is automatic when the index $[M:N]$ is less than $4$, where indeed all bipartite graphs are finite and have norms of the form $2\cos ^2(\pi /n)$, $n\geq 3$.

The objects $\mathcal{G}_{N\subset M}$ have been axiomatised in a number of ways—by Ocneanu with paragroups and connections 95 in the finite depth case, then in the general case by Popa with $\lambda$-lattices 105, and by Vaughan with planar algebras 78.

By Connes fundamental result in 21, the hyperfinite II$_1$ factor $R$, obtained as an inductive limit of finite-dimensional algebras, is the unique amenable II$_1$ factor, so in particular all its finite index subfactors are isomorphic to $R$. In a series of papers 103104106107, Popa identified the appropriate notion of amenability for inclusions of II$_1$ factors $N\subset M$ and for the objects $\mathcal{G}_{N\subset M}$, in several equivalent ways, one being the Kesten-type condition $\|\Gamma _{N\subset M}\|^2=[M:N]$. He proved the important result that for hyperfinite subfactors $N\subset M$ satisfying this amenability condition, $\mathcal{G}_{N\subset M}$ is a complete invariant. In other words, whenever $M\simeq R$ and $\|\Gamma _{N\subset M}\|^2=[M:N]$ (in particular if $N\subset M$ has finite depth), $N\subset M$ can be recovered from the data encoded by the sequence of commuting squares in the Jones tower. In particular, Popa’s theorem allows us to often assimilate/identify a finite-depth hyperfinite subfactor with its commuting square invariant, thus just referring to either of them as a subfactor.

Constructions of interesting commuting squares are related to statistical mechanics through the Yang–Baxter equation and an IRF, vertex or spin model 70. (See the monograph of Baxter 6 for these types of statistical mechanical models. Also see 73 for a general overview by Vaughan on these types of relations.) We choose one edge each from the four diagrams for the four inclusions so that they make a closed square. Then we have an assignment of a complex number to each such square. Ocneanu 95 gave a combinatorial characterisation of this assignment of complex numbers under the name of a paragroup and a flat connection. We also assign a complex number, called a Boltzmann weight, to each square arising from a finite graph in the theory of IRF or vertex models, and we have much similarity between the two notions. The simplest example corresponds to the Ising model built on the Coxeter–Dynkin diagram $A_3$, and a more general case corresponds to the Andrews–Baxter–Forrester model 1 related to the quantum groups $U_q(\mathfrak{sl}_2)$ for $q=\exp (2\pi i/l)$ a root of unity. These fundamental examples correspond to the subfactors generated by the Jones projections alone, and the graphs for these cases are the Coxeter–Dynkin diagrams of type $A_n$. Others related to the quantum groups $U_q(\mathfrak{sl}_n)$ have been studied in 2765.

We give a typical example of a flat connection as follows. Fix one of the Coxeter–Dynkin diagrams of type $A_n$, $D_{2n}$, $E_6$, or $E_8$, and use it for the four diagrams. Let $h$ be its Coxeter number, and set $\varepsilon =\sqrt {-1}\exp (\pi \sqrt {-1}/2h)$. We write $\mu _j$ for the Perron–Frobenius eigenvector entry for a vertex $j$ for the adjacency of the diagram. Then the flat connection is given as in Figure 1 and is essentially a normalisation of the braid element 2.2.

The index value given by this construction is $4\cos ^2 (\pi /h)$. If the graph is $A_n$, then the vertices are labeled with $j=1,2,\dots ,n$ and the Perron–Frobenius eigenvector entry for the vertex $j$ is given by $\sin (j\pi /(n+1))$. The value in Figure 1 in this case is essentially the same as what the Andrews–Baxter–Forrester model gives at a limiting value, and it also arises from a specialisation of the quantum $6j$-symbols for $U_q(\mathfrak{sl}_2)$ at a root of unity in the sense that two of the $6j$’s are chosen to be the fundamental representation of $U_q(\mathfrak{sl}_2)$. These are also related to IRF models by Roche in 111. These subfactors for the Dynkin diagrams $A_n$ are the ones constructed by Vaughan 67 as $N=\langle e_2, e_3,\dots \rangle$ and $M=\langle e_1, e_2, e_3, \dots \rangle$ with the above relations 2.1 with $[M:N]=4\cos ^2(\pi /(n+1))$.

The same formula as in Figure 1 for the Coxter–Dynkin diagrams $D_{2n+1}$ and $E_7$ almost gives a flat connection, but the flatness axiom fails. There are corresponding subfactors but they have principal graphs $A_{4n-1}$ and $D_{10}$, respectively. Nevertheless, the diagrams $D_{2n+1}$ and $E_7$ have interesting interpretations in connection with nonlocal extensions of conformal nets $\mathrm{SU}(2)_k$, as explained below.

The relations 2.1 of the Jones projections $e_j$ are reminiscent of the defining relations of the Hecke algebra $H_n(q)$ of type $A$ with complex parameter $q$, which is the free complex algebra generated by $1,g_1,g_2,\dots , g_{n-1}$ satisfying

$$\begin{equation*} \left\{ \begin{aligned} g_j g_{j+1} g_j&=g_{j+1} g_j g_{j+1},\\ g_j g_k &= g_k g_j,\quad \text{for $j\neq k$},\\ g_j^2&=(q-1)g_j+q. \end{aligned} \right. \end{equation*}$$

This similarity was exploited to construct more examples of subfactors with index values $\frac{\sin ^2 (k\pi /l)}{\sin ^2 (\pi /l)}$ with $1\le k\le l-1$ in the early days of subfactor theory by Wenzl in a University of Pennsylvania thesis supervised by Vaughan 125. He constructed representations $\rho$ of $H_\infty (q)=\bigcup _{n=1}^\infty H_n(q)$ with roots of unity $q=\exp (2\pi i/l)$ and $l=4,5,\dots$ such that $\rho (H_n(q))$ is always semisimple and gave a subfactor as $\rho (\langle g_2,g_3,\dots \rangle )''\subset \rho (\langle g_1,g_2,\dots \rangle )''$ using a suitable trace. The index values converge to $k^2$ as $l\to \infty$. When $k=2$, these subfactors are the ones constructed by Vaughan for the Coxeter–Dynkin diagram $A_{l-1}$. This construction is also understood in the context of IRF models 2765 related to $\mathrm{SU}(k)$. The relation between the Hecke algebras and the quantum groups $U_q(\mathfrak{sl}_n)$ is a “quantum” version of the classical Weyl duality. This duality also connects this Jones–Wenzl approach based on statistical mechanics and type II$_1$ factors with the Jones–Wassermann approach based on quantum field theory and type III$_1$ factors, which are explained below.

It is important to have a spectral parameter for the Boltzmann weights satisfying the Yang–Baxter equation in solvable lattice models, but we do not have such a parameter for a flat connection initially in subfactor theory. We usually obtain a flat connection by a certain specialisation of a spectral parameter for a Boltzmann weight. Vaughan proposed “Baxterization” in 71 for the converse direction in the sense of introducing a parameter for analogues of the Boltzmann weights in subfactor theory. This is an idea to obtain a physical counterpart from a subfactor, and we discuss a similar approach to construct a conformal field theory from a given subfactor at the end of this article. It should be noted that to rigorously construct a conformal field theory at criticality is a notoriously difficult problem—even for the Ising model; see, e.g., 112.

The finite depth condition means that we have a finite graph in this analogy to solvable lattice models. Even from a set of algebraic or combinatorial data similar to integrable lattice models involving infinite graphs, one sometimes constructs a corresponding subfactor. A major breakthrough of Popa 102 was to show that the Temperley–Lieb–Jones lattice is indeed a standard invariant showing for the first time for any index greater than $4$ that there exist subfactors with just the Jones projections as the higher relative commutants. Then, introducing tracial amalgamated free products, Popa 105 could show existence in full generality. These papers 102105 led to important links with free probability theory, leading to more sophisticated free random models to prove that certain amalgamated free products are free group factors that were adapted, by Ueda 120, to prove similar existence/reconstruction statements for actions of quantum groups. Popa and Shlyakhtenko 108 showed that any $\lambda$-lattice acts on the free group factor $L(\mathbb{F}_\infty )$. This involved a new construction of subfactors from $\lambda$-lattices, starting from a commuting square of semifinite von Neumann algebras, each one a direct sum of type I$_\infty$ factors with a semifinite trace, and with free probability techniques showing that the factors resulting from this construction are $\infty$-amplifications of $L(\mathbb{F}_\infty )$. The von Neumann algebras resulting in these constructions are not hyperfinite. A new proof using graphical tools, probabilistic methods, and planar algebras was later found by Guionnet, Jones, and Shlyakhtenko 57. Moreover they and Zinn and Justin 59 use matrix model computations in loop models of statistical mechanics and graph planar algebras to construct novel matrix models for Potts models on random graphs. This is based on the planar algebra machinery developed by Vaughan 78 for understanding higher relative commutants of subfactors. In 58 Guionnet, Jones, and Shlyakhtenko explicitly show that it is the same construction as in the Popa–Shlyakhtenko 108 paper. The paper 78 has been published only very recently in the Vaughan Jones memorial special issue after his passing away, but its preprint version appeared in 1999 and has been highly influential. Note also that Kauffman 8081 had found a diagrammatic construction of the Jones polynomial directly related to the Potts model based on a diagrammatic presentation of the Temperley–Lieb algebra which then has a natural home in the planar algebra formalism. The polynomial was understood by Reshetikhin and Turaev in 110 in the context of representations of the quantum groups $U_q(\mathfrak{sl}_2)$ 3264.

3. Subfactors and quantum field theory

Witten 126 gave a new interpretation of the Jones polynomial based on quantum field theory, the Chern–Simons gauge field theory, and generalised it to an invariant of a link in a compact 3-manifold. However, it was not clear why we should have a polynomial invariant in this way. Taking an empty link yields an invariant of a compact 3-manifold. Witten used a path integral formulation and was not mathematically rigorous. A mathematically well-defined version based on combinatorial arguments using Dehn surgery and the Kirby calculus has been given by Reshetikhin and Turaev 110. In the case of an empty link, we realise a 3-manifold from a framed link with Dehn surgery, make a weighted sum of invariants of this link using representations of a certain quantum group at a root of unity, and prove that this weighted sum is invariant under the Kirby moves. Two framed links give homeomorphic manifolds if and only if they are related with a series of Kirby moves. For the quantum group $U_q(\mathfrak{sl}_2)$, the link invariant is the colored Jones polynomial. A color is a representation of the quantum group and labels a connected component of a link. This actually gives a $(2+1)$-dimensional topological quantum field theory in the sense of Atiyah 5, which is a certain mathematical axiomatisation of a quantum field theory based on topological invariance. Roughly speaking, we assign a finite-dimensional Hilbert space to each closed two-dimensional manifold, and also assign a linear map from one such Hilbert space to another to a cobordism so that this assignment is functorial. It is also easy to extend this construction from quantum groups to general modular tensor categories, as we explain below.

A closely related, but different, $(2+1)$-dimensional topological quantum field theory has been given by Turaev and Viro 119. In this formulation, one triangulates a 3-manifold, considers a weighted sum of quantum $6j$-symbols arising from a quantum group depending on the triangulation, and proves that this sum is invariant under the Pachner moves. Two triangulated manifolds are homeomorphic to each other if and only if we obtain one from the other with a series of Pachner moves. This has been generalised to another $(2+1)$-dimensional topological quantum field theory using quantum $6j$-symbols arising from a subfactor by Ocneanu; see 41, Chapter 12. Here we only need a fusion category structure which we explain below, and no braiding. This is different from the above Reshetikhin–Turaev case. For a given fusion category, we apply the Drinfel′d center construction, a kind of “quantum double” construction, to get a modular tensor category with a nondegenerate braiding. This construction was developed in subfactor theory by Ocneanu 95 through an asymptotic inclusion, by Popa 104 through a symmetric enveloping algebra, through the Longo–Rehren subfactor 92 and Izumi 6263, and in a categorical setting by Müger 94. We then apply the Reshetikhin–Turaev construction to the double. We can also apply the Turaev–Viro–Ocneanu construction to the original fusion category, and these two procedures give the same topological quantum field theory 85. In particular, if we start with $U_q(\mathfrak{sl}_2)$ at a root of unity, the Turaev–Viro invariant of a closed $3$-manifold is the square of the absolute value of the Reshetikhin–Turaev invariant of the same $3$-manifold.

Another connection of subfactors to quantum field theory is through algebraic quantum field theory, which is a bounded operator algebraic formulation of quantum field theory. The usual ingredients for describing a quantum field theory are as follows:

(1)

A spacetime, such as the four-dimensional Minkowski space.

(2)

A spacetime symmetry group, such as the Poincaré group.

(3)

A Hilbert space of states, including the vacuum.

(4)

A projective unitary representation of the spacetime symmetry group on the Hilbert space of states.

(5)

A set of quantum fields, that is, operator-valued distributions, defined on the spacetime acting on the Hilbert space of states.

An ordinary distribution assigns a number to each test function. An operator-valued distribution assigns a (possibly unbounded) operator to each test function. The Wightman axioms give a direct axiomatisation using these, and they have a long history of research, but it is technically difficult to handle operator-valued distributions, so we have a different approach based on bounded linear operators giving observables. Let $O$ be a region within the spacetime. Take a quantum field $\varphi$ and a test function $f$ supported on $O$. The self-adjoint part of $\langle \varphi , f\rangle$ is an observable in $O$ which could be unbounded. Let $A(O)$ denote the von Neumann algebra generated by spectral projections of such self-adjoint operators. This passage from operator-valued distributions to von Neumann algebras is also used in the construction of a conformal net from a vertex operator algebra by Carpi, Kawahigashi, Longo, and Weiner 19, which we explain below. Note that a von Neumann algebra contains only bounded operators.

Locality is an important axiom arising from the Einstein causality which says that if two regions are spacelike separated, observables in these regions have no interactions, hence the corresponding operators commute. In terms of the von Neumann algebras $A(O)$, we require that $[A(O_1), A(O_2)]=0$, if $O_1$ and $O_2$ are spacelike separated, where the Lie bracket means the commutator. This family of von Neumann algebras parameterised by spacetime regions is called a net of operator algebras. Algebraic quantum field theory gives an axiomatisation of a net of operator algebras, together with a projective unitary representation of a spacetime symmetry group on the Hilbert space of states including the vacuum. A main idea is that it is not each von Neumann algebra but the relative relations among these von Neumann algebras that contains the physical contents of a quantum field theory. In the case of two-dimensional conformal field theory, which is a particular example of a quantum field theory, each von Neumann algebra $A(O)$ is always a hyperfinite type III$_1$ factor, which is unique up to isomorphism and is the Araki–Woods factor of type III$_1$. Thus the isomorphism class of a single von Neumann algebra contains no physical information. Each local algebra of a conformal net is a factor of type III$_1$ by 56, Proposition 1.2. It is also hyperfinite because it has a dense subalgebra given as an increasing union of type I algebras, which follows from the split property shown in 93, Theorem 5.4.

Fix a net $\{A(O)\}$ of von Neumann algebras. It has a natural notion of a representation on another Hilbert space without the vacuum vector. The action of these von Neumann algebras on the original Hilbert space itself is a representation, and it is called the vacuum representation. We also have natural notions of unitary equivalence and irreducibility of representations. The unitary equivalence class of an irreducible representation of the net $\{A(O)\}$ is called a superselection sector. We also have a direct sum and irreducible decomposition for representations. If we have two representations of a group, it is very easy to define their tensor product representation, but it is not clear at all how to define a tensor product representation of two representations of a single net of operator algebras. Doplicher, Haag, and Roberts gave a proper definition of the tensor product of two representations 2930. Under a certain natural assumption, each representation has a representative given by an endomorphism of a single algebra $A(O)$ acting on the vacuum Hilbert space for some fixed $O$. This endomorphism contains complete information about the original representation. For two such endomorphisms $\rho$ and $\sigma$, the composed endomorphism $\rho \sigma$ also corresponds to a representation of the net $\{A(O)\}$. This gives a correct notion of the tensor product of two representations. Furthermore, it turns out that the two compositions $\rho \sigma$ and $\sigma \rho$ of endomorphisms give unitarily equivalent representations. If the spacetime dimension is higher than 2, this commutativity of the tensor product is similar to unitary equivalence of $\pi _1\otimes \pi _2$ and $\pi _2\otimes \pi _1$ for two representations $\pi _1$ and $\pi _2$ of the same group. The representations now give a symmetric monoidal $C^*$-category, where a representation gives an object, an intertwiner gives a morphism, and the above composition of endomorphisms gives the tensor product structure. This category produces a compact group from the new duality of Doplicher and Roberts 31. Here an object of the category is an endomorphism, and a morphism in $\mathrm{Hom}(\rho ,\sigma )$ is an intertwiner, that is, an element in

$$\begin{equation*} \{T\in A(O)\mid T\rho (x)=\sigma (x)T\text{ for all } x\in A(O)\}. \end{equation*}$$

In other words, the Doplicher–Roberts duality gives an abstract characterisation of the representation category of a compact group among general tensor categories. The vacuum representation plays the role of the trivial representation of a group, and the dual representation of a net of operator algebras corresponds to the dual representation of a compact group. This duality is related to the classical Tannaka duality but gives a duality more generally for abstract tensor categories.

Using the structure of a symmetric monoidal $C^*$-category, we define a statistical dimension of each representation, which turns out to be a positive integer or infinity 2930. That the Jones index value takes on only discrete values below 4 is reminiscent of this fact that a statistical dimension can take only integer values. Longo 8990 showed that the statistical dimension of the representation corresponding to an endomorphism $\rho$ of $A(O)$ is equal to the square root of the Jones index $[A(O):\rho (A(O))]$. This opened up a wide range of new interactions between subfactor theory and algebraic quantum field theory.

Generalizing the notion of a superselection sector, Longo 8990 introduced the notion of a sector, the unitary equivalence class of an endomorphism of a factor of type III, inspired by Connes theory of correspondences, based on the equivalences between Hilbert bimodules, endomorphisms, and positive definite functions on doubles; see 22 23, VB, 24, and, e.g., Popa 101 for developments. He defined a dual sector using the canonical endomorphism which he had introduced based on the modular conjugation in Tomita–Takesaki theory. Note that in a typical situation of a subfactor $N\subset M$, these von Neumann algebras are isomorphic, so we have an endomorphism $\rho$ of $M$ onto $N$. Then we have the dual endomorphism $\bar{\rho }$, and the irreducible decompositions of $\rho \bar{\rho }\rho \bar{\rho }\cdots \bar{\rho }$ give objects of a tensor category, where the morphisms are the intertwiners of endomorphisms and the tensor product operation is composition of endomorphisms. If we have finitely many irreducible endomorphisms arising in this way, which is equivalent to the finite depth condition, our tensor category is a fusion category, where we have the dual object for each object and we have only finitely many irreducible objects up to isomorphisms. The higher relative commutants $M'\cap M_k$ are described as intertwiner spaces, such as $\mathrm{End}(\rho \bar{\rho }\rho \bar{\rho }\cdots \bar{\rho })$ or $\mathrm{End}(\rho \bar{\rho }\rho \bar{\rho }\cdots \rho )$.

In our setting, for a factor $M$, we have the standard representation of $M$ on the Hilbert space $L^2(M)$, the completion of $M$ with respect to a certain inner product, and this $L^2(M)$ also has a right multiplication by $M$ based on Tomita–Takesaki theory. For an endomorphism $\rho$ of $M$, we have a new $M$-$M$ bimodule structure on $L^2(M)$ by twisting the right action of $M$ by $\rho$. In this setting, all $M$-$M$ bimodules arise in this way, and we have a description of the above tensor category in terms of bimodules. Here the tensor product operation is given by a relative tensor product of bimodules over $M$. For type II$_1$ factors, we need to use this bimodule description to obtain the correct tensor category structures. It is more natural to use type II$_1$ factors in statistical mechanics, and it is more natural to use type III$_1$ factors in quantum field theory, but they give rise to equivalent tensor categories, so if we are interested in tensor category structure, including braiding, this difference between type II$_1$ and type III$_1$ is not important.

4. Subfactors and conformal field theory

A two-dimensional conformal field theory is a particular example of a quantum field theory, but it is a rich source of deep interactions with subfactor theory, so we treat this in an independent section.

We start with the $(1+1)$-dimensional Minkowski space and consider quantum field theory with conformal symmetry. We restrict a quantum field theory onto two light rays $x=\pm t$ and compactify a light ray by adding a point at infinity. The resulting $S^1$ is our spacetime now, though space and time are mixed into one dimension, and our symmetry group for $S^1$ is now $\mathrm{Diff}(S^1)$, the orientation preserving diffeomorphism group of $S^1$. Our spacetime region is now an interval $I$, a nonempty, nondense open connected subset of $S^1$. For each such an interval $I$, we have a corresponding von Neumann algebra $A(I)$ acting on a Hilbert space $H$ of states containing the vacuum vector. Isotony means that we have $A(I_1)\subset A(I_2)$ if we have $I_1\subset I_2$. Locality now means that $[A(I_1),A(I_2)]=0$, if $I_1 \cap I_2=Ø$. Note that spacelike separation gives this very simple disjointness. Our spacetime symmetry group now is $\mathrm{Diff}(S^1)$, and we have a projective unitary representation $U$ on $H$. Conformal covariance asks for $U_g A(I) U_g^*=A(gI)$ for $g\in \mathrm{Diff}(S^1)$. Positivity of the energy means that the restriction of $U$ to the subgroup of rotations of $S^1$ gives a one-parameter unitary group and its generator is positive. In this setting, a family $\{A(I)\}$ of von Neumann algebras satisfying these axioms is called a conformal net.

A representation theory of a conformal net in the Doplicher–Haag–Roberts style now gives a braiding due to the low-dimensionality of the spacetime $S^1$. This is a certain form of the nontrivial commutativity of endomorphisms up to inner automorphisms. That is, two representations give two endomorphisms $\lambda ,\mu$ of a single von Neumann algebra $A(I_0)$ for some fixed interval $I_0$, and we have a unitary $\varepsilon (\lambda ,\mu )\in A(I)$ satisfying $\mathrm{Ad}(\varepsilon (\lambda ,\mu ))\lambda \mu =\mu \lambda$. This unitary $\varepsilon (\lambda ,\mu )$, sometimes called a statistics operator, arises from the monodromy of moving an interval in $S^1$ to a disjoint one and back, and it satisfies various compatibility conditions such as braiding-fusion equations for intertwiners as in 445190. Switching two tensor components corresponds to switching two wires of a braid. For two wires, we have an overcrossing and an undercrossing. They correspond to $\varepsilon (\lambda ,\mu )$ and $\varepsilon (\mu ,\lambda )^*$. In particular, if we fix an irreducible endomorphism and use it for both $\lambda$ and $\mu$, we have a unitary representation of the braid group $B_n$ for every $n$. In the case of a higher-dimensional Minkowski space, $\varepsilon (\lambda ,\mu )$ gives a so-called degenerate braiding, like the case of a group representation where we easily have unitary equivalence of $\pi \otimes \sigma$ and $\sigma \otimes \pi$ for two representations $\pi$ and $\sigma$, but we now have a braiding in a more nontrivial way on $S^1$. It was proved by Kawahigashi, Longo, and Müger in 84 that if we have a certain finiteness of the representation theory of a conformal net, called complete rationality, then the braiding of its representation category is nondegenerate, and hence it gives rise to a modular tensor category by definition. A modular tensor category is also expected to be useful for topological quantum computations, as in the work of Freedman, Kitaev, Larsen, and Wang 47. This is a hot topic in quantum information theory and many researchers work on topological quantum information using the Jones polynomial and its various generalisations.

It is a highly nontrivial task to construct examples of conformal nets. The first such attempt started in a joint project of Vaughan and Wassermann trying to construct a subfactor from a positive energy representation of a loop group. Wassermann 123 then constructed conformal nets arising from positive energy representations of the loop groups of $\mathrm{SU}(N)$ corresponding to the Wess–Zumino–Witten models $\mathrm{SU}(N)_k$, where $k$ is a positive integer called a level. These examples satisfy complete rationality as shown by Xu in 130. The conformal nets corresponding to $\mathrm{SU}(2)_k$ give unitary representations of the braid groups $B_n$ which are the same as the one given by Vaughan from the Jones projections $e_j$. Wassermann’s construction has been generalised to other Lie groups by Loke, Toledano Laredo, and Verrill in dissertations supervised by him; 88118122, see also 124. Loke worked with projective unitary representations of $\mathrm{Diff}(S^1)$ and obtained the Virasoro nets.

A relative version $\{A(I)\subset B(I)\}$ of a conformal net for intervals $I\subset S^1$ called a net of subfactors has been given in 92. Suppose that $\{A(I)\}$ is completely rational. Assuming that we know the representation category of $\{A(I)\}$, we would like to know that of $\{B(I)\}$. The situation is similar to a group inclusion $H\subset G$ where we know representation theory of $H$ and would like to know that of $G$. In the group representation case for $H\subset G$, we have a restriction of a representation of $G$ to $H$ and an induction of a representation of $H$ to $G$. In the case of a net of subfactors, the restriction of a representation of $\{B(I)\}$ to $\{A(I)\}$ is easy to define, but the induction procedure is more subtle. Our induction procedure is now called $\alpha$-induction, first defined by Longo and Rehren in 92 and studied by Xu 127, Böckenhauer and Evans 78910, and Böckenhauer, Evans, and Kawahigashi 1112, also in connection to Ocneanu’s graphical calculus on Coxeter–Dynkin diagrams in the last two papers. (In these two papers, this $\alpha$-induction is studied in the more general context of abstract modular tensor categories of endomorphisms rather than conformal field theory. For an $A$-$A$ bimodule $X$, then the tensor product $X \otimes B$ can be regarded as a $B$-$B$ module if one uses the braiding to let $B$ act on the left.) Take a representation of $\lambda$ of $\{A(I)\}$ which is given as an endomorphism of $A(I_0)$ for some fixed interval $I_0$. Then using the braiding on the representation category of $\{A(I)\}$, we define an endomorphism $\alpha _\lambda ^\pm$ of $B(I_0)$, where $\pm$ represents a choice of a positive or negative braiding, $\varepsilon ^\pm (\lambda ,\theta )$, where $\theta$ represents the dual canonical endomorphism of the subfactor $A(I)\subset B(I)$. This nearly gives a representation of $\{B(I)\}$, but not exactly. It turns out that the irreducible endomorphisms arising both from a positive induction and a negative one exactly correspond to those arising from irreducible representations of $\{B(I)\}$. The braiding of the representation category of $\{A(I)\}$ gives a finite-dimensional unitary representation of $\mathrm{SL}(2,\mathbb{Z})$ through the so-called $S$- and $T$-matrices. Böckenhauer, Evans, and Kawahigashi 11 showed that the matrix $Z_{\lambda ,\mu }=\langle \alpha ^+_\lambda , \alpha ^-_\mu \rangle$, where $\lambda ,\mu$ label irreducible representations of $\{A(I)\}$ and the symbol $\langle \cdot ,\cdot \rangle$ counts the number of common irreducible endomorphisms including multiplicities, satisfies the following properties:

(1)

We have $Z_{\lambda ,\mu }\in \{0,1,2,\dots \}$.

(2)

We have $Z_{0,0}=1$, where the label $0$ denotes the vacuum representation.

(3)

The matrix $Z$ commutes with the image of the representation of $\mathrm{SL}(2,\mathbb{Z})$.

Such a matrix $Z$ is called a modular invariant, because $\mathrm{PSL}(2,\mathbb{Z})$ is called the modular group. For a given completely rational conformal net (or more generally, a given modular tensor category), we have only finitely many modular invariants. Modular invariants naturally appear as partition functions in two-dimensional conformal field theory, and they have been classified for several concrete examples since Cappelli, Itzykson, and Zuber’s paper 16 for the $\mathrm{SU}(2)_k$ models and the Virasoro nets with $c<1$, where $c$ is a numerical invariant called the central charge. It takes a positive real value, and if $c<1$, then it is of the form $1-6/m(m+1)$, $m=3,4,5,\dots$ by Friedan, Qiu, and Shenker 50 and Goddard, Kent, and Olive 54. This number arises from a projective unitary representation of $\mathrm{Diff}(S^1)$ and its corresponding unitary representation of the Virasoro algebra, a central extension of the complexification of the Lie algebra arising from $\mathrm{Diff}(S^1)$. Note that some modular invariants defined by the above three properties do not necessarily correspond to physical ones arising as partition functions in conformal field theory. Modular invariants arising from $\alpha$-induction are physical in this sense.

The action of the $A$-$A$ system on the $A$-$B$ sectors (obtained by decomposing $\{ \iota \lambda =\alpha ^{\pm }_{\lambda } \iota : \lambda \in$ $A$-$A\}$ into irreducibles where $\iota : A \rightarrow B$ is the inclusion) gives naturally a representation of the fusion rules of the Verlinde ring: $G_{\lambda }G_{\mu } = \sum N_{\lambda \mu }^{\nu } G_{\nu }$, with matrices $G_{\lambda } = [G_{\lambda a}^{b} : a, b \in$ $A$-$B$ sectors]. Consequently, the matrices $G_{\lambda }$ will be described by the same eigenvalues but with possibly different multiplicities. Böckenhauer, Evans, and Kawahigashi 12 showed that these multiplicities are given exactly by the diagonal part of the modular invariant: $\text{spectrum} (G_{\lambda }) = \{S_{\lambda \kappa }/S_{0\kappa } : \text{ with multiplicity } Z_{\kappa \kappa } \}.$ This is called a nimrep, a nonnegative integer matrix representation. Thus a physical modular invariant is automatically equipped with a compatible nimrep whose spectrum is described by the diagonal part of the modular invariant. The case of $\mathrm{SU}(2)$ is just the $A$-$D$-$E$ classification of Cappelli, Itzykson, and Zuber 16 with the $A$-$B$ system yielding the associated (unextended) Coxeter–Dynkin graph. Since there is an $A$-$D$-$E$ classification of matrices of norm less than $2$, we can recover independently of Cappelli, Itzykson, and Zuber 16 that there are unique modular invariants corresponding to the three exceptional $E$ graphs.

If we use only positive $\alpha$-inductions for a given modular tensor category, we still have a fusion category of endomorphisms, but no braiding in general. This is an example of a module category. For the tensor category $\mathrm{Rep}(G)$ of representations of a finite group $G$, all module categories are of the form $\mathrm{Rep}(H, \chi )$ for the projective representations with $2$-cocycle $\chi$ for a subgroup $H$ 96. For this reason, module categories have also been called quantum subgroups. Such categories have been studied in a more general categorical context by Ostrik in 98. However, Carpi, Gaudio, Giorgetti, and Hillier 18, have shown that for unitary fusion categories, such as those that occur in subfactor theory or arise from loop groups, that all module categories are equivalent to unitary ones. For the conformal nets corresponding to $\mathrm{SU}(2)_k$, the module categories or quantum subgroups are labeled with all the Coxeter–Dynkin diagrams $A_n$, $D_n$, and $E_{6,7,8}$. Here there is a coincidence with the affine $A$-$D$-$E$ classification of finite subgroups of $\mathrm{SU}(2)$. Di Francesco and Zuber 27 were motivated to try to relate $\mathrm{SU}(3)$ modular invariants with subgroups of $\mathrm{SU}(3)$. There is a partial match, but this is not helpful. In general whilst the number of finite subgroups of $\mathrm{SU}(n)$ grows with $n$, the number of exceptional modular invariants, beyond the obvious infinite series, does not.

If we have a net of subfactors $\{A(I)\subset B(I)\}$ with $\{A(I)\}$ being a completely rational conformal net, then the restriction of the vacuum representation of $\{B(I)\}$ to $\{A(I)\}$ gives a local $Q$-system in the sense of Longo 91. This notion is essentially the same as a commutative Frobenius algebra, a special case of an algebra in a tensor category, in the algebraic or categorical literature. This $Q$-system is a triple consisting of an object and two intertwiners. Roughly speaking, the object gives $B(I)$ as an $A(I)$-$A(I)$ bimodule and the intertwiners give the multiplicative structure on $B(I)$. Our general theory of $\alpha$-induction shows that the corresponding modular invariant $Z$ for the modular tensor category of representations of $\{A(I)\}$ recovers this object. Since we have only finitely many modular invariants for a given modular tensor category, we have only finitely many objects for a local $Q$-system. It is known that each object has only finitely many local $Q$-system structures, and we thus have only finitely many local $Q$-systems, which means that we have only finitely many possibilities for extensions $\{B(I)\}$ for a given $\{A(I)\}$.

For some concrete examples of $\{A(I)\}$, we can classify all possible extensions. In the case of the $\mathrm{SU}(2)_k$ nets, such extensions were studied in the context of $\alpha$-induction in 11 by Böckenhauer, Evans, and Kawahigashi, and it was shown in 82 by Kawahigashi-Longo that they exhaust all possible extensions. (A similar classification based on quantum groups was first given in 86.) They correspond to the Coxeter–Dynkin diagrams $A_n$, $D_{2n}$, $E_6$, and $E_8$. The $A_n$ cases are the $\mathrm{SU}(2)_k$ nets themselves, the $D_{2n}$ cases are given by simple current extensions of order 2, and the $E_6$ and $E_8$ cases are given by conformal embeddings $\mathrm{SU}(2)_{10}\subset \mathrm{SO}(5)_1$ and