Subfactors and mathematical physics

By David E. Evans and Yasuyuki Kawahigashi

This paper is dedicated to the memory of Vaughan Jones


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 Reference 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 be a subfactor of type II, and let be its Jones index, which is a positive real number or infinity. That is, and are infinite-dimensional simple von Neumann algebras with a trace . We only consider the case that is finite. Vaughan Reference 67 constructed a sequence of projections , , called the Jones projections, and discovered the following relations:

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

Vaughan made the substitution

where to yield the Artin relations of the braid group, where is the braid which interchanges the and strands. Vaughan’s representation came equipped with a trace satisfying the Markov trace property in the probabilistic sense , where belongs to the algebra generated by . 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 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 Reference 68Reference 69 in the variable , a polynomial invariant of a link.

Evans pointed out in 1983 that these relations Equation 2.1 appear in similar formalism to one studied by Temperley and Lieb Reference 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 at the vertices of a lattice. Important generalisations include the Potts model, with 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 Reference 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 Equation 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 Reference 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 Equation 2.1 and in this way demonstrated equivalence of the models. Whilst the relations for the Potts model only occur when is integral, the partition function is a Tutte–Whitney dichromatic polynomial. One variable is which can be extrapolated, and the partition function is then related to the Jones polynomial on certain links associated to the lattice Reference 68, page 108. The ice-type representation though has a continuous parameter . In both cases, the Markov trace did not manifest itself.

Pimsner and Popa Reference 100 discovered that the inverse of the index, namely , is the best constant for which for all , which they called the probabilistic index. Here is the conditional expectation of onto 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 Reference 89 identifying statistical dimension with the Jones index, and by Fredenhagen, Rehren, and Schroer Reference 44Reference 45, 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 Reference 25, , for irreducible subfactors.

For a subfactor with finite Jones index, we have the Jones tower construction

where is generated by and . The basic construction from to and its iteration to give the Jones tower of II factors has a fundamental role in subfactor theory and applications in mathematical physics. The higher relative commutants , , give a system of commuting squares of inclusions of finite-dimensional -algebras with a trace, an object denoted by and called the standard invariant of . 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 , of Cayley type, called the principal graph of , with a canonical weight vector , whose entries are square roots of indices of irreducible inclusions in the Jones tower. The weighted graph satisfies the Perron–Frobenius type condition , and also .

Of particular relevance to mathematical physics is when has finite depth, corresponding to the graph being finite, in which case the weights give the (unique) Perron–Frobenius eigenvector, entailing . Finite depth is automatic when the index is less than , where indeed all bipartite graphs are finite and have norms of the form , .

The objects have been axiomatised in a number of ways—by Ocneanu with paragroups and connections Reference 95 in the finite depth case, then in the general case by Popa with -lattices Reference 105, and by Vaughan with planar algebras Reference 78.

By Connes fundamental result in Reference 21, the hyperfinite II factor , obtained as an inductive limit of finite-dimensional algebras, is the unique amenable II factor, so in particular all its finite index subfactors are isomorphic to . In a series of papers Reference 103Reference 104Reference 106Reference 107, Popa identified the appropriate notion of amenability for inclusions of II factors and for the objects , in several equivalent ways, one being the Kesten-type condition . He proved the important result that for hyperfinite subfactors satisfying this amenability condition, is a complete invariant. In other words, whenever and (in particular if has finite depth), 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 Reference 70. (See the monograph of Baxter Reference 6 for these types of statistical mechanical models. Also see Reference 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 Reference 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 , and a more general case corresponds to the Andrews–Baxter–Forrester model Reference 1 related to the quantum groups for 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 . Others related to the quantum groups have been studied in Reference 27Reference 65.

We give a typical example of a flat connection as follows. Fix one of the Coxeter–Dynkin diagrams of type , , , or , and use it for the four diagrams. Let be its Coxeter number, and set . We write for the Perron–Frobenius eigenvector entry for a vertex 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 Equation 2.2.

The index value given by this construction is . If the graph is , then the vertices are labeled with and the Perron–Frobenius eigenvector entry for the vertex is given by . 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 -symbols for at a root of unity in the sense that two of the ’s are chosen to be the fundamental representation of . These are also related to IRF models by Roche in Reference 111. These subfactors for the Dynkin diagrams are the ones constructed by Vaughan Reference 67 as and with the above relations Equation 2.1 with .

The same formula as in Figure 1 for the Coxter–Dynkin diagrams and almost gives a flat connection, but the flatness axiom fails. There are corresponding subfactors but they have principal graphs and , respectively. Nevertheless, the diagrams and have interesting interpretations in connection with nonlocal extensions of conformal nets , as explained below.

The relations Equation 2.1 of the Jones projections are reminiscent of the defining relations of the Hecke algebra of type with complex parameter , which is the free complex algebra generated by satisfying

This similarity was exploited to construct more examples of subfactors with index values with in the early days of subfactor theory by Wenzl in a University of Pennsylvania thesis supervised by Vaughan Reference 125. He constructed representations of with roots of unity and such that is always semisimple and gave a subfactor as using a suitable trace. The index values converge to as . When , these subfactors are the ones constructed by Vaughan for the Coxeter–Dynkin diagram . This construction is also understood in the context of IRF models Reference 27Reference 65 related to . The relation between the Hecke algebras and the quantum groups is a “quantum” version of the classical Weyl duality. This duality also connects this Jones–Wenzl approach based on statistical mechanics and type II factors with the Jones–Wassermann approach based on quantum field theory and type III 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 Reference 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., Reference 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 Reference 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 that there exist subfactors with just the Jones projections as the higher relative commutants. Then, introducing tracial amalgamated free products, Popa Reference 105 could show existence in full generality. These papers Reference 102Reference 105 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 Reference 120, to prove similar existence/reconstruction statements for actions of quantum groups. Popa and Shlyakhtenko Reference 108 showed that any -lattice acts on the free group factor . This involved a new construction of subfactors from -lattices, starting from a commuting square of semifinite von Neumann algebras, each one a direct sum of type I factors with a semifinite trace, and with free probability techniques showing that the factors resulting from this construction are -amplifications of . 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 Reference 57. Moreover they and Zinn and Justin Reference 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 Reference 78 for understanding higher relative commutants of subfactors. In Reference 58 Guionnet, Jones, and Shlyakhtenko explicitly show that it is the same construction as in the Popa–Shlyakhtenko Reference 108 paper. The paper Reference 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 Reference 80Reference 81 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 Reference 110 in the context of representations of the quantum groups Reference 32Reference 64.

3. Subfactors and quantum field theory

Witten Reference 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 Reference 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 , 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 -dimensional topological quantum field theory in the sense of Atiyah Reference 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, -dimensional topological quantum field theory has been given by Turaev and Viro Reference 119. In this formulation, one triangulates a 3-manifold, considers a weighted sum of quantum -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 -dimensional topological quantum field theory using quantum -symbols arising from a subfactor by Ocneanu; see Reference 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 Reference 95 through an asymptotic inclusion, by Popa Reference 104 through a symmetric enveloping algebra, through the Longo–Rehren subfactor Reference 92 and Izumi Reference 62Reference 63, and in a categorical setting by Müger Reference 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 Reference 85. In particular, if we start with at a root of unity, the Turaev–Viro invariant of a closed -manifold is the square of the absolute value of the Reshetikhin–Turaev invariant of the same -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:


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


A spacetime symmetry group, such as the Poincaré group.


A Hilbert space of states, including the vacuum.


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


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 be a region within the spacetime. Take a quantum field and a test function supported on . The self-adjoint part of is an observable in which could be unbounded. Let 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 Reference 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 , we require that , if and 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 is always a hyperfinite type III factor, which is unique up to isomorphism and is the Araki–Woods factor of type III. 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 by Reference 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 Reference 93, Theorem 5.4.

Fix a net 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 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 Reference 29Reference 30. Under a certain natural assumption, each representation has a representative given by an endomorphism of a single algebra acting on the vacuum Hilbert space for some fixed . This endomorphism contains complete information about the original representation. For two such endomorphisms and , the composed endomorphism also corresponds to a representation of the net . This gives a correct notion of the tensor product of two representations. Furthermore, it turns out that the two compositions and 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 and for two representations and of the same group. The representations now give a symmetric monoidal -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 Reference 31. Here an object of the category is an endomorphism, and a morphism in is an intertwiner, that is, an element in

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 -category, we define a statistical dimension of each representation, which turns out to be a positive integer or infinity Reference 29Reference 30. 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 Reference 89Reference 90 showed that the statistical dimension of the representation corresponding to an endomorphism of is equal to the square root of the Jones index . 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 Reference 89Reference 90 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 Reference 22 Reference 23, VB, Reference 24, and, e.g., Popa Reference 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 , these von Neumann algebras are isomorphic, so we have an endomorphism of onto . Then we have the dual endomorphism , and the irreducible decompositions of 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 are described as intertwiner spaces, such as or .

In our setting, for a factor , we have the standard representation of on the Hilbert space , the completion of with respect to a certain inner product, and this also has a right multiplication by based on Tomita–Takesaki theory. For an endomorphism of , we have a new - bimodule structure on by twisting the right action of by . In this setting, all - 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 . For type II factors, we need to use this bimodule description to obtain the correct tensor category structures. It is more natural to use type II factors in statistical mechanics, and it is more natural to use type III 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 and type III 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 -dimensional Minkowski space and consider quantum field theory with conformal symmetry. We restrict a quantum field theory onto two light rays and compactify a light ray by adding a point at infinity. The resulting is our spacetime now, though space and time are mixed into one dimension, and our symmetry group for is now , the orientation preserving diffeomorphism group of . Our spacetime region is now an interval , a nonempty, nondense open connected subset of . For each such an interval , we have a corresponding von Neumann algebra acting on a Hilbert space of states containing the vacuum vector. Isotony means that we have if we have . Locality now means that , if . Note that spacelike separation gives this very simple disjointness. Our spacetime symmetry group now is , and we have a projective unitary representation on . Conformal covariance asks for for . Positivity of the energy means that the restriction of to the subgroup of rotations of gives a one-parameter unitary group and its generator is positive. In this setting, a family 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 . This is a certain form of the nontrivial commutativity of endomorphisms up to inner automorphisms. That is, two representations give two endomorphisms of a single von Neumann algebra for some fixed interval , and we have a unitary satisfying . This unitary , sometimes called a statistics operator, arises from the monodromy of moving an interval in to a disjoint one and back, and it satisfies various compatibility conditions such as braiding-fusion equations for intertwiners as in Reference 44Reference 51Reference 90. 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 and . In particular, if we fix an irreducible endomorphism and use it for both and , we have a unitary representation of the braid group for every . In the case of a higher-dimensional Minkowski space, gives a so-called degenerate braiding, like the case of a group representation where we easily have unitary equivalence of and for two representations and , but we now have a braiding in a more nontrivial way on . It was proved by Kawahigashi, Longo, and Müger in Reference 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 Reference 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 Reference 123 then constructed conformal nets arising from positive energy representations of the loop groups of corresponding to the Wess–Zumino–Witten models , where is a positive integer called a level. These examples satisfy complete rationality as shown by Xu in Reference 130. The conformal nets corresponding to give unitary representations of the braid groups which are the same as the one given by Vaughan from the Jones projections . Wassermann’s construction has been generalised to other Lie groups by Loke, Toledano Laredo, and Verrill in dissertations supervised by him; Reference 88Reference 118Reference 122, see also Reference 124. Loke worked with projective unitary representations of and obtained the Virasoro nets.

A relative version of a conformal net for intervals called a net of subfactors has been given in Reference 92. Suppose that is completely rational. Assuming that we know the representation category of , we would like to know that of . The situation is similar to a group inclusion where we know representation theory of and would like to know that of . In the group representation case for , we have a restriction of a representation of to and an induction of a representation of to . In the case of a net of subfactors, the restriction of a representation of to is easy to define, but the induction procedure is more subtle. Our induction procedure is now called -induction, first defined by Longo and Rehren in Reference 92 and studied by Xu Reference 127, Böckenhauer and Evans Reference 7Reference 8Reference 9Reference 10, and Böckenhauer, Evans, and Kawahigashi Reference 11Reference 12, also in connection to Ocneanu’s graphical calculus on Coxeter–Dynkin diagrams in the last two papers. (In these two papers, this -induction is studied in the more general context of abstract modular tensor categories of endomorphisms rather than conformal field theory. For an - bimodule , then the tensor product can be regarded as a - module if one uses the braiding to let act on the left.) Take a representation of of which is given as an endomorphism of for some fixed interval . Then using the braiding on the representation category of , we define an endomorphism of , where represents a choice of a positive or negative braiding, , where represents the dual canonical endomorphism of the subfactor . This nearly gives a representation of , 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 . The braiding of the representation category of gives a finite-dimensional unitary representation of through the so-called - and -matrices. Böckenhauer, Evans, and Kawahigashi Reference 11 showed that the matrix , where label irreducible representations of and the symbol counts the number of common irreducible endomorphisms including multiplicities, satisfies the following properties:


We have .


We have , where the label denotes the vacuum representation.


The matrix commutes with the image of the representation of .

Such a matrix is called a modular invariant, because 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 Reference 16 for the models and the Virasoro nets with , where is a numerical invariant called the central charge. It takes a positive real value, and if , then it is of the form , by Friedan, Qiu, and Shenker Reference 50 and Goddard, Kent, and Olive Reference 54. This number arises from a projective unitary representation of and its corresponding unitary representation of the Virasoro algebra, a central extension of the complexification of the Lie algebra arising from . 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 -induction are physical in this sense.

The action of the - system on the - sectors (obtained by decomposing - into irreducibles where is the inclusion) gives naturally a representation of the fusion rules of the Verlinde ring: , with matrices - sectors]. Consequently, the matrices will be described by the same eigenvalues but with possibly different multiplicities. Böckenhauer, Evans, and Kawahigashi Reference 12 showed that these multiplicities are given exactly by the diagonal part of the modular invariant: 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 is just the -- classification of Cappelli, Itzykson, and Zuber Reference 16 with the - system yielding the associated (unextended) Coxeter–Dynkin graph. Since there is an -- classification of matrices of norm less than , we can recover independently of Cappelli, Itzykson, and Zuber Reference 16 that there are unique modular invariants corresponding to the three exceptional graphs.

If we use only positive -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 of representations of a finite group , all module categories are of the form for the projective representations with -cocycle for a subgroup Reference 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 Reference 98. However, Carpi, Gaudio, Giorgetti, and Hillier Reference 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 , the module categories or quantum subgroups are labeled with all the Coxeter–Dynkin diagrams , , and . Here there is a coincidence with the affine -- classification of finite subgroups of . Di Francesco and Zuber Reference 27 were motivated to try to relate modular invariants with subgroups of . There is a partial match, but this is not helpful. In general whilst the number of finite subgroups of grows with , the number of exceptional modular invariants, beyond the obvious infinite series, does not.

If we have a net of subfactors with being a completely rational conformal net, then the restriction of the vacuum representation of to gives a local -system in the sense of Longo Reference 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 -system is a triple consisting of an object and two intertwiners. Roughly speaking, the object gives as an - bimodule and the intertwiners give the multiplicative structure on . Our general theory of -induction shows that the corresponding modular invariant for the modular tensor category of representations of 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 -system. It is known that each object has only finitely many local -system structures, and we thus have only finitely many local -systems, which means that we have only finitely many possibilities for extensions for a given .

For some concrete examples of , we can classify all possible extensions. In the case of the nets, such extensions were studied in the context of -induction in Reference 11 by Böckenhauer, Evans, and Kawahigashi, and it was shown in Reference 82 by Kawahigashi-Longo that they exhaust all possible extensions. (A similar classification based on quantum groups was first given in Reference 86.) They correspond to the Coxeter–Dynkin diagrams , , , and . The cases are the nets themselves, the cases are given by simple current extensions of order 2, and the and cases are given by conformal embeddings and