Chirality for operator algebras

Adrian Ocneanu^{∗} Department of Mathematics, Pennsylvania State University University Park, PA 16802, U.S.A.

Abstract

For a finite system of bimodules over II_{1} factors, we define a notion of “nondegenerate braided system of bimodules”. If a system of bimodules is braided and nondegenerate, then the resulting 3-dimensional topological quantum field theory (TQFT) based on triangulation splits as a tensor product of another TQFT based on vertical surgery and its complex conjugate. This is a generalization of a theorem of Turaev that the Turaev-Viro TQFT is a tensor product of the Reshetikhin-Turaev TQFT and its complex conjugate.

1 Introduction

We first explain a general relation between operator algebras and topological quantum field theory. See [4] for more details.

We start with a subfactor N ⊂ M with finite Jones index and finite depth. We use the standard bimodule h = _{N}L^{2}(M)_{M} to make irreducible decompositions of finite tensor products ⋯ ⊗_{N} h ⊗_{M} h̄ ⊗_{N} h ⊗_{M} h̄ ⊗_{N} ⋯, where h̄ is the contragredient bimodule _{M}L^{2}(M)_{N}. The finite depth condition means that we have only finitely many equivalence classes of (four kinds of) bimodules in the decompositions. We denote this system of bimodules by M. Each bimodule x ∈ M has the corresponding Jones index [x]. The global index [M] is defined by ∑_{_{N} x_{N}} [x], which is also equal to ∑_{_{M} x_{M}} [x]. The tensor product operation makes the system M a finite tensor category. With intertwiners, we can define quantum 6j-symbols satisfying unitarity, tetrahedral symmetry, and the pentagon relation as in [4]. This gives another (equivalent) formulation of paragroups introduced in [2]. Then the general method of [8] gives a 3-dimensional TQFT based on triangulation. Roughly speaking, each oriented tetrahedron with bimodules on edges and intertwiners on faces has a value given by the composition of the four intertwiners on the faces (with a suitable normalization). Then we make a state sum, or a partition function, over all the configurations of bimodules and intertwiners for a fixed triangulation. We can prove that this number does not depend on the triangulation. We further study this TQFT with an operator algebraic technique.

∗Recorded by Yasuyuki Kawahigashi (University of Tokyo)

First we look at the asymptotic inclusion M∨(M′∩M_{∞}) ⊂ M_{∞}. (Note that for the subfactor of type A_{n}, the asymptotic inclusion is given by ⟨e_{j}⟩_{j≠0} ⊂ ⟨e_{j}⟩_{j∈Z}, where the projections {e_{j}}_{j∈Z} satisfy the Jones relation for 4 cos^{2} π/(n + 1).) This construction of the asymptotic inclusion can be regarded as an analogue of the “quantum double” construction of Drinfeld for paragroups. The index [M_{∞} : M ∨ (M′ ∩ M_{∞})] is given by the global index [M] of the original inclusion N ⊂ M. The principal graph of the subfactor M ∨(M′ ∩M_{∞}) ⊂ M_{∞} is given by the fusion graph of the original subfactor N ⊂ M as follows. First we take the (finite) system of M-M bimodules arising from the initial inclusion N ⊂ M. Then we make the graph G_{0} so that its even vertices are labeled by pairs (x, y) of M-M bimodules, its odd vertices are labeled by M-M bimodules z, and the number of edges from (x, y) to z is the multiplicity of z in x ⊗_{M} y. We call this graph the fusion graph. The principal graph of the asymptotic inclusion M ∨(M′ ∩M_{∞}) ⊂ M_{∞} is given by the connected component G of the fusion graph containing the even vertex (∗, ∗). (See [3, §III. 1].)

Next we study the system M_{∞} of the M_{∞}-M_{∞} bimodules arising from the subfactor M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞}. We call this system the asymptotic system. In the TQFT, the Hilbert space associated to the torus S^{1} × S^{1} has a special meaning. The system M_{∞} of the M_{∞}-M_{∞} bimodules gives a natural basis of this Hilbert space H_{S^{1}×S^{1}}, if the fusion graph is connected. To see this, we introduce the tube algebra as follows [5].

Now all the bimodules x, x′, y, y′, a, a′, b are N-N bimodules in the system M. We define a finite dimensional C^{∗}-algebra Tube M

Tube M = ⊕_{x,y,a}Hom(x ⊗ a, a ⊗ y),

as in Figure 5 (a), with the product defined by

ξ · ξ′ = δ_{y,x′} ∑_{b,α}(α ⊗ 1_{y′}) · (1_{a} ⊗ ξ′) · (ξ ⊗ 1_{a′}) · (1_{x} ⊗ α^{∗})

where ξ ∈ Hom(x ⊗ a, a ⊗ y), ξ′ ∈ Hom(x′ ⊗ a′, a′ ⊗ y′) and α is in a basis of Hom(a ⊗ a′, b). (See Figure 5 (b).)

The Hilbert space H_{S^{1}×S^{1}} in the TQFT has a commutative multiplicative structure, and the minimal projections in the center of the tube algebra naturally give a basis of this Hilbert space. Denote the label set for the center of the tube algebra by T and call it the tube system. Furthermore this set T is naturally identified with the set of the irreducible M_{∞}-M_{∞} bimodules arising from the asymptotic inclusion, if the fusion graph is connected.

The group SL(2, Z) has a natural unitary representation ρ on the Hilbert space H_{S^{1}×S^{1}} and we define

S = ρ (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )) , T = ρ (( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )) .

Then S diagonalizes the fusion algebra of the M_{∞}-M_{∞} bimodules and T is diagonal. This is an analogue of the Verlinde identity [9]. In particular, the fusion algebra of M_{∞}-M_{∞} bimodules is commutative, which is not trivial at all. We can actually construct all the data satisfying the combinatorial axioms for RCFT in the sense of Moore and Seiberg [1].

We further have a formula for the dimension of the Hilbert space H_{S} associated to a surface S in the case where the fusion graph is connected. We denote the global index of this system ∑_{X∈M_{∞}}[X] by [M_{∞}], and define [[X]]_{∞} = [X]^{−1/4}[M_{∞}]^{1/4} for each M_{∞}-M_{∞} bimodule X. Then the dimension of the Hilbert space H_{S} is given by ∑_{X}[[X]]_{∞}^{−2χ(S)}, where χ(S) is the Euler characteristic of the surface S. This number is also the topological invariant associated to the manifold S × S^{1}. This is also an analogue of the Verlinde formula. Note that if S is the 2-dimensional sphere S^{2}, we get 1 as the dimension, and that if S is the 2-dimensional torus S^{1} × S^{1}, we get the number of the M_{∞}-M_{∞} bimodules as the dimension.

Example 1.1 Start with a subfactor N ⊂ M of type A_{4}. In this case, the principal graph of the asymptotic inclusion is D_{6} as in [3, §III. 1], so the dual principal graph is also D_{6} and its even vertices give the M_{∞}-M_{∞} bimodules. The Perron-Frobenius weight of the even vertices of D_{6} is given by 1, φ^{2}, φ, φ, where we have φ^{2} = φ + 1. Then the global index [M_{∞}] is 1 + φ^{4} + φ^{2} + φ^{2} = 5φ^{2}, and the above formula give the dimension 5φ^{2}(1 + φ^{−4} + φ^{−2} + φ^{−2}) = 25, if χ = −2, for example.

If we start from a subfactor of type A_{n}, n ≥ 4, then the labeling of the M_{∞}- M_{∞} bimodules is given by the following. We take all the pairs (a, b) of integers a, b ∈ {1, 2,…, n} with a ≡ b mod 2 and define an equivalence relation (a, b) ∼ (ā, b̄), where ā = n+1−a. If n is even, the M_{∞}-M_{∞} bimodules are labeled by the equivalence classes of these pairs, and in particular, the number of the M_{∞}-M_{∞} bimodules is n^{2}/4. If n is odd, we take all the equivalence classes of these pairs and split the class ((n + 1)/2, (n + 1)/2) into two, ((n + 1)/2, (n + 1)/2)_{+} and ((n + 1)/2, (n + 1)/2)_{−}. Then the M_{∞}-M_{∞} bimodules are labeled by these, and in particular, the number of the M_{∞}-M_{∞} bimodules is (n^{2} + 7)/4. These are the dimensions of the Hilbert space H_{S^{1}×S^{1}} associated to the two dimensional torus S^{1} × S^{1}.

2 Braided systems of bimodules

We start with a finite system M of N-N bimodules, e.g., arising from a subfactor with finite depth. (Here we consider only one kind of bimodule.) The tensor product operation of the N-N bimodules is not commutative in general, but we assume that it is commutative. Even under such an assumption, we have no distinguished homomorphism (intertwiner) ξ ∈ Hom(a ⊗ b, b ⊗ a) for two N-N bimodules a, b in the system in general. We define a notion of “braiding” as a possibility of systematic choices of intertwiners from a ⊗ b to b ⊗ a.

Definition 2.1 We say that the system M is braided if for any pair a, b of the N-N bimodules in M, there is a system ε of non-zero intertwiners ε_{ab} = ξ ∈ Hom(a ⊗ b, b ⊗ a) denoted by the following picture

[figure: b; b; a; a]

with the following three axioms.

First, we define actions of the two matrices

S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , Γ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

as follows. For an intertwiner ξ : a⊗b → b⊗a denoted by the above picture, we apply the Frobenius reciprocity map [4] twice to get the intertwiner ^{a}ξ^{a} : b ⊗ ā → ā ⊗ b. This is the image of ξ with the action of S, and denoted by the following picture.

[figure: ā; ā; b; b]

Graphically, this action is regarded as a counterclockwise 90 degree rotation. Note that the action of S is a linear unitary. Next we define an action of Γ on ξ by \overline{^{a}ξ^{a}} : a ⊗ b̄ → b̄ ⊗ a. Graphically, this is regarded as a reflection with respect to a vertical axis, and denoted by the following picture.

[figure: b̄; b̄; a; a]

Note that this action is a conjugate linear unitary. The first axiom for ε states that the system ε is invariant under the action S^{2}. The second axiom states that the system ε is invariant under the action SΓ.

The third axiom, the crossing-fusion axiom, is stated graphically as follows.

[figure: b; b; b; a; a; a; a; a; c; d; c; d; c; d; =]

Here the following picture means an intertwiner (labeled by •) from a ⊗ b to c.

[figure: c; a; b]

The choice ε is called a crossing.

In the picture of the crossing-fusion axiom, the intertwiners denoted by • on the both hand sides are equal. The both pictures denote intertwiners from a ⊗ c ⊗ d to b ⊗ a, and we require these two intertwiners are equal.

In the picture of the crossing-fusion axiom, we can take the “tail” of the “Y” shape to be the identity bimodule, we get the following picture.

[figure: a; b; a; b; b; a; a; b; =]

In the picture, the left picture denotes an intertwiner from a ⊗ b to a ⊗ b. The right picture means the identity intertwiner on a ⊗ b. This is regarded as invariance under Reidemeister move II.

The fact that a ⊗ b is decomposed into a direct sum of N-N bimodules c is represented by the following picture,

[figure: b; b; b; b; a; a; a; a; c; = ∑_{c,•}]

and this implies the Yang-Baxter equation together with the crossing-fusion axiom as in the following picture.

[figure: c; c; c; c; c; c; b; a; b; a; a; b; a; b; a; b; b; a; =]

This is invariance under Reidemeister move III.

Reidemeister move I simplifies, for a given label a, to a scalar multiple of the identity τ(a)1_{a}.

[figure: a; a; a; a]

By using Reidemeister II move on the sphere S^{2}, one shows that τ(a)\overline{τ(a)} = 1.

In the crossing-fusion axiom, it is enough to give the crossing coefficients for generators of the fusion algebra, because of the fusion axiom. This is a general form of relations like the one satisfied by the Jones polynomial.

[figure: a; a; a; a; a; a; a; a; a; a; a; a; = q; − q^{−1}]

where the coefficient q satisfies q^{n} = 1.

Definition 2.2 For a crossing ε, we define the opposite, or conjugate crossing by ε̄ = S(ε).

On the commutative fusion algebra given by M, we define a character as follows.

Definition 2.3 For a braided system of bimodules, we define a character χ_{x}(a) ∈ C for N-N bimodules a, x by the following picture, up to a normalization constant. This is the Hopf link.

[figure: a; x]

Note that this picture is regarded as an assignment of bimodules/intertwiners to the boundary of a 3-ball, then the partition function scheme with 6j-symbols produce the number, which is the meaning of the above picture. (We assign a dotted square to each of the two crossing of the above knot. Then two squares glued together make a 2-sphere and intertwiners are assigned on it.) This is also equal to a weighted trace value of an intertwiner from a ⊗ x to itself.

This character is an invariant of the colored Hopf link, and gives an analogue of the S-matrix. We can prove the following.

Proposition 2.4 For three N-N bimodules x, a, b, we have

χ_{x}(a)χ_{x}(b) = ∑_{c} N_{ab}^{c}χ_{x}(c).

Definition 2.5 If a braided system of bimodules satisfies the following identity, then the system is called nondegenerate.

[figure: a; x; ∑_{x}; [x]^{1/2}[M]^{−1} = δ_{a,1}]

With the above, we make the following definition of a killing ring.

Definition 2.6 We define an element in the fusion algebra by the following formula, and call it a killing ring.

[figure: = ∑_{x}; [x]^{1/2}[M]^{−1}; x]

The killing ring satisfies the slide move, as in Figure 5 (c)–(f). The wire a in (c) expands as in (d) to the symmetric expression in (e) which further yields (f). The normalization coefficients have been omitted. This property was introduced (for SU(2)_{q}) by L. Kauffman.

With the above definition, the condition “nondegenerate” can also be stated as follows.

[figure: a; =; {; 0, a ≠ 1.; a = 1.; ·; ·; ·; ·; ·; ·]

In the right hand side of the formula, the dotted vertical line means the identity bimodule _{N}N_{N}.

Here is an example of the character.

Example 2.7 Take a subfactor of type A_{n}. We label even vertices of the graph A_{n} by 1, 3, 5 . . .. Then the character pairing (j, k) is given by sin(jkπ/(n + 1)) up to normalization constant. We list the tables for A_{4} and A_{5} (up to normalization constants).

\begin{matrix}j \ k & 1 & 3 \\ 1 & 1 & −φ \\ 3 & φ & −1 \end{matrix}  \begin{matrix}j \ k & 1 & 3 & 5 \\ 1 & 1 & 2 & 1 \\ 3 & 2 & −2 & 2 \\ 5 & 1 & 2 & 1 \end{matrix}

These show that the system A_{4} is nondegenerate, but A_{5} is degenerate.

The element a satisfying the following identity is called degenerate. The degenerate elements give a sub-fusion algebra.

[figure: =; a; a]

The operator

[figure: a; ↦; a]

is the projection onto the degenerates.

We have the following theorem for nondegenerate braided system of bimodules.

Theorem 2.8 Suppose that a braided system M of bimodules is nondegenerate. Then the center of the corresponding tube algebra is labeled by pairs (a, b) of N-N bimodules a and b.

Graphically, each label is denoted by a link of three components drawn on the surface of the two-dimensional torus; one is a killing ring set as a meridian, and the other two are set longitudinally and labeled by a, b respectively. The point is that the crossing of a and the killing ring is the opposite of the crossing of b and the killing ring. These label the Hilbert space of the two-dimensional torus of the TQFT.

The crossing ε_{a,b} between wires a, b ∈ M lives in the Hilbert space of the square with matching opposite edges, i.e., the space of the torus with one point. In general, ε_{a,b} is not in the center of the tube algebra. However the sum ε_{∗}(a) = [M]^{−1} ∑_{b} ε_{a,b} is central, and thus lives in the intrinsic space of the torus with no points. We represent ε_{∗}(a) by a longitudinal circle labeled a lying on top of the killing ring arranged as a a meridian. For a ∈ M, denote by p_{a} the degenerate tube with two opposite edges labeled a and the other edges labeled 1 given by 1 ∈ Hom(a ⊗ 1, 1 ⊗ a). Then p_{a} is a projection in the tube algebra.

Theorem 2.9 Given a crossing ε on M, for every a ∈ M, the element ε_{∗}(a) is a minimal projection in the tube algebra Tube M of M, which is central and satisfies ε_{∗}(a) ≤ p_{a}.

The map ε_{∗} : M → T gives an embedding of (M, ε) (with fusion and braiding) into the system T with the natural braiding. The vector ε = [M] ∑_{a∈M} ε_{∗}(a) ∈ Tube M satisfies S^{2}(ε) = SΓ(ε) = ε. The image ε_{∗}(M) is closed under the fusion and satisfies ε_{∗}(M)′ = \overline{ε_{∗}(M)} = Γ(ε_{∗}(M)).

Conversely, suppose given a map ρ : M → M with ρ(a) ≤ p_{a} for any a ∈ M such that the vector ξ_{ρ} = ∑_{a∈M} ρ(a) satisfies the properties S^{2}(ξ_{ρ}) = SΓ(ξ_{ρ}) = ξ_{ρ}. Then ρ is an embedding of the system M into T and yields a braiding ε^{ρ} on M, the preimage of the canonical braiding on T . We have ρ = (ε^{ρ})_{∗}.

Definition 2.10 We say that the labels a and b permutes if the following graphically represented equality holds.

[figure: b; b; b; b; a; a; a; a; =]

We define the relative permutant N′ ∩ M of a subset N of M as the elements of M which permute with all the elements of N.

Note that a is degenerate if it permutes with all b and that the degenerates are M′ ∩ M.

For the tube system T , we have the following bipermutant property.

Theorem 2.11 For any subsystem N of T , we have (N′ ∩ T )′ ∩ T = N.

We say that the tube system is reflexive.

Next we study extensions of systems of bimodules.

Definition 2.12 A nondegenerate extension M ⊂ P is called minimal nondegenerate if M′ ∩ P = M′ ∩ M.

We then have the following theorem.

Theorem 2.13 For any braided bimodule system, there is an essentially unique minimal nondegenerate extension M ⊂ P, also called the ghost extension. The elements in the complement of M are called the ghosts.

For example, for the case of A_{2n+1}, the system of N-N bimodules corresponds to the integer spin system. Its minimal nondegenerate extension is the embedding into the system of all the half-integer spins.

Besides the minimal nondegenerate extension,which is quite difficult to construct, there is another nondegenerate extension, the standard one. The intuitive idea of the standard extension is to take the picture of a crossing and drill tubes under each wire, thus obtaining a picture with wires but no crossings on a nontrivial surface. In the latter picture everything is determined by fusion alone. Here is an example of a standard nondegenerate extension.

Example 2.14 Let M denote the system of the M-M bimodules of the Jones subfactor N ⊂ M of index 3 with principal graph A_{5}. Let G be the symmetric group S_{3} with generators a, b of order 3, 2 respectively and irreducible representations 1, σ, ε of dimensions 1, 2, 1 respectively. The conjugacy orbits of G have representatives 1, a, b. The centralizers of 1, a, b are the groups S_{3}, Z_{3}, Z_{2} and have irreducible representations {1, σ, ε}, {1, α, ᾱ},{1, ε} respectively. (Here ε denotes the sign representation and α is a primitive third root of 1). Identify the bimodules in M with Irr G = {1, σ, ε} and the elements of T with the pairs (orbit representative, irreducible of centralizer), i.e., \{(1, 1), (1, σ), (1, ε), (a, 1), (a, α), (a, \overline{α}), (b, 1), (b, ε)\}. Each element of T is selfconjugate, and Γ permutes (a, α), (a, \overline{α}) and leaves the other labels fixed. There are three different braidings of M, one selfconjugate and the other two conjugate to each other. For any braiding, the associated embedding of 1, ε ∈ M can be only (1, 1), (1, ε) respectively. The element σ can be mapped into any of (1, σ), (a, α), (a, \overline{α}), which gives the three crossings. The first map is ρ_{0} : Irr G ∋ π ↦ (1, π). This corresponds to the canonical braiding on Irr G which is completely degenerate, i.e., the canonical homomorphism α ⊗ β → β ⊗ α for α, β ∈ Irr G. The other two maps ρ_{+}, ρ_{−} with images \{(1, 1), (a, α), (1, ε)\}, \{(1, 1), (a, \overline{α}), (1, ε)\} respectively correspond to the identification of M with the integer spin representations of SU(2)_{q}, q = exp(πi/6). Since ρ_{+}(M) and ρ_{−}(M) are the permutant of each other, the degenerate part of each is their intersection {(1, 1), (1, ε)}.

A braided bimodule system in which any two wires permute is called a completely degenerate system. An example is given by the natural permutation of a pair of representations of a finite (or compact) group. Using quantum deformations, this degenerate braiding is deformed to nondegenerate braidings. For completely degenerate systems, the minimal extension is easy to construct.

Theorem 2.15 If M is completely degenerate, then the minimal extension coincides with the standard extension T .

Let M be a braided bimodule system with degenerate subsystem D = M′ ∩ M and minimal extension P.

Consider the finite dimensional Hilbert space K consisting of vectors ξ ∈ ⨁_{x,y∈P,m∈M} Hom(x ⊗ y, m). The space A consisting of vectors

ξ ∈ ⨁_{x_{1},y_{1},x_{2},y_{2}∈P} Hom(x_{1} ⊗ x_{2}, y_{1} ⊗ y_{2})
= ⨁_{x_{1},y_{1},x_{2},y_{2}∈P} ⨁_{p∈P} Hom(x_{1} ⊗ x_{2}, p) ⊗_{C} Hom(p, y_{1} ⊗ y_{2}),

for which

ξ ∈ ⨁_{x_{1},y_{1},x_{2},y_{2}∈P} ⨁_{m∈M} Hom(x_{1} ⊗ x_{2}, m) ⊗_{C} Hom(m, y_{1} ⊗ y_{2}) = K ⊗ \overline{K},

as in Figure 5 (g), and

Sξ = ^{y_{1}}ξ^{x_{2}} ∈ ⨁_{x_{1},y_{1},x_{2},y_{2}∈P} ⨁_{d∈D} Hom(\overline{y_{1}} ⊗ x_{1}, d) ⊗_{C} Hom(d, y_{2} ⊗ \overline{x_{2}}),

as in Figure 5 (h). Then A ⊂ K ⊗ \overline{K} = End(K) is closed under composition of homomorphisms, and has a finite dimensional C^{∗}-algebra structure. The subalgebra End(K) is obtained from the algebra ⨁_{x_{1},y_{1},x_{2},y_{2}∈P} Hom(x_{1} ⊗x_{2}, y_{1} ⊗y_{2}) by reduction with the central projection π in Figure 5 (i), where the dotted killing ring is labeled by the degenerate elements D; it is here that we use the minimality property D′∩P = M. The conditional expectation E_{A} : End(K) → A is shown in Figure 5 (j), where the dashed killing ring is labeled by the observable elements M, and we use the fact that M′ ∩ P = D.

Theorem 2.16 The tube algebra Tube M is isomorphic to the relative commutant A′ ∩ End(K). The tube system T is labeled by the minimal projections in the center of the algebra A.

The algebra A′ ∩ End(K) = \overline{K} ⊗_{A} K is contained into the subset

⨁_{m_{1},m_{2}∈M} ⨁_{x,y∈P} Hom(m_{1}, x ⊗ y) ⊗_{C} Hom(x ⊗ y, m_{2}) ⊂ \overline{K} ⊗ K

and map into Tube M as in the Figure 5 (l)–(m). The killing ring marked by the dashed line is the killing ring of the system M. The Figure 5 (l)–(m) shows that this map is well defined on \overline{K} ⊗_{A} K. Note that in spite of the presence of elements of the minimal extension P, by composing the elements in the tube one obtains an element of Tube M, i.e., the “ghosts” in P are not observable. This motivates our terminology by analogy with the Faddeev-Popov ghosts in which non-observable half integer spin elements are introduced to remove the degeneracy of integer spin elements.

A tube (Figure 6 (a), with x, y, a, b ∈ M) is written in terms of the above basis with coefficients as in the Figure 6 (b), where t_{+}, t_{−} ∈ P, ξ, η ∈ K and the dashed killing ring is labeled by M. The proof of this crucial observation transforms (a) into the intermediate step (c), in which the dashed killing ring (labeled by M) is killed by the dotted killing ring labeled by P. The latter expands as in (d), and a slide move brings it to the form (e), from which (b) follows. Note that the sum in (b) is always in the image of \overline{K} ⊗_{A} K.

The Figure 6 (f) shows that the product of tubes corresponds to the product in \overline{K} ⊗_{A} K,

(\overline{ξ_{1}} ⊗_{A} η_{1}) · (\overline{ξ_{2}} ⊗_{A} η_{2}) = \overline{ξ_{1}} ⊗_{A} ⟨η_{1}, ξ_{2}⟩_{A} · η_{2} = \overline{ξ_{1}} ⊗_{A} E_{A}(η_{1} ⊗ \overline{ξ_{2}}) · η_{2}.

The original system M embeds via the map ε_{∗} into the tube algebra as in the Figure 6 (g). The product in (h) between a tube and ε_{∗}(u) is equal, as in (i), (j) to a scalar multiple of ε_{∗}(u). Thus ε_{∗}(u) is a minimal projection which is central. The general intertwiner in Hom(ε_{∗}(a) ⊗ ε_{∗}(b), ε_{∗}(c)), shown in (k) transforms, using the slide moves in (l), into an element of ε_{∗}(Hom(a ⊗ b, c)) in (m), thus showing that the homomorphism spaces of M and ε_{∗}(M) are naturally isomorphic.

Consider the asymptotic system M_{∞} arising from a subfactor with finite depth. The bimodules (labels of edges) of the asymptotic system are naturally associated to labels of circles and to minimal central projections in the tube algebra in the TQFT of the initial system. The triangles in the asymptotic systems correspond to trinion (trouser) surfaces in the initial theory. The natural braiding is then obtained as follows.

Take a ball in R^{3} and remove two disjoint cylindrically holes through it, one above the other (i.e., make a solid picture of the crossing into a ball). The surface of of this body is made of two tubes together with a four-holed sphere (which decomposes into two trinions). On the tubes, put vectors corresponding to two asymptotic labels. The TQFT gives then a vector in the space of the four-holed sphere. This is a 3dimensional thickening of the right hand side of the equality decomposing a crossing into fusions, after Definition 2.1. Take the boundary of a small tubular neighborhood of each member of this equality. Remark that the crossing tubes in the left member can be put inside inflated letter “H” in the right member, and then the boundaries of the two members can be joined on four circles, to give a 3-dimensional manifold with boundary. The invariant of this manifold gives the coefficient of the crossing. In this way, we get the following theorem.

Theorem 2.17 The asymptotic system has a natural braiding. This braiding is nondegenerate if and only if the fusion graph is connected. The labels of the minimal nondegenerate extension of this system are the minimal projections in the center of the tube algebra.

3 Chirality and TQFT

Given a link L in S^{3}, choose a projection L_{0} on S^{2} for which the writhe is 0 (i.e., the canonical framing of L coincides with the vertical framing — also called blackboard framing — of L_{0}; this can be done from any projection of L by adding twists on each component). Choose arbitrary labels for each link component. Then we have an assignment of a dotted square at each crossing, which means we have intertwiners at squares that make a 2-sphere. (Note that the first two axioms in the definition of the braiding system imply that this assignment of intertwiners is well-defined.) This is regarded as a labeling of the boundary of a 3-ball, thus the state sum gives a complex number associated to L_{0}. This is an invariant of L. For SU(2)_{k} and the spin 1/2 label, this coincides with the value of the Jones polynomial at roots of unity.

Take now an arbitrary link projection and sum the previous invariants, by replacing each link component with the killing ring. Then we obtain an invariant of the 3-manifold given by Dehn surgery with vertical framing on L. The crossing-fusion axiom implies invariance under handle slides, thus our invariant of framed links is invariant under Kirby moves and we get an invariant of compact oriented 3-manifolds. This is a generalization of the Reshetikhin-Turaev TQFT [6], which is essentially a weighted sum of invariants of a framed link over all the colorings for U_{q}(sl_{2}). Thus we have two methods to construct a TQFT from a system M. One is based on triangulation and the quantum 6j-symbols for M [4] and the other is based on vertical surgery as above. The relation between the two is given as follows.

Theorem 3.1 Suppose that the system M of bimodules is braided and nondegenerate. Then the TQFT based on triangulation splits as a tensor product of another TQFT based on vertical surgery and its complex conjugate.

This generalizes a result of Turaev [7], which is based on his theory of shadows and 4-dimensional manifolds.

We give a sketch of our proof of the above theorem. We use Figures 1 – 3 at the end of this paper. We start with a 3-manifold given by a surgery with a link with vertical framing, and compute the invariant based on triangulation first.

In Figure 1, the first picture represents the fact that the manifold obtained by vertical surgery on a link is obtained by joining wedges (called thin wedges, in which the two tubes are contracted to circles) on the adjacent faces; this is a purely topological observation.

In Figure 1, the dashed circles (killing rings) and the 4 wires crossing the edges of the wedge (all pictured thick black) are in the plane of the surface of the wedge. The thick darker wires (one in each hole) are slightly above the killing ring and the thick lighter wires are slightly below the killing ring; both go through the hole of the wedge. The same convention applies in the other pictures. The black wire is in the plane, dark slightly above and light slightly below.

The first one describes a basis of the Hilbert space of the surface of the wedge; the drawings are links with crossings on the surface of the wedge. Compare this situation with that in the remark after Theorem 2.8. (Only the thick wires matter on the wedge; the others merely show its shape.)

The fact used next is that the invariant for a picture on the surface of a ball B^{3} is obtained by composing the respective homomorphisms. For a sphere with handles, if there is a single wire labeled a on a handle, then the result is 0 if a ≠ 1, and if a = 1 then one removes the wire a and cuts the handle. (This expresses the fact that the handle is full).

The slide move is applied to the bottom killing ring, after which there are only killing rings left on the two handles. These are cut as above and the picture remaining on the sphere is simplified to obtain the final picture of Figure 1.

That is, we find a basis in the Hilbert space of an annulus with two entries and exits. The surface of the wedge is made of two such annuli. Take arbitrary labels for the thick wires and intertwiners for the eight triple points. The partition function of the wedge applied to these will be the evaluation of the last picture in Figure 1.

The we have two links with opposite crossings which slide away, and we are done. The pictures in Figure 3 describes a proof of the equivalence between a spin model,

in which there are numbers at each crossing having matching faces (the blobs are arbitrary) and a wire model. There are “bridge” moves and “close” moves, done until all faces of the knot projection are exhausted (a leftover ring cancels a normalization coefficient). The regions of the complement of the link projection must be ordered so that the union of the first k be simply connected for each k. In particular, this gives an easy proof of the equivalence between two models of Reshetikhin for SU(2)_{q}, first proved in a quite complicated way by Viro. If blobs are replaced by our computation of the wedge, one can finish the proof as in Figure 1. The point is that this proof extends easily to the degenerate case.

In summary, the pair of triple points on each face of the wedge must be matched to a pair of triple points on a neighboring wedge, and then summed. This is done in Figure 1 using the theorem proved in Figure 3. The result is two separate copies of the link. This all works in the nondegenerate case. In the degenerate case, the “over” and “under” worlds are connected by degenerates.

Let the closed 3-manifold V be obtained by vertical surgery on a link L with components L_{i}, i ∈ I. Let M be a braided system of bimodules with degenerate subsystem D and minimal nondegenerate extension P. Let λ denote a labeling of the components of L by elements of P, with L_{i} labeled by p_{i} ∈ P, and denote by τ_{λ}(L) the corresponding knot invariant of L. Let P^{I} denote the set of labelings λ above. For p, q ∈ P let

[p, q]_{M} = [M]^{−1}[P]^{−1} ∑_{m∈M} N_{p,\overline{q}}^{m}[m]^{1/2}.

Here the fusion number N_{p,\overline{q}}^{m} denotes the multiplicity of m in p ⊗ \overline{q}. Since m ∈ M induces the trivial character on the degenerate algebra D, the only pairs p, q ∈ P with [p, q]_{M} ≠ 0 are the pairs for which p and q induce the same degenerate character.

A modification of the argument described in the Figures 1–3 proves the following result.

Theorem 3.2 The triangulation invariant Z^{(M)}(V) satisfies

Z^{(M)}(V) = ∑_{λ,μ∈P^{I}} ∏_{i∈I}[λ_{i}, μ_{i}]_{M}τ_{λ}(L)\overline{τ_{μ}(L)}.

A sketch of the proof of this result is the following. In the Figure 2 (e), one obtains a copy of the projection π in Figure 5 (i) for every link segment between two crossings of the original link. Up to this point the computations were done in M; we switch now to the system P. Since we are computing the invariant of the ball B^{3} with marked surface, the invariant is simply multiplied in this process by a normalization factor. We now go backwards to (c), to find that we are computing the invariant of the link (c) with each tube component marked by the projection π. The inner product of π with a minimal central projection of Tube (P) labeled with projections p, q ∈ P (recall that P is nondegenerate) is, up to a factor, precisely [p, q]_{M} described above.

In particular, this result shows that although the subfactor of type E_{6} generates an abelian bimodule system, there exists no braiding on it, since Niţică and Török have shown that the invariant of the lens space L(3, 1) is not real.

At the end, we explain two examples of nondegenerate braided systems of bimodules.

From a finite depth subfactor, we pass to the asymptotic inclusion M ∨ (M′ ∩ M_{∞}) ⊂ M_{∞}. Then the system of M_{∞}-M_{∞} bimodules is braided and nondegenerate as mentioned above, if the fusion graph is connected. The above theorem implies splitting of the corresponding TQFT, but it is easy to see the splitting directly. Instead of the M_{∞}-M_{∞} bimodules, we can use the M ∨ (M′ ∩ M_{∞})-M ∨ (M′ ∩ M_{∞}) bimodules. Then from the description of the corresponding quantum 6j-symbols, the TQFT splits as a tensor product of a TQFT based on the triangulation for the original inclusion N ⊂ M with the system of the M-M bimodules and its complex conjugate.

Consider the subfactors of type A_{n}. The even vertices of the principal graph correspond to N-N bimodules and odd vertices to N-M bimodules. So it is impossible to define multiplication among the odd vertices in general, but in the case of A_{n} subfactors, we can make a fusion algebra in which all the multiplications are possible regardless parity of the vertices by using endomorphisms. In this way, we can construct a TQFT based on triangulation from the A_{n} subfactors, and then this is exactly same as the one considered by Turaev-Viro [8] with q = exp(πi/(n+1)). This system is braided and nondegenerate, and the above theorem gives another proof of a theorem of Turaev [7] that the Turaev-Viro TQFT [8] is a tensor product of the Reshetikhin-Turaev TQFT [6] and its complex conjugate.

References

[1] G. Moore & N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 177–254, (1989).

[2] A. Ocneanu, Quantized group, string algebras and Galois theory for algebras, in “Operator algebras and applications, Vol. 2 (Warwick, 1987),” London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, pp. 119–172, (1988).

[3] A. Ocneanu, “Quantum symmetry, differential geometry of finite graphs and classification of subfactors”, University of Tokyo Seminary Notes 45, (Notes recorded by Y. Kawahigashi), (1991).

[4] A. Ocneanu, An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, unpublished announcement, (1991).

[5] A. Ocneanu, Lectures at Collège de France, Fall 1991.

[6] N. Reshetikhin & V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 547–598, (1991).

[7] V. G. Turaev, Topology of shadows, preprint (1991).

[8] V. G. Turaev & O. Y. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology, 31 865–902, (1992).

[9] E. Verlinde, Fusion rules and modular transformation in 2D conformal field theory, Nucl. Phys. B300 360–376, (1988).

Acknowledgement by Y. K.

This article is based on the talks of Professor A. Ocneanu given in the Taniguchi Symposium and the RIMS meeting on operator algebras in Japan in July, 1993. More details have been supplied by him in correspondences to the recorder, and the figures in the following six pages were made by him. The recorder thanks him for his kind help. The recorder also thanks Professor D. E. Evans for helpful comments during the preparation of this article.