Exotic subfactors of finite depth with Jones indices (5 + 13)/2 and (5 + 17)/2

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1 arxiv:math/98344v5 [math.oa] 7 Jun 999 Exotic subfactors of finite depth with Jones indices (5 + 3/ and (5 + 7/ M. Asaeda Graduate School of Mathematical Sciences University of Tokyo U. Haagerup Institut for Matematik og Datalogi Odense Universitet Abstract We prove existence of subfactors of finite depth of the hyperfinite II factor with indices (5 + 3/ 4.3 and (5 + 7/ The existence of the former was announced by the second named author in 993 and that of the latter has been conjectured since then. These are the only known subfactors with finite depth which do not arise from classical groups quantum groups or rational conformal field theory. Introduction In the theory of operator algebras subfactor theory has been developing dynamically involving various fields in mathematics and mathematical physics since its foundation by V. F. R. Jones in 983 [J]. Above all the classification of subfactors is one of the most important topics in the theory. In the celebrated Jones index theory [J] Jones introduced the Jones index for subfactors of type II as an invariant. Later he also introduced a principal graph and a dual principal graph as finer invariants of subfactors. Since Jones proved in the middle of 98 s that subfactors with index less than 4 have one of the Dynkin diagrams as their (dual principal graphs the classification of the hyperfinite II subfactors has been studied by A. Ocneanu and S. Popa and also by M. Izumi Y. Kawahigashi and a number of other mathematicians. In this process Ocneanu s paragroup theory [O] has been quite effective. He penetrated the algebraic or rather combinatorial nature of subfactors and constructed a paragroup from a subfactor of type II. A paragroup is a set of data consisting of four graphs made of (dual principal graph and assignment of complex numbers to cells arising from four graphs called a connection. Thanks to the generating property for subfactors of finite depth proved by Popa in [P] it has turned out that the correspondence between paragroups and subfactors of the hyperfinite II factor 99 Mathematics Subject Classification 46L37 8T5. Komaba Meguro-ku Tokyo JAPAN asaeda@ms.u-tokyo.ac.jp Campusvej 55 DK-53 Odense M DENMARK haagerup@imada.ou.dk

2 Introduction with finite index and finite depth is bijective therefore the classification of hyperfinite II subfactors with finite index and finite depth is reduced to that of paragroups. By checking the flatness condition for the connections on the Dynkin diagrams Ocneanu has announced in [O] that subfactors with index less than 4 are completely classified by the Dynkin diagrams A n D n E 6 and E 8. (See also [BN] [I] [I] [K] [SV]. After that Popa ([P] extended the correspondence between paragroups and subfactors of the hyperfinite II factor to the strongly amenable case and gave a classification of subfactors with indices equal to 4. (In all the above mentioned cases the dual principal graph of a subfactor is the same as the principal graph. See also [IK]. We refer readers to [EK] [GHJ] for algebraic aspects of a general theory of subfactors. The second named author then tried to find subfactors with index a little bit beyond 4. Some subfactors with index larger than 4 had been already constructed from other mathematical objects. For example we can construct a subfactor from an arbitrary finite group by a crossed product with an outer action and this subfactor has an index equal to the order of the original finite group. Trivially the index is at least 5 if it is larger than 4. We also have subfactors constructed from quantum groups U q (sl(n q e πi/k with index sin (nπ/k as in [W] and these index values do not fall in sin (π/k the interval (4 5. Unfortunately or naturally these subfactors do not contain more information about the algebraic structure than the original mathematical objects themselves such as groups or quantum groups. Does there exist any subfactor not arising from (quantum groups? If it is the case we have a subfactor as a really new object producing new mathematical structures. We expect that subfactors with index slightly larger than 4 would be indeed those with exotic nature and they do not to arise from other mathematical objects. The second named author gave in 993 a list of possible candidates of graphs which might be realized as (dual principal graphs of subfactors with index in ( ( in [H]. We see four candidates of pairs of graphs including two pairs with parameters in 7 of [H]. At the same time the second named author announced a proof of existence of the subfactor with index (5 + 3/ for the case n 3 of ( in the list in [H] but the proof has not been published until now. Ever since nothing had been known for the other cases for some years until D. Bisch recently proved that a subfactor with (dual principal graph (4 in 7 of [H] does not exist [B]. About case (3 in 7 of [H] as in Figure as well as the case n 3 of ( we can easily determine a biunitary connection uniquely on the four graphs consisting of the graphs (3 and we thus have a hyperfinite II subfactor with index (5 + 7/ constructed from the connection by commuting square as in [S]. The problem is whether this subfactor has (3 as (dual principal graphs or not. This amounts to verifying the flatness condition of the connection. In 996 K. Ikeda made a numerical check of the flatness of this connection by approximate computations on a computer in [Ik] and showed that the graphs (3 are very likely to exist as (dual principal graphs. (He also made a numerical verification of the flatness for the case n 7 of ( in 7 of [H].

3 Introduction 3 In this paper we will give the proof of the existence for the case of index (5 + 3/ previously announced by the second named author and give the proof of the existence for the case of index (5 + 7/. The proof in the latter case was recently obtained by computations of the first named author based on a strategy of the second named author. Our main result in this paper is as follows. Theorem ((5 + 3/ case The two graphs in Figure are realized as a pair of (dual principal graphs of a subfactor with index equal to 5+ 3 of the hyperfinite II factor. Theorem ((5 + 7/ case The two graphs in Figure are realized as a pair of (dual principal graphs of a subfactor with index equal to 5+ 7 of the hyperfinite II factor. b σ a σ σ a b c b σ a σ σ a c 4 3 a σ a σ Figure : The case n 3 of the pair of graphs ( in the list of Haagerup In Section we will give two key lemmas to prove our two main results respectively. In Section 3 we will give a construction of generalized open string bimodules which is a generalization of Ocneanu s open string bimodules in [O] [Sa] and we will give a correspondence between bimodules and general biunitary connections on finite graphs. In Sections 4 and 5 we will prove our two main theorems respectively. The first named author acknowledges financial supports and hospitality from Odense University and University of Copenhagen during her visit to Denmark in March/April and September 997. She also acknowledges a financial support from the Honda Heizaemon memorial fellowship. She is much grateful to Y. Kawahigashi and M. Izumi for constant advice and encouragement.

4 Key lemmas to the main results 4 h g f h * a b c d e d c b ã 4 c a c 3 e 5 g 6 ã Figure : The pair of graphs (3 in Haagerup s candidates list Key lemmas to the main results In this section we will give the key lemmas which have been proved by the second named author. First of all we will explain the motivation to the lemmas. Proofs of our main theorems presented in Section are reduced to verifying flatness of the biunitary connections which exist on the four graphs made of the pairs of the graphs in Figure respectively. However it is well known that to verify flatness exactly is almost impossible except for some easy cases such as the biunitary connections arising from the subfactors of crossed products of finite groups ([EK.6]. So far in the history of classification of subfactors several methods have been introduced to prove flatness/nonflatness of biunitary connections. Finding inconsistency of the fusion rule on the graph of a given biunitary connection has been sometimes effective to prove nonflatness e.g. D n+ E 7 ([I] [SV]... On the other hand since consistency of fusion rule does never mean flatness of given biunitary connection several ideas have been introduced to prove flatness ([EK] [I] [IK] [K]... The second named author however inspected the fusion rules of the upper graphs in Figures and and noticed that if we can construct bimodules satisfying a part of the fusion rule we can conclude that there exists a subfactor having the desired principal graph. These are the lemmas which we explain in this section. Now we introduce the notation used in the lemmas. Definition Let N and M be II factors and N X M X be an N-M bimodule (see [EK]. We denote by R X (M the right action of M on X and by L X (N the left action of N. We have the subfactor R X (M L X (N. We denote its Jones index by [X]. We define the principal graph of the bimodule X as that of the subfactor

5 Key lemmas to the main results 5 R X (M L X (N. Definition For bimodules X and Y with common coefficient algebras we define X Y dim Hom(X Y. A formal Z-linear combination Y of bimodules (of finite index will be called positive if it is an actual bimodule i.e. of Y Z for any irreducible bimodule Z which appears in the direct sum decomposition of X. When Y X Z for some positive bimodule Z we write X Y. Hereafter we use the expression as follows so far as it does not cause misunderstanding. N N N N X X X XY X N Y X X N X where N is a II factor and X and Y are suitable bimodules. Lemma Let X N X M be a bimodule with finite Jones index larger than or equal to four. Then XX N and (XX 3XX N are positive N-N bimodules. XXX X and (XX N X 4XXX 3X are positive N-M bimodules. Proof Let G be the principal graph of X. We set G ( even : set of all irreducible components of N (XX n n... G ( odd : set of all irreducible components of X (XXn X n... where G even ( (resp. G ( odd means the even (resp. odd vertices of G and G (G YZ ( Y G to be the incidence matrix for G i.e. evenz G ( odd G YZ Y X Z Y G ( even. Since 4 [X] < We have G <. Put ( G G t

6 Key lemmas to the main results 6 then is the adjacency matrix of G ( G ( even G( odd and G. Let P P P...be the sequence of the polynomials given by P (x P (x x... P n+ (x P n (xx P n (x. Then by [HW] all of P ( P 3 (...have non-negative entries. For n we get in particular that GG t GG t G G (GG t 3GG t + and (GG t G 4GG t G + 3G have non-negative entries. Hence for W G ( even XX N W (GG t N W (XX 3XX N W ((GG t 3GG t + N W namely XX N and (XX 3XX N are positive N-N bimodules. The same argument (with W G ( odd shows that XXX X and (XX X 4XXX 3X are positive N-M bimodules. q.e.d.. Key lemma for the case of index (5 + 3/ In this subsection we present the key lemma given by the second named author to which the construction of the finite depth subfactor with index (5 + 3/ is reduced. Lemma Let M and N be II factors. Assume the following. We have an N-M bimodule X N X M of index (5+ 3 We have an N-N bimodule S N S N of index satisfying S 3 N and S N i.e. S is given by an automorphism of N of outer period 3 3 The six bimodules are irreducible and mutually inequivalent. 4 The four bimodules N S S XX N S(XX N S (XX N are irreducible and mutually inequivalent 5 (the most important assumption X SX S X XXX X S(XX N (XX N S. Then the principal graph of X and bimodules corresponding to the vertices on the graph are as follows;

7 Key lemmas to the main results 7 σ S N X a XX N b c b σ a σ S(XX N XXX X b σ a σ SX S (XX N S X σ S Remark By lemma all the above bimodules are well-defined (i.e. positive in the sense of Def.. Proof We have (XXX XX XX N XXX X (XX N X (by Frobenius reciprocity XXX X XXX X + XXX X X because X and XXX X are irreducible and inequivalent. Hence by irreducibility of XX N we have (i XX N (XXX XX. We have (ii (XXX XX (XX N N and 5 says (iii S(XX N (XX N S. Hence (iv Therefore and Hence by (ii S (XX N S(XX N S (XX N S 4 (XX N S. S(XX N (XX N S (XX N (XX N S S (XX N (XX N S. S(XXX XXS (XXX XXS 3 (XXX XX

8 Key lemmas to the main results 8 and similarly S (XXX XXS (XXX XX. So by (i (v S(XXX XXS (XXX XX (vi S (XXX XXS (XXX XX by (iii and (iv. Hence by (i (v (vi and 3 (XX N S(XX N S (XX N are mutually inequivalent subbimodules of (XXX X i.e. (XXX XX (XX N S(XX N S (XX N Y where Y is an N-N bimodule (possibly zero. Since X N X M is irreducible the subfactor R X (M L X (N has the trivial relative commutant hence extremal (see [P] p.76. Therefore the square root of the Jones index of a bimodule [ ] / is additive and multiplicative on the bimodules expressed in terms of X and X (see [P]. Thus we have [(XXX XX] 3 [(XX N ] + [Y ] where the index of a zero bimodule is defined to be. Hence with λ [X] we get [Y ] λ(λ 3 λ 3(λ λ 4 5λ + 3. We must finally prove that (vii S(XXX X XXX X. To see this we compute S(XXX X XXX X S(XXX X XXX X SX XXX X S(XX N (XXX XX SX XXX X The first bracket is because S(XX N is contained in (XXX XX with multiplicity and the second bracket is because SX and XXX X are irreducible and inequivalent by 4 hence S(XXX X XXX X thus the equality of the irreducible bimodules (vii holds.

9 Key lemmas to the main results 9 From all the above it follows easily that (a N G ( (b G is connected (c Multiplication by X (resp. X from the right (resp. left on any bimodule U in (resp. G( even gives a direct sum of the bimodules in G( to U by edges namely we find that G is the principal graph of X. G ( odd even. Key lemma for the case of index (5 + 7/ (resp. G( even connected q.e.d. In this subsection we present the key lemma similar to the previous one for the construction of the finite depth subfactor with index (5 + 7/. Lemma 3 Let M N be II factors. Assume the following. We have an N-M bimodule X of index We have an N-N bimodule S of index satisfying S N and S N i.e. S is given by an automorphism of N of outer period 3 The eight N-N bimodules N S XX N S(XX N (XX N S S(XX N S (XX 3XX N S((XX 3XX N are irreducible and mutually inequivalent. 4 The six N-M bimodules X SX XXX X S(XXX X (XX X 4XXX 3X (XX N SX are irreducible and mutually inequivalent. 5 (The most important assumption S(XX N SX (XX N SX. Then the principal graph of X and the bimodules corresponding to the vertices on the graph are as follows: S(XX N S (XX N S h g f h S(XX N SX (XX N SX N X a b c d e d c b ã SX S

10 Key lemmas to the main results where b XX N c XXX X d (XX 3XX N e (XX X 4XXX 3X f (XX N S(XX N S(XX N S d S((XX 3XX N c S(XXX X b S(XX N Remark By lemma we know that all the bimodules above except for the bimodule corresponding to f are well-defined (i.e. positive in the sense of Def.. The welldefinedness of the bimodule at f will come out of the proof below. Proof In the proof we will sometimes use formal computations in the Z-linear span of the N-M bimodules or N-N bimodules considered. The symbol for computing the dimension of the space of intertwiners can be extended to Z bilinear maps and Z Z implies Z also for these generalized bimodule. Since N XX we have by 5 that both (XX N S and S(XX N S are (equivalent to subbimodules of (XX N SXX. By 3 (XX N S and S(XX N S are two non-equivalent irreducible bimodules. Hence we can write (XX N SXX (XX N S S(XX N S R where R is an N-N bimodule and we have R (XX N S(XX N S(XX N S. Note that R because R would imply [(XX N ] which is impossible since X is irreducible and [X] > 4. Next we will show the following. (i The bimodule R is irreducible (ii S((XX 3XX N S (XX 3XX N (iii S((XX X 4XXX 3X (XX X 4XXX 3X. We have R R (XX N S(XX N (XX N S(XX N S(XX N S (XX N S(XX N + S(XX N S S(XX N S t + t + t 3

11 Key lemmas to the main results where t S(XX N S (XX N t S(XX N S (XX N S(XX N t 3 (XX N (XX N. Note first that t 3 because (XX N is irreducible. Next t S(XX N S (XX N SXX S(XX N S (XX N S. The last term is because S(XX N S and (XX N S are irreducible and inequivalent by 3. Hence using 4 and 5 we get t S(XX N SX (XX N SX. To compute t set irreducible bimodules Y Z as Then Y XX N Z (XX 3XX N. t S( N Y ZS N Y Z N N + SY S Y + SZS Z + N Y + N Z + SY S Z. By 3 N Y SY S and Z are irreducible and mutually inequivalent. Hence Altogether we have shown that t + SZS Z. R R ( + SZS Z + SZS Z. Since R we have R R. Moreover since Z is irreducible so is SZS. Hence SZS Z. Therefore R R SZS Z which shows that R is irreducible and using that Z and SZS are irreducible we also get that SZS Z. Hence we have verified (i and (ii. To prove (iii put G XXX X E (XX X 4XXX 3X. Note that E ((XX 3XX N X (XXX X then by (ii E S((XX 3XX N SX XXX X S(XX N SX S(XX N SX (XX N X.

12 Key lemmas to the main results Using 5 we have E S(XX N S(XX N SX (XX N SX (XX N X S(XX N SXXSX S(XX N X (XX N SX (XX N X. Hence again using 5 we get E (XX N SXXSX S(XX N X (XX N SX (XX N X (XX N SX(XSX N ( N S(XX N X. From this expression of E and 5 we clearly have SE E which proves (iii. We next prove (iv RX (XX N SX E where R (XX N S(XX N S(XX N S is irreducible by (i. We put By 5 we have E RX (XX N SX. E (XX N S(XX N X (XX N SX (XX N S(XX 3 N X. To prove (iv we just have to show that E E namely Note that E E E E. E E E E s s + s 3 where s E E s E E and s 3 E E. First s 3 because E is irreducible. Next Finally s (XX N S(XX 3 N X (XX N (XX 3 N X S (XX N (XX 3 N XX(XX 3 N (XX N S(XX N (XX 3 N X (XX N (XX 3 N X SE E (by (iii. s E E S(XX N S (XX 3 N XX(XX 3 N S( N Y ZS (XX 3 N XX. Here using SY S S(XX N S and SZS Z by (ii we have s N S(XX N S Z (XX 3 N XX X S(XX N SX ZX (XX 3 N X X (XX N SX ZX (XX 3 N X

13 Key lemmas to the main results 3 where we have used 5 again. We expand ZX and (XX 3 N X in terms of the irreducible bimodules X G XXX X and E (XX X 4XXX 3X and get and Hence ZX G E (XX 3 N X X G E. s X (XX N SX G E X G E. By 4 X G E and (XX N SX are irreducible and mutually inequivalent hence Altogether s X X G G + E E. E E E E s s + s 3 + which proves (iv. We need to prove one more relation (v EX Z SZ R. To prove (v note first that X Z and E all correspond to the vertices in G ( set of the odd vertices of the principal graph of X. (We write G even ( vertices. Hence EE G even ( and since S EE SE E odd the for the even S is an irreducible subbimodule of EE so also S G ( even. Therefore every irreducible N- N bimodule or N-M bimodule that can be expressed in terms of X X and S belong to the principal graph G of X. Therefore by the same argument in the proof of the previous lemma the square root of the Jones index is additive and multiplicative on the N-N bimodules or N-M bimodules which can be expressed in terms of X X and S because it will occur as a submodule of (XX n n or (XX n X n. Since ZX G E we have E ZX and therefore (vi Z EX. By (iii also (vii SZ EX. Moreover in the same way we have (viii R EX by (iv. We know that Z SZ and S are irreducible and Z SZ by 3. Moreover by a simple computation using the additivity and multiplicativity of [ ] / we have [R] / [Z] / [SZ] /.

14 3 Generalized open string bimodules 4 Hence all of R Z SZ are mutually inequivalent. Thus EX Z SZ R T where T is an N-N bimodule. By [X] (5 + 7/ we easily get [EX] / [Z] / + [SZ] / + [R] / hence T. Putting everything together we see that conditions (a (b (c in the proof of the previous lemma hold namely G is the principal graph of X. q.e.d. 3 Generalized open string bimodules In section we have reduced our construction problem to verification of certain fusion rules but we still have a problem of handling bimodules in a concrete way. For example we do not know how to represent X or S or how to verify equalities of infinite dimensional bimodules. In this section we will introduce the item to make full use of the lemmas. Consider a biunitary connection α as in Figure 3 on the V K V α V L V 3 Figure 3: the connection α with four graphs four graphs with upper graph K lower graph L and the sets of vertices V... V 3. Note that by the definition of biunitary connection the graphs K and L should be connected and the vertical graphs are not nessesary to be connected. We fix the vertices K V and L V. We will now construct the bimodule corresponding to α. First we construct AFD II factors from the string algebras K L n n weak String (n K K weak String (n L L by the GNS construction using the unique trace where String (n G G span{(ξ η a pair of paths on the graph G s(ξ s(η G r(ξ r(η ξ η n}.

15 3 Generalized open string bimodules 5 Here for a path ζ we denote the initial vertex the final vertex and the length of the path by s(ζ r(ζ and ζ respectively. We define its -algebra structure as Now we have another AFD II factor { ( L span K K n x (ξ η (ξ η δ ηξ (ξ η (ξ η (η ξ. x a pair of paths x L ( horizontal paths are in L length n. where L ( denotes the set of vertices on L. We identify elements in K with elements in L by the embedding using connection α and then have an AFD II subfactor K L. (See [EK] Chapter. Next we construct the K-L bimodule corresponding to α. Consider a pair of paths as follows: ( K L here the horizontal part of the left (resp. right path consists of edges of the graph K (resp. L the vertical edge is from one of the two vertical graphs of the four graphs of the connection α and the paths have a common final vertex. In general a pair of paths as above with a common final vertex not necessary with a common initial vertex is called an open string. It was first introduced by Ocneanu in [O] in more restricted situations. We embed an open string of length k into the linear span of open strings of length k + in a similar way to the embedding of string algebras as follows: }weak ( K η L ( K ξ η ξ ξ ξ η α η ξ ξ L ξ ( K ξ η L ξ here the square marked with α means the value given by the connection α. We define the vector space spanned by the above open strings with the above embedding as follows. X α span{(ξ η s(ξ K s(η L r(ξ r(η} n ( span{ K }. L

16 3 Generalized open string bimodules 6 We define an inner product of X α as the sesqui-linear extension of the following; (ξ ζ η (ξ ζ η ( K ξ ζ η L µ K(s(ζ µ L (r(ζ δ ζζ tr K(ξ ξ tr L (η η ( K ξ ζ L η where ξ ζ denotes the concatenation of ξ and ζ µ K and µ L denotes the Perron- Frobenius eigen vector of the graphs tr K is the unique trace on K and tr L is as well. We set (ξ ξ and (η η are the elements of L and L respectively if the end points of each pair of paths do not coincide. X α is regarded as a pre-hilbert space and then we complete By this inner product it and denote the completion by X α. We have the natural left action of K and the right action of L as follows; for x k l ( K ξ η X α ( L K σ K σ K ( ρ ρ L L L we have k x ζ δ σ ζξ ζ ( K σ ζ K σ ζ x ( K σ ζ η L ( x l δ K ηρ ξ ρ. L By the extension of this action the Hilbert space X α is considered as a Hilbert K-L bimodule K X α L. Then we have a K-L bimodule KX α L constructed from α. (We call this bimodule made of open strings an open string bimodule. This is a generalization of open string bimodules in [O] and [Sa] which are the bimodules constructed from flat connections. We make the correspondence between direct sums relative tensor products and the contragredient map of bimodules and sums products and the renormalization of connections so that fusion rules on open string bimodules reduced to the operations of connections. ([O3]

17 3 Generalized open string bimodules 7 First we introduce the sum of two connections. Consider α and as connections on the four graphs with upper graph K lower graph L and sets of vertices V...V 3 as in Figure 3 (The side graphs of α and need not to be identical then they give rise two K-L bimodules. We define the sum of the connections as follows: k (α + ( m n l k α( m n if both m n are edges appearing in α l k ( m n if both m n are edges appearing in l otherwise. Obviously it satisfies the biunitarity. We denote the bimodule constructed from a connection γ by X γ. By considering the action of K from the left it is easy to see that KXL α K X L K X α+ L thus we can use the summation of connections instead of the direct sum of bimodules. Next we define the product of connections ([O3] [Sa]. Consider the connections α and as in Figure 4 which give rise to K-L bimodule (resp. L-M bimodule. V K V α V L V 3 V L V 3 V 4 M V 5 Figure 4: Note that L is appears in the both four graphs. Then we can define the product connection α on the four graphs with upper graph K lower graph M and the sets of vertices V V V 4 V 5. The side graphs consist of the edges {p q p V (resp. V q V 4 (resp. V 5 } with multiplicity {p x q a path of length from p to q x V (resp. V 3 }. We have the connection α as follows: k k (α( n o n o (α n m o l m α( n k l l o ( n o m

18 3 Generalized open string bimodules 8 where n and n are edges such that their concatenation n n is n and o and o are as well. We observe that this process corresponds to the following process of composing commuting squares of finite dimension. A C B D C E D F A E B F where these three squares are finite dimensional commuting squares. We will show that K X α L X M is isomorphic to KX α M. We define the map ϕ from KX α L X M to KX α M as follows; For x ( K ξ L η ζ K ξ ζ L η ζ K X α L y ( L ρ M σ L X M we define ϕ(x L y x y δ η ζρ K ξ ζ M σ KX α M. Since (x L y x L y (x(y y L x tr M (x x(y y L (x y x y tr M (y x x y tr M (y (x xy ((x x L ((x xy y tr M (x x(y y L (x L y x L y where x and y means that we reverse the order of the pairs of paths and also take the complex conjugate of their coefficients we see that ϕ is an isometry so it is welldefined as a linear map from K X α L X M to K X α M and it is also injective. Moreover it is surjective because for an element x K ρ M KX α M

19 3 Generalized open string bimodules 9 where we assume without loss of generality that x is long enough that there is a path connecting L and s(ρ x ξ ξ δ ρξ K K ξ L ( ϕ( K L ξ M η η ξ ( L η L ( L η ρ M ρ M for some η with s(η L r(η s(ρ. Therefore we have the isomorphism Next we prove KX α L X M K X α M. KX α L LX α K here we denote the renormalization of the connection α by α. Take an element ( ζ x K ξ η K XL. α L For x we easily see that its image by the contragredient map is given as ( η x L ξ ζ K XL α K here ξ means the upside down edge of ξ. Since x ζ K ξ σ σ η L σ σσ ξ σσ ξ σ ζ ξ α ξ K σ σ ( ξ α ξ K η σ x is regarded as the element of L XK α thus we have KX α L L X α K. σ ξ η L σ σ ξ ζ L σ Now we have a good correspondence between the operations of certain bimodules and those of connections. To complete it we should check that the construction of bimodules from connections is a one to one correspondence of the equivalent classes.

20 3 Generalized open string bimodules Theorem 3 Let α and be two connections as below; V K V S α T V L V 3 V K V S T V L V 3 then the K-L bimodules K X α L and K X L are isomorphic if and only if α and are equivalent to each other up to gauge choice for the vertical edges in particular the pairs (S T and (S T of the vertical graphs must coincide. Remark In [O3] the same correspondence of bimodules and equivalent classes of connections has been introduced for limited objects and there an equivalent class of connections is defined as that of a gauge transform not only by vertical gauges but also horizontal ones. If the horizontal graphs are trees the equivalent class by total gauges is the same as that by vertical gauges however for general biunitary connections we should limit the gauge choices only to vertical ones. Proof. First assume that α and are equivalent up to gauge choice for the vertical edges. Now α and are on the common four graphs namely S S S T T T. From the assumption we have two unitary matrices u S u T corresponding to the graphs S T respectively such that u S αu T where α and represent the matrices corresponding to the connections. Now we define the isomorphism Φ from K X α L to K X L as follows; ( x K L Φ(x ( (id (n u S K ξ L ( (id (n u T K ξ L K XL α x n if n is even if n is odd K X L where id (n represents the identity of String (n K K and id (n u S is the concatenation regarding u S as an element of p V String ( p S and u T is as well. Note that this map changes only the vertical part of the elements of bimodule. Now we check that Φ is

21 3 Generalized open string bimodules a well-defined linear map i.e. does not depend on the length of the expression of x. Here we assume n is even. We have Φ(x ( u ηξ K S η η L u ηξ ( K S η η σ σ σ L η u ηξ S σσ η σ η η σ ( K σ η L σ where u ηξ S denotes the η-ξ entry of the matrix u S. On the other hand we have Φ(x Φ( σ id (n+ u T σσ ξ ( K ξ σ L σσ ξ σ ξ ξ α σ η σ ξ α ξ σ u η ξ T σ ( K σ η L σ ( K σ ξ L σ By u S αu T the above two expressions of Φ(x coincide. When n is odd it follows from the same argument. Therefore Φ is a well-defined linear map. Here Φ is obviously a right L-homomorphism and since id u S (resp. id u T of any length commutes with the element of K of the same length Φ is a left K- homomorphism too. Since u S and u T are unitaries Φ is an isomorphism. Then we have K XL α K XL. Next we prove the converse. Assume K XL α K X L. Then we have a partial isometry such that u End( K X α L KX L End( K X α+ L u : K X α L K X L uu + u u id. Our aim is to prove S S T T and construct a gauge transform between α and from u. Claim Consider a connection γ with four graphs as below. V K V. S γ T V L V 3

22 3 Generalized open string bimodules and three AFD II factors as in the beginning of this section. Then we have End( K X γ L K L where the embedding of K L is given by γ. Proof First we have End( K X γ L (the left action of K on X γ (the right action of L on X γ. We have a natural left action of L on K X γ L. Now we prove (the right action of L on X γ (the left action of L on X γ. ( Obviously we have the inclusion so we prove the equality by comparing dimensions of X γ as modules of both algebras. Take a vertex x on L and consider projections as below; p q ( K K ( x L x We see that p Lp consists of the strings such as ( p x L x x x L L. where means the concatenation. Namely p Lp essentially consists of the strings of L with the initial vertex x. It is the case for px γ q and qlq by similar argument thus we have dim p Lp (px γ q. On the other hand we have dim p Lp (px γ q tr Lq tr Lp dim LX γ then we have By the same argument we have dim LX γ tr Lp tr L q. dim(px γ q qlq

23 3 Generalized open string bimodules 3 and dimx γ L tr Lq tr Lp. Thus we have dim LX γ dimx γ dimx γ L L and the equality in ( holds. q.e.d. By applying this claim to α+ we see that the partial isometry u is in K L and the map X α X is given by the natural left action of L on X α. To construct the gauge matrices which transfer α to we use the compactness argument of Ocneanu. ([O] [EK Section.4] We introduce some necessary notions and facts. Definition 3 (Flat element Flat field Ocneanu [O] [EK] Consider a connection γ on the four graphs as in the previous claim and three AFD II factors K L and L as at the beginning of this section. Take an element ξ p V String ( p S. It is called a flat element if id (l ξ id (l ξ l N under the identification by the connection γ where id (l denotes the string σ l(σ σ on the graph K (resp. L. We use this notation often hereafter under similar conditions. It is known that for a flat element ξ there is the element η p V String ( such that and ξ id ( id ( η id (l η id (l η. by the connection γ. We call η a flat element too. This couple of flat elements represents an element of the string algebra with identification by γ namely K id (k ξ K id (l η for any sufficiently large k:even and l:odd that is large enough that the set of the end points of id (k (resp. id (l coincides with V (resp. V. Now we define z to be a function on V V such that z(p String ( p S (resp. String( p T for p V (resp. V and p V z(p ξ (resp. p V z(p η and call it flat field. Let V n to be a proper subset of V n where n and put ξ p V z(p (resp. η o V z(p. It is known that id (j ξ (resp. η ξ(resp. η id (j for sufficiently large j. We call such elements as ξ and η flat too though they are not flat elements by the definition above. p T

24 3 Generalized open string bimodules 4 Theorem 4 (Ocneanu [O][EK] Let K L be the AFD II subfactor constructed from the connection γ. Then K L {flat field}. The correspondence of elements as follows; Take a flat field z and let ξ p V z(p then id (k ξ K L and conversely for x K L it turns out that x is written as K for some flat field z. x Kz( String ( S L This theorem is proved by the compactness argument of Ocneanu see [O] and [EK]. (Generally the length of flat field/element can be arbitrary. Now we continue the proof of Theorem 3 using the above notions. Let γ α+ and S S S T T T. By the above theorem we consider the partial isometry u K L which gives the isometry X α X as a flat field for the connection γ. Take p V and q V so that they are connected in S. Since uu + u u we have u(p qu(p q + u(p q u(p q in the algebra String ( (pq S span{(σ ρ σ ρ s(σ s(ρ p r(σ r(ρ q} where u(p q String ( (pq S such that q V u(p q u(p. Take an element of X α x ( ζ p K ξ q L From the definition of u we have ux X then (id u(p q x K id u(p q η u(p q ηξ ( K ζ pη q L ε q ξ S. ( ζ p K ξ q ε L q ε X u(p q ηξ C. q Note that η S if u(p q ηξ. Since u gives an isometry of X α and X ( ζ p dim span{ K ξ q L ε q ( ζ p dim span{ K η q L ξ S } ε q η S }

25 3 Generalized open string bimodules 5 for each ζ and ε. This means { p ξ q } { p S η q S }. By seeing all the possible pairs of vertices p and q we have By the same discussion we have also S S. T T then we know that α and are on the same four graphs. Now we see that u(p q gives the gauge matrix for the edges which connect p and q. Let u S and u T be stable flat elements on S and T corresponding to the flat field u. Since the isomorphism x X α u x X is well-defined from the same deformation as we proved the well-definedness of Φ in the first half proof of our main statement here u S αu T follows. Under the identification of S S and T T u S and u T are considered as unitary matrices corresponding to the gauge transform action of α and. Thus we have α up to vertical gauge choice. Corollary K X α L is irreducible if and only if α is indecomposable. Proof Assume α is decomposable i.e. there exist gauge unitaries u S u T γ such that u S αu T + γ. Then we have KX u S αu T L K X L K X γ L K X α L and connections namely K XL α is reducible. Conversely assume K XL α is reducible. then we have bimodules KY L and K Z L such that KXL α K Y L K Z L and a projection p End( K X α L K L with p : K X α L K Y L. Along the same argument as in the proof of the previous theorem we consider p as a flat field and make the projections p S and p T which project elements of K XL α to KY L

26 4 Main theorem for the case of (5 + 3/ 6 at the finite level and they act as the projections of the connection matrix and we have a sub connection of α p S αp T so that thus α is decomposable. KX L K Y L q.e.d. Corollary Let γ be a connection as in Claim i.e. V K V S γ T V L V 3. If there exists a vertex p to which only one vertical edge is connected. Then the bimodule K X γ L is irreducible. Proof Assume p V without missing generality. Let ξ be the only one vertical edge in S connected to p. Assume K X γ L is not irreducible. Then by the above argument we have connections γ and γ with the four graphs V K V V K V respectively so that S γ T V L V 3 S γ T V L V 3 γ γ + γ S S S T T T. ξ should be contained either in S or S. Assume ξ S then no edge in S connects to p. This contradicts to the unitarity of γ. q.e.d. Remark Corollary is a generalization of Wenzl s Criterion for irreducibility of subfactors obtained from a periodic sequence of commuting squares (c.f. [W]. 4 Main theorem for the case of (5 + 3/ In this section we give a proof for our main theorem for the case of index (5+ 3/ due to the second named author.

27 4 Main theorem for the case of (5 + 3/ 7 Theorem 5 A subfactor with principal graph and dual principal graph as in Figure exists. From the key lemma we know that the above theorem follows from the next proposition. We define the connection σ as p σ r q s δ σ(prδ σ(qs where p q r s are the vertices on the upper graph in Figure and we define σ( as σ(x x σ σ(x σ x σ and x σ 3 x. Note that for the vertex c we put σ(c c. Proposition Let α be the unique connection on the four graphs consisting of the pair of the graphs appearing in Figure and σ be the connection defined above. Then the following hold. The six connections σ σ ( σ( σ ( are indecomposable and mutually inequivalent. The four connections α σα σ α α α are irreducible and mutually inequivalent. 3 σ( ( σ. Proof The four graphs of the connection α are as in Figure 5.

28 4 Main theorem for the case of (5 + 3/ 8 V G V G 3 α G V 3 G V V G V G V b b σ b σ σ σ a c a a σ σ 3 4 G V 3 a c a σ a σ G 3 V b b σ b σ σ σ Figure 5: Four graphs of the connection α The Perron-Frobenius weights of the vertices can easily be computed as follows µ( µ(a µ(a σ µ(a σ λ µ(b µ(b σ µ(b σ λ µ(c λ 3 λ µ( µ( λ µ(3 λ µ(4 λ 5+ where λ 3. One can check that the Table defines a connection α on the four graphs (Figure 5 which satisfies Ocneanu s biunitary conditions i.e. p α η ξ ξ is a unitary matrix for each fixed p s (unitarity η s ξ ηξ ξ and x η z ξ ξ µ(yµ(z y η w y η w µ(xµ(w ξ ξ (renormalization x η z see [O] and [EK chapter ]. We see that such a biunitary connection α on these four graphs is determined uniquely up to complex conjugate arising from the symmetricity of the graphs namely it is essentially unique. The connection α is as in Table. Note x z (xy zw-entry in the table α y w

29 4 Main theorem for the case of (5 + 3/ 9 where we note that since all the graphs which consist the four graphs in Figure 5 are tree all the edges are expressed by the both ends. For example in the Table one can find a ( a a-entry α a. We also note that blank entries are all s and ρ ( λ 4 + i 8 λ τ ( λ i 5 λ ρ τ. ( τ 3 ρ. Now we display the table of the connection α computed by renormalization in a a c c3 c4 a σ 4 a σ 4 a ba bc λ λ λ λ λ λ λ λ b σ c ρ τ b σ a σ b σ c ρ τ b σ a σ λ λ λ λ λ λ λ λ λ λ λ λ σ a σ σ a σ Table : Connection α Table for use of the later computations. First we check condition 3 namely we prove σ( ( σ up to vertical gauge choice. It is enough to show ( σ( σ so now we will prove this equivalence. First we compute the connection. The four graphs on which the connection exists are as in Figure 7. The vertical graphs are constructed as in Figure 6 where we explain it only by GG t.

30 4 Main theorem for the case of (5 + 3/ 3 a a c 3c 4c 4a σ 4a σ λ a λ ab λ λ λ cb cb σ cb σ λ λ(λ 3 λ ρ 3 τ λ(λ 3 τ λ ρ λ(λ λ λ λ λ a σ b σ a σ σ a σ b σ a σ σ Table : Connection α V G V t G V a c a σ a σ 3 4 a c a σ a σ V G G t V a c a σ a σ a c a σ a σ Figure 6: construction of the vertical graphs of.

31 4 Main theorem for the case of (5 + 3/ 3 G V V G 3 α G V G V 3 t G 3 α t G V G V V G V t G 3 G 3 t G G V G V V b b σ b σ σ σ G V G G t V a c a σ a σ a c a σ a σ G V G 3 G 3 t V b b σ b σ σ σ b b σ b σ σ σ Figure 7: the four graphs of To obtain the connection we multiply the entries of the connections α and α properly (We call this sort of computations of the multiplication of the connections actual multiplication. transform it by vertical gauge so that the entries corresponding to the trivial connection are and subtract. (In Figure 7 the broken lines corresponds to this trivial summand. Here in the Table 3 we show the landscape of with s in the entries corresponding to. First we will compute the entries marked in the Table 3. We assume that gauge transform unitaries corresponding to single vertical edges which connect different vertices in the graph G to be without losing generality because they are not involved in the trivial connection. We compute such entries by actual multiplication. a a b c α a a α b c b c b c a α a a α a λ λ λ λ λ λ λ.

32 4 Main theorem for the case of (5 + 3/ 3 aa aa ca accc cc cc 3 a σ ca σ c ca σ a σ a σ a σ a σ ca σ a σ a σ a σ a σ b b bb bb bb σ bb σ b σ b b σ b σ b σ b σ b σ b σ b σ σ b σ b b σ b σ b σ b σ b σ b σ b σ σ σ b σ σ σ σ b σ σ σ Table 3: Landscape of the connection after a gauge transform From here we only write the result of multiplication. b a b a b σ c α c c α b σ c ρ λ b c b c α b σ a σ c 4 c 4 α b σ a σ b a b a b σ c α c c α b σ c ρ λ b c b σ a σ b c α c 4 c 4 α b σ a σ

33 4 Main theorem for the case of (5 + 3/ 33 b σ c b σ c b a α c c α b a ρ b σ a σ b σ b c α c 4 a σ b σ a σ b σ b σ c α c 4 a σ c 4 α b c λ c 4 α b σ c λ 4 λ b σ c b σ a σ b σ c α c 4 c 4 α b σ a σ λ b σ a σ b σ a σ a σ b σ α c 4 c 4 α b σ a σ λ λ b σ c σ a σ b σ c α a σ 4 a σ 4 α σ a σ λ λ b σ a σ σ a σ a σ b σ α a σ 4 b σ a σ b σ a σ b σ c α c 4 a σ 4 α σ a σ c 4 α b σ c λ b σ c b σ c c b a α α c b a ρ b σ a σ a σ c c 4 b c b σ α c 4 b c λ λ 4 λ b σ c b σ c α b σ a σ c 4 b σ a σ b σ a σ α b σ a σ c 4 b σ c b σ c α σ a σ a σ 4 c 4 α b σ a σ λ c 4 λ α b σ a σ λ a σ 4 λ α σ a σ λ

34 4 Main theorem for the case of (5 + 3/ 34 b σ a σ b σ a σ α σ a σ a σ 4 σ a σ σ b σ c α a σ 4 σ a σ σ a σ 4 α σ a σ λ a σ a σ a σ 4 α b σ c a σ 4 α b σ a σ α b σ a σ a σ 4 σ a σ σ a σ a σ 4 b σ c α α a σ 4 b σ c σ a σ σ a σ a σ 4 α α. b σ a σ a σ 4 b σ a σ Next we will obtain the entries marked. We have two vectors of connection concerning to b-b double edges by actual multiplication as follows: b c a b c α α λ b a a b a λ b c c λ α α c b a b a a b a b c α α a b c λ b a c. (λ α α λ c b c Since these two vectors are proportional they are transformed to two proportional vectors by a left vertical gauge transform for the double edges b-b i.e. multiplication from the left by an element of U(. Since we should have s in the (bb aa -entry and the (bb cc -entry they can be transformed into the following pair ( λ and ( λ 4 λ then we have (bb ca λ (bb ac λ 4 λ respectively where we have omitted -entry. The same procedure for the entries with vertical double edges b σ -b σ and b σ -b σ gives two pairs of vectors as follows. b σ a σ b σ c b σ a σ α c 4 b σ a σ α a σ 4 c 4 α b σ c a σ 4 α c b σ (λ (λ λ

35 4 Main theorem for the case of (5 + 3/ 35 b σ c c 4 b σ c α α c 4 b σ a σ λ b σ a σ b σ c a σ 4 λ α α λ a σ 4 b σ a σ b σ a σ c 4 b σ a σ α α b σ c c 4 b σ c (λ (λ b σ a σ a σ 4 α α λ a σ 4 b σ c b σ a σ c 4 b σ c α α c 4 b σ a σ λ b σ a σ b σ c a σ 4. λ α α λ a σ 4 b σ a σ The first pair concerning to b σ -b σ double edges can be transformed by gauge unitary into the pair ( ( and λ where we note (b σ b σ cc (b σ b σ a σa σ by this gauge. Since second pair is equal to the first pair it can be transformed the same pair of vectors by the left gauge unitary of the double edges b σ -b σ. Thus we have and Along the same argument we have from the actual multiplications (b σ b σ a σc (b σ b σ a σ c λ (b σ b σ ca σ (b σ b σ ca σ. ( b aa (b aa a b a b a a ( λ ( λ λ λ λ λ λ by the right gauge unitary of double edges a-a. So far we have the entries of as in Table 4 where the entries g s mean that they have not been determined so far.

36 4 Main theorem for the case of (5 + 3/ 36 aa aa ca accc cc cc 3 a σ ca σ c ca σ a σ a σ a σ a σ ca σ a σ a σ a σ a σ b λ λ λ b bb bb λ λ 4 λ λ λ ρ bb σ g bb λ g bbσ ρ bb σ λ g bbσ b σ b ρ g bσb λ b σ b σ b σ b σ g bσb σ λ b σ b σ g bσb σ b σ σ λ 4 λ λ λ λ b σ b ρ g bσ b λ λ b σ b σ g 4 bσ b σ λ λ λ λ b σ b σ b σ b σ g bσ b σ λ b σ σ λ λ λ σ b σ σ σ σ b σ σ σ λ λ λ Table 4: Connection (λ n λ n Now we will obtain the entries marked in Table 3. Denote the vectors of entries of actual multiplication corresponding to ( g?? by f??. We use the following data of f?? s. f bbσ b c c b σ ( b c α c ρ λ(λ τ 3 c b c α b σ c α c 3 λ ( ρ f bbσ λ(λ τ 3 λ ( ρ f bσb λ(λ τ 3 λ c 3 b c α b σ c α c 4 c 4 α b σ c

37 4 Main theorem for the case of (5 + 3/ 37 f bσb σ f bσ b σ ( (λ ρ λ(λ τ 3 ( f bσ b ρ λ(λ τ 3 (λ λ λ ( (λ ρ λ(λ τ 3 (λ λ Note that f bbσ f bσb and f bσ b f bbσ so they are transformed with keeping equality by the gauge transform of the triple edges c-c. Therefore we see only f bbσ f bσb f bσb σ and f bσ b σ. We have the following lemma. Lemma 4 The three vectors ( u 3 λ λ ( λ λ u 3 λ 3 3 λ 3 and ( λ λ 3 λ 3 form an orthonormal basis for C 3 and f bσ b f bbσ λ u + ( 3 λ + λ 3 λ iu 3 f bbσ f bσb λ u + ( λ 3 3 λ λ iu 3 f bσb λ 4 σ u + ( λ λ 3 iu 3 f bσ b σ λ 4 3 u + ( λ 3 λ 3 iu 3.. Proof Checked by elementary but heavy computations using λ 4 5λ + 3. q.e.d.

38 4 Main theorem for the case of (5 + 3/ 38 From Lemma 4 we have g?? s as the expression of f?? s by the orthonormal basis u and u 3 as follows ( λ g bbσ g bσ b + λ 3 3 λ λ i ( λ g bσb g bbσ λ 3 3 λ λ i g bσb σ g bσ b σ λ 4 3 λ 4 3 λ 3 + λ 3 λ 3 i λ 3 i. g bb g bσb σ and g bσ b σ are uniquely determined so that the matrices b c λ 4 λ ρ λ ρ λ g bb g bbσ g bbσ and are unitaries hence we have b σ b σ c c g bb g bσb σ g bσ b σ g bσb λ g bσb σ λ g bσb σ g bσ b g bσ b σ g bσ b σ λ 4 λ λ λ 4 λ λ ( λ 3 ( λ 3 3 λ ( λ 3. 3 Now the connection ( is as in the Table 5. Our aim is to show ( σ( σ. For this purpose an expression of with symmetry up to σ is useful. We will re-choose another gauge as in Table 6 where s τ λ n λ n and numbers beside the name of edges denote unitaries corresponding to the edges namely we have multiplied these numbers to the corresponding rows (resp. columns in the previous table and g s

39 4 Main theorem for the case of (5 + 3/ 39 aa ca accc cc 3 a σ ca σ c ca σ a σ a σ ca σ a σ a σ b λ b λ λ bb λ λ 4 λ λ λ g bb ρ bb σ λ g bbσ ρ bb σ λ g bbσ b σ b ρ g bσb λ b σ b σ g bσb σ λ λ b σ b σ g 4 λ bσb σ λ λ λ b σ σ λ b σ b ρ g bσ b λ b σ b σ g 4 λ bσ b σ λ λ λ b σ b σ g bσ b σ b σ σ λ λ λ λ λ λ σ b σ - σ b σ - λ Table 5: Connection (λ n λ n denote the vectors corresponding to g s after being multiplied by suitable gauge numbers respectively. By seeing this table we easily see that entries other than g s are invariant to the transformation of ( σ( σ which acts on the table as the relabeling xy σ(xσ(y. The remaining problem is whether we have a gauge unitary matrix u ( c corresponding to double edges c-c c such that u( c c g σ( σ( g or not. We can check by a simple computation that ( c u c gives rise to the transformation 3 (λ λ i 6 λ 3 λ λ 3 i + (λ 3 6 i g bb σ g b σb σ g b σ b g bb σ g b σb g b σ b σ g bb σ g b σb

40 4 Main theorem for the case of (5 + 3/ 4 s s s s s s aa ca accc cc 3 a σ c a σ c ca σ a σ a σ ca σ a σ a σ b λ λ λ b bb λ λ 4 λ λ λ s s bb σ s bb σ g bb λ g bb σ s s λ g bb σ s b σ b s g b σb b σ b σ g b σb σ s s b σ b σ g b σb σ s b σ σ s λ s λ λ 4 λ λ λ λ s b σ b s g b s σ b λ s b σ b σ g b λ 4 λ σ b σ λ λ λ b σ b σ g b σ b σ s λ λ λ s λ s λ s b σ σ λ λ σ b σ σ b σ and Table 6: Connection after taking symmetric gauge choice g bb g b σb σ g b σ b σ. Thus we have proved the equivalence of connections σ( σ. Finally we will check conditions and. Mutual inequivalence is obvious by seeing four graphs of the connections appearing there. Namely connections producing the bimodules of different indices are trivially mutually inequivalent. To prove inequivalence of connections which produce the bimodules of the same index it is sufficient x to show the existence of the unitary matrices of the connection of the form y which have different sizes in each connection. We can check it only by seeing the four graphs. About the indecomposability since it was irreducibility of the bimodules in our original lemma all we must see is the irreducibility of bimodules made of connections here. The bimodule X N N N is trivially irreducible and indecomposability of σ and σ follows. To see the irreducibility of X α consider the subfactor N M constructed from the connection α. Then X α N M M. By Ocneanu s compactness argument (see Section 3 Theorem 4 End( N X α M End( NM M

41 5 Main theorem for the case of (5 + 7/ 4 N M String ( G C where G is the upper graph in Figure thus irreducibility of X α X σα and X σ α follows. Similarly we have End( N X N End( NM M M N N M String ( G C C where N M M is Jones tower of N M thus irreducibility of X X σ( and X σ ( follows. Irreducibility of X α α follows in the same way using String (3 G C M (C. Now the proposition holds and thus we have proved the theorem. 5 Main theorem for the case of (5 + 7/ q.e.d. In this section we will give a proof for our main theorem for the case of index (5 + 7/ due to the first named author. Theorem 6 A subfactor with principal graph and dual principal graph as in Figure exists. From the key lemma we know that the above theorem follows from the next proposition. We define the connection σ as p σ r q s δ p r δ q s here p q r s are the vertices on the upper graph in Figure and we consider x as x and if x is one of e f g x x. Proposition Let α be the unique connection on the four graphs consisting of the pair of the graphs appearing in Figure and σ be the connection defined above. Then the following hold. The eight connections σ σ ( σ( ( σ σ( σ are indecomposable and mutually inequivalent. The six connections ( 3 + σ(( 3 + α σα α α σ(α α ( α 4α + 3α ( σα

42 5 Main theorem for the case of (5 + 7/ 4 are irreducible and mutually inequivalent. 3 σ( σα ( σα. Proof The four graphs of the connection α and the Perron-Frobenius weights are as in Figure 8. V G V G 3 α G V 3 G V V G V G V G V 3 G 3 V * b d f d h b h a c e c g ã A C E C G Ã * b d f d h b h The Perron-Frobenius weights: µ( µ( µ(a µ(ã µ(b µ( b µ(h µ( h µ(c µ( c 3 µ(d µ( d µ(e 3 + µ(f µ(g 3 µ( µ(3 µ(4 + µ(5 3 µ(6. Figure 8: Four graphs of the connection α Note that the Perron-Frobenius weights of the vertices in V 3 are the same as that of the vertices in V and here we used The biunitary connection α on these four graphs is determined uniquely as in Table 7 as in (5 + 3/ case. We will also display the Table of the connection α for use of later computations.

43 5 Main theorem for the case of (5 + 7/ 43 a a c c3 e3 e4 e5 c5 g5 g6ã6 A ba bc dc de γ γ γ γ fe fg hg h C hã de d C γ γ γ γ bg à A Ab a a c 3c 3e 4e 5e 5 c 5g 6g 6ã Cb Cd Ed Ef E d γ C d C h Gf Gh G b à h à where n n n n γ γ. Table 7: Connections α (upper and α (lower

44 5 Main theorem for the case of (5 + 7/ 44 First we check condition 3 namely we prove σ( σα ( σα. up to vertical gauge choice. Now we compute the connection. The four graphs on which the connection exists are as in Figure 9. V G V 3 G 3 G3 t G G t V G V 3 V G V G G t V G V G 3 G t 3 V * b d f d h b h a c e c g ã a c e c g ã * b d f d h b h * b d f d h b h Figure 9: Four graphs of the connection The broken edges correspond to the trivial connection. We will now compute the connection which is determined only up to vertical gauges. As in the (5 + 3/-case we assume that gauge transfrom unitaries corresponding to single vertical edges which connect different vertices in the graph G G t G 3G3 t to be. Then we easily find 38 entries of by actual multiplication of the connections α and α. Next as in the (5 + 3/-case we can compute all the entries of which involve the double edges in the graph of by a simple gauge transform then we have 4 entries listed in the Tables 8 9 other than the entries marked (? where n n and γ.