Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings

Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obt...

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Datum:	2009
Hauptverfasser:	Coquereaux, R., Schieber, G.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2009
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	nasplib_isofts_kiev_ua-123456789-1491622025-02-09T23:31:58Z Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings Coquereaux, R. Schieber, G. Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs E4, E6 and E8 describing exceptional quantum subgroups of type SU(4). We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupoïds. This research was supported in part by the ANR program “Geometry and Integrability in Mathematical Physics”, GIMP, ANR-05-BLAN-0029-0. One of us (R.C.) thanks the Centre de Recerca Matem`atica (CRM), Bellaterra, Universitat Aut`onoma de Barcelona, where part of this work was done, for its support. 2009 Article Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings / R. Coquereaux, G. Schieber // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 43 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R50; 16W30; 18D10 https://nasplib.isofts.kiev.ua/handle/123456789/149162 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we determine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs E4, E6 and E8 describing exceptional quantum subgroups of type SU(4). We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupoïds.
format	Article
author	Coquereaux, R. Schieber, G.
spellingShingle	Coquereaux, R. Schieber, G. Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Coquereaux, R. Schieber, G.
author_sort	Coquereaux, R.
title	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
title_short	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
title_full	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
title_fullStr	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
title_full_unstemmed	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings
title_sort	quantum symmetries for exceptional su(4) modular invariants associated with conformal embeddings
publisher	Інститут математики НАН України
publishDate	2009
url	https://nasplib.isofts.kiev.ua/handle/123456789/149162
citation_txt	Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings / R. Coquereaux, G. Schieber // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 43 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT coquereauxr quantumsymmetriesforexceptionalsu4modularinvariantsassociatedwithconformalembeddings AT schieberg quantumsymmetriesforexceptionalsu4modularinvariantsassociatedwithconformalembeddings
first_indexed	2025-12-01T18:21:48Z
last_indexed	2025-12-01T18:21:48Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 044, 31 pages Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings Robert COQUEREAUX and Gil SCHIEBER Centre de Physique Théorique (CPT), Luminy, Marseille, France? E-mail: coque@cpt.univ-mrs.fr, schieber@cpt.univ-mrs.fr Received December 24, 2008, in final form March 31, 2009; Published online April 12, 2009 doi:10.3842/SIGMA.2009.044 Abstract. Three exceptional modular invariants of SU(4) exist at levels 4, 6 and 8. They can be obtained from appropriate conformal embeddings and the corresponding graphs have self-fusion. From these embeddings, or from their associated modular invariants, we deter- mine the algebras of quantum symmetries, obtain their generators, and, as a by-product, recover the known graphs E4, E6 and E8 describing exceptional quantum subgroups of type SU(4). We also obtain characteristic numbers (quantum cardinalities, dimensions) for each of them and for their associated quantum groupöıds. Key words: quantum symmetries; modular invariance; conformal field theories 2000 Mathematics Subject Classification: 81R50; 16W30; 18D10 Foreword General presentation. A classification of SU(4) graphs associated with WZW models, or “quantum graphs” for short, was presented by A. Ocneanu in [31] and claimed to be completed. These graphs generalize the ADE Dynkin diagrams that classify the SU(2) models [7], and the Di Francesco–Zuber diagrams that classify the SU(3) models [13]. They describe modules over a ring of irreducible representations of quantum SU(4) at roots of unity. A particular partition function associated with each of those quantum graphs is modular invariant. According to [31], the SU(4) family includes the Ak series (describing fusion algebras) and their conjugates for all k, two kinds of orbifolds, the D(2) k = Ak/2 series for all k (with self- fusion when k is even), and members of the D(4) k = Ak/4 series when k is even (with self-fusion when k is divisible by 8), together with their conjugates. The orbifolds are constructed by using the Z4 action on weigths generated by ε{λ1, λ2, λ3} = {k − λ1 − λ2 − λ3, λ1, λ2} or the Z2 action generated by ε2. The SU(4) family also includes an exceptional case, D(4)t 8 , without self-fusion (a generalization of E7), and three exceptional quantum graphs with self-fusion, at levels 4, 6 and 8, denoted E4, E6 and E8, together with one exceptional module for each of the last two. The modular invariant partition functions associated with E4, E6 and E8 can be obtained from appropriate conformal embeddings, namely from SU(4) level 4 in Spin(15), from SU(4) level 6 in SU(10), and from SU(4) level 8 in Spin(20). There exists also a conformal embedding of SU(4), at level 2, in SU(6), but this gives rise to D(2) 2 = A2/2, the first member of the D(2) k series. This exhausts the list of conformal embeddings of SU(4). The other SU(4) quantum graphs, besides the Ak, can either be obtained as modules over the exceptional ones, or are associated (possibly using conjugations) with non-simple conformal embeddings followed by contraction, SU(4) appearing only as a direct summand of the embedded algebra. ?UMR 6207 du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var, affilié à la FRUMAM (FR 2291) mailto:coque@cpt.univ-mrs.fr mailto:schieber@cpt.univ-mrs.fr http://dx.doi.org/10.3842/SIGMA.2009.044 2 R. Coquereaux and G. Schieber Vertices a, b, . . . of a chosen quantum graph (denoted generically by Ek) describe boundary conditions for a WZW conformal field theory specified by SU(4)k. These irreducible objects span a vector space which is a module over the fusion algebra, itself spanned, as a linear space, by the vertices m,n, . . . of the graph Ak(SU(4)), or Ak for short since SU(4) is chosen once and for all, the truncated Weyl chamber at level k (a Weyl alcove). Vertices of Ak can be understood as integrable irreducible highest weight representations of the affine Lie algebra ŝu(4) at level k or as irreducible representations with non-zero q-trace of the quantum group SU(4)q at the root of unity q = exp(iπ/(k + g)), g being the dual Coxeter number (for SU(4), g = 4). Edges of Ek describe action of the fundamental representations of SU(4), the generators of Ak. To every quantum graph one associates an algebra of quantum symmetries1 O, along the lines described in [30]. Its vertices x, y, . . . can be understood, in the interpretation of [34], as describing the same BCFT theory but with defects labelled by x. To every fundamental representation (3 of them for SU(4)) one associates two generators of O, respectively called “left” and “right”. Multiplication by these generators is described by a graph2, called the Ocneanu graph. Its vertices span the algebra O as a linear space, and its edges describe multiplication by the left and right fundamental generators (we have 6 = 2× 3 types of edges3 for SU(4)). The exceptional modular invariants at level 4 and 6 were found by [39, 2], and at level 8 by [1]. The corresponding quantum graphs4 E4, E6 and E8 were respectively obtained by [35, 36, 31]. There are several techniques to determine quantum graphs. One of them, probably the most powerful but involving rather heavy calculations, is to obtain the quantum graph associated5 with a modular invariant as a by-product of the determination of its algebra of quantum symmetries. This requires in particular the solution of the so-called modular splitting equation, which is a huge collection of equations between matrices with non-negative integral entries, involving the known fusion algebra, the chosen modular invariant, and expressing the fact that O is a bi-module over Ak. Because of the heaviness of the calculation, a simplified method using only the first line of the modular invariant matrix was used in [31] to achieve this goal, namely the determination of SU(4) quantum graphs (some of them, already mentioned, were already known) but the algebra of quantum symmetries was not obtained in all cases. To our knowledge, for exceptional modular invariants of SU(4) at levels 4, 6 and 8, the full modular splitting system had not been solved, the full torus structure had not been obtained, and the graph of quantum symmetries was not known. This is what we did. We have recovered in particular the structure of the already known quantum graphs; they now appear, together with their modules, as components of their respective Ocneanu graphs. Categorical description. Category theory offers a synthetic presentation of the whole subject and we present it here in a few lines, for the benefice of those readers who may find appealing such a description. However, it will not be used in the body of our article. The starting point is the fusion category Ak associated with a Lie group K. This modular category, both monoidal and ribbon, can be defined either in terms of representation theory of an affine Lie algebra (simple objects are highest weight integrable irreducible representations), or in terms of representation theory of a quantum group at roots of unity (simple objects are irreducible representations of non-vanishing quantum dimension). In the case of SU(2), we refer to the 1Sometimes called “fusion algebra of defect lines” or “full system”. 2One should not confuse the quantum graph (or McKay graph) that refers to Ek with the graph of quantum symmetries (or Ocneanu graph) that refers to O. 3Actually, since those associated with weights {100} and {001} are conjugated and {010} is real (self- conjugated), we need only 2 types of edges (the first is oriented, the other is not) for Ek or Ak, and 4 = 2 × 2 types of edges for O. 4We often drop the reference to SU(4) since no confusion may arise: we are not discussing in this paper the usual E6 = E10(SU(2)) or E8 = E28(SU(2)) Dynkin diagrams! 5We use the word “associated” here in a rather loose sense, since the relation between both concepts is not one-to-one. Quantum Symmetries for Exceptional SU(4) Modular Invariants 3 description given in [33, 16]. One should keep in mind the distinction between this category (with its objects and morphisms), its Grothendieck ring (the fusion ring), and the graph describing multiplication by its generators, but they are denoted by the same symbol. The next ingredient is an additive category Ek, not modular usually, on which the previous one, Ak, acts. In general this module-category Ek has no-self-fusion (no compatible monoidal structure) but in the cases studied in the present paper, it does. Again, the category itself, its Grothendieck group, and the graph (here called McKay graph) describing the action of generators of Ak are denoted by the same symbol. The last ingredient is the centralizer (or dual) category O = O(Ek) of Ek with respect to the action of Ak. It is monoidal and comes with its own ring (the algebra of quantum symmetries) and graph (the Ocneanu graph). One way to obtain a realization of this collection of data is to construct a finite dimensional weak bialgebra B, which should be such that Ak can be realized as Rep(B), and also such that O can be realized as Rep(B̂), where B̂ is the dual of B. These two algebras are finite dimensional, actually semisimple in our case, and one algebra structure (say B̂) can be traded against a coalgebra structure on its dual. B is a weak bialgebra, not a bialgebra, because ∆1l 6= 1l ⊗ 1l, the coproduct in B being ∆, and 1l its unit. B is not only a weak bialgebra but a weak Hopf algebra: one can define an antipode, with the expected properties. Remark 1. Given a graph defining a module over a fusion ring Ak for some Lie group K, the question is to know if it is a “good graph”, i.e., if the corresponding module-category indeed exists. Using A. Ocneanu’s terminology [31], this will be the case if and only if one can associate, in a coherent manner, a complex number to each triangle of the graph (when the rank of K is ≥ 2): this defines, up to some kind of gauge choice, a self-connection on the set of triangular cells. Here, “coherent manner” means that there are two compatibility equations, respectively nicknamed the small and large pocket equations, that this self-connection should obey. These equations reflect properties that hold for the intertwining operators of a fusion category, they are sometimes called “compatibility equations for Kuperberg spiders” (see [26]). The point is that exhibiting a module over a fusion ring does not necessarily entail existence of an underlying theory: when the graph (describing the module structure) does not admit any self-connection in the above sense, it should be rejected; another way to express the same thing is to say that a particular family of 6j symbols, expected to obey appropriate equations, fails to be found. Such features are not going to be discussed further in the present paper. Historical remarks concerning E8(SU(4)). Not all conformal embeddings K ⊂ G corre- spond to isotropy-irreducible pairs and not all isotropy-irreducible homogeneous spaces define conformal embeddings. However, it is a known fact that most isotropy irreducible spaces G/K (given in [41]) indeed define conformal embeddings. This is actually so in all examples stu- died here, and in particular for the SU(4) ⊂ Spin(20) case which can also be recognized as the smallest member (n = 1) of a D2n+1 ⊂ D(n+1)(4n+1) family of conformal embeddings appear- ing on table 4 of the standard reference [3], and on table II(a) of the standard reference [38], since SU(4) ' Spin(6). This embedding, which is “special” (i.e., non regular: unequal ranks and Dynkin index not equal to 1), does not seem to be quoted in other standard references on conformal embeddings (for instance [24, 27, 40]), although it is explicitly mentioned in [1] and although its rank-level dual is indirectly used in case 18 of [37], or in [43]. The corresponding SU(4) modular invariant was later recovered by [31], using arithmetical methods, and used to determine the E8(SU(4)) quantum graph, but since the existence of an associated conformal em- bedding had slipped into oblivion, it was incorrectly stated that this particular example could not be obtained from conformal embedding considerations. Structure of the article. The technique relating modular invariants to conformal embed- dings is standard [12] but the results concerning SU(4) are either scattered in the literature, or unpublished; for this reason, we devote the main part of the first section to it. In the same 4 R. Coquereaux and G. Schieber section, we obtain characteristic numbers (quantum cardinality, quantum dimensions etc.) for the Ek graphs. In the second section, after a description of the structures at hand and a general presentation of our method of resolution, we solve, in a first step, for the three exceptional cases E4, E6 and E8 of the SU(4) family, the full modular splitting equation that determines the corresponding set of toric matrices (generalized partition functions) and, in a second step, the general intertwining equations that determine the structure of the generators of the algebra of quantum symmetries. The size of calculations involved in this part is huge (quite intensive computer help was required) and, for reasons of size, we can only present part of our results. On the other hand, each case being exceptional, there are no generic formulae. For each case, we encode the structure of the algebra of quantum symmetries by displaying the Cayley graphs of multiplication by the fundamental generators, whose collection makes the Ocneanu graph. We also give a brief description of the structure of the corresponding quantum groupöıds. In the appendices, after a short description of the Kac–Peterson formula, we gather several explicit re- sults, providing quantum dimensions for those irreducible representations of the various groups used in the text. The interested reader may also consult the article [11] which provides more information on the general theory and gives a more complete description of the E4(SU(4)) case. Properties of quantum graphs of type SU(3) and their quantum symmetries are summarized in [10], see also [20] and references therein. Those of type SU(2) are certainly well known but many explicit results, like the explicit structure of toric matrices for exceptional diagrams, can be found in [9]. 1 Conformal embeddings of SU(4) 1.1 Homogeneous spaces G/K We describe the embeddings of K = SU(4) in G = Spin(15), SU(10), Spin(20). The reduction of the adjoint representation of G with respect to K reads Lie(G) = Lie(K)⊕T (G/K). The isotropy representation of K on the tangent space at the origin of G/K has dimension dim(G)−dim(K). In all three cases, the space G/K is isotropy irreducible (but not symmetric): the isotropy representation is real irreducible. After extension to the field of complex numbers it may stay irreducible (strong irreducibility) or not. The following are known results, already mentioned in [41]. SU(4) ⊂ SU(6). This embedding leads to the lowest member of an orbifold series (the D(2) 2 = A2/2 graph) and, in this paper, we are not interested in it. SU(4) ⊂ Spin(15). Reduction of the adjoint representation of G with respect to K reads [105] 7→ [15]+[90]. After complexification, [90] is recognized as the reducible representation with highest weight {0, 1, 2}⊕{2, 1, 0} = [45]⊕[45] so that G/K is not strongly irreducible. SU(4) ⊂ SU(10). Reduction of the adjoint representation of G with respect to K reads [99] 7→ [15] + [84]. After complexification, [84] is recognized as the irreducible representa- tion with highest weight {2, 0, 2} so that G/K is strongly irreducible. SU(4) ⊂ Spin(20). Reduction6 of the adjoint representation of G with respect to K reads [190] 7→ [15] + [175]. After complexification, [175] is recognized as the irreducible representation with highest weight {1, 2, 1}, so that G/K is strongly irreducible. 6The inclusion SU(4)/Z4 ⊂ SO(20) ⊂ GL(20, C) is associated with a representation of SU(4), of dimension 20, with highest weight {0, 2, 0}. Quantum Symmetries for Exceptional SU(4) Modular Invariants 5 1.2 The Dynkin index of the embeddings The Dynkin index k of an embedding K ⊂ G defined by a branching rule µ 7→ ∑ j αjνj , where µ refers to the adjoint representation of G (one of the νj on the right hand side is the adjoint representation of K), αj being multiplicities, is obtained in terms of the quadratic Dynkin indices Iµ, Iνj of the representations: k = ∑ j αjIνj/Iµ with Iλ = dim(λ) 2 dim(K) 〈λ, λ + 2ρ〉. Here ρ is the Weyl vector and 〈 , 〉 is defined by the fundamental quadratic form. For the three embeddings of SU(4) that we consider, into Spin(15), SU(10) and Spin(20), one finds respectively k = 4, 6, 8. 1.3 Those embeddings are conformal An embedding K ⊂ G, for which the Dynkin index is k, is conformal if the following equality is satisfied7: dim(K)× k k + gK = dim(G)× 1 1 + gG , where gK and gG are the dual Coxeter numbers of K and G. One denotes by c the common value of these two expressions. In the framework of affine Lie algebras, c is interpreted as a central charge and the numbers k and 1 denote the respective levels for the affine algebras corresponding to K and G. The above definition, however, does not require the framework of affine Lie algebras (or of quantum groups at roots of unity) to make sense. Using dim(G) = 105, 99, 190, for G = Spin(15), SU(10), Spin(20), dim(K = SU(4)) = 15 and the corresponding values for the dual Coxeter numbers gG = 13, 10, 18 and gK = 4, we see immediately that the above equality is obeyed, for the levels k = 4, 6, 8, with central charges c = 15/2, c = 9, and c = 10. The conformal embeddings of SU(4) into SU(6), SU(10) and Spin(15) belong respectively to the series of embeddings of SU(N) into SU(N(N − 1)/2), SU(N(N + 1)/2 and Spin(N2 − 1), at respective levels N − 2, N + 2 and N (provided N is big enough), whereas the last one, namely SU(4) into Spin(20), is recognized as the smallest member of the Spin(N) ⊂ Spin((2N + 2)(4N + 1)) series, since SU(4) ' Spin(6). 1.4 The modular invariants Here we reduce the diagonal modular invariants of G = Spin(15), SU(10), Spin(20), at level k = 1, to K = SU(4), at levels k = 4, 6, 8, and obtain exceptional modular invariants for SU(4) at those levels. The previous section was somehow “classical” whereas this one is “quantum”. Since there is an equivalence of categories [14, 25] between the fusion category (integrable highest weight representations) of an affine algebra at some level and a category of representations with non-zero q-dimension for the corresponding quantum group at a root of unity determined by the level, we shall freely use both terminologies. From now on, simple objects will be called i-irreps, for short. 7Warning: it is not difficult to find embeddings K ⊂ G, and appropriate values of k for which this equality is satisfied, but where k is not the Dynkin index! Such embeddings are, of course, non conformal. 6 R. Coquereaux and G. Schieber 1.4.1 The method • One has first to determine what i-irreps λ appear at the chosen levels. Given a level k, the integrability condition reads 〈λ, θ〉 ≤ k, where θ is the highest root of the chosen Lie algebra. This is the simplest way of determining these representations. One may notice that they will have non vanishing q-dimension when q is specialized to the value q = exp(iπ/κ), with κ = gG+k (use the quantum Weyl formula together with the property 〈ρ, θ〉 = g − 1, ρ being the Weyl vector, and the fact that κq = 0, see footnote 10). When k = 1, the i-irreps of G = SU(n) are the fundamental representations, and the trivial. For other Lie groups G, not all fundamental representations give rise to i-irreps at level 1 (see Appendix). • To an i-irrep λ of G or of K, one associates a conformal weight defined by hλ = 〈λ, λ + 2ρ〉 2(k + g) , (1) where g is the dual Coxeter number of the chosen Lie algebra, k is the level (for G, one chooses k = 1), ρ is the Weyl vector (of G, or of K). The scalar product is given by the inverse of the Cartan matrix when the Lie algebra is simply laced (A3 ' SU(4), A9 ' SU(10) or D10 ' Spin(20)), and is the inverse of the matrix obtained by multiplying the last line of the Cartan matrix by a coefficient 2 in the non simply laced case B7 ' Spin(15). Note that hλ is related to the phase mλ of the modular t matrix by mλ = hλ − c/24. One builds the list of i-irreps λ of G at level 1 and calculate their conformal weights hλ; then, one builds the list of i-irreps µ of K at level k and calculate their conformal weights hµ. • A necessary – but not sufficient – condition for an (affine or quantum) branching from λ to µ is that hµ = hλ + m for some non-negative integer m. So we can make a list of candidates for the branching rules λ ↪→ ∑ n cnµn, where cn are positive integers to be determined. • There exist several techniques to determine the coefficients cn (some of them can be 0), for instance using information coming from the finite branching rules. An efficient possibility8 is to impose that the candidate for the modular invariant matrix should commute with the generators s and t of SL(2, Z) (modularity constraint). • We write the diagonal invariant of type G as a sum ∑ s λsλs. Its associated quantum graph is denoted J = A1(G). Using the above branching rules, we replace, in this expression, each λs by the corresponding sum of i-irreps for K. The modular invariant M of type K that we are looking for is parametrized by Z = ∑ s∈J (∑ n cn(s)µn(s) )(∑ n cn(s)µn(s) ) . In all three cases we shall need to compute conformal weights for SU(4) representations. In the base of fundamental weights9, an arbitrary weight reads λ = (λn), the Weyl vector is ρ = {1, 1, 1}, the scalar product of weights is 〈λ, µ〉 = (λm)Qmn(µn). At level k, i-irreps λ = {λ1, λ2, λ3} are such that 0 ≤ n=3∑ n=1 λn ≤ k; they build a set of cardinality rA = (k + 1)(k + 2)(k + 3)/6. We order the irreducible representations {i, j, k} of SU(4) as follows: first of all, 8A drawback of this method is that it may lead to several solutions (an interesting fact, however). 9We use sometimes the same notation λi to denote a representation or to denote the Dynkin labels of a weight; this should be clear from the context. We never write explicitly the affine component of a weight since it is equal to k − 〈λ, θ〉. Quantum Symmetries for Exceptional SU(4) Modular Invariants 7 they are ordered by increasing level i+j+k, then, for a given level, we set {i, j, k} < {i′, j′, k′} ⇔ i + j + k < i′ + j′ + k′ or (i + j + k = i′ + j′ + k′ and i > i′) or (i + j + k = i′ + j′ + k′, i = i′ and j > j′). We now consider each case, in turn. 1.4.2 SU(4) ⊂ Spin(15), k = 4 • At level 1, there are only three i-irreps for B7, namely {0} , {1, 0, 0, 0, 0, 0, 0} and {0, 0, 0, 0, 0, 0, 1}, namely the trivial, the vectorial and the spinorial. From equation (1) we calculate their conformal weights: { 0, 1 2 , 15 16 } . • At level 4, we calculate the 35 conformal weights for SU(4) i-irreps and find (use ordering defined previously): 0, 15 64 , 5 16 , 15 64 , 9 16 , 39 64 , 1 2 , 3 4 , 39 64 , 9 16 , 63 64 , 1, 55 64 , 71 64 , 15 16 , 55 64 , 21 16 , 71 64 , 1, 63 64 , 3 2 , 95 64 , 21 16 , 25 16 , 87 64 , 5 4 , 111 64 , 3 2 , 87 64 , 21 16 , 2, 111 64 , 25 16 , 95 64 , 3 2 . • The difference between conformal weights of B7 and A3 should be an integer. This selects the three following possibilities: 0000000 ? ↪→ 000 + 210 + 012 + 040, 1000000 ? ↪→ 101 + 400 + 121 + 004, 0000001 ? ↪→ 111. The above three possibilities give only necessary conditions for branching. Imposing the modularity constraint implies that the multiplicity of (111) should be 4, and that all the other coefficients indeed appear, with multiplicity 1. This is actually a particular case of general branching rules already found in [22, 24, 17]. The partition function obtained from the diagonal invariant \|0000000\|2+\|1000000\|2+\|0000001\|2 of B7 reads: Z(E4) = \|000 + 210 + 012 + 040\|2 + \|101 + 400 + 121 + 004\|2 + 4\|111\|2. It introduces a partition on the set of exponents, defined as the i-irreps corresponding to the nine non-zero diagonal entries of M: {000, 210, 012, 040, 101, 400, 121, 004, 111}. To our knowledge, this invariant was first obtained in [39]. 1.4.3 SU(4) ⊂ SU(10), k = 6 • At level 1, there are ten i-irreps for A9, namely {0, 0, 0, 0, 0, 0, 0, 0, 0}, and {0, . . . 0, 1, 0, . . . , 0}. From equation (1) we calculate their conformal weights: { 0, 9 20 , 4 5 , 21 20 , 6 5 , 5 4 , 6 5 , 21 20 , 4 5 , 9 20 } . • At level 6, we calculate the 84 conformal weights for SU(4) i-irreps and find (use ordering defined previously): 0 3 16 1 4 3 16 9 20 39 80 2 5 3 5 39 80 9 20 63 80 4 5 11 16 71 80 3 4 11 16 21 20 71 80 4 5 63 80 6 5 19 16 21 20 5 4 87 80 1 111 80 6 5 87 80 21 20 8 5 111 80 5 4 19 16 6 5 27 16 33 20 119 80 27 16 3 2 111 80 9 5 127 80 29 20 111 80 159 80 7 4 127 80 3 2 119 80 9 4 159 80 9 5 27 16 33 20 27 16 9 4 35 16 2 11 5 159 80 37 20 183 80 41 20 151 80 9 5 49 20 35 16 2 151 80 37 20 43 16 12 5 35 16 41 20 159 80 2 3 43 16 49 20 183 80 11 5 35 16 9 4 0 • The difference between conformal weights of A9 and A3 should be an integer. This con- straint selects ten possibilities that give only necessary conditions for branching. Imposing 8 R. Coquereaux and G. Schieber the modularity constraint eliminates several entries (that we crossed-out in the next tab- le). One finds actually two solutions but only one is a sum of squares (the other solution corresponds to the “conjugated graph” Ec 6, see our discussion in Section 2.5): 000000000 ? ↪→ 000 + 202 + 501/////+ 222 + 105/////+ 060 100000000 ? ↪→ 200 + 002/////+ 212 + 240/////+ 042 010000000 ? ↪→ 210/////+ 012 + 230 + 032/////+ 303 001000000 ? ↪→ 030 + 301/////+ 103 + 321 + 123///// 000100000 ? ↪→ 400 + 121 + 004/////+ 420/////+ 024 000010000 ? ↪→ 010/////+ 220 + 022 + 050/////+ 600 + 006 000001000 ? ↪→ 400/////+ 121 + 004 + 420 + 024///// 000000100 ? ↪→ 030 + 301 + 103/////+ 321/////+ 123 000000010 ? ↪→ 210 + 012/////+ 230/////+ 032 + 303 000000001 ? ↪→ 200/////+ 002 + 212 + 240 + 042///// The partition function obtained from the diagonal invariant of A9 reads: Z(E6) = \|000 + 060 + 202 + 222\|2 + \|042 + 200 + 212\|2 + \|012 + 230 + 303\|2 + \|030 + 103 + 321\|2 + \|024 + 121 + 400\|2 + \|006 + 022 + 220 + 600\|2 + \|004 + 121 + 420\|2 + \|030 + 123 + 301\|2 + \|032 + 210 + 303\|2 + \|002 + 212 + 240\|2. It introduces a partition on the set of exponents, which are, by definition, the 32 i-irreps cor- responding to the non-zero diagonal entries of M. To our knowledge, this invariant was first obtained in [2]. 1.4.4 SU(4) ⊂ Spin(20), k = 8 • At level 1, there are only four i-irreps for D10, namely {0, 0, 0, 0, 0, 0, 0, 0; 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0; 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0; 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0; 0, 1}; the last two entries refer to the fork of the D graph. These i-irreps correspond to the trivial, the vectorial and the two half-spinorial representations. From equation (1) we calculate their conformal weights:{ 0, 1 2 , 5 4 , 5 4 } . • At level 8, we calculate the 165 conformal weights for SU(4) i-irreps and find (use ordering defined previously): 0 5 32 5 24 5 32 3 8 13 32 1 3 1 2 13 32 3 8 21 32 2 3 55 96 71 96 5 8 55 96 7 8 71 96 2 3 21 32 1 95 96 7 8 25 24 29 32 5 6 37 32 1 29 32 7 8 4 3 37 32 25 24 95 96 1 45 32 11 8 119 96 45 32 5 4 37 32 3 2 127 96 29 24 37 32 53 32 35 24 127 96 5 4 119 96 15 8 53 32 3 2 45 32 11 8 45 32 15 8 175 96 5 3 11 6 53 32 37 24 61 32 41 24 151 96 3 2 49 24 175 96 5 3 151 96 37 24 215 96 2 175 96 41 24 53 32 5 3 5 2 215 96 49 24 61 32 11 6 175 96 15 8 77 32 7 3 69 32 223 96 17 8 191 96 19 8 69 32 2 61 32 239 96 9 4 199 96 47 24 61 32 8 3 77 32 53 24 199 96 2 191 96 93 32 21 8 77 32 9 4 69 32 17 8 69 32 77 24 93 32 8 3 239 96 19 8 223 96 7 3 77 32 3 93 32 65 24 23 8 85 32 5 2 93 32 8 3 239 96 19 8 3 263 96 61 24 77 32 7 3 101 32 23 8 85 32 5 2 77 32 19 8 27 8 295 96 17 6 85 32 61 24 239 96 5 2 117 32 10 3 295 96 23 8 263 96 8 3 85 32 65 24 4 117 32 27 8 101 32 3 93 32 23 8 93 32 3 • The difference between conformal weights of D10 and A3 should be an integer. This selects four possibilities that give only necessary conditions for branching. Imposing the modularity constraint implies eliminating entries 400, 004, 440, 044 from the first line. 0000000000 ? ↪→ 000 + 400/////+ 121 + 004/////+ 141 + 412 + 214 + 800 + 440/////+ 080 + 044/////+ 008, 1000000000 ? ↪→ 020 + 230 + 032 + 303 + 060 + 602 + 323 + 206, 0000000010 ? ↪→ 311 + 113 + 331 + 133, 0000000001 ? ↪→ 311 + 113 + 331 + 133. Notice that the contribution comes from 0 (the trivial representation), from the first vertex of D10 and from the two vertices of the fork (they have identical branching rules). Quantum Symmetries for Exceptional SU(4) Modular Invariants 9 The partition function obtained from the diagonal invariant \|0000000000\|2 + \|1000000000\|2 + \|0000000010\|2 + \|0000000001\|2 of D10 reads: Z(E8) = \|000 + 121 + 141 + 412 + 214 + 800 + 080 + 008\|2 + 2\|311 + 113 + 331 + 133\|2 + \|020 + 230 + 032 + 060 + 303 + 602 + 323 + 206\|2. It introduces a partition on the set of exponents, which are, by definition, the 20 i-irreps cor- responding to the non-zero diagonal entries of M. To our knowledge, this invariant was first obtained in [1]. 1.4.5 Quantum dimensions and cardinalities Quantum dimensions for Ak(SU(4)). Multiplication by its generators (associated with fundamental representations of SU(4)) is encoded by a fusion matrix that may be considered as the adjacency matrix of a graph with three types of edges (self-conjugated fundamental rep- resentation corresponds to non-oriented edges). Its vertices build the Weyl alcove of SU(4) at level 8: a tetrahedron (in 3-space) with k floors. It is convenient to think that Ak is a quantum discrete group with \|Âk\| = rA representations. The quantum dimension dim(n) of a repre- sentation n is calculated, for example, from the quantum Weyl formula. For the fundamental representations f = {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, (with classical dimensions 4, 6, 4) one finds: dim(f) = {4q, 3q4q/2q, 4q}. In particular10, β = 4q = 4 cos ( π κ ) cos ( 2π κ ) with κ = k + 4. The square of β is the Jones index. The quantum cardinality (also called quantum mass, quantum order, or “global dimension” like in [15]) of this quantum discrete space, is obtained by summing the square of quantum dimensions for all rA simple objects: \|Ak\| = ∑ n dim(n)2. Details are given in the Appendix. • If k = 4, rA = 35, dim(f): { β = √ 2 ( 2 + √ 2 ) , 2 + √ 2, √ 2 ( 2 + √ 2 )} , \|A4\| = 128 ( 3 + 2 √ 2 ) . • If k = 6, rA = 84, dim(f): { β = √ 5 + 2 √ 5, 2 + √ 5, √ 5 + 2 √ 5 } , \|A6\| = 800 ( 9 + 4 √ 5 ) . • If k = 8, rA = 165, dim(f): { β = √ 3 ( 2 + √ 3 ) , 3 + √ 3, √ 3 ( 2 + √ 3 )} , \|A8\| = 3456 ( 26 + 15 √ 3 ) . Quantum dimensions for Ek(SU(4)), {k = 4, 6, 8}. First method. Action of Ak on Ek is encoded by matrices generically called “annular matrices” (they are also called “nimreps” in the literature, but this last term is sometimes used to denote other types of matrices with non negative integer entries). In particular, action of the generators is described by annular matrices that we consider as adjacency matrices for the graph Ek itself. Once the later is obtained, one calculates quantum dimensions dim(a) for its rE vertices (the simple objects) by using for instance the Perron–Frobenius vector of the annular matrix associated with the genera- tor F{1,0,0}. Its quantum cardinality is then defined by \|Ek\| = ∑ a dim(a)2. The problem is that we do not know, at this stage, the values dim(a) for all vertices a of Ek, since this graph will only be determined later. Quantum dimensions for Ek(SU(4)), {k = 4, 6, 8}. Second method. It is convenient to think that Ak/Ek is a homogenous space, both discrete and quantum. Like in a classical situation, we have11 restriction maps Ak 7→ Ek and induction maps Ek 7→ Ak. One may think that vertices of the quantum graph Ek do not only label irreducible objects a of E but also space of sections of quantum vector bundles Γa which can be decomposed, using induction, into irreducible objects of Ak: we write Γa = ⊕ n↑Γa n. This implies, for quantum dimensions, 10We set nq = (qn − q−n)/(q − q−1), with qκ = −1, κ = k + g and g = 4 for SU(4). 11These maps (actually functors) are described by the square annular matrices Fn or by the rectangular essential matrices Ea with (Ea)nb = (Fn)ab that we shall introduce later. 10 R. Coquereaux and G. Schieber the equality dim(Γa) = ⊕ n↑Γa dim(n). The space of sections F = Γ0, associated with the identity representation, is special since it can be considered as the quantum algebra of functions over Ak/Ek. Its dimension dim(Γ0) = \|Ak/Ek\| is obtained by summing q-dimensions (not their squares!) of the n ↑ Γ0 representations. We are in a type-I situation (the modular invariant is a sum of blocks) and in this case, we make use of the following particular feature – not true in general: the irreducible representations n ↑ Γ0 that appear in the decomposition of Γ0 are exactly those appearing in the first modular block of the partition function. From the property \|Ak/Ek\| = \|Ak\|/\|Ek\|, we finally obtain \|Ek\| by calculating \|Ak\| \|Ak/Ek\| . • When k = 4, we have F = Γ0 = 000 ⊕ 210 ⊕ 012 ⊕ 040 so that dim(F) = dim(Γ0) = \|A/E\| = 8 + 4 √ 2. Using the known value for \|A\| one obtains \|E\| = 16 ( 2 + √ 2 ) . • When k = 6, we have F = Γ0 = 000 ⊕ 060 ⊕ 202 ⊕ 222 so that dim(F) = dim(Γ0) = \|A/E\| = 20 + 8 √ 5. Using the known value for \|A\| one obtains \|E\| = 40 ( 5 + 2 √ 5 ) . • When k = 8, we have F = Γ0 = 000⊕121⊕141⊕412⊕214⊕800⊕080⊕008 so that dim(F) = dim(Γ0) = \|A/E\| = 12(9 + 5 √ 3). Using the known value for \|A\| one obtains \|E\| = 48(9 + 5 √ 3). Quantum dimensions for Ek(SU(4)), {k = 4, 6, 8}. Third method. The third method (which is probably the shortest, in the case of quantum graphs obtained from conformal em- beddings) does not even use the expression of the first modular block but it uses some gen- eral results and concepts from the structure of the graph of quantum symmetries O(Ek) that will be discussed in a coming section. In a nutshell, one uses the following known results: 1) \|O(Ek)\| = \|E\| × \|E\|/\|JO\| where JO denotes the set of ambichiral vertices of the Ocneanu graph, 2) \|Ak\| = \|O(Ek)\|, and 3) \|JO\| = \|JE \| where JE denote the sets of modular vertices of the graph Ek. Finally, one notices that for a case coming from a conformal embedding K = SU(4) ⊂ G, one can identify vertices c ∈ JE ⊂ E with vertices c ∈ J = A1(G). The conclusion is that one can first calculate \|J \| = ∑ s dim(s) as the mass of the small quantum group A1(G), and finally obtain \|Ek\| from the following relation12: \|Ek(K)\| = √ \|Ak(K)\| × \|A1(G)\|. Values for quantum cardinality of the (very) small quantum groups \|A1(G)\| at relevant13 va- lues of q are obtained in the appendix. One finds14 \|A1(Spin(15))\| = 4, \|A1(SU(10))\| = 10, \|A1(Spin(20))\| = 4 and we recover the already given results for \|Ek\|. Incidentally this provides another check that obtained branching rules are indeed correct. Remark 2. We stress the fact that the calculation of \|Ek\| can be done, using the second or the third method, before having determined the quantum graph Ek itself, in particular without using any knowledge of the quantum dimensions dim(a) of its vertices. Once the graph is known, one can obtain these quantum dimensions from a Perron–Frobenius eigenvector, then check the consistency of calculations by using induction, from the relation dim(a) = dim(Γa)/ dim(Γ0), and finally recover the quantum cardinality of E by a direct calculation (first method). 2 Algebras of quantum symmetries 2.1 General terminology and notations We introduce some terminology and several notations used in the later sections. 12In this paper G = SU(4) and K = Spin(15), SU(10), Spin(20) for k = 4, 6, 8, but this relation is valid for any case stemming from a conformal embedding. 13For a conformal embedding K ⊂ G, the value of q used to study Ak(K) is not the same as the value of q used to study A1(G), since q is given by exp(iπ/(k + g)): for instance one uses q12 = −1 for A8(SU(4)) but q19 = −1 for A1(Spin(20)). 14One should not think that \|J \| is always an integer: compute for instance \|A1(G2)\| which can be used to determine E8 = E28(SU(2)). However, \|A1(SU(g))\| = g. Quantum Symmetries for Exceptional SU(4) Modular Invariants 11 Fusion ring Ak: the commutative ring spanned by integrable irreducible representations m,n, . . . of the affine Lie algebra of SU(4) at level k, of dimension rA = (k + 1)(k + 2)(k + 3)/3!. Structure constants are encoded by fusion matrices Nm of dimension rA × rA: m · n =∑ p(Nm)np p. Indices refer to Young Tableaux or to weights. Existence of duals implies, for the fusion ring, the rigidity property (Nm)np = (Nm)pn, where m refers to the conjugate of the irreducible representation m. In the case of SU(4), we have three generators (fundamental irre- ducible representations): one of them is real (self-conjugated) and the other two are conjugated from one another. Ak acts on the additive group spanned by vertices a, b, . . . of the quantum graph Ek. This module action is encoded by annular matrices Fm: m · a = ∑ b(Fm)ab b. These are square matrices of dimension rE × rE , where rE is the number of simple objects (i.e., vertices of the quantum graph) in Ek. To the fundamental representations of SU(4) correspond particular annular matrices which are the adjacency matrices of the quantum graph. The rigidity property of Ak implies15 (Fn)ab = (Fn)ba. It is convenient to introduce a family of rectangular matrices called “essential matrices” [8], via the relation (Ea)mb = (Fm)ab. When a is the origin16 0 of the quantum graph, E0 is usually called “the intertwiner”. In general there is no multiplication in Ek, with non-negative integer structure constants, compatible with the action of the fusion ring. When it exists, the quantum graph is said to possess self-fusion. This is the case in the three examples under study. The multiplication is described by another family of matrices Ga with non negative integer entries: we write a · b =∑ c(Ga)bc c; compatibility with the fusion algebra (ring) reads m · (a · b) = (m · a) · b, so that (Ga · Fm) = ∑ c(Fm)ac Gc. The additive group Ek is not only a Z+ module over the fusion ring Ak, but also a Z+ module over the Ocneanu ring (or algebra) of quantum symmetries O. Linear generators of this ring are denoted x, y, . . . and its structure constants, defined by x · y = ∑ z(Ox)yz z are encoded by the “matrices of quantum symmetries” Ox. To each fundamental irreducible representation f of SU(4) one associates two fundamental generators of O, called chiral left fL and chiral right fR. So, O has 6 = 2× 3 chiral generators. Like in usual representation theory, all other linear gen- erators of the algebra appear when we decompose products of fundamental (chiral) generators. The Cayley graph of multiplication by the chiral generators (several types of lines), called the Ocneanu graph of Ek, encodes the algebra structure of O. Quantum symmetry matrices Ox have dimension rO× rO, where rO is the number of vertices of the Ocneanu graph. Linear generators that appear in the decomposition of products of left (right) chiral generators span a subalgebra called the left (right) chiral subalgebra. These two subalgebras are not necessarily commutative but the left and the right commute. Intersection of left and right chiral subalgebras is called the ambichiral subalgebra. The module action of O on Ek is encoded by “dual annular matrices” Sx, defined by x · a = ∑ b(Sx)ab b. From general results obtained in operator algebra by [29] and [4, 5, 6], translated to a cate- gorical language by [33], one shows that the ring of quantum symmetries O is a bimodule over the fusion ring Ak. This action reads, in terms of generators, m · x · n = ∑ y(Vmn)xy y, where m, n refer to irreducible objects of Ak and x, y to irreducible objects of O. Structure constants are encoded by the “double-fusion matrices” Vmn, with matrix elements (Vmn)xy, again non negative integers. To the fundamental representations f of SU(4) correspond particular double fusion matrices encoding the multiplication by chiral generators in O (adjacency matrices of the Ocneanu graph): Vf0 = OfL and V0f = OfR , where 0 is to the trivial representation of SU(4). One also introduces the family of so-called toric matrices Wxy, with matrix elements (Wxy)mn = (Vmn)xy. When both x and y refer to the unit object of O (that we label 0), one recovers the modular invariant M = W00 encoded by the partition function Z of the corresponding confor- 15For the SU(2) theory, this property excludes non-ADE Dynkin diagrams. 16A particular vertex of E is always distinguished. 12 R. Coquereaux and G. Schieber mal field theory. As explained in [34], when one or two indices x and y are non trivial, toric matrices are interpreted as partition functions on a torus, in a conformal theory of type Ak, with boundary type conditions specified by E , but with defects specified by x and y. Only M is modular invariant (it commutes with the generators s and t of SL(2, Z) given by Kac–Peterson formulae). Toric matrices were first introduced and calculated by Ocneanu (unpublished) for theories of type SU(2). Various methods to compute or define them can be found in [18, 8, 34]. Reference [9] gives explicit expressions for all Wx0, for all members of the SU(2) family (ADE graphs). Left and right associativity constraints (m ·(n ·x ·p) ·q) = (m ·n) ·x ·(p ·q) for the A×A bimodule structure of O can be written in terms of fusion and toric matrices. A particular case of this equality reads17 ∑ x(W0x)mn Wx0 = NmMN tr n . It was presented by A. Ocneanu in [31] and called the “modular splitting equation”. Another particular case of the bimodule associa- tivity constraints gives the following “intertwining equations”: ∑ y(Wxy)mn Wy0 = Nm Wx0 N tr n . A practical method to solve this system (matrix elements should be non-negative integers) is discussed in [21], with several SU(3) examples. Given fusion matrices Nm, known in general, and a modular invariant matrix M = W00, solving the modular splitting equation, i.e., finding the Wx0, and subsequently solving the intertwining equations, allows one to construct the chiral generators of O and obtain the graph of quantum symmetries (and the graph Ek itself, as a by- product). This is what we do in the next section, starting from the three exceptional partition functions of type SU(4) obtained previously. 2.2 Method of resolution (summary) The following program should be carried out for all examples: 1. Solve the modular splitting equation (find toric matrices Wx0). 2. Solve the intertwining equations (i.e., find generators Ox for the Ocneanu algebra and its graph O(Ek)), obtain the permutations describing chiral transposition. 3. Determine the quantum graph Ek (find its adjacency matrices). 4. Determine the annular matrices Fn describing Ek as a module over the fusion algebra Ak. 5. Describe the self-fusion on Ek (find matrices Ga). 6. Reconstruct O(Ek) in terms of Ek. 7. Determine dual annular matrices Sx describing Ek as a module over O(Ek). 8. Checks: reconstruct toric matrices from the previous realization of O(Ek), verify the rela- tion18, expressing Fn in terms of Sx, check identities for quantum cardinalities, etc. 9. Describe the two multiplicative structures of the associated quantum groupöıd B. 10. Matrix units and block diagonalization of Ek and O(Ek). 11. Check consistency equations (self-connection on triangular cells). Determination of the toric matrices Wz0. These matrices, of size rA× rA, are obtained by solving the modular splitting equation. This is done by using the following algorithm. For 17Equivalently, one can write (Nσ⊗Nτ )Mστ = P1324 ∑ x Wx⊗Wx where Wx = Wx0 and P1324 is a permutation. 18 Since O(Ek) is an Ak bimodule, we obtain in particular two algebra homomorphisms from the later to the first (this should coincide with the notion of “alpha induction” introduced in [28, 42] and used in [5]): αL(m) = m 0Oc 0A and αR(m) = 0A 0Oc m, for m ∈ Ak. They can be explicitly written in terms of toric matrices: αL(m) = ∑ y(W0y)m0 y and αR(m) = ∑ y(W0y)0m y. If we compose these two maps with the homomorphism S from O(Ek) to Ek, described by dual annular matrices, we obtain a morphism from Ak to Ek that has to coincide with the one defined by annular matrices, so that we obtain the identity Fm = ∑ y(W0y)m0 Sy = ∑ y(W0y)0m Sy that implies, in particular dm = ∑ y(W0y)m0 dy. Quantum Symmetries for Exceptional SU(4) Modular Invariants 13 each choice of the pair (m,n) (i.e., r2 A possibilities), we first define and calculate the matrices Kmn = NmMN tr n . The modular splitting equation reads: Kmn = rO−1∑ x=0 (W0x)mn Wx0. (2) It can be viewed as the linear expansion of the matrix Kmn over the set of toric matrices Wx0, where the coefficients of this expansion are the non-negative integers (W0x)mn, and rO = Tr(MM†) is the dimension of the quantum symmetry algebra. This set of equations has to be solved for all possible values of m and n. In other words, we have a single equation for a huge tensor K with r2 A × r2 A components viewed as a family of r2 A vectors Kmn, each vec- tor being itself a rA × rA matrix. In general the family of toric matrices is not free: the rO toric matrices Wz0 are not linearly independent and span (like matrices Kmn) a vector space of dimension rW < rO; this feature (related to the possible non-commutativity of O(Ek)) ap- pears whenever the modular invariant M has coefficients bigger than 1. Toric matrices Wz0 are obtained by using the following iterative algorithm already used in [21, 11]. The alge- bra of quantum symmetries comes with a basis, made of the linear generators that we called x, which is special because structure constants of the algebra, in this basis, are non negative integers. We define a scalar product in the underlying vector space for which the x basis is orthonormal, and consider, for each matrix Kmn, and because of equation (2), the vector∑ x(W0x)mn, x ∈ O, whose norm, abusively19 called norm of Kmn and denoted \|\|Kmn\|\|2, is equal to ∑ x \|(W0x)mn\|2. The relation Vmn = (Vmn)tr (see later) and the modular splitting equation imply that \|\|Kmn\|\|2 = (Kmn)mn = ∑ p,q(Nm)mp (M)pq (N tr n )qn, i.e., for each m, n, this norm can be directly read from the matrix Kmn itself. The next task is therefore to calculate the norm of the matrices Kmn. The toric matrices Wx0 themselves are then obtained by considering matrices Kmn of increasing norms 1, 2, 3, . . .. For example those of norm 1 immediately define toric matrices (since the sum appearing on the r.h.s. of the modular splitting equation involves only one term), in particular one recovers W00 = M. Then we analyse those of norm 2, and so on. The process ultimately stops since the rank rW is finite. A case dependent complication, leading to ambiguities in the decomposition of Kmn, stems from the fact that the family of toric matrices is not free (see our discussion of the specific cases). In order to ease the discussion of the resolution of the equation of modular splitting, it is convenient to introduce the following notations and definitions. We order the i-irrep as in Section 1.4.1 and call m# the position of m, so that {0, 0, 0}# = 1, {1, 0, 0}# = 2, . . .. For each possible square norm u, we set Ku = {Kmn/\|\|Kmn\|\|2 = u}; notice that this is defined as a set: it may be that, for a given M ∈ Ku, there exist distinct pairs (m,n), (m′, n′) such that M = Kmn = Km′n′ , but this matrix appears only once in Ku. The tensor K is a square array of square matrices of dimension r2 A × r2 A. Lines and columns are ordered by using the previously given ordering on the set of irreducible representations. We scan K from left to right and from top to bottom. This allow us to order the sets Ku: for a given M ∈ Ku we take note of its first occurrence, i.e., the number Inf{(m# − 1)rA + n#} over all {(m,n)} such that M = Kmn, this defines a strict order on the set Ku (use the fact that m# < rA and n# < rA). We can therefore refer to elements M of Ku by their position v, and we shall write M = Ku[v]. Conjugations. Complex conjugation is defined on the set of irreducible representations of SU(4) which, in terms of fusion matrices, reads Nm = N tr m. At the level of the tensor square, this star representation defined by (Vmn) = Vm n reads Vm n = (Vmn)tr since all matrices have non negative integral entries (no need to take conjugate of complex numbers). In terms of toric matrices, this implies (Wxy)m n = (Wyx)mn. We have also a conjugation (“bar operation”) 19Indeed, it may happen that two toric matrices Wx0, Wy0 appearing on the r.h.s. of (2) are equal, even though x 6= y. 14 R. Coquereaux and G. Schieber x 7→ x on the algebra of quantum symmetries, that maps toric matrices to toric matrices, Wx = Wx0 7→ Wx = W0x. More generally we set Wx y = Wyx. Real generators are defined by the property x = x, in that case we have Wx0 = W0x, i.e., Wx = Wx. Toric matrices are usually not symmetric: transposition is not trivial, but it leaves in- variant the set of toric matrices and induces an operation called “chiral transposition”20, de- noted x 7→ xc on the algebra of quantum symmetries. It reads Wxcyc = (Wxy)tr. In par- ticular (Wxc0)m n = (Wx0)nm. Using αL,R morphisms introduced in footnote 18, it reads αL(m)c = αR(m). Symmetric generators are defined by the property x = xc, in that case the corresponding toric matrices are symmetric: Wx = (Wx)tr. It is this operation that maps chiral left to chiral right generators: (fL)c = fR. The operation x 7→ x† obtained by composing the above two operations is called “chiral adjoint”. It is such that (m xn)† = n x† m. In terms of toric matrices, it reads (Wx†y†)m n = (Wyx)nm. Self-adjoint (or hermitian) generators are defined by the property x = x†, in that case Wx0 = (W0x)tr, i.e., Wx = (Wx)tr. Ambichiral generators (remember that they span a subalgeb- ra J defined as intersection of the left and right chiral subalgebras) can be recognized as those self-adjoint generators whose corresponding vertices belong to the first connected component21 in the graph of quantum symmetries. We summarize the above discussion by the following collection of equalities: Wx†y† = Wycxc = (Wx y)tr = (Wyx)tr. For each of the above three conjugations, one can introduce a permutation matrix acting on the set of generators of O, intertwining between x and x, xc or x†. Determination of the conjugations is not straightforward when rW < rO. What we do is to parametrize the solutions found after analysis of the set of toric matrices and we use them to solve the set of intertwining equations (see next paragraph). Imposing that the obtained generators of O obey the expected constraints (see later) restrict the possible choices for the conjugations, and ultimately fixes all free parameters, up to possible graph automorphisms. In many cases, and in particular in the three exceptional cases that we consider in this article, one can realize the algebra of quantum symmetries O as a quotient, over the ambichiral subalgebra, of an algebra defined in terms of the tensor square of the algebra of the quantum graph E (in simple cases, O can be identified with E ⊗E/J ). Using this realization, i.e., writing x = a ⊗ b, the above three operations read: x = a ⊗ b, xc = b ⊗ a and x† = b ⊗ a. Actually the conjugation a 7→ a in E (we could very well choose E = A) can be deduced from the same operation in O via the identification a ' a⊗ 1l. Solving the intertwining equations. The family of toric matrices “with one twist”, i.e., the Wx0 matrices, was determined in a previous step, but we should determine all the matri- ces Wxy. For each triplet (m,n, x), we define the matrices Kx mn = Nm Wx0 Nn tr and calculate them. The intertwining equations (one matrix equation for each triplet) read: Kx mn = ∑ y (Wxy)mn Wy0. It can be viewed as the linear expansion of the matrix Kx mn over the set of toric matrices Wx0, where the coefficients of this expansion are the non-negative integers (Wxy)mn = (Vmn)xy, that we want to determine. In order to find the algebra of quantum symmetries and its graph, it 20Chiral transposition may be related to the “conjugation of defect lines”, as in [19]. This operation is not a priori defined for all CFT’s, but its existence, along with the invariance property of the set of toric matrices under transposition is, for the cases studied here, an observational fact, consequence of our explicit determination of toric matrices, and quantum symmetry generators. 21It is defined as the connected component of the graph of quantum symmetries, using the generator {100}, that contains the identity of the algebra O, whose corresponding toric matrix is the modular invariant M = W00. Quantum Symmetries for Exceptional SU(4) Modular Invariants 15 is enough to solve only those equations involving the six chiral generators22, i.e., to determine the matrices Vf0 = OfL and V0f = OfR , where f refer to the three fundamental representations of SU(4), f = {100}, f = {010}, f = {001}. In other words, we solve the intertwining equations Nf Wx0 N0 tr = ∑ y (OfL)xy Wy0 and N0 Wx0 Nf tr = ∑ y (OfR)xy Wy0. When the toric matrices Wy0 are linearly independent, the linear expansions of Kx f0 and Kx 0f are unique and the determination of the chiral generators OfL and OfR is straightforward (for any chosen f one sets the elements of matrices Of to unknown parameters and solves a system of linear equations in rO 2 unknowns). But in general the family of toric matrices is not free, and even after imposing that matrix coefficients of OfL and OfR should be non-negative integers, we are still left with a solution with many free parameters. Some of them are determined by imposing the bar-conjugation relations: O f L = (OfL)tr (respectively O f R = (OfR)tr). In particular, for the real generator f = {010} = f , matrices of the corresponding left and right chiral generators should be symmetric. Other parameters are determined by imposing the chiral transposition relations (fL)c = fR, so that OfR = P OfL P−1 where P is the permutation matrix implementing chiral transposition (so P 2 = 1); it can be obtained from our knowledge of toric matrices since an equality y = xc among the generators of O implies Wy = (Wx)tr. The operation c (or the matrix P ) is usually not fully determined at that stage since it may happen that two distinct generators x and y are represented by identical toric matrices Wx = Wy. One can then enforce the fact that left and right fundamental generators Of should commute and that they should commute with their complex conjugates (this does not imply that the algebra O is commutative (see remark in Section 2.1). However, some free parameters can still remain. In order to determine their value, we proceed as follows. First of all, we notice that fusion matrices N1, N2 and N3 obey non-trivial polynomial relations (see below) reflecting the fact that the fusion ring Ak(SU(4)) is a quotient of the representation ring of SU(4). Since O and Ek are modules over the fusion ring, the same relations have to be satisfied by the corresponding generators OfL,R and Gf . In general these equations allow us to determine the remaining parameters but it may be (see for instance our discussion of the E8 case in Section 2.6) that the final solution, after that last step, is not unique; however, in our cases, this reflects the existence of possible automorphisms23 of the graph of quantum symmetries. In the case of SU(4) at level k, one can use three non-trivial polynomial relations yk+1 = 0, yk+2 = 0, yk+3 = 0 expressing the fact that irreducible representations associated with weights {k + 1, 0, 0}, {k + 2, 0, 0} and {k + 3, 0, 0} do not exist in Ak(SU(4)) (in terms of quan- tum groups at roots of unity, they correspond to representations with vanishing quantum dimension). Setting x1 for {100}, x2 for {010}, x3 for {001}, the polynomial ys can be ex- pressed as the determinant of a square matrix s × s, whose line number j is given by the vector . . . , 0, 1, x1, x2, x3, 1, 0, . . ., which should be truncated in such a way that x1 belongs to the diagonal (for instance, line number 1 of y6 is (x1, x2, x3, 1, 0, 0), line number 6 of y7 is (0, 0, 0, 0, 1, x1, x2), etc). This property (Giambelli formula) is a consequence of the Littlewood– Richardson rule. Remark: One can always eliminate x2 between yk+1 = 0, yk+2 = 0, yk+3 = 0 and express x3 as a (rational) polynomial in x1; one can instead eliminate x3 and find a poly- nomial relation between x1 and x2 but one cannot express polynomially x2 in terms of x1 (it is known [12] that this is never possible for a fusion ring of SU(g) when g and the chosen level k are both even). We therefore use the vanishing of y5, y6, y7 for E4, of y7, y8, y9 for E6 and of y9, 22Remember that the notation OfL does not refer to the chiral adjoint of OfR but to its chiral transpose. 23These are permutations π on vertices of O such that for all vertices, (π(x), π(y)) is an edge iff (x, y) is an edge. 16 R. Coquereaux and G. Schieber y10, y11 for E8 as a tool to determine the remaining parameters. In the case of E4 for instance, these polynomial relations read as follows: y5 = x1 5 − 4x2x1 3 + 3x3x1 2 + 3x2 2x1 − 2x1 − 2x2x3 = 0, y6 = x1 6 − 5x2x1 4 + 4x3x1 3 + 6x2 2x1 2 − 3x1 2 − 6x2x3x1 − x2 3 + x3 2 + 2x2 = 0, y7 = x1 7 − 6x2x1 5 + 5x3x1 4 + 10x2 2x1 3 − 4x1 3 − 12x2x3x1 2 − 4x2 3x1 + 3x3 2x1 + 6x2x1 + 3x2 2x3 − 2x3 = 0. Eliminating for example x3, one finds that x1x2 should be equal to 1 55611516017584128 (x1 3(1572913848761 x1 28 − 101219273794784 x1 24 + 1519972607520288 x1 20 − 10071512027614400 x1 16 − 12849609824079344 x1 12 + 189817789697417216 x1 8 − 183010445962251264 x1 4 − 16377652617161728)). Once the matrices describing the fundamental generators have been fully determined, up to possible graph automorphisms, we want to be able of giving explicitly the permutations describ- ing the complex conjugation, the chiral transposition and the chiral adjoint operations (the last one being the composition of the first two). We remember, however, that there is still some freedom in the determination of these permutations. For example if P is a matrix implementing the chiral transposition (so OfR = POfLP−1), and if U is a permutation matrix commuting both with OfL and OfR and such that UŨ = 1, we find another acceptable “chiral matrix” P ′ by setting P ′ = P U . We shall restrict as follows the possible choices for P : whenever x 6= y are two distinct vertices of the graph O for which the associated toric matrices are both symmetric and equal, we decide that x and y should be fixed (rather than interchanged) by the operation c. From the knowledge of the six chiral generators, we can draw the two chiral subgraphs making the Ocneanu graph of quantum symmetries. There are at least three ways to draw such a graph. The first one uses rO vertices and one type of line for each chiral generator; this is still readable in the SU(2) situation but not in our case, where we have six types of lines (actually four: two oriented ones, and two unoriented ones); another method (see the article [11] as an example) draws only the left graph that describes multiplication of an arbitrary vertex by a chiral left generator; chiral conjugated vertices are then related by a dashed line so that multiplication by chiral right generators is obtained by conjugating the left multiplication. In this paper, we shall use a third solution, that we find more readable: we only display the graphs describing the multiplication by left generators (see Figs. 1, 2 and 3), with some arbitrary labeling of the vertices, but we give, for each case, the permutation describing the chiral adjoint operation. This allows the reader to obtain easily the multiplication by the bar-conjugated of the right generators, from the relations O f R = Q−1 OfL Q where Q is the matrix implementing the chiral adjoint operation. About 4-ality. We have Z4 grading τ (4-ality) defined on the set of irreps, such that τ(λ) = −τ(λ) mod 4 given by τ(λ1, λ2, λ3) = λ1 + 2λ2 + 3λ3 mod 4. It is also obtained from the corresponding Young tableau by calculating the number of boxes modulo 4. This 4-ality defined on vertices of Ak induces a Z4 grading in the modules Ek, and in O. It will be used to display their corresponding graphs. Determination of the quantum graph E. In all three cases, it is obtained as one particu- lar component of the left (or right) graph of quantum symmetries O, where it coincides with the left (or right) chiral subgraph (this property is not generic but holds for those quantum graphs obtained from direct24 conformal embedding). Other components of O describe other quantum graphs (that do not have self-fusion in general) but are modules for E , and of course for Ak as well. The graph E is obtained as the union of three graphs Gf (sharing the same vertices but with different types of edges) defined by (three) adjacency matrices also denoted Gf read from the adjacency matrices OfL (or OfR) of O. The graph Gf is connected for f = {100} (or {001}) 24i.e., not followed by a contraction with respect to some simple component of the possibly non simple group K under study. Quantum Symmetries for Exceptional SU(4) Modular Invariants 17 but not, in general, for {010}. The fact that E has self-fusion, not necessarily commutative since it is isomorphic with the chiral subalgebras, follows from the multiplicative structure of O. Obtaining annular matrices Fn is now straightforward since they obey the same recurrence relations as the fusion matrices Nn of SU(4), but with a different seed, namely F000 = IrE and Ff = Gf . We shall not give explicitly these matrices for reasons of size (only Gf will be given), but the fact that their calculated matrix elements turn out to be non-negative integers, as they should, provides a compatibility check of the previous determination of the quantum graphs: indeed, any mistake in one of the adjacency matrices Gf usually induces the appearance of some negative integer coefficients in one or several of the Fn’s. What else is to be found, or not to be found, in the coming sections. We have determined the toric structure (i.e., all toric matrices Wx0) for all three cases, using the modular splitting equation. This was a necessary step towards the determination of chiral generators for the graph of quantum symmetries. However, displaying for instance these 192 matrices of size 165×165 (the case of E8(SU(4)) in a printed form is out of question. In order to keep the size of this paper reasonable, we shall only describe the structure of the chiral generators, by displaying the graphs of OfL and the permutation P that implements chiral transposition and allows one to reconstruct the graphs of OfR . Matrices OfL,R are adjacency matrices of those graphs. We shall not describe the full multiplicative structure of O in terms of linear generators; this was done for E4 in [11]. For the same reason we shall not give the dual annular matrices Sx. Once the quantum graph itself is known (adjacency matrices Gf ), it is possible to “reverse the machine” and realize explicitly the algebra O in terms of the algebra E : it is a particular quotient of its tensor square. Using then the annular matrices Fn and the realization of generators of O as tensor products, there is a way to check that our determination of toric matrices Wx0 was indeed correct. This was done explicitly [11] in the case of E4, and can be done for the other graphs along the same lines. This analysis will not be repeated here. Along general lines discussed in [30], one can associate a quantum groupöıd B to every quan- tum graph E . More precisely, B is a finite dimensional weak Hopf algebra which is simple and co-semisimple. One can think of the algebra B as a direct sum of rA matrix simple components, and of its dual, the algebra B̂, as a sum of rO matrix simple components. The dimensions dn (and dx) of these blocks, called horizontal or vertical dimensions, or dimensions of generalized spaces of essential paths, or spaces of admissible triples or generalized triangles, etc., can be obtained from the annular (or dual annular) matrices. We denote by dH = ∑ n dn the total horizontal dimension. We shall not provide more details about the structure of this quantum groupöıd in the present paper but its total dimension dB = ∑ n d2 n = ∑ x d2 x will be given in each case. We calculated the quantum dimensions of simple objects of Ek in two possible ways: using spectral properties of the adjacency matrix F{100}, obtained as a by-product of the determination of the graph of quantum symmetries, and using induction/restriction from the fusion algebra A4. The quantum cardinality \|Ek\|, already obtained at the end of the previous section, is then recovered by summing the squares of these quantum dimensions. This provides a non trivial check of the calculations. Real-ambichiral partition functions: As it was recalled already, all vertices of an Ocneanu graph are associated with partition functions. Among them, only one (Z1, associated with the origin) is modular invariant: it commutes with s and t. The others are not, although they all commute with s−1ts. It would be rather heavy to give tables for all of them, and the reader can certainly obtain these results by using the provided information (they can also be obtained from the authors, if needed). However we shall give explicit expressions for partition functions associated with those vertices that are both ambichiral (i.e., x is such that it belongs to the first connected component of the graph O and such that x = x†) and real (i.e., x = x). There are only three vertices of that type for E4, two for E6 (not ten25) and four for E8. 25They coincide with ambichiral vertices for E4 and E8, but not for E6. 18 R. Coquereaux and G. Schieber In some cases, exceptional modules can be found among the connected components of the graph of quantum symmetries of a quantum graph with self-fusion. They provide new quan- tum graphs, in general without self-fusion, and they can be themselves associated with modular invariant partition functions (they may be new or not). At this point we also discuss possible conjugate invariants by using the permutation matrix C of size rA × rA that intertwines repre- sentations n and n of Ak(SU(4)). This matrix can also be considered as the modular invariant matrix associated with the conjugated quantum graph Ak c. See our discussion in the different cases. 2.3 Tables for Ek From the modular invariant, we read immediately the following: rA = (k + 1)(k + 2)(k + 3)/3!, rE = Tr(M), rO = Tr(M†M), rW = #{(i, j)/Mij 6= 0}. We gather in the following table the values of rA (number of i-irreps i.e., number of vertices of the graph Ak ), rE (number of vertices of the graph Ek), rO (number of vertices of the graph of quantum symmetries), rW (rank of the family of toric matrices, in general rW < rO), ν(Kmn) (number of distinct norms for the matrices Kmn relative to the equations of modular splitting), dH (total horizontal dimension), dB (dimension of the associated quantum groupoid), and quantum cardinalities \|Ek\|. rA rE rO rW ν(Kmn) dH dB \|Ek\| E4 35 12 48 33 17 2572 2527131 16(2 + √ 2) E6 84 32 112 100 46 2651731 2952131531 40(5 + 2 √ 5) E8 165 24 192 144 142 26711791 21369971 48(9 + 5 √ 3) Specific details concerning the different cases are given in the following subsections. 2.4 Oc(E4(SU(4)) We consider the matrices Kmn, keeping only those that are distinct, corresponding to each one of the possible norms26 u. For instance there are 8 (distinct) matrices in norm 1 (i.e., #(K1) = 8), 11 in norm 2 (i.e., #(K2) = 11), then 8, 5, 6, 12, 3, 2, 4, 2, 4, 6, 4, 2, 2, 4, 1 of them for the next possible norms. A first analysis gives immediately 8 toric matrices in norm 1, therefore all elements of K1, in particular K1[1] = W00 = M , then we find 11 new ones in norm 2 (with multiplicity 2), 4 others in norm 3 (with multiplicity 2). Elements of matrices K of norm 4 are multiple of 4, so these matrices are either a sum of 4 toric matrices (the same toric matrix but with multiplicity 4), or 2 times a toric matrix with elements multiple of 2. As the total number of toric matrices is limited (equal to 48), we select the second possibility, and therefore we find 5 toric matrices in norm 4 (with elements multiple of 2 and multiplicity 1), then with the same arguments we find 4 others in norm 6 (with elements multiple of 2 and multiplicity 1) and finally the last toric matrix in norm 8 (again with elements multiple of 2 and multiplicity 1). All other equations, for the 17 possible norms, are then satisfied, and we check that the equation of modular splitting, itself, holds. Altogether we have therefore 18 = 8 + 5 + 4 + 1 toric matrices with multiplicity 1 and 15 = 11+4 toric matrices with multiplicity 2. The total number of toric matrices is 18 + 15 + 15 = 48, as it should, but the rank is only 18 + 15 = 33, as expected. The next step is to solve the intertwining equations that determine the 6 matrices expressing the generators of O, using the methods described in the previous section. As the family of toric 26u = 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 16, 18, 32, 40, 48, 128. Quantum Symmetries for Exceptional SU(4) Modular Invariants 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 4-ality 0 1 2 3 * Figure 1. The left chiral graph of quantum symmetries Oc(E4). Multiplication by the left chiral generator 100 labeled 5 (resp. 001 labeled 10) is encoded by oriented red edges (thick lines), in the direction of increasing (resp. decreasing) 4-ality. Multiplication by 010 labeled 8 is encoded by unoriented blue egdes (thin lines). matrices is not free (rw < rO), there are some free parameters left in these matrices. Elementary considerations bring their number down to 4 for OL 100, and 4 for OL 010; each of them (say α) could a priori have values equal to 0, 1 or 2 because matrix elements such as α, 1−α, 2−α do appear in matrices Of , but requiring that polynomials y5, y6 and y7 should vanish imposes that all of these coefficients α are equal to 1. Generators OL,R f are then fully determined. We display in Fig. 1 the graph (with 48 vertices) describing the multiplication by the chiral left generators. Multiplication by 100 (resp. 001) is encoded by oriented red edges (thick lines), in the direction of increasing (resp. decreasing) 4-ality, and multiplication by 010 is encoded by unoriented blue edges (thin lines). The identity in O is marked with a star on the graph. The vertex representing the fundamental left generator of type f is the neighbour of the identity along the corresponding edge (of type f) in the left chiral graph of quantum symmetry. The chiral adjoint operation (that interchanges matrices OfL and O f R) is given by the following table, where we list only the non trivial pairs that are interchanged by this operation: x 3 4 5 6 7 8 10 11 12 17 18 19 22 23 24 30 31 34 x† 13 14 33 37 38 21 45 25 26 32 39 40 44 27 28 47 48 41 There are 12 self-adjoint generators (x = x†), ambichiral ones are 1, 2 and 9. 20 R. Coquereaux and G. Schieber Adjacency matrices of E4. We order vertices x of the left quantum graph in such a way that connected components are separated in blocks. With this choice, the left chiral generator ma- trices take the following block diagonal form (see also [11]): OL 100 = diag(G100, G100, G100, G100), OL 010 = diag(G010, G010, G010, G010). G100 is obtained from the decomposition of OL 100 (or of OR 100) in connected components, and G010 from the decomposition of OL 010 (or of OR 010). For the Gf matrices, we order the basis elements by increasing 4-ality G100 =  . . . . 1 . . . . . . . . . . . 1 . . . . . . . . . . . 1 1 1 . . . . . . . . . 1 1 1 . . . . . . . . . . . . 2 . . . . . . . . . . . 1 1 . . . . . . . . . . 1 1 . . . . . . . . . . . . 2 1 1 . . . . . . . . . . 1 1 1 1 1 1 . . . . . . . . . . 1 1 . . . . . . . . . . 1 1 . . . . . . . .  , G010 =  . . . . . . . 1 . . . . . . . . . . . 1 . . . . . . . . . . . 2 1 . . . . . . . . . . 2 1 . . . . . . . . . . . . 2 1 1 . . . . . . . . . 1 1 1 . . . . . . . . . 1 1 1 1 1 2 2 . . . . . . . . . . 1 1 . . . . . . . . . . . . 2 1 1 . . . . . . . . . 1 1 1 . . . . . . . . . 1 1 1 . . . . .  . It is now easy to determine the annular matrices Fn, the horizontal dimensions dn, the total horizontal dimension dH and the dimension dB of the quantum groupöıd B(E4). One can also calculate the quantum dimensions of simple objects of E4 and check the already obtained value for the quantum cardinality \|E4\|. Results are summarized in the table of Section 2.3. Real-ambichiral partition functions of E4 (i.e., x = x, x = x† and x ∈ J ). Here, such x’s coincide with ambichiral ones. Setting U = 004+101+121+400, V = 000+012+040+210, W = U + V and X = 111, we get Z1 = 4X X + UU + V V (the modular invariant partition function Z), Z2 = 4X X + UV + UV, Z9 = 2XW + 2 WX. No exceptional module for E4. Since the permutation matrix C commutes with s and t, we may think of considering the new modular invariant matrix Mc = CM. However, in this particular case, Mc = M, and we do not discover any new invariant in this way. Moreover the graph O of quantum symmetries contains only copies of the quantum graph E4. So we do not find any exceptional module in this case. 2.5 Oc(E6(SU(4)) We consider the matrices Kmn, keeping only those that are distinct, corresponding to each one of the possible norms27 u. For instance there are 30 (distinct) matrices in norm 1 (i.e., #(K1) = 30), then 104 in norm 2 (i.e., #(K2) = 104), then 32, 130, 26, 50, 70, 64, 44, . . . , 2 of them for the other possible norms. A first analysis gives immediately 30 toric matrices in norm 1, therefore all elements of K1, in particular K1[1] = W00 = M, then we find 36 new ones in norm 2 (many cases remaining unsettled at that stage), 4 other in norm 3 and 2 in norm 12 (but with multiplicity 5), therefore a total of 72 independent ones. A refined analysis of the norm 2 case gives us 28 more toric matrices, so that we have now reached the expected rank rW = 100. We are still missing 12 = rO − rW others that should not be independent of those already found. 8 (= 2× 5− 2) among them are immediately obtained from the fact that those coming from the norm 12 analysis had multiplicity 5. The last four are harder to find and are obtained after a deeper analysis of the norm 2 case: they can be expressed as differences between a matrix belonging to the set K2 and a toric matrix previously determined in our analysis of the norm 3 case (their matrix elements are nevertheless non-negative integers, of course). We then verify that all equations for the 46 possible norms can be satisfied with the obtained 112 toric matrices, and that the equation of modular splitting, itself, holds. 27u = 1, 2, 3, 4, 5, 6, 8, 10, 12, . . . , 326. Quantum Symmetries for Exceptional SU(4) Modular Invariants 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 6263 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 9495 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 4-ality 0 1 2 3 * Figure 2. The left chiral graph of quantum symmetries Oc(E6). Multiplication by the left chiral generator 100 labeled 11 (resp. 001 labeled 27) is encoded by oriented red edges (thick lines), in the direction of increasing (resp. decreasing) 4-ality. Multiplication by 010 labeled 17 is encoded by unoriented blue edges (thin lines). The next step is to solve the intertwining equations that determine the 6 matrices expres- sing the generators of O using the methods described previously. Here again there are several free parameters still remaining after resolution of these equations, but they are subsequently determined by using non negativity and integrality of coefficients, commutation properties of generators, and imposing that the SU(4) polynomials y7, y8, y9 should vanish. We display in Fig. 2 the graph (with 112 vertices) describing the multiplication by the chiral left generators, where we adopt the same conventions as for the E4 case. The graph of quantum symmetries contains 4 connected components, the quantum graph E6 appears 3 times, and a module, that we call E c 6, appears once. The chiral adjoint operation (that interchanges matri- ces OfL and O f R) is given by the following table, where we list only the non trivial pairs that are interchanged by this operation, x 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 27 28 29 30 x† 86 87 88 89 90 58 54 55 56 57 97 65 66 67 68 69 33 34 35 36 x 31 32 38 39 40 41 42 48 49 50 51 52 53 64 80 96 x† 37 106 76 77 78 79 75 107 91 92 93 94 95 99 105 98 There are 40 self-adjoint vertices (x = x†), ambichiral ones are (1, 2, 3, 4, 5, 22, 23, 24, 25, 26). 22 R. Coquereaux and G. Schieber Adjacency matrices of E6 and Ec 6. We order vertices x of the left quantum graph in such a way that connected components are separated in blocks. The left chiral gene- rator matrices take the following block diagonal form: OL 100 = diag(G100, G100, G100, G100 c), OL 010 = diag(G010, G010, G010, G010 c). Matrices G100 and G100 c are obtained from the decompo- sition of OL 100 (or of OR 100) in connected components, G010 and G010 c from the decomposition of OL 010 (or of OR 010). For the Gf matrices, we order the basis elements by increasing 4-ality. For the E6 graph we obtain: G100 =  . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 . . . . 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . .  , G010 =  . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 2 1 . . . . 1 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . . 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 2 . . . . . . . . . . . . . . . .  and for the E c 6 graph: Gc 100 =  . . 1 . . . . . . . . . . . . . . . 2 1 1 1 1 1 . . . . . . . . . . . . . . . . 2 1 . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 2 1 1 1 1 1 . . . . . . . . . . 1 . . . . . 1 2 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . .  , Gc 010 =  . . . . . . . . 1 . . . . . . . . . . . . . . . 4 1 . . . . . . . . . . . . . . . . 2 1 1 1 1 1 . . . . . . . . . . 1 . . 1 1 . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . 1 . 1 1 . . 1 4 . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 2 1 1 1 1 1 . . . . . . . . . . 1 . . 1 1 . . . . . . . . . . . 1 . . . 1 1 . . . . . . . . . . 1 1 . . . 1 . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . 1 . 1 1 . . . . . . . . . .  . Quantum Symmetries for Exceptional SU(4) Modular Invariants 23 It is now easy to determine the annular matrices Fn, the horizontal dimensions dn, the total horizontal dimension dH and the dimension dB of the quantum groupöıd B(E6). One can also calculate the quantum dimensions of simple objects of E6 and check the already obtained value for the quantum cardinality \|E6\|. Results are summarized in the table of Section 2.3. Real-ambichiral partition functions of E6 (i.e., x = x, x = x† and x ∈ J ). Here, there are only two such x’s, but 10 ambichiral vertices. Setting U1 = 000 + 060 + 202 + 222, U2 = 006 + 022 + 220 + 600, V1 = 042 + 200 + 212, V2 = 012 + 230 + 303, V3 = 030 + 103 + 321, V4 = 004 + 121 + 420, V5 = 030 + 123 + 301, V6 = 024 + 121 + 400, V7 = 032 + 210 + 303, V8 = 002 + 212 + 240 we obtain Z1 = U1U1 + U2U2 + V1V1 + V2V2 + V3V3 + V4V4 + V5V5 + V6V6 + V7V7 + V8V8, Z24 = U1U2 + U2U1 + V1V4 + V4V1 + V2V5 + V5V2 + V3V7 + V7V3 + V6V8 + V8V6 and in particular we recognize the modular invariant Z = Z1. An exceptional module for E6. Since the permutation matrix C commutes with s and t, we may think of considering the new modular invariant matrix Mc = CM. Here, Mc 6= M and we find a new invariant. Using the above notations, the corresponding partition function Zc = ∑ λ χλMc λµ χ̄µ, which is of type-II, (the modular invariant is not a sum of blocks) reads: Z1 c = U1U1 + U2U2 + V1V8 + V2V7 + V3V5 + V4V6 + V5V3 + V6V4 + V7V2 + V8V1. Its own quantum graph, denoted E6 c appears as a module in the graph of quantum symmetries of O(E6). It has 16 = Tr(Mc) = rE/2 vertices. One can then study it directly, i.e., determine its own annular matrices, its own algebra of quantum symmetries etc. Since the quantum graph E6 c is known from the very beginning (its adjacency matrices Gf c, given previously, are obtained from the connected components of O(E6) which are not of type E6), the analysis is much easier than for E6. The A module structure of E6 and E6 c differ (not the same annular matrices, of course). One finds dH = ∑ n dn = 11456 = 261791 and dB = ∑ n dn 2 = 2152960 = 2951292. Notice that E6 c has no self-fusion. As expected, O(E6) and O(E6 c) are isomorphic algebras, but their realizations, in terms of the graph algebra of E6, are different. 2.6 Oc(E8(SU(4)) We consider the matrices Kmn, keeping only those that are distinct, corresponding to each one of the possible norms28 u. For instance there are 63 (distinct) matrices in norm 1 (i.e., #(K1) = 63), then 48 in norm 2 (i.e., #(K2) = 48), then 38, 71, 25, 36, 26, 60, 16, 18, 32, 38, 30, 36, 9, . . . of them for the other possible norms. A first analysis gives immediately 63 toric matrices in norm 1, therefore all elements of K1, in particular K1[1] = W00 = M , then we find 48 new ones in norm 2 (among them, 23 appear with multiplicity 2), 8 others in norm 3 (all of them with multiplicity 2), 16 in norm 4 (among them, 12 with multiplicity 2, and 8 entries remain unsettled cases at that stage), 3 in norm 5 (among them, 3 with multiplicity 2), 4 in norm 6 (no multiplicities), 1 in norm 7 (with multiplicity 2), nothing in norms 8, 9, 10, 11, 12, 14, but 1 in norm 15 (with multiplicity 2) so that we reach a total of 63+48+8+16+3+4+1+1 = 144 = rW , the expected rank. After having checked that the 8 unsettled cases remaining in length 4 can be reexpressed in terms of the others, we take into account the already determined multiplicities and see that 28 u = 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, . . . , 4096. 24 R. Coquereaux and G. Schieber 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 * 4-ality 0 1 2 3 Figure 3. The left chiral graph of quantum symmetries Oc(E8). Multiplication by the left chiral genera- tor 100 labeled 7 (resp. 001 labeled 19) is encoded by oriented red edges (thick lines), in the direction of increasing (resp. decreasing) 4-ality. Multiplication by 010 labeled 13 is encoded by unoriented blue edges (thin lines). 23+8+12+3+1+1 = 48, so that we can easily complete the family since 144+48 = 192 = rO. We then verify that all equations for the 142 possible norms can be satisfied with the obtained 192 toric matrices, and that the equation of modular splitting, itself, holds. This case E8 involves huge computations, compared with the previous cases (the tensor K contains 40869849 non-zero entries), but, fortunately, the resolution of the equation of modular splitting is somehow easier than in the E6 case (in particular, the determination of the rO − rW matrices needed to complete the family). The next step is to solve the intertwining equations that determine the 6 matrices expressing the fundamental generators of O. Eliminating arbitrary coefficients in those matrices is actu- ally a rather hard and tedious task involving simultaneously all the constraints and methods described previously (integrality, positivity, intertwining equations, commutation with complex conjugates and with chiral partners, polynomial identities in degrees 9, 10, 11). At the very end we obtain, up to reordering of the 192 vertices, a single solution O010L,R for the matri- ces of the left and right symmetric generators, a single solution for the left generator O100L (that we block diagonalize in the form given below) but several solutions for the corresponding right generator O100R . However, the non unicity of the solutions for the pair of fundamental generators (O100L , O100R) reflects the existence of graph automorphisms (see our discussion in Quantum Symmetries for Exceptional SU(4) Modular Invariants 25 Section 2.2); for instance one of them exchanges vertices 17 with 18, or 122 with 123 (see Fig. 3). For each of these choices (for definiteness we select one of these equivalent solutions and just call it O100R) one can find several distinct permutation matrices Q, with Q = Q̃ = Q−1 such that O100R = QO001L Q−1. We then restrict the possible choices for this matrix in the manner discussed at page 16. The permutation associated with the chiral adjoint matrix Q that inter- changes matrices OfL and O f R , up to graph isomorphisms, is given below (as before, we list only the non trivial pairs that are interchanged by this operation). x 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 x† 121 126 25 26 174 28 29 169 97 49 50 102 52 53 73 74 150 76 77 145 x 27 30 33 36 37 38 39 40 41 42 43 44 45 46 47 48 51 54 57 60 x† 127 132 180 175 103 55 56 108 58 59 79 80 156 82 83 151 135 136 184 183 x 61 64 67 68 69 70 71 72 75 78 81 84 85 88 93 96 98 99 100 101 x† 111 112 86 87 160 89 90 159 139 144 192 187 115 120 168 163 133 134 138 137 x 104 105 106 107 116 117 118 119 128 129 130 131 140 141 142 143 152 153 154 155 x† 185 186 181 182 162 161 158 157 173 172 170 171 148 149 147 146 189 188 191 190 There are 32 self-adjoint vertices (x = x†), ambichiral ones are 1, 2, 4, 5 (the toric matrices associated with vertices 4 and 5 are actually equal). Adjacency matrices of E8. We order vertices x of the left quantum graph in such a way that connected components are separated in blocks. The left chiral generator matrices take the following block diagonal form: OL 100 = diag(G100, G100, G100, G100, G100 c, G100 c, G100 c, G100 c), OL 010 = diag(G010, G010, G010, G010, G010 c, G010 c, G010 c, G010 c). Matrices G100 and G100 c are ob- tained from the decomposition of OL 100 (or of OR 100) in connected components, G010 and G010 c from the decomposition of OL 010 (or of OR 010). For the Gf matrices, we order the basis elements by increasing 4-ality, G100 =  . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 2 . . 1 . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 2 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . . 1 . . . . . . . . . . . . . . . . . . 1 . 1 . . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 . 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . 2 . . 1 . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . 1 . . 2 . . . . . . . . . . . . . . . . . .  , G010 =  . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 . 1 1 . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 2 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 . . 2 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . 2 1 1 2 1 1 2 . . . . . . . . . . . . . . . . . . . . . . . 2 . . 1 . . . . . . . . . . . . . . . . . . . . 2 . . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 . . . . . . . . . . . . . . . . . . . . 1 . . 2 . . . . . . . . . . . . . . . . . . . . 1 . . 2 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 . . 2 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . 2 1 1 2 . . . . . . . . . . . .  and for the E c 8: Gc 100 =  . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 . 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . .  , Gc 010 =  . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 2 1 . . . . . . . . . . . . . . . . . . . . 1 1 2 1 . . . . . . . . . . . . . . . . . . . . . . 1 2 1 1 . . . . . . . . . . . . . . . . . . . . 1 2 1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . 2 2 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 2 2 . . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . 1 1 2 . . . . . . . . . . . .  . 26 R. Coquereaux and G. Schieber It is now easy to determine the annular matrices Fn, the horizontal dimensions dn, the total horizontal dimension dH and the dimension dB of the quantum groupöıd B(E8). One can also calculate the quantum dimensions of simple objects of E8 and check the already obtained value for the quantum cardinality \|E8\|. Results are summarized in the table of Section 2.3. Real-ambichiral partition functions of E8 (i.e., x = x, x = x† and x ∈ J ). Here such x’s coincide with ambichiral ones. With U1 = 113 + 133 + 311 + 331, U2 = 020 + 032 + 060 + 206 + 230 + 303 + 323 + 602, U3 = 000 + 008 + 080 + 121 + 141 + 214 + 412 + 800, one finds: Z1 = 2U1U1 + U2U2 + U3U3 (the modular invariant Z), Z2 = 2U1U1 + U2U3 + U3U2, Z4 = Z5 = U1U2 + U2U1 + U1U3 + U3U1. An exceptional module for E8. Since the permutation matrix C commutes with s and t, we may think of considering the new modular invariant matrix Mc = CM. However, in this case, and like for E4, we find that Mc = CM is equal to M. Therefore, we do not find a new modular invariant in this way. However, the graph O of quantum symmetries of E8 contains not only four copies of E8 but also four copies of a module that we call E8 c, with 24 vertices as well. One can then study it directly, like we did in the previous case (its adjacency matrices Gf c, given previously, are obtained from the connected components of O(E8) which are not of type E8). In particular, we can determine its own annular matrices, since, although they are associated with the same invariant, theAmodule structure of E8 and E8 c differ. One can deduce, from this study, the dimensions of the different blocks dn of the associated bialgebra B(E8 c), for its first multiplication, and find dH = ∑ n dn = 95040 = 263351111 and dB = ∑ n dn 2 = 80547840 = 2123251191231. In the present case O(E8) and O(E8 c) are identical. Afterword Although this was already discussed in our introduction, we conclude this article by comparing what was already known and what, to our knowledge, is new in the present paper. What was already known: • The modular invariant partition functions corresponding to the three exceptional cases discussed in this paper [39, 2, 1]. • The corresponding quantum graphs [35, 36, 31]. • The fact that every such graph passes the self-connection test [31], see also our remark on page 3. Concerning this last point, we believe that checking this condition should not be necessary for those quantum subgroups obtained, as here, from direct conformal embeddings, at least in those cases where the solution, obtained after resolution of the equations of modular splitting and intertwining, is unique (the solution should exist, and if the one obtained is unique . . . it is it!). What was not known29: • The full resolution of the equation of modular splitting for these three cases (only the chiral part of it30 was used to obtain the SU(4) graphs in [31]). 29Notice however that a detailed presentation of the k = 4 case is given in our paper [11]. 30The chiral equations of modular splitting is a simplified form of the full system of equations described in Section 2.2 reflecting the fact that the following equality, for m, n ∈ A, a ∈ E , holds: (m(na)) = (m · n)a. For cases where (Fp)00 = Mp0, a condition that holds in the cases studied in this paper, this associativity constraint implies immediately ∑ p(Nn)mp Mp0 = ∑ b (Fn)0b (Fn)b0. The left hand side (that involves only the first line of the modular invariant matrix) is known and the right hand side (the annular matrices) can be determined thanks to methods analog to those used in this paper, but this is technically simpler since the previous identity describes only r2 A equations instead of r4 A. Adjacency matrices Gf can then be obtained, but not the quantum symmetry matrices OfL,R . Quantum Symmetries for Exceptional SU(4) Modular Invariants 27 • The structure of the algebra of quantum symmetries and the graphs of its fundamental generators for these three cases. One can also find in the previous sections many details concerning quantum dimensions and cardinalities for quantum graphs obtained from conformal embeddings, as well as a description of general techniques that, to our knowledge, are not discussed elsewhere. Appendix We first remind the reader what are the expressions for representatives of the generators s and t of the modular group in the particular case of SU(g) groups and we give recurrence formulae for the fusion matrices of SU(4). Then, we give the quantum dimensions of the spaces of sections associated with the modular blocks of the partition function, for the three exceptional cases studied in this article. The quantum dimensions of SU(4) irreducible representations needed in this calculation are obtained from the quantum version of the Weyl formula. Using the Kac– Peterson formula for modular generators, we also obtain a general expression for the quantum cardinality of Ak when qk+4 = −1. Quantum dimensions for irreducible representations of B7 ' Spin(15), A9 ' SU(10) and D10 ' Spin(20) at level k can be calculated in a standard way from the quantum version of the Weyl formula; those at level 1 have been used in the text (Section 1.4.5). In the case of SU(10), or of SU(g) in general, the calculation at level 1 is very simple, and one finds dim(n) = 1 for all n ∈ A1(SU(g)). In the case of Spin(15) we refer to a discussion in [11]. Our last appendix gives some details concerning the case of Spin(20). A Generators s and t for SL(2, Z) Expressions for representatives of the generators s and t of SL(2, Z) are given, for any simple Lie algebra, and for a given level k, by the Kac–Peterson formulae [23]. These general expressions, that we recall below in the case of SU(g), involves a summation over the elements of the Weyl group, which, for SU(4), is the symmetric group on four objects, and scalar products between fundamental weights shifted by the Weyl vector. These explicit expressions for s and t are matrices of size rA × rA. Indices m,n, . . . range over all SU(g) Young tableaux (including the trivial) corresponding to i-irreps with levels up to k. As usual, s and t are unitary and such that (s t)3 = s2 = C, the “charge matrix” satisfying C2 = l1. The diagonal t matrix obeys t2 g κ = l1, where κ is the altitude, defined as κ = k + g, (in our SU(4) case, the dual Coxeter number is g = 4). Using these expressions, we checked that M indeed commutes with the modular generators s and t. smn = irg/2 √ ∆(r) (g + k)r/2 ( g!∑ s=1 εw(s)e − 2iπ〈w(s)(m+ρ),n+ρ〉 g+k ) , tmn = e 2iπ [ 〈m+ρ,m+ρ〉 2(g+k) − 〈ρ,ρ〉 2g ] δmn, where w(s) runs over the g! permutations of the Weyl group of SU(g), εw(s) is its signature, r = g − 1 is the rank, ρ is the Weyl vector, and ∆(r) is the determinant of the fundamental quadratic form. For SU(g), each entry smn of the s matrix can be written in terms of the determinant of a g × g matrix A(m,n), with matrix elements A(m,n)a,b=1,...,g, as follows [24]: smn = irg/2 √ g(g + k)r/2 det A(m,n), with A(m,n)ab = exp[− 2iπ k + g φa(m)φb(n)] and φa(m) = g−1∑ s=a (ms + 1)− 1 g g−1∑ s=1 s(ms + 1), 28 R. Coquereaux and G. Schieber where ms are the components of m along the basis of fundamental weights, and with the under- standing that the first sum gives 0 if a = g. B Recurrence formulae for fusion and annular matrices of SU(4) Fusion matrices for the fundamental irreducible representations can be obtained in several ways. One possibility is to use the previous expressions for s and t together with the Verlinde formulae, but in the case of SU(g) groups, and in particular for SU(4), it is much simpler to use standard Young tableaux techniques (at a fixed level, the horizontal size of the tableaux is bounded) or, equivalently, the Pieri rules describing the reduction of tensor products of arbitrary irreducible representations by the fundamental ones (see for instance [12, p. 695]). The same rules can be used to obtain the recursion formulae given below. Once the fusion matrices of the fundamental irreducible representations are known, the others can be determined from the truncated recursion formulae of SU(4) irreps, applied for increasing level `, up to k (2 ≤ ` ≤ k): N(`−p,p−q,q) = N(1,0,0)N(`−p−1,p−q,q) −N(`−p−2,p−q+1,q) −N(`−p−1,p−q−1,q+1) −N(`−p−1,p−q,q−1) for 0 ≤ q ≤ p ≤ `− 1, N(0,`−q,q) = (N(q,`−q,0)) tr for 1 ≤ q ≤ `, N(0,`,0) = N(0,1,0)N(0,`−1,0) −N(1,`−2,1) −N(0,`−2,0). Annular matrices Fn = (Fn)ab relative to a quantum graph Ek are calculated with exactly the same recurrence relations. Only the seed used to start the recurrence is different: F(1,0,0), F(0,1,0) and F(0,0,1) = F tr (1,0,0) are the adjacency matrices of the chosen graph. C Quantum dimensions Quantum dimensions of the modular blocks. As discussed at the end of Section 1, every vertex a of Ek has a dimension dim(a) = dim(Γa)/ dim(Γ0), where the dimension of the quantum space of sections Γa is calculated by using induction from Ak(SU(4)): dim(Γa) = ⊕ n↑Γa dim(n). In particular, the dimensions of spaces of sections Γs associated with the modular vertices s of Ek (or with the various modular blocks of the partition function) are obtained by summing the quantum dimensions of the irreducible representations that appear in the modular blocks for the different cases: When k = 4, i.e., q = exp(iπ/8), the first two modular blocks have dimension 4 ( 2 + √ 2 ) and the third has dimension 4 ( 1 + √ 2 ) . When k = 6, i.e., q = exp(iπ/10), the ten modular blocks have dimension 20 + 8 √ 5. When k = 8, i.e., q = exp(iπ/12), the four modular blocks (remember that the last one appears twice in the modular invariant) have dimension 12 ( 9 + 5 √ 3 ) . Complements and checks. Quantum dimensions of irreducible representations of SU(4), when k = 4, 6, 8, that are needed in the previous calculation, have been calculated in two ways: from the Perron–Frobenius eigenvector of the adjacency matrix of the graph Ak associated with the defining representation, and from the quantum version of the Weyl formula, dim(m) = ∏ α>0 〈m + ρ, α〉q 〈ρ, α〉q taking qk+4 = −1 at the end. Scalar products between a chosen weight m = {m1,m2,m3} and the six positive roots α of A3 are displayed below in the half-ribbon diagram (seen as a generalized root set [32]) stemming from the A3 = A2(SU(2)) graph. We also give the Weyl Quantum Symmetries for Exceptional SU(4) Modular Invariants 29 vector ρ m1 0 m1 0 m1 0 + 0 0 m2 m2 m2 m2 + 0 m3 m3 m3 0 0 , ρ = 1 1 3 2 2 1 . The quantum Weyl denominator is 13 q2 2 q3q (equal to 2 when q8 = −1, to 5 + 2 √ 5 when q10 = −1 and to 5 + 3 √ 3 when q12 = −1). The q-dimensions of the fundamental irreducible representations have been already given in the text. The quantum cardinality (quantum mass) \|Ak\| of SU(4) at level k. This quantity is obtained by summing the square of the quantum dimensions of the rA irreducible representations. One can also use the property \|Ak\| = 1/s2 00 where s is the first generator of the modular group, together with the Kac–Peterson formula [23], therefore expressing s00 in terms of weighted Weyl group averages of the norm of the Weyl vector. In this way, one finds: \|Ak(SU(4))\| = 4(k + 4)3 216 cos4 ( π k+4 )( 2 cos ( 2π k+4 ) + 1 )2 sin12 ( π k+4 ) and one recovers in particular the given expressions directly calculated for specific values of k. Quantum dimensions for D10 ' Spin(20) at level 1. We already know what the integrable representations of D10 at level 1 are, namely the trivial, the vectorial, and the two half-spinorial, but it is nice to recover this from a direct calculation of quantum dimensions. We use the quantum version of the Weyl formula and, like we did with A3, we display the scalar products between an arbitrary weight and the 90 positive roots of D10 in the half-ribbon diagram (seen as a generalized root set [32]) stemming from the D10 = D16(SU(2)) graph. We only display the Weyl vector ρ and, to its right, the q-dimensions for the fundamental irreducible representations. The quantum Weyl denominator is 110 q 29 q3 9 q4 8 q5 8 q6 7 q7 7 q8 6 q9 6 q104 q114 q123 q133 q142 q152 q16q17q, ρ = 1 1 1 1 1 1 3 3 3 4 2 5 5 6 3 3 4 7 8 8 2 6 1 1 5 5 4 9 12 12 2 7 11 14 7 7 5 9 13 16 3 7 11 15 9 9 5 9 13 17 2 7 11 15 8 8 4 9 13 14 2 6 11 12 6 6 4 8 1 1 2 6 7 8 4 4 4 5 5 6 2 3 3 3 2 2 1 1 1 1 , 10q18q 1q9q 10q16q19q 1q2q8q 10q14q18q19q 1q2q3q7q 10q12q17q18q19q 1q2q3q4q6q 102q16q17q18q19q 1q2q3q4q52q 8q10q15q16q17q18q19q 1q2q3q42q5q6q 10q14q15q16q17q18q19q 1q2q32q4q5q7q 10q13q14q15q16q17q18q19q 1q22q3q5q6q7q8q 10q12q14q16q18q 1q3q5q7q9q 10q12q14q16q18q 1q3q5q7q9q With an altitude of κ = 18+1 = 19, so that q = exp(iπ/19), one finds that the q-dimensions of the fundamental irreducible representations are 1, 0, 0, 0, 0, 0, 0, 0, 1, 1. Taking into account the trivial representation, one finds \|A1(Spin(20))\| = 1 + 3 = 4, as expected. 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[43] Xu F., An application of mirror extensions, arXiv:0710.4116. http://arxiv.org/abs/hep-th/9411077 http://msri.org/publications/ln/msri/2000/subfactors/ocneanu/1/index.html http://arxiv.org/abs/math.QA/0111139 http://arxiv.org/abs/hep-th/0101151 http://arxiv.org/abs/hep-th/9510175 http://arxiv.org/abs/hep-th/9701103 http://arxiv.org/abs/hep-th/9205072 http://arxiv.org/abs/0710.4116 Foreword 1 Conformal embeddings of SU(4) 1.1 Homogeneous spaces G/K 1.2 The Dynkin index of the embeddings 1.3 Those embeddings are conformal 1.4 The modular invariants 1.4.1 The method 1.4.2 SU(4) \subset Spin(15), k=4 1.4.3 SU(4) \subset SU(10), k=6 1.4.4 SU(4) \subset Spin(20), k=8 1.4.5 Quantum dimensions and cardinalities 2 Algebras of quantum symmetries 2.1 General terminology and notations 2.2 Method of resolution (summary) 2.3 Tables for E_k 2.4 Oc(E_4(SU(4)) 2.5 Oc(E_6(SU(4)) 2.6 Oc(E_8(SU(4)) Afterword Appendix A Generators s and t for SL(2,Z) B Recurrence formulae for fusion and annular matrices of SU(4) C Quantum dimensions References

Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings

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