Non-unitary fusion categories and their doubles via endomorphisms

We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup-Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang-Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman-Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matrices of the doubles of these fusion categories. We would expect all of these doubles to be realised as the category of modules of a rational VOA and conformal net of factors. We expect our approach will also suffice to realise the non-semisimple tensor categories arising in logarithmic conformal field theories.


Introduction
The chiral part of a unitary rational conformal field theory (CFT) can be represented as either a completely rational conformal net of factors on a circle or a rational vertex operator algebra (VOA). Whilst conformal nets and subfactors theory focus and exploit the analytic aspects, vertex operator algebras focus on the algebraic aspects. The relation between these approaches is studied in [5]; at the simplest level, they both must give rise to the same modular tensor category (MTC) if they are to correspond to the same CFT.
Conformal nets of factors are a particularly rich framework, with connections with twisted equivariant K-theory and non-commutative geometry. Subfactor methods have proved to be much more effective than VOA methods in many ways. For example, structure theorems such as rationality of orbifolds or cosets is much easier in the conformal nets of subfactors picture (see e.g. [31]) than in the VOA picture. Also, the factor setting captures in a natural way the full CFT as an inclusion of (local) nets [3], [30].
However, the VOA setting for the chiral CFT is apparently more flexible in allowing non-unitary examples. For example, the Virasoro minimal models are parametrised by pairs p > q of coprime numbers; they are unitary if and only if p = q + 1. The simplest of these is the Yang-Lee model V (2, 5) (see e.g. section 7.4.1 of [8]), which Cardy [4] showed arises as the Yang-Lee edge singularity in the Ising model in an imaginary magnetic field. Other non-unitary statistical mechanical examples are the scaling limit of critical dense polymers, and critical perculation, both with central charge c = −2. An unrelated non-unitary example crucial to string theory is the (super-)ghost CFT; what must be unitary is space where the physical states lie, namely the BRST cohomology of the ghosts coupled to a matter CFT. Wess-Zumino-Witten models on Lie supergroups provide other non-unitary examples important to string theory. In the VOA setting, realising non-unitary CFTs presents no special problems, whereas subfactors and nets of factors have unitarity built in.
A fundamental question is whether there are any rational CFTs beyond those constructed from loop groups or quantum groups, using standard methods such as orbifolds and cosets (see e.g. [31] for a discussion on this point). It is known that all unitary fusion categories, hence all unitary MTCs, can be realised by endomorphisms on a factor. These methods have produced countless 'exotic' examples of unitary MTCs [10,13]. Indeed, the relative abundance of these examples suggests that most modular tensor categories may be 'exotic'. Finding conformal net and VOA realisations of these 'exotic' MTCs is an important but difficult challenge -we expect most or all of them to have such realisations. The situation for the (double of the) Haagerup subfactor is discussed in detail in [10]. In any case, the effectiveness of these subfactor methods in constructing new unitary MTCs provides another compelling reason for extending these methods to the non-unitary setting.
The main purpose of this paper is to provide a broader context, dropping the requirement of unitarity, in which the subfactor methods can be applied. After all, most rational CFT are non-unitary, and one would like to exploit the powerful methods of subfactors and nets of factors in the general case.
In the remainder of the Introduction we sketch in more detail some of the terms used earlier, as well as the content of the paper.
The sectors of a rational CFT, or modules of a rational VOA, give rise to a tensor category of a very special type, namely an MTC. More generally, we are interested in fusion categories, which roughly speaking are MTCs without the braiding (we review their definition in section 3). Given a fusion category, the double or centre construction canonically associates an MTC. Unitarity in a category can be defined as follows. A * -operation on a C-linear category C is a conjugate-linear involution Hom(X, Y ) → Hom(Y, X) satisfying (f g) * = g * f * for all f ∈ Hom(X, Y ), g ∈ Hom(Z, Y ). If the category is tensor (and strict), we also require (f ⊗ g) * = f * ⊗ g * for all f ∈ Hom(X, Y ), g ∈ Hom(Z, W ). A * -operation is called positive if f * f = 0 implies f = 0. A category equipped with a (positive) * -operation is called hermitian (resp. unitary).
Associated to an MTC is a representation of SL 2 (Z) called modular data. It is generated by a symmetric unitary matrix S which gives the fusion coefficients (structure constants of the Grothendieck ring of the category) through Verlinde's formula, together with a diagonal matrix T of finite order. Some column of S must be strictly positive -e.g. in a unitary MTC that Perron-Frobenius column corresponds to the unit. In a rational CFT, the characters χ M (τ ) = q h M −c/24 ∞ n=0 dim M n q n of the irreducible modules M = n M n form a vector-valued modular function for SL 2 (Z) with modular data as its multiplier. The minimal conformal weight h M corresponds to the positive column of S. The conformal weights and central charge c must be rational, but in a unitary theory they will also be non-negative. For more comparisons between the modular data of non-unitary versus unitary theories, see [15].
A very convenient realisation of tensor categories is through endomorphisms on an algebra, where objects are algebra endomorphisms and morphisms are intertwiners.
The tensor product of objects corresponds to composition and of morphisms to the (twisted) product in the underlying algebra. However, it is awkward to realise other properties in the category, such as additivity or rigidity, without assuming special structures on the algebra. When the underlying algebra is a C * -algebra such as the Cuntz algebra, these other properties arise naturally. Indeed, any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra (see section 7 of [20]).
A natural question is, can we find systematic realisations by endomorphisms of non-unitary fusion categories? We will see that the answer is yes.
Our approach was influenced by recent work of Phillips [28], who studies nonunitary analogues of the Cuntz algebra. But all of our calculations are within a polynomial algebra (the Leavitt algebra). Rather than completing that algebra as studied by Phillips, we have found it sufficient to work exclusively within the Leavitt algebra itself.
For concreteness we focus on the Haagerup-Izumi family of fusion rings, but our method works more generally. Let G be any finite abelian group. The (isomorphism classes of) simple objects in these fusion rings are [α g ] and [α g ρ] as g ranges over G. The fusions are given by (1.1) In the following sections we explain explicitly how to construct, using endomorphisms on the Leavitt algebra, fusion categories (not necessarily unitary) which realise the Haagerup-Izumi fusions when G has odd order. We compute the corresponding tube algebras and from that obtain the modular data S, T of the double of the system. We give several examples and explicitly classify these systems for small G. The (unitary) Haagerup-Izumi fusions (1.1) for |G| odd was introduced by Izumi in [23]. His motivation was to construct the Haagerup subfactor [19,2], so he focussed on the special class of systems of Cuntz algebra endomorphisms, called Q-systems, which arise as the even subsystem of a subfactor with canonical endomorphism 1 + ρ. Q-systems correspond to especially constrained ρ; their fusion categories are always unitary. He showed that there was a unique Q-system satisfying (1.1) for the group G = Z 3 , and comparing indices observed that it must correspond to the Haagerup subfactor. Likewise, he showed that there is a unique Q-system for G = Z 5 . He also computed the modular data for the doubles of his systems (modulo a technicality discussed shortly). Evans-Gannon [10] pushed this further, finding Q-systems in this class for all G with |G| ≤ 19 (including the complete lists for |G| ≤ 9), and simplifying considerably Izumi's expressions for the modular data. Thanks to this work, it is now expected that there are subfactors (usually several) for each odd order, and they are all expected to correspond through their doubles to rational VOAs etc. Grossman-Snyder [17] found new systems of endomorphisms realising (1.1) (unitary but not Q-systems), for G = Z 3 and Z 5 , which are Morita equivalent to Izumi's systems (and thus have the same doubles). This treatment has been extended to even order G, and to all unitary systems (not only Q-systems) realising (1.1), by Evans-Gannon [11], [12] and independently Izumi [24].
In this paper, as an illustration of our method, we characterise all realisations by endomorphisms (not necessarily Q-systems nor unitary) of the Haagerup-Izumi fusions (1.1) for |G| odd. We show they all yield fusion categories. Like [23], our systems correspond to solutions of finitely many equations in finitely many variables, but unlike [23] our equations are all polynomials (those of [23] involve complex conjugates). In broad strokes the method we use is analogous to that of [23], but the absence of unitarity introduces several complications and our argument is required to be much more subtle. We find the doubles and modular data of our systems.
For example, there are precisely 2,4,4 inequivalent fusion categories realised by endomorphisms, of Haagerup-Izumi type for G = Z 1 , Z 3 , Z 5 respectively (of course we recover all of them). Precisely 1,2,2 of these, respectively, are unitary: 1,1,1 are Qsystems, and 0,1,1 are the aforementioned Grossman-Snyder systems. The Yang-Lee system is the unique non-unitary one corresponding to G = Z 1 .
Every fusion category C is defined over some number field [9]. An automorphism σ of that field acts on the quantities of that category in the natural way, defining a new fusion category C σ . These categories may or may not be equivalent -e.g. a Galois associate of a unitary fusion category may not be unitary. In general, C and C σ will have identical fusion rings, but their modular data for example will be Galois associates. Our construction, unlike that of e.g. Izumi, is closed under this Galois action.
It turns out that all 5 non-unitary fusion categories for G = Z 1 , Z 3 , Z 5 are Galois associates of unitary categories. We expect though that this is an accident of small G. Our system of equations involve twice as many variables as in the unitary case, and approximately the same number of equations. For these reasons, we would expect typically many more non-unitary categories than unitary ones.
In any case, it is easy to construct non-unitary fusion categories, all of whose Galois associates are also non-unitary. A simple example is the tensor product of affine G 2 at level 1 (a unitary MTC) with the Yang-Lee model (a non-unitary one).
Actually, the equations in [23] are not sufficient to determine the half-braidings, and hence the modular data, for most odd abelian G, even in the Q-system case. In section 6 below we supply additional equations which are both necessary and sufficient.
Incidentally, another interesting class of CFTs and VOAs are the so-called logarithmic or C 2 -cofinite non-rational ones [6], for example the symplectic fermions [7]. Unlike the rational CFTs, their category of modules will not be semisimple and so direct (sub)factor realisations of them wouldn't be possible. Logarithmic theories appear to be intimately connected with non-unitarity: all known ones are conformal embeddings of non-unitary rational VOAs (with states of negative conformal weight). In any case, although we address in this paper only fusion categories (which are semisimple by definition), modelling non-semisimple systems is also possible by our methods and we would expect we could realise with endomorphisms these loga-rithmic theories.

The Yang-Lee model
This section illustrates the ideas developed in the following sections, with the simplest non-unitary example: the Yang-Lee model (this CFT is described e.g. in section 7.4.1 of [8]). It consists of two simple objects 1 and ρ, which obey the fusion rule (2.1) Let us try to realise (2.1) as a system of algebra endomorphisms on some algebra A. To motivate our solution though, let's reverse the logic and derive the consequences of such a realisation. It would require the relation where s, s ′ , t, t ′ ∈ A satisfy the Leavitt-Cuntz relations More precisely, these relations say that (2.2) expresses ρ • ρ as a direct sum of objects id and ρ in the category EN D(A) (we describe this category in detail next section). These elements s, s ′ , t, t ′ generate by definition a copy of the Leavitt algebra L 2 inside A; we will see shortly that ρ restricts to an endomorphism of L 2 . In order to identify the restriction of ρ to L 2 , it is necessary and sufficient to determine the values ρ(s), ρ(s ′ ), ρ(t) and ρ(t ′ ) of ρ on the generators. For * -maps, we would have ρ(s ′ ) = ρ(s) ′ etc, but we cannot require that here if we hope to realise the Yang-Lee model. We require that both endomorphisms id and ρ be simple, equivalently that the intertwiner spaces Hom(id,id) and Hom(ρ, ρ) in the algebra A be C1, and that Hom(ρ, id) = Hom(id, ρ) = 0. (The definition of intertwiners is given next section.) From (2.2) we obtain (2.4) The first means s ∈ Hom(id, ρ 2 ). Conversely, suppose r ∈ Hom(id, ρ 2 ), i.e. rx = ρ 2 (x)r for all x. Then s ′ rx = s ′ ρ 2 (x)r = xs ′ r and t ′ rx = t ′ ρ 2 (x)r = ρ(x)t ′ r. Thus by simplicity of id and ρ we have s ′ r ∈ C and t ′ r = 0, so r = (ss ′ + tt ′ )r = ss ′ r ∈ Cs using the Leavitt-Cuntz relation ss ′ + tt ′ = 1. We have shown Hom(id, ρ 2 ) = Cs. In the same way (see Lemma 3 below for details and the generalisation), we can identify the intertwiner spaces Ct = Hom(ρ, ρ 2 ), Cs ′ = Hom(ρ 2 , id) and Ct ′ = Hom(ρ 2 , ρ). These observations are crucial for what follows. Note, using (2.4), that In other words, s ′ ρ(s) ∈ Hom(ρ, ρ) = C, so s ′ ρ(s) equals some complex number a. Likewise, t ′ ρ(s) ∈ Hom(ρ, ρ 2 ) so t ′ ρ(s) = bt for some b ∈ C. The point is that ρ(s) = (ss ′ + tt ′ )ρ(s) = s(s ′ ρ(s)) + t(t ′ ρ(s)) = as + btt .
(2.6) Similar calculations (see section 4 for details and the generalisation) give for some a ′ , b ′ , c, c ′ , d, d ′ , e, e ′ ∈ C. Because ρ sends the generators of L 2 into L 2 , this means ρ is actually an endomorphism of L 2 . If we required ρ to be a * -map, then we would have a ′ = a etc, but again we shouldn't do that if we are to recover Yang-Lee.
We can now use the constraints on ρ to solve for those 10 parameters. First, ρ is required to be an algebra endomorphism, so it must respect the Leavitt-Cuntz relations (2.3). One relation requires 1 = ρ(s ′ )ρ(s), i.e.
. In particular, aa ′ cannot be 1, so we must have a = a ′ and thus b = √ da and b ′ = db. We eliminate the possibility that d = −1 by considering the st ′ t ′ coefficient of t ′ ρ 2 (t) = ρ(t)t ′ , which gives c = ab ′ d 2 + cdee ′ . So we have determined that a = a ′ = −e = −e ′ = (−1 ± √ 5)/2, b = b ′ = c = c ′ = √ a, and d = d ′ = 1, where we can take the square-root for b so that b ∈ R >0 ∪ iR >0 . So we have 2 possible solutions, corresponding to the choice of signs in a = (−1 ± √ 5)/2. In section 4 we generalise this argument to arbitrary odd order abelian G in (1.1).
Much more delicate is to associate a (strict) fusion category to both of these ρ. The biggest challenge here for arbitrary G is to define arbitrary (but finite) sums of endomorphisms using the Leavitt algebra, in the sense of the right-side of (2.2). We are lucky here with the Yang-Lee: because its Leavitt algebra has 2×2 generators, we can capture arbitrary sums -e.g. ρ ⊕ ρ 3 ⊕ ρ 5 can be written sρs ′ + tsρ 3 s ′ t ′ + ttρ 5 t ′ t ′ , to choose a random example. The resulting fusion category for the solution with a = (−1 + √ 5)/2 is the unitary category associated to e.g. the integrable modules of the affine G 2 algebra at level 1, whilst for a = (−1 − √ 5)/2, we obtain the Yang-Lee fusion category. These two fusion categories are inequivalent even though they share the same fusions (2.1) -indeed, it can be shown that the categorical dimension of ρ (defined next section) is 1/a = (1 ± √ 5)/2, so is positive in one and negative in the other. Nevertheless they are clearly related by the Galois automorphism interchanging a = (−1 ± √ 5)/2. To realise the fusions (1.1) for general G, we will need a Leavitt algebra L with (1 + |G|) × 2 generators (one pair for each term on the right of (1.1)), but for such an algebra only direct sums with n ≡ 1 (mod |G|) terms can be realised. When ρ is a * -map (e.g. the case studied in [23], [10]), we can extend ρ to an endomorphism of an infinite von Neumann factor N [23]; semisimplicity is then automatic, since N contains copies of the Leavitt algebras of arbitrary rank, so arbitrary sums of endomorphisms can be made. On the other hand, when ρ is not a * -map, we obtain semisimplicity by first forming the idempotent completion. In section 5 we show that any solution to the various consistency equations yields a (usually non-unitary) fusion category.
MTC structures can be placed on both of the G = 1 fusion categories constructed in this section, though in more than one way -e.g. the a = (−1 + √ 5)/2 category is realised by both affine G 2 level 1 and affine F 4 level 1, which which are inequivalent as MTC since they have different central charges mod 8. This behaviour too is special to G = 1: the fusion categories for larger G never come with a braiding (this is clear from (1.1), as [α g ][ρ] = [ρ][α g ] when |G| is odd and > 1). For these other G, we realise in section 6 the associated MTC through the centre of the tube algebra. Incidentally, this construction applied to e.g. the fusion category of affine G 2 at level 1, would yield the MTC of affine G 2 ⊕ F 4 at level (1,1).
In these cases, the first character given is that of the VOA V = ∞ n=0 V n itself, and so lists the dimensions of its graded spaces V n , so we see that there appears little relation between the Yang-Lee VOA and that of say the G 2 one. On the other hand, the naive inner product of the G 2 and F 4 character vectors is j(τ ) 1/3 , reflecting the fact that the VOA V(G 2 , 1) ⊗ V(F 4 , 1) is a conformal subalgebra of the E 8 lattice VOA. Note also that the first column of the matrix S is strictly positive for the VOAs V(G 2 , 1) and V(F 4 , 1) (as it must be for unitary VOAs), and isn't for the Yang-Lee (as is typical for non-unitary VOAs).

Leavitt algebras and categories of endomorphisms
For each n > 1 define the Leavitt algebra L n to be the associative * -algebra freely generated over C by x 1 , . . . , x n , x ′ 1 , . . . , x ′ n , modulo the Leavitt-Cuntz relations The elements of L n are polynomials in the non-commuting variables and obeys (cyz) ′ = cz ′ y ′ for all c ∈ C and x, y ∈ L n . It has an obvious grading by Z n . The Leavitt algebra L n can be regarded as the polynomial part of the Cuntz algebra O n , its C * -algebra completion.
The Leavitt algebras L n are all non-isomorphic for n = 2, 3, 4, . . ., since the inclusion of L n in O n induces an isomorphism on K-theory with the cyclic group Z n−1 [1]. The only obstruction to embedding L m unitally in L n is given by the K-theory [29]. More precisely L m embeds unitally in L n if and only if m − 1 divides n − 1. In the Cuntz framework of Izumi [21,23,24] and Evans-Gannon [10,13,12], one constructs endomorphisms on a fixed Cuntz algebra O n , with prescribed fusion rules and then extends these to a completion as an infinite von Neumann factor N. Any Cuntz algebra O m can be unitally embedded in the factor N for any m, even though usually it cannot be unitally embedded in O n . The fusion category will then be realised as a system of endomorphisms of N, since addition of any number m of endomorphisms can be expressed in N.
We will realise fusion categories through endomorphisms of L n . But we do not require that our endomorphisms be * -maps, so they need not extend to the completion, the Cuntz algebra O n or the Banach algebras of Phillips [28].
Note that if ρ is any algebra endomorphism on L n , then so isρ defined bỹ Throughout this paper we distinguish an algebra endomorphism from a * -algebra endomorphism. The latter must obey f (y) ′ = f (y ′ ) (equivalentlyρ = ρ) while the former may not. There is a canonical way to write any element of L n . Call any monomial in the generators x i , x ′ j reduced if no primed variable appears to the left of any unprimed variable, and x 1 is not adjacent to x ′ 1 in the monomial. Call any linear combination over C of finitely many distinct reduced monomials, a reduced sum. Lemma 1. [25] Any y ∈ L n can be written in one and only one way as a reduced sum.
This simple observation has several easy consequences, as we'll see. It easily implies the centre of L n is trivial [25]. Moreover: Corollary 1. An algebra endomorphism ρ on L n is uniquely defined by its values ρ(x i ), ρ(x ′ j ) on the generators, and these can be assigned arbitrarily provided they respect the Leavitt-Cuntz relations (3.1).
There are several complications caused by avoiding the completion and working exclusively with L n . In particular, two serious challenges are how to add endomorphisms, and how to get rigidity. We accomplish the former through the idempotent completion (described below), and the latter by hand.
Recall that because L n is a unital algebra over C, by general nonsense its algebra endomorphisms define a C-linear preadditive strict tensor category EN D(L n ). More precisely, the objects in EN D(L n ) are algebra (but not * -algebra) endomorphisms of L n . The morphisms r ∈ Hom(β, γ) are intertwiners, i.e. r ∈ L n for which rβ(x) = γ(x)r for all x ∈ L n ; composition of morphisms is multiplication in L n . EN D(L n ) is C-linear, i.e. each Hom(β, γ) is a vector space over C; it is also preadditive, i.e. composition of morphisms is bilinear. The tensor product of objects is composition: A fusion category [9] is a C-linear semisimple rigid tensor category with finitely many isomorphism classes of simple objects and finite dimensional spaces of morphisms, such that the unit object 1 is simple. A simple object X is one with End(X) = C id X ; amongst other things, every object in a semisimple category is a direct sum of simple ones. We say object X has a right-dual X ∨ iff there is a pair of morphisms evaluation e X ∈ Hom(X ∨ ⊗X, 1) and co-evaluation b X ∈ Hom(1, X ⊗X ∨ ) for which (where we assume the category is strict, for convenience). Left-dual ∨ X is defined similarly. In particular in EN D(L), an object β ∈ End(L) has a right-dual A tensor category is called rigid if every object X has a right-and left-dual. In a (strict) rigid category, we can define the right-dual In particular in EN D(L), the right-dual of r ∈ Hom(α, β) is defined by A rigid tensor category is pivotal if it is equipped with a natural isomorphism from the identity functor to the double-dual functor X → X ∨∨ . In a pivotal category we can take ∨ X = X ∨ . In a rigid category the (left-)dimension of object X is e X ∨ b X ; a semisimple pivotal category is called spherical if X and X ∨ have the same dimension for all objects X (it suffices to check this for simple X). See e.g. [27] for the remaining terminology not explained here. Let E be a collection of algebra endomorphisms of L n closed under composition. We require the identity to be in E. Let C(E) denote the subcategory of EN D(L n ) restricted to E. Then like EN D(L n ), C(E) is a C-linear tensor category, and the endomorphism algebra of the unit object 1 is C. By its idempotent completion we mean the category C(E) whose objects consist of pairs (p, β) where β ∈ E and p ∈ End(β) is an idempotent, i.e. p 2 = p, and whose morphism spaces are Hom((p, β), (q, γ)) = qHom(β, γ)p with composition again given by multiplication.
Lemma 2. Let E be a collection of L n -endomorphisms as above, and recall (3.2). Suppose Hom(β, γ) = Hom(β,γ) in L n for all β, γ ∈ E, and that these are all finitedimensional. Then C(E) ds , the idempotent completion extended by direct sums, is a semisimple strict C-linear tensor category with finite-dimensional hom-spaces. If C(E) is rigid, then so is C(E) ds .
Moreover, suppose C(E) is rigid. Then Lemma 3.1 of [32] says that its idempotent completion C(E) is also rigid: e.g.
This condition Hom(β, γ) = Hom( β, γ) is crucial for extending the (unitary) Cuntz algebra methods to the (not necessarily unitary) Leavitt setting. We show near the end of section 5 that this condition holds for the Haagerup-Izumi systems considered here, and the same argument should work for the near-group systems constructed in [13]. Nevertheless, Lemma 2 emphasises that semisimplicity in the Leavitt picture is not automatic, and this is very good: it means our context should be flexible enough to include non-semisimple examples such as those corresponding to the logarithmic CFTs discussed in the Introduction.

Non-unitary Haagerup-Izumi: deconstruction
Let G be any abelian group of odd order ν = 2n+1, and define δ ± = (ν ± √ ν 2 + 4)/2, the two roots of x 2 = 1 + νx. Recall the Haagerup-Izumi fusions (1.1). A main result (Theorem 1) of this paper associates to any system of algebra endomorphisms realising these fusions, a set of numerical invariants. The converse, which associates a system of endomorphisms and a fusion category to these same numerical invariants, is given next section.
Suppose α g , ρ are algebra endomorphisms of an algebra A which realise the Haagerup-Izumi fusions. More precisely, this means We do not assume A is a * -algebra. Equation (4.1) implies that each α g is invertible. Note that we have the freedom to rescale the ν +1 elements s, t g arbitrarily and independently, provided we then rescale s ′ , t ′ g inversely. We also require α g and α g ρ = α g • ρ to be simple, i.e. that their intertwiners in A are Hom(α g ρ, α h ρ) = Hom(α g , α h ) = Cδ g,h and Hom(α g , α h ρ) = Hom(α g ρ, α h ) = 0. This implies for instance that A has trivial centre, and that the representation g → α g of G is faithful.
Unless G is cyclic (in which case H 2 G (pt; T) = 1), (4.1) can be generalised by twisting by 2-cocycles ξ ∈ Z 2 G (pt; T) and (1.1) will still hold, as explained e.g. in the proof of Theorem 1 in [13]. We will ignore this generalisation, as it is conceptually straightforward and merely makes the arithmetic a little messier, and our primary purpose with this paper is to explain how to capture non-unitary fusion categories by endomorphisms. [23] also ignored these cocycles, but the unpublished notes [24] introduces them (though of course in the unitary setting). Theorem 1. Let G, α g , ρ and s, s ′ , t g , t ′ g ∈ A be as above. Then for some fixed sign ±, where b ∈ {1/ ωδ ± } and ω 3 = 1. In particular, α g and ρ restrict to algebra endomorphisms of the Leavitt algebra L = L ν+1 with generators s, s ′ , t g , t ′ g . Moreover, A g,h ∈ C satisfy We will show in Proposition 2 below that in fact for all g, h, k, l ∈ G. We expect that this can be used to derive the more complicated (4.10), but we haven't established this yet. According to Izumi [23], a (unitary) Q-system corresponds to the special case of Theorem 1 with ω = 1, δ ± = δ + , A g,h = A h,g , A g,0 = δ g,0 − 1/(δ + − 1). In this case the quartic identity (4.10) can be replaced with the cubic identity (4.11). This special case corresponds to fusion categories coming from one of the even subsystems of a finite depth finite index subfactor.
Incidentally, it doesn't matter which square-root is chosen for b in Theorem 1: This means that we can require without loss of generality that b lies on the positive halves of the real or imaginary axes. Which triples (±, ω, A) yield isomorphic fusion categories is answered below in Theorem 2, as is the question of unitarity.
. Thus we know all α g restrict to endomorphisms of the Leavitt algebra L ν+1 generated by the s, s ′ , t g , t ′ g . Since s ∈ Hom(id, ρ 2 ) and ρ is an endomorphism, ρ(s) ∈ Hom(ρ, ρ 3 ). Hence Hitting Thus we also know ρ restricts to an endomorphism of the Leavitt algebra L ν+1 generated by the s, s ′ , t g , t ′ g . Thus the A-endomorphism ρ is determined from the 2ν 2 + 8 parameters h,k , as well as the character ψ ∈ G. However there are several consistency conditions, coming from (4.2) and also the fact that ρ being an endomorphism must preserve the Leavitt-Cuntz relations. To compute various expressions in L ν+1 , it is convenient to collect our equations Implicit in the following is Lemma 1, which permits us to compare corresponding coefficients of an expression in L ν+1 in reduced form (i.e. replace any occurrence of ss ′ with 1 − g t g t ′ g ). Because ρ satisfies (4.2), we must have s ′ ρ(ρ(x)) = xs ′ . But if instead we compute s ′ ρ(ρ(s)) = ss ′ directly from (4.13) and (4.14), using (4.16) and (4.17), we obtain Comparing these expressions for s ′ ρ 2 (s), and performing the analogous calculation for ρ(ρ(s ′ ))s = ss ′ , we obtain . Substituting this into (4.19), we obtain −a 2 = νa − 1 and so a ∈ {1/δ ± } (similarly for a ′ ).
In particular, a, a Thus, since the order ν of G is odd, we have that ψ is identically 1. We thus recover (4.6). Other Note that we still have the freedom to rescale s → λs and s ′ → s/λ; choose λ so that c = b. Then bb ′ = cc ′ (obtained by comparing (4.22) with (4.24) Moreover, comparing (4.23) and (4.20) gives b ′ = bd.
Theorem 2. Choose any finite abelian group G of odd order ν.
(a) Let b ∈ {1/ ωδ ± } and ω 3 = 1, and choose any solution A g,h to (4.7)-(4.10). Define the values of ρ and α g on the generators s, s ′ , t g , t ′ g by (4.3)-(4.6). Then these extend to algebra endomorphisms ρ, α g on the Leavitt algebra L generated by s, s ′ , t g , t ′ g . Then C({α g ρ n }) ds , the idempotent completion extended by direct sums as described in section 3, is a strict spherical fusion category we'll denote by C(G; ±, ω, A). The simple objects of this category are α g = (1, α g ) and α g ρ = (1, α g ρ) up to equivalence, and they satisfy the Haagerup-Izumi fusions (1.1). The categorical dimensions of α g are 1 and of α g ρ are δ ± .
We will learn below that the simple objects are all of the form (uu ′ , α g ρ n ) or (vv ′ , α g ρ n ) for certain monomials u = u g,n h,i , v = v g,n h,j recursively constructed below. The modular data S, T associated to the double of C(G; ±, ω, A) is computed next section.
By Corollary 1, it is trivial that the α g defined by (4.6) are algebra endomorphisms of L. Similarly, to show that ρ satisfying (4.3)-(4.5) extends to an algebra endomorphism of L, it suffices to verify that the values of ρ(s) etc preserve the Leavitt-Cuntz relations. It is readily verified that these all reduce to the identities b 4 + νωb 2 = 1, (4.8), (4.9), and (the latter follows from (4.8) and (4.7)). Thus ρ is an algebra endomorphism.
The equation s ′ ρ(ρ(t g )) = t g s ′ yields the equations The former follows from h,k A h,k A k,h = ν − νb 2 , which in turn follows from (4.9). The latter follows directly from (4.9). The equation t ′ 0 ρ(ρ(t g )) = ρ(t g )t ′ 0 gives (4.10) as well as which follow from (4.9) and (4.7). The simplicity of ρ etc is established by the following proposition.
Proof. Write a = δ −1 ± . Choose any x ∈ L commuting with ρ(s) = as + b g t g t g . We will begin by proving that such an x must be a polynomial in ρ(s). Write x in reduced form (recall Lemma 1). We can assume without loss of generality that no term in x is a scalar times a power of s, i.e. cs l , since otherwise we could replace x with x − c(ρ(s)/a) l (the result will still lie in L and commute with ρ(s), and will be in C[ρ(s)] iff x is). Suppose for contradiction that x = 0.
Assume first that not all terms in x begin with s ′ . Amongst those terms, let w = s l w ′ = 0 be the sum of all terms with the maximal leading string of s's (l may be 0). Then ρ(s)x contains the terms asw = as l+1 w ′ , and these are reduced and have longer leading strings of s's than any other terms in ρ(s)x − xρ(s) (since no term can be a pure power of s). Being reduced, these terms asw cannot cancel anything, contradicting ρ(s)x = xρ(s).
It remains to consider x = s ′ x ′ . Then every term in x ′ = 0 involves only s ′ 's and t ′ g 's (since x is reduced). Then ρ(s)x − xρ(s) when reduced contains terms −a h t h t ′ h s ′ x ′ with leading factors t h t ′ h . Again, these terms cannot cancel, which contradicts ρ(s)x = xρ(s).
These contradictions mean x = 0. Thus any x ∈ L commuting with ρ(s) must be a polynomial in ρ(s), and hence can contain no s ′ , t ′ k . Likewise, any x ∈ L commuting with ρ(s ′ ) must be a polynomial in ρ(s ′ ), and thus contains no s, t k . Together, they tell us that any x commuting with both ρ(s) and ρ(s ′ ) must be a scalar. Now suppose xα g ρ(y) = α h ρ(y)x for all y. Then taking y = s tells us xρ(s) = ρ(s)x, since α g ρ(s) = ρ(α −g s) = ρ(s), while taking y = s ′ tells us xρ(s ′ ) = ρ(s ′ )x. Therefore x ∈ Hom(α g ρ, α h ρ) must again be a scalar λ ∈ C. Now, Now turn to x ∈ Hom(α g , α h ), i.e. xα g (y) = α h (y)x for all y ∈ L. In particular, xs = sx and xs ′ = s ′ x. By the identical argument as above, the former requires x ∈ C[s] while the latter requires x ∈ C[s ′ ], and thus x is a scalar λ ∈ C. Of course, λ = 0 intertwines α g and α h iff g = h, by evaluating at y = t 0 . Hence Hom(α g , α h ).
Finally, suppose x ∈ Hom(α g ρ, α h ) and x = 0 is reduced. Then e.g. xρ(s) = sx. Assume first that at least one term in x does not begin with s ′ . Amongst those terms, let y be one with a maximal string of leading s's (this string may be empty, if no term in x begins with s). Then sy will be a reduced term in sx, and the only reduced terms in xρ(s) with a leading string of s's of similar length are those which are pure monomials in s. So y = rs n for some n ≥ 0 and some non-zero scalar r. But even those y won't work: the reduced terms in sx − xρ(s) corresponding to y are rs n+1 − ars n+1 , which can never vanish because a = 1. If instead all terms in x begin with an s ′ , then none of them end with an s, so repeat this argument with xρ(s ′ ) = s ′ x. The proof that Hom(α g , α h ρ) = 0 is identical.
We have α g ρ n = α gρ n . An easy induction from (4.2) (replacing x there with ρ n−2 (x) and hitting with α g ) verifies where u g,n h,i ∈ Hom(α h , α g ρ n ) ∩ Hom(α h , α gρ n ) and v g,n k,j ∈ Hom(α k ρ, α g ρ n ) ∩ Hom(α kρ , α gρ n ) are (finitely many) monomials in the Leavitt generators s, t l and (for each fixed pair g, n) the collection {u g,n h,i , u g,n ′ h,i , v g,n k,j , v g,n ′ k,j } together satisfy the Leavitt-Cuntz relations u g,n ′ h,i u g,n k,j = δ i,j δ h,k etc. More precisely, {u g,n+1 h,⋆ } = {v g,n h,i } i and {v g,n+1 Certainly Hom(α g ρ n , α g ′ ρ n ′ ) contains all u g ′ ,n ′ h,i ′ u g,n ′ h,i and v g ′ ,n ′ k,j ′ v g,n ′ k,j . In fact, we will show now using simplicity (Proposition 1) that together they span that Hom-space. To see this, choose any x ∈ Hom(α g ρ n , α g ′ ρ n ′ ). Then u g,n ′ h The identical argument shows Hom(α gρ n , α g ′ρ n ′ ) is also spanned by the same elements, and so those Homspaces are identical (and finite-dimensional). Thus Lemma 2 applies.
Recall C(E) ds , the idempotent completion of C(E) extended by direct sums. Note that all u g,n h,i u g,n ′ h,i , v g,n k,j v g,n ′ k,j are idempotents in End(α g ρ n ), thanks to the Leavitt-Cuntz relations. Enumerate these p 1 , . . . , p N . Then p i End(α g ρ n )p j = δ ij Cp i and i p i = id, using the above spanning set (in fact basis) for End(α g ρ n ), so the p i form a complete set of minimal idempotents in End(α g ρ n ). All (p i , α g ρ n ) are objects in C(E) ds . Since End(p i , α g ρ n ) := p i End(α g ρ n )p i = Cp i is 1-dimensional, the (p i , α g ρ n ) are simple in C(E) ds . These (p i , α g ρ n ) (as i, g, n vary) exhaust all simple objects in C(E) ds , as any other idempotent in End(α g ρ n ) is a disjoint sum of the p i . Moreover, (u g,n h,i u g,n ′ h,i , α g ρ n ) and (1, α h ) are isomorphic, with isomorphism u g,n h,i and inverse u g,n ′ h,i , since u g,n h,i u g,n ′ h,i is the identity in End((u g,n h,i u g,n ′ h,i , α g ρ n )). Likewise, (v g,n k,j v g,n ′ k,j , α g ρ n ) and (1, α k ρ) are isomorphic. We thus get a fusion category, because there are only finitely many isomorphism classes of simple objects, namely the [(1, α g )], [(1, α g ρ)].
Finally, let us turn to part (c) of Theorem 2. Suppose A is hermitian. Define a conjugate-linear map on L by s * = ±s ′ , s ′ * = ±s, t * g = t ′ g and t ′ * g = t g , extended so that (cxy) * =cy * x * for all c ∈ C and x, y ∈ L, where the sign in these expressions is as in ω 2 b −2 = δ ± . Then b = ±ωb so (ρ(x)) * = ρ(x * ) for the 2 + 2ν generators x, and hence that relation holds for all x ∈ L. It is easy to see that this determines a * -operation on C(G; ±, ω, A), in the sense defined in the introduction. If in addition ± = +, then this conjugate linear map is the usual * -operation on L, and so taking completions we get a system of endomorphisms on the Cuntz algebra which extend to the infinite factor N and thus we possess a unitary category.
Conversely, suppose C(G; ±, ω, A) possesses a * -operation. Again, we don't know a priori whether the * -operation (which by definition is defined only on individual Homspaces) lifts to a well-defined * -operation on the Leavitt algebra, so again write (ξ|x|η) for x ∈ Hom(ξ, η) as before. Note that the * -operation must take the identity (ξ|1|ξ) in End(ξ) to itself. From simplicity (Proposition 1), we may write t * g = β g t ′ g and t ′ * g = β ′ g t g for some non-zero β g , β ′ g ∈ C. Then taking * of (5.11) gives µ h µ ′ 2k+h = 1, i.e. that µ g = µ ′−1 h = µ is independent of g, h ∈ G. Now taking * of (5.13), we get A h,k = A k,h , and we see that for the category to be hermitian, the matrix A must be hermitian.
Finally, in a unitary category the categorical dimensions must all be positive. But d ρ = δ ± , and δ − < 0. This concludes the proof of Theorem 2.
6 The tube algebra and modular data 6.1 The tube algebra and its centre We will now determine the quantum double or centre of our categories C(G; ±, ω, A) using the tube algebra approach of [22]. That approach assumes unitarity, but [26] categorises the method, generalising it beyond the context we need, and all of our equations come from there.
Let ∆ = {α g , α h ρ} g,h∈G be a finite system of endomorphisms associated to a solution of our equations (4.7)-(4.10). Write Σ∆ for the objects in C(G; ±, ω, A), and write [σ] for the sector or equivalence class of an object (where the conjugation now need not be by a unitary). The categorical dimension d σ = d [σ] of any object σ ∈ Σ∆ was computed last section. We found there the dimensions d [αg] = 1 and d [αgρ] = δ ± for the simple objects (note that δ + > 0 > δ − , so these dimensions can be negative). The global dimension is then λ ± = ν(1 + δ 2 ± ) = 2ν + ν 2 δ ± , which is strictly positive as it must be.
The multiplicative structure of Tube ∆ is given by where we continue to write α g ρ = gρ and g for α g , and T (ν) denotes whichever 1, s, t l lies in Hom(ν, ζζ). In particular, we obtain: where we've only written the non-zero products. Note that we have G actions, by multiplying by A or C, so for simplicity we restrict to subscripts equal to 0 when this G-action can yield the other values. Unless A is hermitian, we can't expect Tube ∆ to have a natural structure as a * -algebra.
Let σ ∈ Σ∆. A half-braiding for σ is a choice of invertible E σ (ξ) ∈ Hom(σξ, ξσ) for each ξ ∈ ∆, such that for every η, ζ ∈ ∆ and any X ∈ Hom(ζ, ξη), In general, σ will be a formal direct sum of L-endomorphisms, so the values E σ (ξ) will be matrices with entries in L. In this case, by σ(X) in (6.3) we mean the diagonal matrix with entries η(X) as η runs over all simples in σ, with multiplicities, and by ξ(E σ (ζ)) we mean to evaluate each entry of the matrix E σ (ζ) by ξ. This equation makes sense as the morphisms (matrices over L) on the left side are intertwiners for σζ → ζσ → ξησ while the right side intertwines σζ → σξη → ξση → ξησ -composition is just matrix multiplication. Invertibility of E σ (ξ) is equivalent to E σ (id) = 1, the identity matrix. There may be more than one half-braiding associated to a given σ; in that case we denote them by E j σ . The quantum double or centre of the fusion category C = C(G; ±, ω, A) is a strict modular tensor category (MTC) with objects (σ, E σ ) where σ ∈ Σ∆ and E σ is a half-braiding. The morphisms x ∈ Hom((σ, E σ ), (τ, E τ )) are x ∈ Hom C (σ, τ ) satisfying σ(x)E σ (ζ) = E τ (ζ)x ∀ζ ∈ Σ∆; composition is as in C. The tensor product of objects is given by ( ; the tensor product of morphisms is multiplication as in C. The unit is (id, 1). The braiding is c (σ,Eσ ),(τ,Eτ ) = E σ (τ ). Duals are (σ, ); (co-)evaluation is as in C. If C is hermitian (resp. unitary), one should require the E σ (ξ) to be unitary and not merely invertible, in which case the resulting category will be a hermitian (resp. unitary) MTC. See [26] for details.
We will determine the possible half-braidings E j σ , by determining the matrix units in Tube ∆ of the corresponding simple summand M k×k . Matrix units e i,j of M k×k are a basis satisfying e i,j e m,l = δ j,m e i,l . The relation between the matrix units and the corresponding half-braidings is [22] where the sum is over ξ ∈ ∆, and again η,η run through the simples {g i , h i ρ} in σ. The corresponding central projection (the unit of that simple summand) is then z(σ j ) = η e(σ j ) η,η . Our primary interest this section is in determining the modular data of the double, and for this purpose the diagonal matrix units are all that we need. As a C-algebra, Tube ∆ decomposes as a direct sum [gρ] , and E l

Σg[gρ]
respectively, where ψ, φ ∈ G but ψ is non-trivial and ψ,ψ give the same half-braiding, h ∈ G but h is non-trivial and ±h give same half-braiding, and 1 ≤ l ≤ ν 2 +3 2 is some parameter to be interpreted later.
The fourth class of half-braidings is parametrised by all characters ψ ∈ G and nontrivial pairs ±h ∈ G, and has diagonal matrix units Compare with Proposition 8.2 of [23] (where there is a minor typo there for e iv;h,ψ −h,−h ). This corresponds to The matrix units for the final summand of (6.5) are addressed next subsection.

The half-braidings for σ = α h ρ
Define n, µ, m by ν = 2n + 1 and µ = ν 2 + 4 = 2m + 1. The analysis of [23] is not complete in determining the matrix units for the final summand of (6.5), even if one is only interested in unitary Q-systems as in [23]. Because of this, it is not possible to determine the modular S, T matrices in general. (To be fair, [23] was mainly interested in the solution for ν = 3 corresponding to the Haagerup subfactor, and for this solution his equations do uniquely determine the matrix units.) In this subsection we supplement the equations given in [23], Lemma 8.3.
Generalising Lemma 8.3 of [23] to our context, we learn that the matrix units corresponding to the final summand, i.e. to the half-braidings with [σ] = h [α h ρ], are of the form e v;j gρ,hρ = ν λ ± C gh + w j δ ± F gh + δ ± k,l C j k,l G k+g/2+h/2,l+g/2+h/2 gh , (6.13) for 1 ≤ j ≤ (ν 3 +3)/2, where the m(ν 2 +1) variables w j , C j k,l ∈ C satisfy the m(ν 2 +1) equations: for all g, h, k ∈ G. This half-braiding corresponds to E v;j (k) gρ,gρ = δ k,0 and E v;j ((k + g)ρ) gρ,gρ = δ k,0 w j ss ′ + l C j k,l t l+k+g t ′ l−k+g , (6.15) for all g, k ∈ G. The w j are the corresponding diagonal entries of the modular matrix T , and so must be roots of unity. Some solutions to (6.14), occurring for w j of small order, are redundant (i.e. correspond to the previous summands of (6.5)) and should be dropped.
Note that, when there is more than one half-braiding E j Σ gρ with the same value of w j , say w j = w j ′ , then there will be infinitely many different solutions to (6.14) with w j , namely tC j g,h + (1 − t)C j ′ g,h for any t ∈ C. This is because (6.14) are linear, for fixed w j . Such w j can indeed occur -in [10], 6 fusion categories (in fact Qsystems) C(Z ν ; +, 1, A) were found with w j of higher multiplicities. Those examples correspond to ν = 9, 11, 19; for reasons explained in [10], we expect there to be higher multiplicities, and hence ambiguities, whenever µ is composite. Whenever we cannot uniquely determine the C j g,h , we cannot uniquely determine e.g. the modular S, T matrices.
The situation will only get worse as we generalise the context beyond Q-systems to not-necessarily-unitary fusion categories. For this reason, we need to supplement Izumi's (6.14) with non-linear constraints. This is done next.
Proposition 2. Let C(G; ±, ω, A) be any category in Theorem 2. Then (4.11) holds. Moreover, in addition to (6.14), w j , C j g,h must satisfy: for all ψ ∈Ĝ, g ∈ G. Conversely, these equations uniquely determine C j g,h and w j . Proof. Consider the subalgebra A ρ = span{C 00 , F 00 , G gh 00 } of Tube ∆. From the products calculated in the previous subsection, we find that A ρ is commutative with unit C 00 . Now, the diagonal matrix unit e j := e v;j ρ,ρ in (6.13) is a minimal projection in A ρ , and hence for any P ∈ A ρ , e j P e j (which equals P e j by commutativity) must be a scalar multiple of e j . Write e j G gh 00 = x g,h e j for scalars x g,h ∈ C. We compute λ ± ν x s,p e j = G sp 00 e j = G sp 00 + w j δ s,0 Therefore x s,p = ωw j C j p,s and we recover (6.17). Similarly, we compute e j C 00 = e j and e j F 00 = δ −1 ± w j e j . (6.18) and (6.20) now immediately follow from e j e j ′ = δ jj ′ e j ′ and e j e iii;ψ ρ,ρ = e j e iv;g,ψ ρ,ρ = 0 respectively. Comparing the G mh 00 coefficients of the associativity of F 00 F 00 G kl 00 gives 20) which is equivalent to (4.11) using (4.7). Conversely, the matrix units e j are uniquely determined by their orthogonality to those for the other half-braidings, as well as the relations e j e j ′ = δ jj ′ e j . These are equivalent to (6.18) and (6.20), once we know e j C 00 = e j , e j F 00 = δ −1 ± w j e j , e j G gh 00 = ωw j C j h,g e j . These latter equations follow from (6.17) and (6.14). QED Curiously, the right-side of (6.18) isn't manifestly symmetric in j ↔ j ′ , even though the left-side is. We know we have a complete list of identities satisfied by A, ω and δ ± , so (6.20) (equivalently (4.11)) is redundant, but it doesn't seem to be trivially redundant. Conversely, we expect that it, in conjunction with (4.7)-(4.9), implies the more complicated (4.10), and so can replace it in Theorems 1 and 2, but we haven't verified this yet. (iii) S 1i ∈ R \ {0} and some index 1 ′ has S 1 ′ i > 0, for all i; (iv) for each i, j, k, the numbers N k ij defined by are nonnegative integers.
Any MTC has modular data. The index i parametrises the simple objects (primaries) X i . The entries T i,i of the diagonal matrix T (up to normalisation) are eigenvalues of the twist θ X i = (tr X i ⊗ id End X i )(c X i ,X i ) ∈ C id X i while those of the symmetric matrix S are associated to the Hopf link: up to normalisation, S i,j = tr X i ⊗X j (c X i ,X j • c X j ,X i ). By Proposition 2.12 of [9], the matrices S and T will be unitary in any MTC, even when the category is not unitary (or even hermitian). '1' corresponds to the tensor identity X 1 and the permutation S 2 sends i to 1 , but it is always possible to choose a normalisation so that it defines a linear (i.e. true) representation of SL 2 (Z). This choice uniquely determines S up to a sign and then T up to a third root of 1. Property (iv) says the S matrix diagonalises the fusion coefficients N k ij , so some column of S (a Perron-Frobenius eigenvector) will have constant phase. We require that column (which we call the 1 ′ th) to be strictly positive, as this is necessary for the existence of a character vector, as explained in section 7.3. This will be the case e.g. in a rational CFT.
From the point of view of modular data, there is little difference between unitary and non-unitary MTCs. In a unitary category, 1 ′ must equal 1. Now specialise to the MTCs which are the doubles of the fusion categories C(G; ±, ω, A) of Theorem 2. Write as before ν = |G| = 2n + 1, µ = ν 2 + 4 = 2m + 1, δ ± = (ν ± √ µ)/2 and λ ± = 2ν + ν 2 δ ± . The main reason for introducing the tube algebra in section 6.1 is to construct its modular data. The simple objects of the MTC are in one-to-one correspondence with the simple summands in (6.5), or equivalently with the irreducible half-braidings. As mentioned earlier, in the tube algebra picture, the braidings are given by the half-braidings, and (co-)evaluations hence traces are as in C. In particular, we obtain the normalised S, T matrices from the diagonal entries E j σ (ξ) η,η : for any j in (6.22) and (6.23), and any simple η ≺ σ in (6.23). In (6.22), ξ can be any simple in σ, and in (6.23) the sum is over all simple ξ in σ while η is any (fixed) simple in σ. The standard left inverse φ ξ of the endomorphism ξ is (6.23) have the desired normalisation built in -as computed in section 5.3 of [26], the normalisation of T is trivial (i.e. T 1,1 = 1) for the double of any (not necessarily unitary) fusion category. The derivation of (6.22),(6.23) is as in [22], except that the complex conjugate in (6.23) replaces the * 's in his Lemma 5.3: his formula assumes φ ξ is a * -map; equation (5.6) of [26] writes this as S σ i ∨ ,σ j , which is equivalent to our complex conjugation.
Breaking S and T into 16 blocks, we get T = diag(1, 1; 1, . . . , 1; φ(h); w 1 , . . . , w m ) , where k a×b for any k ∈ C is the a × b matrix with constant entry k, 1 ∓ y 1 ± y 1 ± y 1 ∓ y and C = ±y 1 1 · · · 1 −1 −1 · · · −1 for y = ν √ µ . We denote transpose with 't'. Much more difficult is to identify the m × m matrix F and the phases w l . Once the solutions C j k,l and w l to (6.14),(6.17)-(6.20) have been found, we conclude Incidentally, (6.17) gives an alternate expression for the diagonal entries of S: where n 3 is the number of g ∈ G with order dividing 3.
We have thus identified the S and T matrices for any fusion category C(G; ±, ω, A), although the numbers w j and the submatrix F seem at this point completely opaque. However, in the following section we list all known fusion categories (unitary or otherwise) of type C(G; ±, ω, A), and identify their modular data. We will find that the mysterious matrix F and phases w l always seem to take a remarkably simple form. For this reason we conjecture: Conjecture 1. Choose any finite abelian group G of odd order ν, and choose any fusion category C = C(G; ±, ω, A). Then there is an abelian group H of order µ = ν 2 +4 and a nondegenerate bilinear form β on H, which determines the submatrix F and the phases w j for the double of C explicitly. In particular, the m = (ν 2 + 3)/2 primaries d l of class (iv) are parametrised by pairs ±l of elements in H, l = 0, and w l = exp[2πım β(l, l)] , (6.26) By a nondegenerate bilinear form β on H, we mean β : and for any non-zero g ∈ H there is an h ∈ H such that β(g, h) ≡ 0 (mod 1).
It is possible that not all G and H arise in Conjecture 1. For example, we know of no fusion categories of type C(Z 3 × Z 3 ; ±, ω, A) ( [10] showed there are no Q-systems for Z 3 × Z 3 ), and we know of no fusion categories C(G; ±, ω, A) whose corresponding modular data has H = Z 5 × Z 5 × Z 5 (it would necessarily have G = Z 11 ). But in both cases, we haven't come close to an exhaustive search.
This conjecture fits into the grafting framework of section 3.3 of [10]. This modular data can be twisted by H 3 (G×Z 2 ; T), as explained in section 3 of [10], and indeed as explained there in section 3.3, non-unitarity is the natural context for some of these twists. We have nothing more to add to this discussion. As mentioned earlier, the method of this paper can be generalised to even-order G [11,12,24], and a very small number of solutions are known at present. Although the corresponding elements of S and T also appear to be surprisingly simple, they do not fit into Conjecture 1, and we are not yet prepared to extend the conjecture to cover them. for c i , d j , f k explicitly defined below; the modular data for all four has H = Z 13 , and β(k, l) = kl/13 resp. β(k, l) = 2kl/13 for '+' resp. '-'; (iii) for G = Z 5 : two inequivalent unitary ones with '+', and two inequivalent hermitian but non-unitary ones with '-'; all four have ω = 1 and for c i , d j , h k explicitly defined below; the modular data for all four has H = Z 29 , and β(k, l) = kl/29 resp. β(k, l) = 2kl/29 for '+' resp. '-'.
The two fusion categories for ν = 1 are realised by affine G 2 at level 1 ('+'), and Yang-Lee ('-'). The first two fusion categories for ν = 3 are realised by an even subsystem of the Grossman-Snyder system H 3 [17] and an even subsystem of the Haagerup subfactor. The other two are their Galois associates. The first two fusion categories for ν = 5 are realised by an even subsystem of the Haagerup-Izumi subfactor for G = Z 5 found in [23], and to one of the even subsystems of the Grossman-Snyder system described in section 6.6 of [17]. The other two are their Galois associates.
Our proof of Theorem 3 uses Gröbner basis techniques as implemented in Maple 17.02. First, we find a basis for the ideal generated by the identities of Theorem 1. Using it, the eigenvalues are found corresponding to multiplication by each of the variables in the quotient of the polynomial ring by our ideal. The eigenvalues are the possible values of the variables. All of these steps are completed in a fraction of a second for ν = 3, 5. We then have to determine (by trial and error) which eigenvalues go together to form solutions. ν = 1 was worked out in section 2, so turn to G = Z 3 . Consider first ω = 1. The order-3 symmetry (4.7) gives us (7.1). These variables (c, d, e, f, g) satisfy (4.8)- where the zeros arise for any entry of A fixed by the order-3 symmetry. The quadratic identities (4.9) give e.g. ω(d 2 +e 2 ) = 1−1/δ ± and d 2 +e 2 = 1, which are incompatible. Thus there are no solutions for G = Z 3 with ω = 1. Now turn to G = Z 5 , with ω = 1. (4.7) gives (7.2). The Gröbner basis method tells us (4.8)-(4.10) have exactly 16 solutions (as always, half with '+' and half with '-'). As before, we must identify solutions related by the action of Aut G ∼ = Z 4 , which sends (c, d, e, f, g, h, i, j, k) → (c, e, g, d, f, i, j, k, h). This yields the 4 inequivalent fusion categories given in Theorem 3. Explicitly, c 1 = (13 + √ 29)/10, c 2 = (13 − √ 29)/10, c 3 = (7 + √ 29)/5, and c 4 = (7 − √ 29)/5.
Finally, turn to G = Z 5 and ω = 1 a nontrivial third root of 1. Write Using the Gröbner basis method, it can be shown that (4.8) and (4.9) with h = 0, 1 are inconsistent. This concludes the proof of Theorem 3.
As is curious aside, the linear and quadratic identities (4.8),(4.9) suffice to fix A for G = Z 5 , but for G = Z 3 there are 8 spurious solutions which run afoul of the quartic (4.10) (or cubic (4.11)) identities.
We know of no examples of fusion categories with ω = 1. Of course, the set of all fusion categories C(G; ±, ω, A) for fixed G is closed under Galois actions. Theorem 3 is disappointing, in that all fusion categories for G = Z 1 , Z 3 , Z 5 are Galois associates of known unitary fusion categories. But we see no reason at all to expect this to continue for larger G, and expect it is an accident of small G.

Some Q-systems and their doubles
Q-systems are unitary fusion categories coming from an even part of a subfactor. After Theorem 1 we explained they correspond here to ω = 1, '+', and A with specified values for A 0,g , A g,0 , A g,g . [10] found several new Q-systems of type C(G; ±, ω, A), although was unable to identify the modular data of some of them. In this subsection we use Proposition 3 to explain how they all fit into Conjecture 1.
A convenient way to express the matrix A of a Q-system, for G = Z ν , is in terms of numbers j 2 , j 3 , . . . , j n+1 ∈ R (recall ν = 2n + 1): for 0 < g < h < ν we have where j 1 = 0 and j n+1+i = j n+1 + j n − j n−i for 1 ≤ i < n (see Lemma 7.3 of [23]). The Q-systems found in [10] correspond to (j 2 , j 3 , j where the superscript (7) etc refers to the value of ν. These approximate values suffice to determine the exact algebraic values of the A g,h , and to verify that these do indeed satisfy all the equations (4.8)-(4.10), using the method described in section 3.5 of [13]. (The solutions for 11 ≤ ν ≤ 19 were announced as conjectural in [10], but using [13] have now been shown to yield exact solutions.) This list constitutes the complete classification of Q-systems for Z 7 , Z 9 , up to equivalence. There is no Q-system solution for G = Z 3 × Z 3 (more precisely, any such solution would require nontrivial 2-cocycle twists of (4.1)).
Given a nondegenerate bilinear form β on some abelian group of order ν 2 + 4, let S β , T β denote the modular data described in Conjecture 1. Section 4.1 of [10] proved this proposition for these Q-systems at ν = 7, 13, 15, 17, and conjectured the correct H and β for 5 of the 6 remaining. It was unable to determine the modular data for the 2 + 2 + 2 Q-systems at G = Z 9 , Z 11 , Z 19 , because of the ambiguity described in section 6.2 above. It had no guess for the modular data for j (11)′ because it did not think of trying noncyclic H.
Our proof of Proposition 3 followed very closely what we used in [10], section 4.1. In particular, a floating point proof is possible and effective, since the integrality of the fusion coefficients N k ij in (6.21) serves as error-correction. More precisely, equation (1.3) of [10] shows S in modular data is uniquely determined from the fusion coefficients, T and the entries S 1,i . Our strategy here is to guess at a phase w j consistent with Conjecture 1, use the linear equations (6.14) and (6.20) to determine the corresponding C j g,h up to a small number of parameters (for almost all choices of w j , this linear system will be inconsistent and we can throw away that choice). For typical examples, the choice w j = e 2πi182/365 for solution j (19) identifies C j g,h up to 1 parameter, while the choice w j = e 2πi2/5 for solution j (11)′ needs 4 parameters. Then we chose at random some nonlinear equations from (6.17) to fix those parameters.

Character vectors
A natural question is to realise the doubles of these fusion categories by completely rational nets of factors and/or by rational vertex operator algebras (VOAs). As a first step, one should consider the corresponding character vectors. This is quite accessible, and provides considerable information.
Choosing any λ ′ k so that T kk = e 2πiλ ′ k , it is clear from holomorphicity and the transformation law (7.3) at 1 0 1 1 , that e −2πiλ ′ k τ χ k (τ ) is holomorphic in the punctured disc 0 < |q| < 1 with an isolated singularity at q = 0, so (7.4) should be regarded as a meromorphicity condition at the so-called cusp τ = i∞. Any holomorphic X : H → C d obeying (7.3) and (7.4) is called a weakly-holomorphic vector-valued modular function (vvmf) for SL 2 (Z) with multiplier ρ. The characters of the irreducible modules M j for any completely rational conformal net of factors on S 1 , or for any strongly rational VOA, form a character vector, where ρ is the modular data coming from the corresponding MTC. The label 1 is the vacuum module M 1 = V 1 (the VOA or net itself), and T 11 = e −πic/12 for a parameter c called the central charge. We can assume without loss of generality that all χ k;0 = 0, in which case h k = λ k + c/24 is called the conformal weight of the module M j . Because T is only determined by the MTC up to a third root of 1, the category determines the central charge only mod 8. For the doubles of fusion categories, as mentioned previously, the central charge c is known to be in 8Z.
The existence of a character vector is not at all automatic. For one thing, it requires that all λ j ∈ Q, but that holds in any MTC. Moreover, given any character vector X(τ ), the vector v := X(i) exists and is strictly positive (since at τ = i we have q = e −2π > 0); then (7.3) says v = Sv and hence S must have a strictly positive eigenvector with eigenvalue 1. But we know that in any modular data, some column (equivalently row, since S = S t ) of S, namely the common Perron-Frobenius eigenvector of the fusion matrices N i = (N k ij ), must have constant phase. This is why we demanded that a column of S be strictly positive, in section 6.3.
When the MTC is unitary, we must have c ≥ 0 and h k > h 1 = 0 for k = 1. The only unitary VOA or net at c = 0 is the trivial theory. In the unitary case, the positive row of S must be the first (=vacuum) row. When unitarity is dropped, then h k ≥ h 1 ′ for all k. The quantity c eff = −24h 1 ′ is called the effective central charge, and must be nonnegative. Again, c eff = 0 can only occur for the trivial VOA and conformal net. To our knowledge, all known examples have h j > h 1 ′ for j = 1 ′ , but this is not yet a theorem.
The Hauptmodul j(τ ) = q −1 + 744 + 196884q + · · · of SL 2 (Z) is a weaklyholomorphic modular function for the trivial multiplier. For any ρ in Definition 2, the space M ! (ρ) of weakly-holomorphic vvmfs is trivially a module over the polynomial ring C[j(τ )]. It turns out that this module is always free of rank d (Theorem 3.3(a) of [16]). Put another way, there is a d × d matrix Ξ(τ ) = q Λ n=0 Ξ n q n , with coefficients Ξ n ∈ M n×n (C), with the property that X(τ ) ∈ M ! (ρ) iff there is a vector-valued polynomial p(x) ∈ C d [x] such that X(τ ) = Ξ(τ )p(j(τ )). So knowing all weakly-holomorphic vvmfs for ρ is equivalent to knowing Ξ(τ ). We can and will require Ξ 0 = I d×d . The matrix Λ will be diagonal, with entries satisfying T kk = e 2πiΛ kk . There is a recursion uniquely determining each Ξ n from the complex matrices Λ and Ξ 1 (equation (36) of [16]). In short, knowing all weakly-holomorphic vvmfs for ρ is equivalent to knowing the exponents Λ and the first nontrivial coefficient matrix Ξ 1 .
Once Ξ(τ ) (or equivalently Λ, Ξ 1 ) are known, it is then just combinatorics to find all character vectors for a given effective central charge (since c eff directly gives bounds for the degrees of all component polynomials p k in p(x) ∈ C d [x]). In [10], this procedure was done for several doubles, including the double of the Haagerup fusion categories, for central charges 8, 16, 24. To illustrate this for a non-unitary example, in this subsection we give Ξ(τ ) for the non-unitary cousin of the Haagerup (G = Z 3 ). Its fusion category and MTC is a Galois associate of that of the Haagerup. By contrast, Ξ(τ ) and hence the corresponding VOA or conformal net, are not at all related in an obvious way to those of the Haagerup, as we'll see.
The double of either of the non-unitary fusion categories for G = Z 3 , at effective