Contents lists available at ScienceDirect

Advances in Mathematics

www.elsevier.com/locate/aim

Non-unitary fusion categories and their doubles via endomorphisms

David E. Evans a'*, Terry Gannon b

a School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, Wales, UK

b Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

I CrossMark

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 March 2016 Accepted 17 January 2017 Available online 4 February 2017 Communicated by Dan Voiculescu

Keywords:

Modular tensor categories Non-unitary Leavitt algebra Quantum double Conformal field theory Subfactor

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.

© 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

* Corresponding author.

E-mail addresses: EvansDE@cf.ac.uk (D.E. Evans), tgannon@math.ualberta.ca (T. Gannon).

http://dx.doi.org/10.10167j.aim.2017.01.015

0001-8708/© 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction..............................................................................................................2

2. The Yang—Lee model..................................................................................................6

3. Leavitt algebras and categories of endomorphisms ..........................................................10

4. Non-unitary Haagerup—Izumi: deconstruction..................................................................13

5. Non-unitary Haagerup—Izumi: reconstruction..................................................................18

6. The tube algebra and modular data..............................................................................24

6.1. The tube algebra and its centre..........................................................................24

6.2. The half-braidings for a = ^ ahp........................................................................28

6.3. Modular data for the double of C(G; ±, u, A)........................................................30

7. Explicit solutions........................................................................................................33

7.1. The fusion category classification for small G........................................................33

7.2. Some Q-systems and their doubles......................................................................36

7.3. Character vectors..............................................................................................38

Acknowledgments................................................................................................................42

References..........................................................................................................................42

1. 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 ^-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. [32]) than in the VOA picture. Also, the factor setting captures in a natural way the full CFT as an inclusion of (local) nets [3,31].

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. [32] 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 [11,12]. 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 [11]. 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 canoni-cally 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 (fg)* = g* f * for all f G Hom(X, Y), g G Hom(Z, Y). If the category is tensor (and strict), we also require (f ® g)* = f * ® g* for all f G Hom(X, Y), g 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 SL2(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 \m(t) = qhM^L0 dim Mn qn of the irreducible modules M = ]Jn Mn form a vector-valued modular function for SL2(Z) with modular data as its multiplier. The minimal conformal weight hM 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 [19]).

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 [29], who studies non-unitary 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 [ag] and [agp] as g ranges over G. The fusions are given by

[ag}[ah] = [ag+h] , [ag}[ahp] = [ag+hp] = [ahp][a-g] ,

[agp][ahp] = [ag-h] + k [akp] ■ (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 [22]. His motivation was to construct the Haagerup subfactor [18,2], so he focused 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+ p. Q-systems correspond to especially constrained p; their fusion categories are always unitary. He showed that there was a unique Q-system satisfying (1.1) for the group G = Z3, and comparing indices observed that it must correspond to the Haagerup subfactor. Likewise, he showed that there is a unique Q-system for G = Z5. He also computed the modular data for the doubles of his systems (modulo a technicality discussed shortly). Evans-Gannon [11] 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 = Z3 and Z5, 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 [14,13] and independently Izumi [23].

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 (though to keep the accommodations demanded by nonunitarity as clear as possible, we impose a simplifying assumption (4.1) — see the discussion in the paragraph before Theorem 1). We show they all yield fusion categories. Like [22], our systems correspond to solutions of finitely many equations in finitely many variables, but unlike [22] our equations are all polynomials (those of [22] involve complex conjugates). In broad strokes the method we use is analogous to that of [22], 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, we find precisely 2,4,4 inequivalent fusion categories realised by endomorphisms, of Haagerup-Izumi type for G = Zi,Z3, Z5 respectively (of course we recover all of them). Precisely 1,2,2 of these, respectively, are unitary: 1,1,1 are Q-systems, and 0,1,1 are the aforementioned Grossman-Snyder systems. The Yang-Lee system is the unique non-unitary one corresponding to G = Zi .

Every fusion category C is defined over some number field [10]. An automorphism a of that field acts on the quantities of that category in the natural way, defining a new fusion category Ca. 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 Ca 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 we have found for G = Zi , Z3, Z5 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 G2 at level 1 (a unitary MTC) with the Yang-Lee model (a non-unitary one).

Actually, the equations in [22] 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 C2-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 logarithmic theories. The 'logarithmic' analogue of the fusion category is the finite tensor category of [9], and the analogue of the modular

version thereof, in particular regarding mapping class group representations including modular data, is explored in [25].

2. 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 p, which obey the fusion rule

[p][p] = [1] + [p] ■ (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

p(p(x)) = sxs' + tp(x)t', (2.2)

where s, s', t, t' G A satisfy the Leavitt-Cuntz relations

ss' + tt' = 1, s's = t't =1, s't = t's = 0 ■ (2.3)

More precisely, these relations say that (2.2) expresses p o p as a direct sum of objects id and p in the category END(A) (we describe this category in detail next section). These elements s, s', t, t' generate by definition a copy of the Leavitt algebra L2 inside A; we will see shortly that p restricts to an endomorphism of L2. In order to identify the restriction of p to L2, it is necessary and sufficient to determine the values p(s), p(s'), p(t) and p(t') of p on the generators. For *-maps, we would have p(s') = p(s)' etc, but we cannot require that here if we hope to realise the Yang-Lee model.

We require that both endomorphisms id and p be simple, equivalently that the inter-twiner spaces Hom(id, id) and Hom(p, p) in the algebra A be C1, and that Hom(p, id) = Hom(id, p) = 0. (The definition of intertwiners is given next section.) From (2.2) we obtain

p2(x)s = sx , p2(x)t = tp(x) , s'p2(x) = xs', t'p2(x) = p(x)t' ■ (2.4)

The first means s G Hom(id, p2). Conversely, suppose r G Hom(id, p2), i.e. rx = p2(x)r for all x. Then s'rx = s'p2(x)r = xs'r and t'rx = t'p2(x)r = p(x)t'r. Thus by simplicity of id and p we have s'r G C and t'r = 0, so r = (ss' + tt')r = ss'r G Cs using the Leavitt-Cuntz relation ss' + tt' = 1. We have shown Hom(id, p2) = Cs. In the same way (see Lemma 3 below for details and the generalisation), we can identify the intertwiner spaces Ct = Hom(p, p2), Cs' = Hom(p2, id) and Ct' = Hom(p2, p). These observations are crucial for what follows.

Note, using (2.4), that

s'p(s)p(x) = s'p(sx) = s'p(p2(x)s) = s'p2(p(x))p(s) = p(x)s'p(s) . (2.5)

In other words, s'p(s) G Hom(p, p) = C, so s'p(s) equals some complex number a. Likewise, t'p(s) G Hom(p, p2) so t'p(s) = bt for some b G C. The point is that

p(s) = (ss' + tt')p(s) = s(s'p(s)) + t(t'p(s)) = as + btt. (2.6)

Similar calculations (see section 4 for details and the generalisation) give

p(s') = a's' + b't't', p(t) = cst' + dtss' + ettt', p(t') = c'ts' + d'ss't' + e'tt't', (2.7)

for some a', b', c, cC, d, d', e, e' G C. Because p sends the generators of L2 into L2, this means p is actually an endomorphism of L2. If we required p 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 p to solve for those 10 parameters. First, p is required to be an algebra endomorphism, so it must respect the Leavitt-Cuntz relations (2.3). One relation requires 1 = p(s')p(s), i.e.

1 = (a's' + b't't')(as + btt) = a'a + bb'. (2.8)

Similarly, p(s)p(s') + p(t)p(t') = 1 gives the identities 1 = aa' + cc' (hence b'b = c'c), aa' + cc! = dd' (hence d'd = 1), and ab' = —ce', amongst others. More precisely, Lemma 1 below gives a unique form for any element of a Leavitt algebra, so once we expand out p(s)p(s') + p(t)p(t') = 1 and put it into reduced form (e.g. replacing ss' by 1 — tt'), the identities fall out by comparing corresponding coefficients.

We also require that p satisfy (2.2). It implies for instance that s'p2(s) = ss'. We can compute s'p(p(s)) directly from (2.6), (2.7), and we find

s'p2(s) = as'p(s) + bs'p(t)p(t) = a2 + bct'(cst' + dtss' + ettt') = a2 + bcdss' + bcett'.

This must equal ss', which (using 1 = ss' + tt') gives 1 = a2 + bcd and 1 = bcd — bce (hence a2 = —bce). Likewise, p2(s')s = ss' gives 1 = a'2 + b'c'd' and a'2 = —b'c'e'. Similarly, (2.2) implies t'p2(s) = p(s)t'; its t and st' coefficients give ab = — bde and a = bcd respectively. Likewise, p2(s')t = tp(s') gives a'b' = —b'd'e' and a' = b'c'd'.

Plugging a = bcd into 1 = a2 + bcd (and likewise for the primed quantities) gives 1 = a2 + a = a'2 + a', which means a, a' G {(—1 ± a/5)/2}. Note that if a = a' then aa' =1 — we will use this shortly. Since a = bcd, a' = b'c'd' are both non-zero, so are all b, c, d, b', c!, d!. Note that we are free to rescale s by A G Cx (hence s' by 1/A) without affecting (2.2) nor the Leavitt-Cuntz relations. Choosing A appropriately we can simultaneously force b = c and also 0 < Arg(b) < n, and then bb' = cc! also gives b' = c!. Comparing a2 = —bce, a = bcd, and ab = —bde give e = —da and d G {±1} (and

likewise e! = -d'a' and dd G {±1}). But we knew ddd = 1, so we have dd = d. Putting aa' = (bcd)(b'c'd') = b2b'2 into (2.8) gives bb' G {(-1 ±V5)/2}. In particular, aa' cannot be 1, so we must have a = a' and thus b = \fda and b' = db.

We eliminate the possibility that d = — 1by considering the st't' coefficient of t'p2 (t) = p(t)t', which gives c = ab'd? + cdee'. So we have determined that a = a' = — e = —e! = (—1 ± a/5)/2, b = b' = c = c = yfa, and d = dd = 1, where we can take the square-root for b so that b G R>0 U £R>0. So we have 2 possible solutions, corresponding to the choice of signs in a = (—1 ± a/5)/2. In section 4 we generalise this argument to arbitrary odd order abelian G in (1.1).

Conversely, given either solution a = (—1 ± a/5)/2, we can define p on the generators s, s', t, t' of L2 by (2.6)-(2.7). Using Corollary 1 below, this choice extends to an algebra endomorphism p on L2 iff it respects the Leavitt-Cuntz relations: i.e. 1 = p(s')p(s) = p(t')p(t) = p(s)p(s') + p(t)p(t') and 0 = p(s')p(t) = p(t')p(s). It is straightforward to verify this (this is done in full generality in section 5). To show p satisfies (2.2), note that both sides of (2.2) are manifestly endomorphisms, so it suffices to verify it for each of the four generators x G {s, s', t, t'}. If we can show s'p2 (x) = xs' and t'p2(x) = p(x)t' for x = s, t (these must hold if (2.2) is to hold), then p2(x) = (ss' + tt')p2(x) shows (2.2) holds for x = s, t. Likewise, if p2(y)s = sy and p2(y)t = tp(y) for y = s', t', then (2.2) holds for x = s', t'. Again, the details are given in full generality in section 5. Thus p defined by (2.7) obeys the Yang-Lee fusions (2.1). Finally, we can confirm that the endomorphism p we have just constructed is indeed simple, i.e. Hom(p, p) = C as well as Hom(id, id) = C and Hom(p, id) = Hom(id, p) = 0 (this is done in full generality in Proposition 1 below).

Much more delicate is to associate a (strict) fusion category to both of these p. The biggest challenge here for arbitrary G is to define arbitrary (but finite) sums of endo-morphisms 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 x 2 generators, we can capture arbitrary sums — e.g. p © p3 © p5 can be written sps' + tsp3s't' + ttp5t't', to choose a random example. The resulting fusion category for the solution with a = ( — 1 + a/5)/2 is the unitary category associated to e.g. the integrable modules of the affine G2 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 p (defined next section) is 1/a = (1 ± a/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|) x 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 p is a *-map (e.g. the case studied in [22,11]), we can extend p to an endomorphism of an infinite von Neumann factor N [22]; 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 p 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 + a/5)/2 category is realised by both affine G2 level 1 and affine F4 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 [ag][p] = [p][ag] 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 G2 at level 1, would yield the MTC of affine G2 0 F4 at level (1, 1).

Although the fusion (or modular tensor) categories of Yang-Lee and affine G2 or F4 at level 1 are merely related by a Galois automorphism, the corresponding VOAs do not seem related in any simple way. For example, the characters of Yang-Lee are

gll/60(1+ q2 + q3 + q4 + g5 +2g6 + ...) ,

q—1/60(1 + q + q2 + q3 +2q4 + 2q5 +3q6 + ••• )

with modular data

S =_L_ (- sin(2n/5) sin(n/5) \ = (e2"11/60 o N S = V5 V sin(n/5) sin(2n/5) J , 1 V 0 e—2ni/60j ,

while those for affine G2 at level 1 are

q—7/60(1 + 14q + 42q2 + 140q3 + 350q4 + 840q5 + •••) ,

q17/60(7 + 34q + 119q2 + 322q3 + 819q4 + 1862q5 + •••)

with modular data

S =_i_ ( sin(n/5) sin(2n/5) N = (e—2ni7/60 0 N S = V5 \sin(2n/5) - sin(n/5^ , 1 ^ 0 e2ni17/60 J

and those for affine F4 at level 1 are

q—13/60(1 + 52q + 377,+ 1976q3 + 7852q4 + •••) , q23/60(26 + 299q + 1702q2 + 7475q3 + 27300q4 + •••)

with modular data

S =_i_ ( sin(n/5) sin(2n/5) N = (e—2li13/60 0 N S = V5 \sin(2n/5) -sin(n/5)y , 1 ^ 0 e2ni23/60) '

In these cases, the first character given is that of the VOA V = ]JVn itself, and so lists the dimensions of its graded spaces Vn, so we see that there appears little relation

between the Yang-Lee VOA and that of say the G2 one. On the other hand, the naive inner product of the G2 and F4 character vectors is j (t)1/3, reflecting the fact that the VOA V(G2, 1) < V(F4, 1) is a conformal subalgebra of the E8 lattice VOA. Note also that the first column of the matrix S is strictly positive for the VOAs V(G2, 1) and V(F4, 1) (as it must be for unitary VOAs), and isn't for the Yang-Lee (as is typical for non-unitary VOAs).

3. Leavitt algebras and categories of endomorphisms

For each n > 1 define the Leavitt algebra Ln to be the associative *-algebra freely generated over C by x1, ■ ■ ■, xn, x[, ■ ■ ■, x'n, modulo the Leavitt-Cuntz relations

i j — ^ i j ^ ^ ^ i i — 1 ■ (3.1)

The elements of Ln are polynomials in the non-commuting variables xi, xj. The *-operation sends xi ^ xi, xi ^ xi, and obeys (cyz)' = cz'y' for all c G C and x, y G Ln. It has an obvious grading by Zn. The Leavitt algebra Ln can be regarded as the polynomial part of the Cuntz algebra On, its C*-algebra completion.

The Leavitt algebras Ln are all non-isomorphic for n = 2, 3, 4, ■ ■■, since the inclusion of Ln in On induces an isomorphism on K-theory with the cyclic group Zn-1 [1]. The only obstruction to embedding Lm unitally in Ln is given by the K-theory [30]. More precisely Lm embeds unitally in Ln if and only if m — 1 divides n — 1. In the Cuntz framework of Izumi [20,22,23] and Evans-Gannon [11-13], one constructs endomorphisms on a fixed Cuntz algebra On, with prescribed fusion rules and then extends these to a completion as an infinite von Neumann factor N. Any Cuntz algebra Om can be unitally embedded in the factor N for any m, even though usually it cannot be unitally embedded in On. 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 Ln. But we do not require that our endomorphisms be *-maps, so they need not extend to the completion, the Cuntz algebra On or the Banach algebras of Phillips [29].

Note that if p is any algebra endomorphism on Ln, then so is p defined by

p(y) = p(y')' ■ (3.2)

Throughout this paper we distinguish an algebra endomorphism from a *-algebra endomorphism. The latter must obey f(y)' = f(y') (equivalently p = p) while the former may not.

There is a canonical way to write any element of Ln. Call any monomial in the generators xi, xj reduced if no primed variable appears to the left of any unprimed variable, and x1 is not adjacent to x[ in the monomial. Call any linear combination over C of finitely many distinct reduced monomials, a reduced sum.

Lemma 1. ([24]) Any y G Ln 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 Ln is trivial [24]. Moreover:

Corollary 1. An algebra endomorphism p on Ln is uniquely defined by its values p(xi), p(xj) 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 Ln. 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 Ln is a unital algebra over C, by general nonsense its algebra endomorphisms define a C-linear preadditive strict tensor category £ND(Ln). More precisely, the objects in £ND(Ln) are algebra (but not *-algebra) endomorphisms of Ln. The morphisms r G Hom(3, 7) are intertwiners, i.e. r G Ln for which rf3(x) = Y(x)r for all x G Ln; composition of morphisms is multiplication in Ln. £ND(Ln) is C-linear, i.e. each Hom(3, 7) is a vector space over C; it is also preadditive, i.e. composition of morphisms is bilinear. The tensor product of objects is composition: 3 < 7 = 3 o 7, whilst of morphisms is: r < s = rfi(s) = Y(s)r G Hom(3 o p, 7 o a) when r G Hom(3, 7), s G Hom(p, a).

A fusion category [10] 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 idx; amongst other things, every object in a semisimple category is a direct sum of simple ones. We say object X has a right-dual Xv iff there is a pair of morphisms evaluation eX G Hom(Xv < X, 1) and co-evaluation bX G Hom(1, X < Xv) for which

(idx < ex) o (bx < idx) = idx , (ex < idxv) ◦ (idxv < bx) = idxv (3.3)

(where we assume the category is strict, for convenience). Left-dual vX is defined similarly. In particular in END(L), an object 3 G End(L) has a right-dual 3v G End(L) if there are elements ep G Hom(3 o ¡v, id) and bp G Hom(id, 3v o 3) in L such that

3(e^)bp = 1 = ep3v(bp) • (3.4)

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 fv G Hom(yv, Xv) of a morphism f G Hom(X, Y) by

fv = (ey < idxv) o (idyv < f < idxv) o (idyv < bx) • (3.5)

In particular in END(L), the right-dual of r G Hom(a,3) is defined by

rv = ep pw(rba) . (3.6)

Then (f og)v = gv ofv when the composition is defined. Left-dual vf is defined similarly. A rigid tensor category is pivotal if it is equipped with a natural monoidal isomorphism from the identity functor to the double-dual functor X ^ Xvv. In a pivotal category we can take vX = Xv. In a rigid category the (left-)dimension of object X is eXvbX; a semisimple pivotal category is called spherical if X and Xv have the same dimension for all objects X (it suffices to check this for simple X). See e.g. [28] for the remaining terminology not explained here.

Let E be a collection of algebra endomorphisms of Ln closed under composition. We require the identity to be in E. Let C(E) denote the subcategory of END(Ln) restricted to E. Then like END(Ln), 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, /3) where /3 G E and p G End(/) is an idempotent, i.e. p2 = p, and whose morphism spaces are Hom((p, ¡3), (q, y)) = qHom(/, y)p with composition again given by multiplication. C(E) is a tensor category using (p, /) ® (q, y) := (p <g) q, / <g) y) = (p/(q), 3 o y), and the tensor product of qrp G Hom((p, /), (q, y)) with q'r'p' G Hom((p', /'), (q', y')) is (qrp) <g) (q'r'p') = qrpfl(q'r'p'). We can introduce direct sums into C(E) as follows. Objects in this new category consist of ordered n-tuples ((p1, ¡1),..., (pn, ¡n)) = (p1, ¡1) © ••• © (pn, ¡n), and the morphism spaces are

Hom(((Pl,¡l),..., (Pn,¡n)),((ql,Yl),..., (qm,Ym)))

/ q1 Hom(/1,Y1)p1

\qm Hom(/1,Ym)p1

Composition is matrix multiplication. Then ((p1, ¡1),..., (pn, ¡n)) ® ((q1, y1), ..., (qm, Ym)) is the direct sum of (pi, fa) ® (qj, Yj), while

/ q1rnp1 \qmr1mp1

is the Kronecker product with (ij, ¿/j')-entry qirjipj ® q'i,rj,vpj,. We will write C(E)ds for the idempotent completion C(E) extended by direct sums in this way.

Lemma 2. Let E be a collection of Ln-endomorphisms as above, and recall (3.2). Suppose Hom(/, y) = Hom(/, Y) in Ln for all ¡3, y G E, and that these are all finite-dimensional. Then C(E)ds, the idempotent completion extended by direct sums, is a semisimple strict

q1 Hom(3n, Y1 )pn qmHom(f3n,Ym)pn

q1 r1 p

qm rnm pn

^qm' r1 m' p1

qm, rn

C-linear tensor category with finite-dimensional hom-spaces. If C(E) is rigid, then so is

Proof. The category C(E)ds is manifestly C-linear and strict. Since all Hom(3, 7) are finite-dimensional, so are all Hom-spaces (3.7) in C(E)ds. Since the anti-linear involution x ^ xX sends Hom(3, 7) to Hom(Y, 3), and Hom(3, 7) = Hom(3, 7) by hypothesis, then x ^ x' bijectively maps Hom(3, 7) to Hom(Y, 3). This implies that the (finite-dimensional) algebra End(((p1, 31), •••, (pn, 3n))) is a *-algebra, and hence is semisimple. Then Corollary 2.3 of [33] tells us C(E)ds is a semisimple category.

Moreover, suppose C(E) is rigid. Then Lemma 3.1 of [33] says that its idempotent completion C(E) is also rigid: e.g. (p, 3)v = (pv, 3v) where the dual morphism pv is defined in (3.6), and (co-)evaluation is epp = pv3v(p) ep and b(p p) = p3(pv) bp. Hence

C(E )ds is also rigid: take ((p1, 31), •••, (pn, 3n))v = ((p1, 31)v, •••, (pn, 3n)v) with diag-

onal (co-)evaluations e(. , (pi pi) ,= diag(e(pi pi)) etc. □

This condition Hom(3, 7) = Hom(3, 7) 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 [12]. 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.

4. Non-unitary Haagerup-Izumi: deconstruction

Let G be any abelian group of odd order v = 2n + 1, and define S± = (v±%/v2 + 4)/2, the two roots of x2 = 1 + vx. 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 ag, p are algebra endomorphisms of an algebra A which realise the Haagerup-Izumi fusions. More precisely, this means

where s, s', tg, t'g G A satisfy s's = 1, s'tg = t'gs = 0, t'gth = Sgyh, and 1 = ss' + J2g tgt'g. We do not assume A is a *-algebra. Equation (4.1) implies that each ag is invertible. Note that we have the freedom to rescale the v +1 elements s, tg arbitrarily and independently, provided we then rescale s', t' inversely. We also require ag and agp = ag o p to be

C(E )ds.

ag o ah = ag+h , ag o p = p o a—g ,

simple, i.e. that their intertwiners in A are Hom(agp, ahp) = Hom(ag, ah) = CSghh and Hom(ag, ahp) = Hom(agp, ah) = 0. This implies for instance that A has trivial centre, and that the representation g ^ ag of G is faithful.

Unless G is cyclic (in which case HG(pt; T) = 1), (4.1) can be generalised by twisting by 2-cocycles £ G ZG(pt; T) and (1.1) will still hold, as explained e.g. in the proof of Theorem 1 in [12]. 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. Izumi [22 also ignored these cocycles, but the unpublished notes [23] introduces them (though of course in the unitary setting).

Equation (4.1) can be generalised to ag oah = ad(Ug^h)ag+h for invertible Ughh. When A is a C*-algebra, these U's can be absorbed into the a's, but there is no reason to expect this to hold for more general A. We will also ignore this generalisation of (4.1) here, for the same reasons as given last paragraph. But we return to this possibility in Section 7.1.

Theorem 1. Let G, ag, p and s, s', tg, t' gA be as above. Then

for some fixed sign ±, where b G {1/^wS±} and w3 = 1. In particular, ag and p restrict to algebra endomorphisms of the Leavitt algebra L = Lv+1 with generators s, s', tg, t'g. Moreover, Agh G C satisfy

p(tg) = bst—g + wt—gss' + ^^ h,k Ah+g,k+gthth+k+gt'k ,

p(t'g ) = wbt—g s' + wss/t'—g + ^2 h,k Ak+g,h+gtkt'g+h+kt'l ag(s) = s, ag (s') = s', ag(th) = th+2g , ag(t'h) = t'h

h+2g ,

Agh = wA — hg — h = wAh — g,

Ah—g,i—gSk,g - wS^ShfiAi^ - wS—1AghSift • (4.10)

We will show in Proposition 2 below that in fact

for all g, h, k, l G 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 [22], a (unitary) Q-system corresponds to the special case of Theorem 1 with w = 1, 5± = 5+, Aghh = Ahg, Agfi = 5gfi — 1/(5+ — 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: replacing s ^ —s, s' ^ —s' shows b is equivalent to —b. 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 (±, w, A) yield isomorphic fusion categories is answered below in Theorem 2, as is the question of unitarity.

Lemma 3. Let p be any algebra endomorphism on A satisfying (4.1) and (4.2), and assume ag and agp are all simple. Then Hom(ag, p2) = Cs5gfi, Hom(agp, p2) = Ctg, Hom(p2, ag) = Cs'5g o, and Hom(p2, agp) = Ct'g. Moreover, Hom(p2, agp2) = Css'5g o +

spanh {th+g 4}.

Proof. Directly from (4.2) we find p2(x)s = sx, p2(x)tg = tgagp(x), s'p2(x) = xs', and t'gp2(x) = agp(x)t'g. In other words, s G Hom(id, p2), tg G Hom(agp, p2), s' G Hom(p2, id), and t'g G Hom(p2, agp).

Now suppose r G Hom(ag, p2). Then s' G Hom(p2, id) and t'h G Hom(p2, ahp) immediately imply s'r G Hom(ag, id) = C5g,o and t'hr G Hom(ag, ahp) = 0 by simplicity. Therefore r = ss'r + ^h tht'hr G Cs5g,0, hence Hom(ag, p2) = Cs5g,0.

Next, suppose r G Hom(p2, ag). Then rs G Hom(id, ag) = C5g0 and rth G Hom(ahp, ag) = 0, which forces r G Cs'5g,0 as before, and thus Hom(p2, ag) = Cs'5g,0.

Now consider r G Hom(agp, p2). Then s'r G Hom(agp, id) = 0 and t'hr G Hom(agp, ahp) = C5g h, and thus Hom(agp, p2) = Ctg.

Similarly, let r G Hom(p2, agp). Then rs G Hom(id, agp) = 0 and rth G Hom(ahp, ag p) = 5hgg C, which gives us Hom(p2, ag p) = Ct'g.

Finally, suppose r G Hom(p2, agp2). Then, using the invertibility of a and the calculation agp2 = pa-gp = p2ag, we get rs G Hom(id, agp2) = Hom(ag, p2ag) = Cs5g,0. Similarly, rth G Hom(ahp, agp2) = Hom(ah+gpag, p2ag) = Cth+g. This suffices to identify Hom(p2, agp2) in the usual way. □

Note that because ag is an algebra endomorphism and s G Hom(id, p2), ag(s) G Hom(ag, agp2). But Hom(ag, agp2) = Hom(ag, p2ag) = Hom(id, p2) since ag is invert-ible. By Lemma 3 this means ag(s) = ^(2g)s for some ^(2g) G C (the 2 is introduced for later convenience; because the order of G is odd, 2 is invertible). Because agah = ag+h, we see ^ G G. From the Leavitt-Cuntz relation s's = 1, we obtain ag(s') = ^(—2g)s'. Likewise, ah(tg) G Hom(ah+gp, ahp2) = Hom(ag+2hp, p2) = Ctg+2h, and hence

ah(tg) = £h(g)tg+2h

for some eh(g) G C. Again, agah = ag+h implies these numbers eh(g) are non-zero and satisfy

eh+k(g) = th(g)ek(g + 2h) • (4.12)

We can rescale t1, • ••, tv—1 so that eh(0) = 1 for all h. But from (4.12) with g = 0 this implies ek(2h) = 1 for all h, k G G, and invertibility of 2 then implies all ek(h) = 1. From t'gtg = 1 we likewise get ah(t'g) = t'g+2h. Thus we know all ag restrict to endomorphisms of the Leavitt algebra Lv+1 generated by the s, s', tg, t'.

Since s G Hom(id, p2) and p is an endomorphism, p(s) G Hom(p, p3). Hence s'p(s) G Hom(p, p) = C and t'0p(s) G Ct0. Write s'p(s) = a and t'0p(s) = bt0 for some a, b G C. Hitting the latter equation with ah, we get t'2hah(p(s)) = bt2h, i.e. t' p(s) = ^(2g)btg. Likewise, p(s')s = a' and p(s')tg = b't' for some a', b' G C. We thus obtain from p(s) = ss'p(s) + J2g tgVgp(s) that

p(s)= as + b^ g ^(g)tg tg , p(s') = a's' + b'J2 g V>(-g) t'g t'g • (4.13)

The computation of p(tg) is similar. First note that p(t0) G Hom(p2, p3), so s'p(t0) G Hom(p2, p) = Cf0 and t'hp(t0) G Hom(p2, ahp2) = span{Shj0ss', tkt'k_h}, using Lemma 3. Write s'p(t0) = ct'0 and t'hp(t0) = Shj0dss' + ^k Ahkkth+kt'k, for complex numbers

c, d, Ah,k. Then p(t0) = cst'0 + dt0ss' + h k Ah,kthth+kt'k. The calculation for p(t'0)th is identical, and involves complex numbers cC, dd, A'h k. Hitting these with a—g/2 yields

p(tg) = ^(-g)cst'—g + dt—gss' + ^2 h,k Ah+g, k+gthtg+h+kt'k , (4.14)

p(t'g ) = V(,g)c't — g s' + d'ss't'—g ^ h , k A'h+g k k+g tk t'g+h+kt'h • (4.15)

Thus we also know p restricts to an endomorphism of the Leavitt algebra Lv+1 generated by the s, s', tg, t'g.

Thus the A-endomorphism p is determined from the 2v2 +8 parameters a, a', b, b', c, c',

d, dd, Ahkk, A'h k, as well as the character ^ G G. However there are several consistency conditions, coming from (4.2) and also the fact that p being an endomorphism must preserve the Leavitt-Cuntz relations. To compute various expressions in Lv+1, it is convenient to collect our equations

s'p(s) = a,tgp(s) = b^(g)tg , (4.16)

s'p(tg) = ^(-g)ct'—g , t'gp(th) = dSg—hss' + kAg+hk+htg+h+kt'k , (4.17)

s'p(fg ) = d's' f—g , thp(t'g ) = ^ ^ j S g^ ^ k +g , h+, ^^k ^fc • (4.18)

Implicit in the following is Lemma 1, which permits us to compare corresponding coefficients of an expression in Lv+1 in reduced form (i.e. replace any occurrence of ss' with

1 - Eg tg trg ).

Because p satisfies (4.2), we must have s'p(p(x)) = xs'. But if instead we compute s'p(p(s)) = ss' directly from (4.13) and (4.14), using (4.16) and (4.17), we obtain

s'p(p(s)) = a2 + bc (vdss' + ^ g A0,g ^ k tktk) .

Comparing these expressions for s'p2(s), and performing the analogous calculation for p(p(s'))s = ss', we obtain

bc^ g A0g = —a2 = vbcd — 1, b'c'Y^ g A0 , g = —a'2 = vb'c'd' — 1. (4.19)

Likewise, the st'0 coefficient of t'0p(p(s)) = p(s)t'0 becomes a = bcd (and similarly we get a' = b'c'd'). Substituting this into (4.19), we obtain —a2 = va — 1 and so a G {1/5±} (similarly for a').

In particular, a, a' = 0, so also b, b', c, c', d, dd = 0. Hitting a = s'p(s) with ag, we obtain

a = ag(s' )p(a-g(s)) = ^(—2g)s'p(^(—2g)s) = ^(—4g)a

for all g G G. Thus, since the order v of G is odd, we have that ^ is identically 1. We thus recover (4.6).

Other coefficients of t'0p(p(s)) = p(s)t'0 we need now give

dX^ h Ah , 0 = —a, d'^2 h Ah , 0 = —a , (4.20)

y^ h Ah , kAk , k'+h — d5k ,0 h Ah,0 = 5k', 0 . (4.21)

From p(s')p(s) = 1 we obtain 1 = (a's' + b'J2g t'gt'g)(as + b J2g tgtg), i.e.

aa' + vbb' = 1. (4.22)

The st'gt'g, tgtgs', constants, tgt'g, and thth+kt'l+kt'l terms of the Leavitt-Cuntz relation

1 = p(s)p(s0 + Eg p(tg)p(t'g) give respectively

^ h A'h,0 = — ab', ¿Yl, h Ah,0 = — ba', (4.23)

aa' + cc'v =1, dd' = 1 , (4.24)

XI g Ah+g,kA'g k = 5h,0 — bb'5k,0 . (4.25)

Note that we still have the freedom to rescale s ^ As and s' ^ s/X; choose A so that c = b. Then bb' = cc! (obtained by comparing (4.22) with (4.24)) implies b' = c!. Now, aa' = (bcd)(b'c'd') = (bb')2, so (4.22) implies bb' G {1/5±}. However, if a = a', then aa' = 1/(5+5_) = —1, contradicting our value for bb'. Thus a = a' = bb'. Moreover, comparing (4.23) and (4.20) gives b' = bd.

Multiplying (4.25) by Ak, h+m and summing over h using (4.21) gives

K, h + Sk,0d(£ gA'g00)^2 hAh,0) = Ah, g - Sk,0bb'Y, hA0,h • (4.26)

But the terms proportional to Sk,0 are d (-a'/d')(-a/d) = da2 and -bb' (-a2/bc) = a2b'/b, which we now know are equal. Thus A'g h = Ahg for all g, h G G. The st'h—gth coefficient of t'0p2(tg) = p(tg)t'0 is

cSh,0 = d2ab'So,g + c^J^ kAg, fe+sAh+k,-g . (4.27)

Multiplying (4.27) by Ah+l,-g and summing over h using (4.25) collapses to Ai,-g = dAgj+g, which recovers (4.7); because the permutation (l, -g) ^ (g, l + g) is order 3, d must be a 3rd root w of 1.

We obtain (4.8) and (4.21) from (4.20) and (4.9). Finally, (4.10) arises from the thti-g+ht'k+jt'k coefficient of t'0p(p(t-g)) = p(t-g)t'0. This completes our derivation of Theorem 1.

5. Non-unitary Haagerup-Izumi: reconstruction

This section is devoted to a proof of the following theorem, another main result of our paper. Recall S± = (v ± %/v2 + 4)/2.

Theorem 2. Choose any finite abelian group G of odd order v.

(a) Let b G {1/^/wS±} and w3 = 1, and choose any solution Aghh to (4.7)-(4.10). Define

the values of p and ag on the generators s, s', tg, t' by (4.3)-(4.6). Then these extend

to algebra endomorphisms p, ag on the Leavitt algebra L generated by s, s', tg, t'g. _ds

Then C({agp"}) , the idempotent completion extended by direct sums as described in section 3, is a strict spherical fusion category we'll denote by C(G; ±, w, A). The simple objects of this category are ag = (1, ag) and agp = (1, agp) up to equivalence, and they satisfy the Haagerup-Izumi fusions (1.1). The categorical dimensions of ag are 1 and of agp are S±.

(b) Two such fusion categories C(G(i); ±(i), w(i), A(i)) are equivalent as tensor categories iff ±(1) = ±(2), w(1) = w(2) and there is a group isomorphism n : G(1) ^ G(2) such that Agh = A^g>, for all g, h G G(1).

(c) C(G; ±, u, A) is unitary iff ± = + and A is a hermitian matrix: Ag h = Ahg for all g, h € G. C(G; ±, u, A) is hermitian iff Agh is hermitian.

We will learn below that the simple objects are all of the form (uu', agpn) or (vv', agpn) for certain monomials u = u^, v = vh'n recursively constructed below. The modular data S, T associated to the double of C(G; ±, u, A) is computed next section.

By Corollary 1, it is trivial that the ag defined by (4.6) are algebra endomorphisms of L. Similarly, to show that p satisfying (4.3)-(4.5) extends to an algebra endomorphism of L, it suffices to verify that the values of p(s) etc preserve the Leavitt-Cuntz relations. It is readily verified that these all reduce to the identities b + vub2 = 1, , , and

^g A0 ,g = -W5-1 (5.1)

(the latter follows from (4.8) and (4.7)). Thus p is an algebra endomorphism.

To verify that agp = pa-g, we need to show that ag(p(x)) = p(a-g(x)) for x = s, s', th, t'h. This is trivial to verify: e.g.

ag (p(t'l )) = bt-l+2gs' + ss/t'-l+2g +Y1 Ak+l, h+ltk+2gt'l+h+k+2gth+2g = P(t'l-2g ) • (5.2)

To see that p satisfies (4.2), it suffices to verify that s'p(p(x)) = xs', t'gp(p(x)) = agp(x)tg, p(p(y))s = sy and p(p(y))tg = tgagp(y) for all g G G, x G {s, th} and y G |s', th}. This is because those equations imply using p2(x) = (ss' + Eg tgt'g)p2(x) = p2(x)(ss' + J2g tgt'g) that (4.2) holds when x is any generator, and this suffices to prove (4.2) for all x because both sides of (4.2) are manifestly endomorphisms. In fact, by ag-equivariance, it suffices to establish these for g = 0. All of these equations reduce to b4 + vwb2 = 1, (4.8), (4.9), and (5.1), except for the following. The equation s'p(p(tg)) = tgs' yields the equations

1 = 2wb4 + ub2J2 h,k Ah,kAk,h , (5.3)

—^b2Ah,k - ul^Sh,0 = ^^ i, m At, m Am,£+hAh,k+m • (5.4)

The former follows from J2h k Ahkk Akhh = v — vb2, which in turn follows from (4.9). The latter follows directly from (4.9). The equation t'0p(p(tg)) = p(tg)t'0 gives (4.10) as well as

y^ l,m Al,mAl+g,kAk+m,l = —b2Sk,0 — utfAgk , (5.5)

Y k Ak+g,hAk,-h = uShg — ub2Sh,0 , (5.6)

y^ m Agm+g A-g,m+k = uSkfi—b"2 S0,g , (5.7)

which follow from (4.9) and (4.7).

The simplicity of p etc is established by the following proposition.

Proposition 1. Let p be as above. Then for each g, h G G, Hom(agp, ahp) = Hom(ag, ah) = CSgih and Hom(ag, ahp) = Hom(agp, ah) = 0.

Proof. Write a = S- . Choose any x gL commuting with p(s) = as + b J2g tgtg. We will begin by proving that such an x must be a polynomial in p(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. csl, since otherwise we could replace x with x — c(p(s)/a)1 (the result will still lie in L and commute with p(s), and will be in C[p(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 = slw' = 0 be the sum of all terms with the maximal leading string of s's (l may be 0). Then p(s)x contains the terms asw = asl+1w/, and these are reduced and have longer leading strings of s's than any other terms in p(s)x — xp(s) (since no term can be a pure power of s). Being reduced, these terms asw cannot cancel anything, contradicting p(s)x = xp(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 p(s)x — xp(s) when reduced contains terms —a J2h tht'hs'x' with leading factors tht'h. Again, these terms cannot cancel, which contradicts p(s)x = xp(s).

These contradictions mean x = 0. Thus any x G L commuting with p(s) must be a polynomial in p(s), and hence can contain no s', t'k. Likewise, any x G L commuting with p(s') must be a polynomial in p(s'), and thus contains no s, tk. Together, they tell us that any x commuting with both p(s) and p(s') must be a scalar.

Now suppose xagp(y) = ahp(y)x for all y. Then taking y = s tells us xp(s) = p(s)x, since agp(s) = p(a-gs) = p(s), while taking y = s' tells us xp(s') = p(s')x. Therefore x G Hom(agp, ahp) must again be a scalar A G C. Now, for A = 0, Xagp(t0) = ahp(t0)X iff Ap(t-2g) = Ap(t-2h), iff g = h (since b = 0). Thus Hom(agp, ahp) = Sg,hC.

Now turn to x G Hom(ag, ah), i.e. xag(y) = ah(y)x for all y G L. In particular, xs = sx and xs' = s'x. By the identical argument as above, the former requires x G C[s] while the latter requires x G C[s'], and thus x is a scalar A G C. Of course, A = 0 intertwines ag and ah iff g = h, by evaluating at y = t0. Hence Hom(ag, ah).

Finally, suppose x G Hom(agp, ah) and x = 0 is reduced. Then e.g. xp(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 xp(s) with a leading string of s's of similar length are those which are pure monomials in s. So y = rsn for some n > 0 and some non-zero scalar r. But even those y won't work: the reduced terms in sx, — xp(s) corresponding to y are rsn+1 — arsn+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 xp(s') = s'x. The proof that Hom(ag, ahp) = 0 is identical. □ to Proposition 1

Recall the category END(L) defined in section 3. Let E consist of all monomials of the form agpn. Since (agpm)(ahpn) = ag±hpm+n, the set E is closed under composition. Let C(E) be the subcategory of END(L) with objects agpn. We want to show C(E) is

rigid. Define (ag p2k+1)v = ag p2k+1 and (ag p2k )v = a-g p2k. Then (ag pn)v (ag pn) = (agpn)(agpn)v = p2n for all g G G, n > 0. Define eagpn = wnb-ns'p(s') • • • pn-1(s') and bagpn = wnb-npn-1(s) • • • p(s)s. Since s' G Hom(pk+2, pk) (this is a special case of s'p2(x) = is'), pm(s') G Hom(pn+2, pn) for any m < n follows because p is an endomor-phism. Therefore, eagpn g Hom(p2n, id) as required. Likewise, bagpn g Hom(id, p2n). To see that eagpn, bagpn satisfy (3.4), first note that for any k > l,

( 1 k = l

pk(s')pl(s)= pl(pk-l(s')s) = i wb2 k = l + 1 . (5.8)

i Pl(s)pk-2(s') k > l + 2

Using this, it is easy to see that for any n > 2, we have

pn(s')pn+1(s')^ • • p2n-1(s>n-1(s)- • • p(s)s

= wb2 pn-1(s')pn(s') • • • p2n-V)pn-2 (s) • • • p(s)s ,

which by an easy induction on n gives the first equation of (3.4). The second equation in (3.4) is handled analogously. Thus C(E) is rigid, with (co)evaluations e, b.

We want to apply Lemma 2. That means we must verify first that Hom(agpm, ahpn) = Hom(agpm, ahpn) in END(L), where f3(x) = ¡3(xJ)' is defined by (3.2). Note that ag = ag (i.e. ag is a *-map), but p is defined by (4.3)-(4.5) using the adjoint Ahg in place of Agh, b in place of b, and w in place of w. It is manifest that p is an endomorphism of L satisfying (4.2).

We have agpn = agpn. An easy induction from (4.2) (replacing x there with pn-2(x) and hitting with ag) verifies

ag pn(x) = ^2 h,i ugh'n ah (x)ugh'n' + ^2 k,j Vgjak p(x)v9gj ', (5.9)

agpn(x) = ^2 hi uh'!i ah (x)uh'n' + ^2 k,j vkj ak P(x)v9gj', (5.10)

where ugh^ G Hom(ah, agpn) n Hom(ah, agpn) and vhj G Hom(akp, agpn) n Hom(akp, agpn) are (finitely many) monomials in the Leavitt generators s, tl and (for each fixed pair g, n) the collection {u^ ,ug^"n', v9g'n ,v9g'n'} together satisfy the Leavitt-Cuntz relations ughn 'ugkn = 5ij5h, k etc. More precisely, {u^^1} = {vh'n}i

and {vg ,:+1} = {uM}i U {vhjtk-h}j,h.

Certainly Hom(agpn, ag>pn ) contains all ugh n ugh'n' and vgk j v9g'n'. In fact, we will show now using simplicity (Proposition 1) that together they span that Hom-space. To see this, choose any x G Hom(agpn, ag>pn ). Then ughn 'xugh,n G Hom(ah, ah') = C6h, h; when h = h', call this number qh;i, i'. Likewise, vh'n'xvgh,'rn, G Hom(ahp, ah<p) = C6h, h'; when h = h', call this number rh;i, i/. Moreover, ughn 'xvgh,' ^ = vh'n 'xugh,'n, = 0

since Hom(«h, ah'p) = Hom(ahp, ah') = 0. Thus x = h, i ugh, f ugh^' + vgh, f vgh,n')x g, n, g n g, n, g n g, n, g n g, n, g n

(Eh',i' uh,, n ul'i' + vh', i, vh',i') = Eh, i , i,(qh;i',iuh, i ul, i + rh;i', ivh, i vh,i ). Thus

Hom(agpn, ag>pn ) = spanh i {ugh n ughn', vgh 'P v9hn'}. The identical argument shows Hom(agpn, agrpn ) is also spanned by the same elements, and so those Hom-spaces 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 uhn uhn', Vk'jvg j' are idempotents in End(ag pn), thanks to the Leavitt-Cuntz relations. Enumerate these p1, • ••, pN. Then pjEnd(ag pn)pj = Sij Cpi and p^ = id, using the above spanning set (in fact basis) for End(agpn), so the pi form a complete set of minimal idempotents in End(agpn). All (pi, agpn) are objects in C(E)ds. Since End(pi, agpn) := piEnd(agpn)pi = Cpi is 1-dimensional, the (pi, agpn) are simple in C(E)ds. These (pi, agpn) (as i, g, n vary) exhaust all simple objects in C(E)ds, as any other idempotent in End(agpn) is a disjoint sum of the pi. Moreover, (u^nughn', agpn) and (1, ah) are isomorphic, with isomorphism u^ and inverse u^', since u^u^' is the identity in End((uhnughn', agpn)). Likewise, (vg jjvk j', agpn) and (1, akp) are iso-morphic. We thus get a fusion category, because there are only finitely many isomorphism classes of simple objects, namely the [(1, ag)], [(1, agp)].

To show that C(E)ds is pivotal, note first that (agpn)vv = agpn. We want to show also that the double-dual on all intertwiner spaces Hom(agpn, ag<pn ) is also the identity map. We must be careful here (and elsewhere) to keep track of the Hom-space we are working in by writing (£|x|n) for x G Hom(£, rj). For convenience abbreviate 1« = (£|1|0, s = (id|s|p2), s' = (p2|s'|id), tg = (agp|tg|p2) and tg = (p2|tg|agp). We can compute directly from (3.6) that (1«)v = 1«v, sv = s', tg = t'g, s/v = s, and tg/ = tg, and so the double-dual leaves unchanged all of these. But the double-dual is a monoidal functor, so it will also leave unchanged the morphisms (akpl|s|akpl+2) = s ® 1akpi, (ak+hPl+1|tk+2h|ahPl+2) = tk+2h ® 1ahpi, (akpl+2 |s'|akpl+2) = s' <g> 1akpi,

and ^hp^^'k^hWk+h pl + 1) = tk+2h ® 1ahpl . By writing G ^Ka^ ag pn) and

vgh,n G Hom(ahp, agpn) as monomials in s, tk, they can be written as a sequence of compositions of these morphisms (akpl |s|akpl+2) and (ak+hpl+1 \tk+2h\ahpl+'2) (this is manifest in the recursions given earlier). Hence the double-dual also leaves unchanged ughn and vh'n. Identical conclusions applies to ughn' and vh'n', and hence to the compositions ugh in ughnn and vh vh'n'. But those compositions span Hom(agpn, ag>pn ). Thus the double-dual fixes every morphism r G Hom(agpn, ag>pn ). From this we get that the double-dual functor X ^ Xvv is the identity functor on C(E)ds, and so C(E)ds is pivotal. The dimension calculation is now trivial: eag bav = 1 and e(agpn)v bagpn = wnb-2n = S±, from which we read off that X and Xv have the same dimension for any simple X. This means that C is spherical.

Now turn to the proof of part (b) of Theorem 2. Suppose there is a tensor category equivalence between C(G(i); ±(i), w(i), A(i)). Because a(1) = (1, a(1)) is simple, the equivalence must send ah1 to (p(2), a^p(2)m) for some (minimal) idempotent p(2) and some a(2)p(2)m. Then id(1) = aig1) v^(x, a^^p(2) mv) for some x G L(2), k G G(2). But if m > 0, this can never equal id(2) = (1, i d(2)) . Similarly, if p(1) ^ (p, a^2^p(2) m) for some m > 1, then no object in C(G(1); ±(1), w(1), A(1)) can get sent to p(2) = (1, p(2)).

So our tensor equivalence defines a bijection n : G« —> and an element r G G(2) by agg1 ^ a^g) and p(1) ^ a^f(2). Thanks to the fusion rules, n must be a group

isomorphism, and the tensor equivalence must send ag^p(1) ^ a^2()g)+rp(2).

Although the tensor equivalence will map Hom-spaces to Hom-spaces, we don't know a priori whether it lifts to a well-defined algebra homomorphism between the Leavitt algebras, so as above we will be careful to keep track of the Hom-space we are working in by using the (£\x\n) notation. For convenience abbreviate = (£\1(i)\£), s(i) = (id\s(i)\p(i)2), and s/(i), tg\ tgl) similarly. Note if the tensor equivalence sends object £ to object f, then it must take the identity in End(£) to the identity 1® in End(f).

By simplicity (Proposition 1), we know s(1) (which spans Hom(id(1), f(1)2)) is sent to As(2) (which spans Hom(id(2), (ai2) p(2))2) = Hom(id(2), p(2)2)) and likewise t{g] G Hom(agx^p(1), p(1)2) to vgtr+ng G Hom^^g^p(2), p(2)2) for some non-zero A, vg G C. Since is sent to \ the relations s' o s = and t'g o th = 1asp5g>h give s'(1) ^

A-1s'(2) and tg(1) ^ v-^'r+lg. From 1afc <g> th = (afe+hp|afe(th)|afep2) and t^+h ® 1afc = (akp2\t/2k+h\ak+hp) we obtain

(t2k+h ® 1ak ) o (1ak ® th) = 1ah+kp i (5.11)

hence 1(1()1) gets sent to both 1(2()2) (2) and v-k+hVh1(2(2) (2). Thus Vg = v

ah + kP(1) an(h + k) + r P(2) +h an(h + k) + rP(2)

is independent of g. Comparing dimensions of p(1) and aT2p(2), we get c(1)b(1)2 = c(2)b(2)2, i.e. we must have b(1) = ±b(2) (hence b(1) = b(2) and the signs ±(1) and ±(2) are equal) and c(1) = c(2). The calculation

t'o o (t'o ® p) o (1p <g> s) = t0 o (p3|t/0|p2) o (p|p(s)|p3) = b 1p (5.12)

means, computing the image of the tensor equivalence in two ways, b(1) = v-2Ab(2), which fixes the value of A. Similarly, the calculation

(th+k ® 1ah )(th ® 1p)(1p ® to)tk = KfX+kK )(fPi\t'h\ahp2)(p2 |p(to )|p3)tk = Ah,k 1ak p

(5.13)

gives Ahk = Anhnk.

Note that s, s', tg, t'g obey the Leavitt-Cuntz relations (3.1), iff ±v2s, ±v-2s', vtg+r, Vt'g+r do, for any sign ±, v G Cx and r G G. These choices leave unchanged the algebra Lv+1 and its endomorphisms p, ag. Part (b) follows.

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* = t' and t'g* = tg, extended so that (cxy)* = cy*x* for all c G C and x, y G L, where the sign in these expressions is as in c2b-2 = S±. Then b = ±cb so (p(x))* = p(x*) for the 2 + 2v generators x, and hence that relation holds for all x gL. It is easy to see that this determines a *-operation on C(G; ±, cc, 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; ±, u, A) possesses a *-operation. Again, we don't know a priori whether the *-operation (which by definition is defined only on individual Hom-spaces) lifts to a well-defined *-operation on the Leavitt algebra, so again write (£|x|n) for x G Hom(£, n) as before. Note that the *-operation must take the identity (£|1|£) in End(^) to itself. From simplicity (Proposition 1), we may write t* = /3gt'g and t'g* = ¡3'gtg for some non-zero /3g, ¡3'g G C. Then taking * of (5.11) gives (h(2k+h = 1, i.e. that (g = p'f-1 = ( is independent of g, h G G. Now taking * of (5.13), we get Ahkk = Akhh, 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 dp = S±, and S- < 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; ±, u, A) using the tube algebra approach of [21]. That approach assumes unitarity, but [27] categorises the method, generalising it beyond the context we need, and all of our equations come from there.

Let A = {ag, ahp}gheG be a finite system of endomorphisms associated to a solution of our equations (4.7)-(4.10). Write £A for the objects in C(G; ±, u, A), and write [ct] for the sector or equivalence class of an object (where the conjugation now need not be by a unitary). The categorical dimension da = d^j of any object ct G £A was computed last section. We found there the dimensions d[ag] = 1 and d[agP] = S± for the simple objects (note that S+ > 0 > S-, so these dimensions can be negative). The global dimension is then A± = v (1 + S2) = 2v + v 2S±, which is strictly positive as it must be.

The tube algebra Tube A is a finite-dimensional algebra over C, defined as a vector space by

TubeA = ©«,^,ceAHom(£C,Cn) • (6.1)

It will be semisimple even if the fusion category is non-unitary [27]. As in section 5, given an element X of Tube A, it is convenient to write (£C|X|Zn) for the restriction to Hom(£Z, Zn), since the same operator may belong to distinct inter-twiner spaces. For readability we will often write g and gp for ag and agp, respectively. In our case the intertwiner spaces are computed by Lemma 3. Then a basis for Tube A consists of Agh = (g, h|1|h, g), Bgh = (g, hp|1|hp, —g), Cgh =

(gp, (g — h)/2|1|(g — h)/2, hP), Dgkh = (g,kp|t2k+g-h|kp, hp), Egkh = (gp,kp\t'g-h|kP, Fgh = (gp, (g + h)/2p|ss'|(g + h)/2P, hP), and ^h = (gp, kp^l-h+k^i+g-k^ hp) (note

that the vector space structure of Tube A given at the bottom of p. 655 of [22] is incomplete). Thus Tube A is v4 + 2v3 + 4v2-dimensional. The multiplicative structure of Tube A is given by

(£Z\X\Zn)(CC\Y\Cn) = Snt E №\T(v)'Z(Y)X£(T(v))\vC) , (6.2)

where we continue to write agp = gp and g for ag, and T(v) denotes whichever 1, s, tl lies in Hom(v, ZC). In particular, we obtain: AghAgl = Ag,h+l, AghBgl =

BglA—g, — h = Bg,h+l, BghB—gl = Ag,h-l + 5g0 E m B0m, CghChk = Cg^ AghDgkh' = Dg,h+k,h', Bg0D—gk0 = El Al + k-g,l + k+gDg,l,0, DgkhChk' = Dg,k+(k'-h)/2,k', £gkhAhl = £g,к — l,h, £0khBh0 =J2m Am+k-h,m+k+h£0m,

— h, Cgh£hkg' £g, (g-h+2k)/2,g', Dg0h£h0l =

JglW Ag ,0 + — Am+g+h,2g Bgm,

E0khDhl0 = tX5l,kC00 + tSk,l + h^00 + ^^ g,m Am—k+l + h,g — k+lAm — h+k—l,g+k—lGm0g ,

CghFhk Fgk, FghChk Fgk,

Dk0hFh0 = Dk, — k — h/2^ F0h£h0l = £0,l — h/2,l, r r _ r—1 . ,r—2 v1 r?l0 r.kl — r.(g — h+2k)/2,(2l+g — h)/2 r.kl P —

F0hFh0 = C00 + l G00, CghGhh' = G gh' , G°h'Ch'g' =

n(2k+g' — h')/2,(2l+g' — h')/2 ^ r.k'l — ^ A A T> r.k'lp —

Ghg' ' Dk0hGh0 — Am+l,m+k+k' Am+k+k',m+l Dkm0, G0g £g0k —

Am+g — k — k' ,l — k—k' Al+m,m — k+k' £0mk,

Fg0 = 5—2^2k,gC00 + ^^ m Al+m—g/2,2k—gGCTOk g/2 ,

F0kG°0l = 2^2k',kC00 + 5—1 Em Am+l — k/2,2k' — kG00,k k/2 , and G°h G°0 =tC5—15k,k' Al+l' — h,2k — hC00 + k CSk'lF00

\ A A A nmm:

T ?^m,m' Am+l'— k,m'+ k'— k Am' — k' +h — k,l' — k' — k+l Am+l — k',m' + k—k' G00 ,

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 A to have a natural structure as a *-algebra.

Let a G £A. A half-braiding for a is a choice of invertible £a(£) G Hom(a(, £a) for each £ G A, such that for every n, Z G A and any X G Hom(Z, £n),

X£a(Z)= £(£,(n)) £x (£) a(X) . (6.3)

In general, a will be a formal direct sum of L-endomorphisms, so the values £a (£) will be matrices with entries in L. In this case, by a(X) in (6.3) we mean the diagonal matrix with entries n(X) as n runs over all simples in a, with multiplicities, and by £(£a(Z)) we mean to evaluate each entry of the matrix £a(Z) by £. This equation makes sense as

the morphisms (matrices over L) on the left side are intertwiners for aZ ^ (a ^ £na while the right side intertwines aZ ^ a£n ^ £an ^ £na — composition is just matrix multiplication. Invertibility of £a (£) is equivalent to £a (id) = 1, the identity matrix. There may be more than one half-braiding associated to a given a; in that case we denote them by £j.

The quantum double or centre of the fusion category C = C(G; ±, cc, A) is a strict modular tensor category (MTC) with objects (a, £a) where a G £A and £a is a half-braiding. The morphisms x G Hom((a, £a), (t, £t )) are x G Home (a, t ) satisfying a(x)£a (Z) = £T(Z)x VZ G £A; composition is as in C. The tensor product of objects is given by (a, £a) (g (t, £t) = (aT, £aT) where £aT(Z) = £a(Z)£T(Z); the tensor product of morphisms is multiplication as in C. The unit is (id, 1). The braiding is c(),(T,£T) = £a(t). Duals are (a, £a)v = (av, £av) where £av (Z) = e av ( v (av(£a(Z)-i)avZ(&a)); (co-)evalua-tion is as in C. If C is hermitian (resp. unitary), one should require the £a(£) to be unitary and not merely invertible, in which case the resulting category will be a hermitian (resp. unitary) MTC. See [27] for details.

Tube A, being a finite-dimensional semisimple algebra over C, decomposes into a direct sum ©iMkiXki(C) of matrix algebras. The (indecomposable) half-braidings £j make this explicit. Decompose the sector [a] into a sum Ei=1[ffi] +Ei=i[hiP] of simples, repetitions allowed. In C, a is the formal direct sum

a = ((1,9i),---,(1,9k' ), (1,hip^.(1, hk" p)) ,

where the 1's denote the identity idempotent (and will be dropped for readability). Let k = k' + k". Then by (3.7), for each simple £ £j(£) will be a k x k matrix with entries £j(£)n,n G Hom(n£, £n) C L, as n, n run over all simples {gi, hip} in a, repetitions included. The resulting k x k matrix algebra {£j(£)} (with entries contained in L) is isomorphic as a C-algebra to an irreducible summand of Tube A, and all irreducible summands are of that form.

We will determine the possible half-braidings £j, by determining the matrix units in Tube A of the corresponding simple summand Mkxk. Matrix units eitj of Mkxk are a basis satisfying eijj emil = Sj^mei^l. The relation between the matrix units and the corresponding half-braidings is [21]

e(aj)n,n = —d= E € ^ (n£|£j(£knl£n) , (6.4)

where the sum is over £ G A, and again n, n run through the simples {gi, hip} in a. The corresponding central projection (the unit of that simple summand) is then z(aj) = e(aj)n n. 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 A decomposes

as a direct sum

Tube A = Mixi©Mv+ixv+i© 1 M„+2x„+2© V — VMv+2xv+2® V :3MvXv (6.5)

c°rresp°nding to half-braidings ^¡d^ £[id] + Ss gp^ £2Viid] + Es [g^ £[h] + [—h] + ^g [gp], and 4s [gp]

respectively, where 1, << G G but 1 is non-trivial and 1, give the same half-braiding, h G G but h is non-trivial and ±h give same half-braiding, and 1 < l < v +3 is some parameter to be interpreted later.

The proof of (6.5) for C(G; ±, cc, A), in particular the determination of the associated matrix units, follows the analysis in section 8 of [22], which does this for the (unitary) Q-systems. The main differences are the presence of t and the absence of a *-structure. The central projection of the unique half-braiding of a = id is again given by zi = A—1 J2 A0g + SA^1 ^ B0g, and so

£i(g)0,0 = £ i(gp)0,0 = 1. (6.6)

The matrix units corresponding to the second summand of (6.5) are

eo,o = g A°,g g B°,g) '

eii = in + A jr + A A rk+g/2+h/2,l+g/2+h/2\

egp,hp = ^ I Cgh + gh + Ak+m,l+mAk-m,l-mGgh

\ k,l,m I

Compare with Proposition 8.2(2) of [22]. This corresponds to

£ii(h)0,0 = 1, £ii(h)gp,gp = Sh,0 , £ii(hp)0,0 = -S—2, (6.8)

£ii((k + g)p)gp,gp = Sk,gss + C ^ ] l,m Ak+m,l+m Ak—m,l—mtg + k+ltg — k+l .

The third class of half-braidings is parametrised by pairs {1, 1} of non-trivial characters 1 G G, and has diagonal matrix units

e0,0^ = v —1 Y1 g 1(g)A0,g , e0'fi' = v —1Yl g 1(g)A0,g , (6.9)

e7p',tp = VS^ ( Cgg + S±Fgg + tS± J2 1(m)Ak+m,l+mAk—m,l—mGkk+g,l+g I .

y k,l,m J

This corresponds to

^(g)°,° = 1>(g), £iii;^(g)°',°' = 1>(g), £iii;^(g)kp,kp = 5g,°, £iii;^ (gp)°,° = Eiii;^ (gp)o' ° = 0, (6.10)

E iii'^ ((g + k)P)kp,kp = + w ^^ l,m ^(m)Ak+m,l+mAk-m,l-mtk+l+g t'k+l-g ■

The fourth class of half-braidings is parametrised by all characters ^ G G and non-trivial pairs ±h G G, and has diagonal matrix units

= vg , e-ZX = g ^(9)A-h,g , (6.11)

eivphg'p = -F" [Cgg + 6±^(h)Fgg + "s± I] ^(m)Ak+h+mJ+mAk-h-m,i-mGkg+9'l+9

Compare with Proposition 8.2 of [22] (where there is a minor typo there for e^-h). This corresponds to

£iv;hX(g)h,h = 1>(g), £iv;hX(g)-h-h = 1>(g), £w(g)kp,kp = Sg,o,

£iv;hX (gp)h,h = £iv;hX (gp)-h-h = 0,

. , , _ (6.12)

£iv;h • X ((g + k)p)kp , kp = Sg , o^(h)ss' V 7

+ 1,m ^(m)Ak+h+m,l+mAk-h-m, l-mtk+l+gttk+l-g ■

The matrix units for the final summand of (6.5) are addressed next subsection.

6.2. The half-braidings for a = ahP

Define n, m by v = 2n + 1 and ^ = v2 + 4 = 2m + 1. The analysis of [22] 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 [22]. Because of this, it is not possible to determine the modular S, T matrices in general. (To be fair, [22] was mainly interested in the solution for v = 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 [22], Lemma 8.3.

Generalising Lemma 8.3 of [22] to our context, we learn that the matrix units corresponding to the final summand, i.e. to the half-braidings with [a] = h[ahp], are of the form

j = £ (Cgh + *jS±Fgh + s± E kll chgk+g/2+h/2,l+g/2+h/2) , (6.13)

for 1 < j < (v3 + 3)/2, where the m(v2 + 1) variables wj, C3k l G C satisfy the m(v2 + 1) equations:

^g C0 , g = wj - W S-1 , wj cg, h k Ag+k,2hCh , k = Sh,o UWjj S-1 , (6.14)

for all g, h, k G G. This half-braiding corresponds to

£V'j (k)gp,gp = Sk,0 and £v'j ((k + g)P)gp,gp = SkfiW~jss' + ^ l Ck,ltl+k+g tl-k+g , (6.15) for all g, k G G.

The Wj are the corresponding diagonal entries of the modular matrix T, and so must be roots of unity. Some solutions to (6.14), occurring for Wj 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 Ejgp with the same value of Wj, say Wj = Wj¡, then there will be infinitely many different solutions to (6.14) with Wj, namely tCj h + (1 — t)Cj h for any t G C. This is because (6.14) are linear, for fixed Wj. Such Wj can indeed occur — in [11], 6 fusion categories (in fact Q-systems) C(Zv; +, 1, A) were found with Wj of higher multiplicities. Those examples correspond to v = 9, 11, 19; for reasons explained in [11], we expect there to be higher multiplicities, and hence ambiguities, whenever ¡ is composite. Whenever we cannot uniquely determine the Cj 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; ±, u, A) be any category in Theorem 2. Then (4.11) holds. Moreover, in addition to (6.14), Wj, Cj h must satisfy:

uWj Cp , sCh r r = Ss , hSr,p + W]Ap+h , 2SSr,s (6.16)

+ à ±2., k , l Ck lAh+l-s , r+k-oAr-k-o , l-k-s+p Ah+p-k ,r+s-k , (6.17)

Vsj,j' = 1 + Wj'Wj + s±UWjYl tq CjqCjt, (618)

0=1+ ^(g)wj + 5±UWj E ^(m) CjtAt+m+g,q+mAt-m-g,q-m , (6.19)

for all ^ G G, g G G. Conversely, these equations uniquely determine CJg h and Wj.

Proof. Consider the subalgebra Ap = span{C00, F00, Goo } of Tube A. From the products calculated in the previous subsection, we find that Ap is commutative with unit C00. Now, the diagonal matrix unit ej := epp in (6.13) is a minimal projection in Ap, and hence for any P G Ap, ejPej (which equals Pej by commutativity) must be a scalar multiple of ej. Write ejGgh = xghej for scalars xg h G C. We compute

xs,pej = Çso0pej = G0p + Wj ^ oà0 + E k Ap+k,2sGkS^ + à± E kl CJk>l

x (u5o,kAp+l,2o^--+ uSojSkpF

+ ^ ] h,r Ah+l-s,r+k — o Ar—k — o,l — k—o+pAh+p—k,r+o — kGoo

Therefore xop = uWjCj o and we recover (6.17).

Similarly, we compute ejCoo = ej and ejFoo = S—1wjej. (6.18) and (6.20) now immediately follow from ejej = Sjjr ej and eje1™^ = ejepVpg'^ = 0 respectively. Comparing the G0oh coefficients of the associativity of FooFooGoo gives

Ah+l, 2k Am+k,2h = --:--+ ^^ r Am-k, h+r-kAh-r-k, l-r-kAm+l-r,h+k-r , (6.20)

which is equivalent to (4.11) using (4.7).

Conversely, the matrix units ej are uniquely determined by their orthogonality to those for the other half-braidings, as well as the relations ej ej = 5jj> ej. These are equivalent to (6.18) and (6.20), once we know ejCoo = ej, ejFoo = J-1 Wjej, ejGgo = wWjC3h gej. These latter equations follow from (6.17) and (6.14). □

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, w and S±, 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.

6.3. Modular data for the double of C(G; ±, w, A)

Definition 1. Modular data consists of a pair S, T of unitary matrices satisfying:

(i) S is symmetric (i.e. Sf = S) and T is diagonal and of finite order (TN = I);

(ii) S2 is a permutation matrix of order < 2, and (ST)3 = S2;

(iii) S1i € R \ {0} and some index 1' has S1ri > 0, for all i;

(iv) for each i, j, k, the numbers Nfj defined by

SilSjiSki

Nk ,= y °il0jl°kl (6 21)

1 v Sil

are nonnegative integers.

Any MTC has modular data. The index i parametrises the simple objects (primaries) Xi. The entries Tiyi of the diagonal matrix T (up to normalisation) are eigenvalues of the twist QXi = (trXi <8> idEnd )(cXi,Xi) € C idXi while those of the symmetric matrix S are associated to the Hopf link: up to normalisation, Sitj = trXi®Xj (cXiyXj o cXj yXi). By Proposition 2.12 of [10], 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 Xi and the permutation S2 sends i to iv, where [X/] = [Xiv]. (6.21) is called Verlinde's formula, and the numbers Nfj are the structure constants [Xi ® Xj] = Nfj [Xk] of the Grothendieck ring of the MTC.

Ignoring the normalisation, those matrices S and T in a MTC define through (11 ) ^ S, (o1) ^ T a projective representation of the modular group SL2(Z) = ((1 -j1) , (¿1)}, but it is always possible to choose a normalisation so that it defines a linear (i.e. true) representation of SL2(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 Nij, 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; ±, w, A) of Theorem 2. Write as before v = |G| = 2n +1, p = v2 +4 = 2rn + 1, S± = (v±^fp)/2 and A± = 2v + v2S±. 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 Ej(£)v,v:

Tjj = d^ fa(Ej(£)«), (6.22)

= A± E « d( fa(Ei(nkc Ei(£)„,„), (6.23)

for any j in (6.22) and (6.23), and any simple n ^ a in (6.23). In (6.22), £ can be any simple in a, and in (6.23) the sum is over all simple £ in a while n is any (fixed) simple in a. The standard left inverse of the endomorphism £ is (x) = R^£v(x) R, where Rz € Hom(1, (v() and Rz € Hom(1, CCV) are normalised by R^((Rz) = d-1 = R'zZV(RZ). Note that for x € End(n£), fa(x) € End(n) = C1. (6.22), (6.23) have the desired normalisation built in — as computed in section 5.3 of [27], the normalisation of T is trivial (i.e. T11 = 1) for the double of any (not necessarily unitary) fusion category. The derivation of (6.22), (6.23) is as in [21], except that the complex conjugate in (6.23) replaces the *'s in his Lemma 5.3: his formula assumes is a *-map; equation (5.6) of [27] writes this as Sai v ^j, which is equivalent to our complex conjugation.

In our case, Rag = Rag = 1 and Ragp = RagP = s, so $ag = a-g and 0agp(x) = s'ag(p(x))s. We see from (6.22), (6.23) and the matrix units computed earlier this section that the modular data is formally identical to that of [22] (e.g. w doesn't explicitly appear), except for a trivial dependence on the sign ±. In particular, using (6.5), the primaries fall into four classes:

(i) two primaries, denoted 0 and b;

(ii) n primaries, denoted a^ = tt^ for non-trivial ^ € G;

(iii) nv primaries, denoted chj0 = for h G G, h = 0 and 0 G G;

(iv) m primaries, denoted dl.

Breaking S and T into 16 blocks, we get

1nvx2 nt

1; 0(h); W!^ . .,Wm)

12xn 12xnv C

2 nx n D 0nxm

Dt E 0nvxm

0mxn 0mxnv F

(6.24)

where kaxb for any k G C is the axb matrix with constant entry k, = ^(h)+^(h),

= 0'(h)0(h') + 0'(h)0(h'), ' '

2(1 ±y 1 *y) andc=-1 -1 • • : -1

for y = . We denote transpose with 't'.

Much more difficult is to identify the m x m matrix F and the phases w;. Once the solutions Cj l and w; to (6.14), (6.17)-(6.20) have been found, we conclude

Fdj d = ^ (wj wl + g,p C—g,p Cg,p+g) • (6.25)

Incidentally, (6.17) gives an alternate expression for the diagonal entries of S: SVj= t— (uwjn3 + w2 (1 - 6±)+ 5±uwj

X ^^ Cl,lAl-p-2g,k-g A-k-g,l-k-g A-k,2p+g-k g,h,k,l

where n3 is the number of g G G with order dividing 3.

We have thus identified the S and T matrices for any fusion category C(G; ±, cc, A), although the numbers wj 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; ±, cc, A), and identify their modular data. We will find that the mysterious matrix F and phases w; always seem to take a remarkably simple form. For this reason we conjecture:

Conjecture 1. Choose any finite abelian group G of odd order v, and choose any fusion category C = C(G; ±, cc, A). Then there is an abelian group H of order ^ = v2 +4 and a nondegenerate bilinear form / on H, which determines the submatrix F and the phases

Wj for the double of C explicitly. In particular, the m = (v2 + 3)/2 primaries di of class (iv) are parametrised by pairs ±l of elements in H, l = 0, and

wl = exp[2nim ¡(l, l)

(6.26)

Fi,i' = T-^cos(2n0(l,l')).

(6.27)

By a nondegenerate bilinear form ¡3 on H, we mean ¡3 : H x H ^ Q/Z obeys (3(g + g', h + h') = ¡3(g, h) + ¡3(g, h') + ¡3(g', h) + ¡3(g', h') (mod 1) for all g, g', h, h! € H, and for any non-zero g € H there is an h € H such that ¡3(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(Z3 x Z3; ±, w, A) ([11] showed there are no Q-systems for Z3 x Z3), and we know of no fusion categories C(G; ±, w, A) whose corresponding modular data has H = Z5 x Z5 x Z5 (it would necessarily have G = Z11). 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 [11]. In particular, associated to H and 3 is a pointed modular tensor category C(H, ¡3), and the role of the affine algebra Br2 in [11] could be played by a Z2-orbifold of C(H, 3). The modular data of Conjecture 1 can be twisted by H3(G x Z2; T), as explained in section 3 of [11], 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 [14,13,23], 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.

7. Explicit solutions

7.1. The fusion category classification for small G

This subsection obtains all fusion categories C(G; ±, w, A) for |G| < 5. Recall S± = (v ± %/v2 +4)/2, where |G| = v, and Conjecture 1 from section 6.3.

Theorem 3. The complete list of fusion categories C (G; ±, w, A) appearing Theorem 2 for G = Z1,Z3, Z5 are (up to equivalence):

(i) for G = Z1: exactly one for either sign; A = ( —1/5±); both have w = 1; their modular data has H = Z5 and ¡3(k, l) = kl/5 (for '+'), 3(k, l) = 2kl/5 (for '— ');

(ii) for G = Z3: two inequivalent unitary ones with '+', and two inequivalent hermitian but non-unitary ones with '— '; all four have w = 1 and

where the parameters for these four solutions are

+ : (c,d,e,f,g) = (ci,di,d2,f5,f5) ,

+ : (c,d,e,f,g) = (c2,d5,d5,fl,f2) ,

- : (c,d,e,f,g) = (c3,d6,d6,f3,f4),

- : (c,d,e,f,g) = (c4,d3,d4,f6 ,f6) ,

for ci, dj, fk explicitly defined below; the modular data for all four has H = Zi3, and (3(k, l) = kl/13 resp. (3(k, l) = 2kl/13 for '+' resp. '-'; (iii) for G = Z5: two inequivalent unitary ones with '+', and two inequivalent hermitian but non-unitary ones with '— '; all four have c = 1 and

(c d e f g\

d g h i h

A = e j f i i (7.2)

f k k e h

\g j k j d)

where the parameters for these four solutions are

+ : (c,d,e, f,g,h,i,j,k) =(c2,di,di,di,di,hr ,hii,hg,hio) , + : (c,d,e, f,g,h,i,j,k) =(c4,d4,d3,d6,d5,h4 ,h2,h4,h2) ,

- : (c,d,e,f,g,h,i,j,k) =(ci,d2,d2,d2,d2,h5,hu,hg,h6),

- : (c,d,e,f,g,h,i,j,k) =(c3,dr,dio,dg,d8,h3,hi,h3,hi) ,

for ci, dj, hk explicitly defined below; the modular data for all four has H = Z29, and 3(k, l) = kl/29 resp. 3(k, l) = 2kl/29 for '+' resp. '-'.

The two fusion categories for v =1 are realised by affine G2 at level 1 ('+'), and YangLee ('-'). The first two fusion categories for v = 3 are realised by an even subsystem of the Grossman-Snyder system H3 [17] and an even subsystem of the Haagerup subfac-tor. The other two are their Galois associates. The first two fusion categories for v = 5 are realised by an even subsystem of the Haagerup-Izumi subfactor for G = Z5 found in [22], 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.

As explained after Proposition 7.5 in [26], Ostrik constructed the two twisted Haagerup categories conjectured in Section 3.2 of [11] by de-equivariantising the two Z9 near-group categories constructed in [12]. Two others arise as their Galois associates. These four fusion categories possess Haagerup-Izumi fusions (1.1) for G = Z3, but don't appear in Theorem 3(ii) presumably because they involve the generalisation of (4.1) discussed before Theorem 1. Ostrik's construction emphasises the desirability of extending

Theorems 1-3 in this paper to that generalisation of (4.1). This extension should now be straightforward but perhaps messy.

Our proof of Theorem 3 uses Grobner 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 v = 3, 5. We then have to determine (by trial and error) which eigenvalues go together to form solutions.

v = 1 was worked out in section 2, so turn to G = Z3. Consider first w = 1. The order-3 symmetry (4.7) gives us (7.1). These variables (c, d, e, f, g) satisfy (4.8)-(4.10). The Grobner basis method tells us there are precisely 8 solutions. However by Theorem 2, two solutions A(1), A(2) yield equivalent fusion categories if they can be obtained from each other by the action of Aut(Z3) = {±1}, i.e. if A(1j = A- - for all i, j € Z3. In other words, the 5-tuples (c, d, e, f, g) and (c, e, d, g, f) are equivalent. Up to this equivalence, we then get 4 solutions, as given in Theorem 3. There, C1 = (2 - -13)/3, C2 = (7 - -13)/6, C3 = (7 + -13)/6, C4 = (2 + -13)/3. d1, ..., d4 « -0.321, 0.554, 0.717 - 0.329i, 0.717 + 0.329i respectively are the roots of 9x4 - 15x3 + 7x2 + x - 1, while d5 = (1 - —13)/6 and d6 = (1 + —13)/6. Finally, f1,..., f4 « 0.217 + 0.758i, 0.217 - 0.758i, -0.954, 0.186 respectively are the roots of 9x4 +3x3 + x2 +5x - 1, and f5 = (1 + -13)/6, f6 = (1 - —13)/6.

Now consider G = Z3 with w = 1, a nontrivial third root of 1. Then (4.7) gives

where the zeros arise for any entry of A fixed by the order-3 symmetry. The quadratic identities (4.9) give e.g. w(d2 + e2) = 1 - 1/5± and d2 + e2 = 1, which are incompatible. Thus there are no solutions for G = Z3 with w = 1.

Now turn to G = Z5, with w = 1. (4.7) gives (7.2). The Grobner 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 = Z4, 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, c1 = (13 + —29)/10, c2 = (13 - —29)/10, c3 = (7 + -29)/5, and c4 = (7 - -29)/5. Also, d1 = (3 - -29)/10, d2 = (3 + -29)/10, d3 « -0.537, d4 « -0.426, d5 « -0.032, d6 « 0.480, d7 « 0.400 - 0.282i, d8 « 0.400 + 0.282i, d9 « 0.957 - 0.983i, d10 « 0.957 + 0.983i, where the final 8 of these di are the roots of the irreducible polynomial 625x8 - 1375x7 + 1275x6 + 245x5 - 654x4 + 152x3 + 75x2 - 29x - 1. Finally, h1 « -0.675, h2 « 0.218, h3 « 0.437, h4 « 0.620, h5 « -1.270, h6 « -0.095, h7 « 0.084 - 0.536i, h8 « 0.084 + 0.536i, h9 « 0.106, h10 « 0.534 - 0.099i, h11 « 0.534 + 0.099i, h12 « 1.420, where h1 to h4 are solutions to the irreducible polynomial 25x4 - 15x3 - 9x2 +7x - 1, while h5 to h12 are solutions to the irreducible 625x8 - 875x7 - 525x6 + 1110x5 - 789x4 + 402x3 - 95x2 - 3x + 1.

Finally, turn to G = Z5 and w = 1 a nontrivial third root of 1. Write

<0 d e f 9 \

u) d üg h i ü h

üj e j uf ui ü i

u f k ü k ue uh

g uk ü j ud y

Using the Grobner 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 = Z5, but for G = Z3 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 c = 1.

Of course, the set of all fusion categories C(G; ±, cc, A) for fixed G is closed under Galois actions. Theorem 3 is disappointing, in that all fusion categories for G = Zi, Z3, Z5 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.

7.2. 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 c = 1, '+', and A with specified values for A0,g, Ag,0, Ag,g. Evans and Gannon [11] found several new Q-systems of type C(G; ±, cc, 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 = Zv, is in terms of numbers j2, j3, • • •, jn+i G R (recall v = 2n +1): for 0 < g < h < v we have

Vs s-1

exp[i(jh - jg - jh-g

where j1 = 0 and jn+i+i = jn+i + jn — jn-i for 1 < i < n (see Lemma 7.3 of [22]). The Q-systems found in [11] correspond to

Ü27)>j37)j47)) ~ (2.471228, 0.51685555, 0.2137724);

(j(9)'

,j(9)) ra (2.396976693, 2.079251103, -0.2079168419, -2.508673987); ,j(9)/) ra (-2.364737070, 1.031057162, 1.569692175, 0.3383837765) ;

(0.9996507, 2.7258434, -0.5714203, -1.7797340, 1.2675985) , (-2.6444397, -1.7629598, -2.6444440, 2.7572657, 0.1128260) ;

j(ii)/

j(13) ra (-3.1050384,0.5993399, -0.111708, -0.969766, 1.336848,1.00483129) ;

(-1.0777623, -.7748018, -2.171863, -1.6068402, -0.257508, 2.092502, 0.72289565) ;

j(17) ra (-1.466074, .291489, 3.130735, -2.693185, 1.398153, -0.611938,

- 1.667078, -1.754821) ;

j(19) ra (-2.677465, 1.088972, -0.899442, 0.015448, -1.240928, -0.493394, 1.839879,

-1.525884, -2.084374) ; j(19)/ ra (0.896858, -0.882585, -2.369855, -1.873294, -1.711620, -0.119360, 2.972018,

- 2.460652, 0.041334),

where the superscript (7) etc refers to the value of v. These approximate values suffice to determine the exact algebraic values of the and to verify that these do indeed

satisfy all the equations (4.8)-(4.10), using the method described in section 3.5 of [12]. (The solutions for 11 < v < 19 were announced as conjectural in [11], but using [12] have now been shown to yield exact solutions.) This list constitutes the complete classification of Q-systems for Z7, Z9, up to equivalence. There is no Q-system solution for G = Z3 x Z3 (more precisely, any such solution would require nontrivial 2-cocycle twists of (4.1)).

Proposition 3. The modular data for the 10 Q-systems listed above, is given by Conjecture 1 with abelian group H and bilinear form 3 given by:

/7): H = Z53, 3(l, l') = W/53; j(9): H = Z§5, /(l, l') = ll'/85;

j(9)/: H= Z85, 3(l, V) = 12W/85;

j(11): H= Z125, 3(l, l/) = ll1 /125;

j(11)/ : H= - Z25 x Z5, 3((l1, l2), (l/, l2)) = 2l 111 /25 + 2l2l2/5;

j(13): H= Z173, 3(l, l/) = ll'/173;

j(15): H= Z229, 3(l, l/) = ll'/229;

j(17): H= Z293, 3(l, l/) = ll'/293;

j(19): H= Z365, 3(l, l/) = ll'/365;

j(19)/ : H= = Z365 , 3(l, l') = 22W/365.

Given a nondegenerate bilinear form 3 on some abelian group of order v2 + 4, let S3, T3 denote the modular data described in Conjecture 1. Section 4.1 of [11] proved this proposition for these Q-systems at v = 7, 13, 15, 17, and conjectured the correct H and 3 for 5 of the 6 remaining. It was unable to determine the modular data for the 2 + 2 + 2 Q-systems at G = Z9, Z11, Z19, 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 [11], section 4.1. In particular, a floating point proof is possible and effective, since the integrality of the fusion coefficients Nk in (6.21) serves as error-correction. More precisely, equation (1.3) of [11] shows S in modular data is uniquely determined from the fusion coefficients, T and the entries S1ji. Our strategy here is to guess at a phase Wj consistent with

Conjecture 1, use the linear equations (6.14) and (6.20) to determine the corresponding Cg h up to a small number of parameters (for almost all choices of wj, this linear system will be inconsistent and we can throw away that choice). For typical examples, the choice Wj = e2ni182/365 for solution j(19) identifies Cg h up to 1 parameter, while the choice Wj = e2ni2/5 for solution j(11)/ needs 4 parameters. Then we chose at random some nonlinear equations from (6.17) to fix those parameters.

7.3. 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.

Definition 2. Let p be a d-dimensional representation of SL2(Z), such that T := p( J 1) is diagonal. By a character vector X(t) = (x1(T), • • •, Xd(T)) for p, we mean a holomorphic function X from the upper half-plane H = {t G C | Im t > 0} to Cd, which obeys

X (^) = P ( Cd) X(t ) (7.3)

for all t G H and (^Cd^ G SL2(Z), and for which there exist exponents Xk G R and coefficients \k;n G Z>0, such that

e-2niXkT Xk (t ) = E Xk;nqn (7.4)

converges absolutely for |q| < 1, for k = 1, • ••, d, where we write q = e2niT. We also require X1;o = 1.

Choosing any \'k so that Tkk = e2niXk, it is clear from holomorphicity and the transformation law (7.3) at (J , that e-2niXkTxk(T) 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 t = i(x>. Any holomorphic X : H ^ Cd obeying (7.3) and (7.4) is called a weakly-holomorphic vector-valued modular function (vvmf) for SL2(Z) with multiplier p. The characters of the irreducible modules Mj for any completely rational conformal net of factors on S1, or for any strongly rational VOA, form a character vector, where p is the modular data coming from the corresponding MTC. The label 1 is the vacuum module M1 = V1 (the VOA or net itself), and T11 = e-nic/12 for a parameter c called the central charge. We can assume without loss of generality that all xk;0 = 0, in which case hk = Xk + c/24 is called the conformal weight of the module Mj. 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 Xj € Q, but that holds in any MTC. Moreover, given any character vector X(t), the vector v := X(i) exists and is strictly positive (since at t = i we have q = e-2n > 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 = S4) of S, namely the common Perron-Frobenius eigenvector of the fusion matrices N = (Nk), 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 hk > h1 = 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 hk > h1> for all k. The quantity ceff = — 24h1> is called the effective central charge, and must be nonnegative. Again, ceg = 0 can only occur for the trivial VOA and conformal net. To our knowledge, all known examples have hj > hy for j = 1', but this is not yet a theorem.

The Hauptmodul j(T) = q-1 + 744 + 196884q + ••• of SL2(Z) is a weakly-holomorphic modular function for the trivial multiplier. For any p in Definition 2, the space M!(p) of weakly-holomorphic vvmfs is trivially a module over the polynomial ring C[j(T)]. 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 x d matrix

S(t) = qA ^ S„q" ,

with coefficients S„ € Mnxn(C), with the property that X(t) € M!(p) iff there is a vector-valued polynomial p(x) € Cd[x] such that X(t) = S(t)p(j(t)). So knowing all weakly-holomorphic vvmfs for p is equivalent to knowing S(t). We can and will require 20 = Idxd. The matrix A will be diagonal, with entries satisfying Tkk = ¿2nlAkk. There is a recursion uniquely determining each S„ from the complex matrices A and S1 (equation (36) of [16]). In short, knowing all weakly-holomorphic vvmfs for p is equivalent to knowing the exponents A and the first nontrivial coefficient matrix S1.

Once S(t) (or equivalently A, S1) are known, it is then just combinatorics to find all character vectors for a given effective central charge (since ceg directly gives bounds for the degrees of all component polynomials pk in p(x) € Cd[x]). In [11], 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 S(t) for the non-unitary cousin of the Haagerup (G = Z3). Its fusion category and MTC is a Galois associate of that of the Haagerup. By contrast, S(t) 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 unitary fusion categories for G = Z3, at any (effective) central charge c = ceff = 8 (mod 24) (one of the three possibilities), was found in [11] to have A resp. Si be

diag(-1/3, -1/3, -1/3, -1/3, -1, -2/3, -34/39, -19/39, -5/39, -37/39, -31/39, -28/39) ,

( 6 80 81 81 8748 1215 3549 273 13 5538 2275 1378

80 6 81 81 8748 1215 -3549 -273 -13 -5538 -2275 -1378

81 81 167 -81 -8748 -1215 0 0 0 0 0 0

81 81 -81 167 -8748 -1215 0 0 0 0 0 0

3 3 -3 -3 -12 18 0 0 0 0 0 0

27 27 -27 -27 1458 -152 0 0 0 0 0 0

7 -7 0 0 0 0 -88 -14 -1 50 63 64

42 -42 0 0 0 0 -1484 92 16 2940 -192 -1041

119 -119 0 0 0 0 -2142 987 11 -24990 -6035 4641

5 -5 0 0 0 0 17 13 -3 -2 35 -14

13 -13 0 0 0 0 174 -1 -5 294 -147 51

14 -14 0 0 0 0 448 -77 7 -343 125 -24

(we are following the conventions of [16], which has A shifted by the identity from the A used in [11]). Here, the positive row of S is 1' = 1, the vacuum 0. At (effective) central charge c = 8, the polynomial p(x) will be (a, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0)4 for constants a, 3 G C (otherwise Ai would not be the unique minimum). But a =1, since \i;0 = 1. Thus the only possible character vectors at central charge c = 8 are

XoM Xb[r )

Xa(T) = Xco (t)

XC2 XD: Xd2

XD, Xd. . . XD6 (t) 7-l/3 (1

= S(t )

o 0 0 0 0 o

(6 + 13/3)q + (120 + 78,%2 + (956 + 351,%3 + (6010 + 1235^)q4 + • • q2/3 ((80 - 13/) + (1250 - 78/)q + (10630 - 351/)q2 + (65042 - 1235/)q3 + ••• q2/3 (81 + 1377q + 11583q2 + 71037q3 + ••• ) 3 + 243q + 2916q2 + 21870q3 + ••• q1/3 (27 + 594q + 5967q2 + 39852q3 + •••) q5/39 ((7 - /) + (292 - 6/)q + (3204 - 43/)q2 + (23010 - 146/)q3 + •••) q20/39 ((42 + 16/) + (777 + 121/)q + (7147 + 547/)q2 + (45367 + 2000/)q3 + ••• q-7/39 (/ + (11/3 + 119)q + (73/3 + 1623)q2 + (300/ + 12996)q3 + (76429 + 1063/)q4 q2/39 ((5 - 3/) + (229 - 50/)q + (2738 - 252/)q2 + (19942 - 1032/)q3 + •••) q8/39 ((13 - 5/) + (34 7 - 37/)q + (3804 - 212/)q2 + (26390 - 794/)q3 + •••) V q11/39 ((14 + 7/) + (441 + 61/)q + (4445 + 303/)q2 + (30329 + 1167/)q3 + •••)

The first coefficient of Xa3(t) (i.e. X9;0) tells us ¡3 G Z>0, while the first coefficient of Xd4 (t) (i.e. x10;0) then implies 3 = 0, 1. Thus there are only two possible character vectors for the Haagerup modular data at c =8, as stated in [11].

The double of either of the non-unitary fusion categories for G = Z3, at effective central charge ceff = 8 (mod 24) (one of three possibilities), has A resp. S1 equal to

diag(-1/3, -1/3, -1/3, -1/3, -1, -2/3, -16/39, -25/39, -40/39, -22/39, -49/39, -4/39) ,

( 110 -24 81 81 -4374 1215 -390 -1820 -16770 -910 -53872 52

-24 110 81 81 -4374 1215 390 1820 16770 910 53872 -52

81 81 167 -81 4374 -1215 0 0 0 0 0 0

81 81 -81 167 4374 -1215 0 0 0 0 0 0

-6 -6 6 6 -12 -36 0 0 0 0 0 0

27 27 -27 -27 -729 -152 0 0 0 0 0 0

-28 28 0 0 0 0 143 -405 -9580 -518 3654 -1

-1/2 1/2 0 0 0 0 -81 -262 1457 56 -3832 26

-1/2 1/2 0 0 0 0 -7 7 -12 6 -7 1

-28 28 0 0 0 0 -35 120 1820 -314 7224 -27

-1/2 1/2 0 0 0 0 2 2 1 -1 0 -1

-57/2 57/2 0 0 0 0 399 2660 8436 -854 -204212 79

Here, the positive row of S is l = 2, the primary b. At effective central charge ceg = 8 for this p, the polynomial p(x) will be (a, 3, Y, S, 0, 0, 0, 0, 0, 0, 0, e)4 for constants a, 3, Y, S, e G C (otherwise A2 would not be the unique minimum). Thus the only possible character vectors at effective central charge ceg = 8 are

/ q-1/3 (a + (110a + 52e - 24, + 817 + 815)? + (1589a - 219,5 + 1377y + 13775 + 650e)q2 \ + (12721a - 1135, + 115837 + 115835 + 4108e)q3 +----)

q-1/3 (, + (110, - 24a + 817 + 815 - 52e)q + (1589, + 13777 + 13775 - 650e - 219a)q2 + (12721, + 115837 + 115835 - 4108e - 1135a)q3 +----)

q-1/3 (y + (167y + 81a - 815 + 81,)q + (27477 - 13775 + 1377a + 1377,)q2 + (23169y - 115835 + 11583a + 11583,)q3 +----)

q-1/3(5 + (1675 + 81a + 81, - 81y)? + (27475 + 1377a + 1377, - 1377y)?2

+ (231695 + 11583a + 11583, - 11583y)?3 +----)

-6a - 6, + 6y + 65 + (-486a - 486, + 486y + 4865)q + (-5832a - 5832, + 5832y + 58325)q2 +----

q1/3 (27a + 27, - 27y - 275 + (594a + 594, - 594y - 5945)q

+ (5967a + 5967, - 5967y - 59675) q2 +----)

q23/39 (-28a + 28, - e + (-1025a/2 + 1025,/2 - 52e)q + (-4359a + 4359, - 378e)q2 +----)

q14/39 (-a/2 + ,/2 + 26e +(-95a + 95, + 352e)q + (-1416a + 1416, + 2431e)q2 +----)

q-1/39 (-a/2 + ,/2 + e + (-67a + 67, + 53e)q + (-932a + 932, + 431e)q2 +----)

q17/39 (-28a + 28, - 27e + (-512a + 512, - 378e)q + (-8585a/2 + 8585,/2 - 2510e)q2 +----)

q—10/39 (-a/2 + ,/2 - e + (-67a + 67, - 53e)q + (-904a +904, - 457e)q2 +----)

V q-4/39 (e + (79e - 57a/2 + 57,/2)q + (756e - 579a + 579,)q2 + (4513e - 5196a + 5196,)q3 +----) '

We see that a, 3, Y, S, e G Z>0 and a = 3 (mod 2); in fact 3 > 0 since ceg = 8. Comparing the leading terms of xs(t) and xs(t), we must have a + 3 = Y + S and hence also y = S

(mod 2). This means that the q, q2, q3 coefficients of q1/3xi(T) are all even and thus cannot equal 1. Hence either c < -88 or a = 1. Assume a = 1. Then all coefficients of e.g. x5(t) up to at least q4 vanish. We don't have a proof yet that there is no character vector with ceff = 8 for this p, but it seems highly likely.

This calculation is meant to give further evidence that, even though these unitary and non-unitary fusion categories and hence MTCs are related simply by a Galois automorphism, the relation if any between corresponding VOAs or conformal nets will be far from straightforward.

Acknowledgments

The authors thank Cardiff School of Mathematics, University of Alberta and University of Warwick for generous hospitality while researching this paper. They also benefited greatly from Research-in-Pairs held at Oberwolfach and Research in Teams at BIRS. Their research was supported in part by EPSRC EP/J003352/1, PIMS and NSERC. The first named author would like to thank Chris Phillips for information and references to the ^-theoretic aspects of Leavitt algebras. We would also like to thank Noah Snyder for a discussion of [26].

References

[1] P. Ara, M. Brustenga, G. Cortinas, K-theory of Leavitt path algebras, Münster J. Math. 2 (2009) 5-34.

[2] M. Asaeda, U. Haagerup, Exotic subfactors of finite depth with Jones indices (5 + vT3)/2 and (5 + vTf)/2, Comm. Math. Phys. 202 (1999) 1-63.

[3] J. Böckenhauer, D.E. Evans, Modular invariants from subfactors: type I coupling matrices and intermediate subfactors, Comm. Math. Phys. 213 (2000) 267-289.

[4] J.L. Cardy, Conformal invariance and the Yang-Lee edge singularity in two dimensions, Phys. Rev. Lett. 54 (1985) 1354-1356.

[5] S. Carpi, Y. Kawahigashi, R. Longo, M. Weiner, From vertex operator algebras to conformal nets and back, arXiv:1503.01260.

[6] T. Creutzig, D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (49) (2013) 494006.

[7] A. Davydov, I. Runkel, A braided monoidal category for symplectic fermions, in: Symmetries and Groups in Contemporary Physics, Proceedings, Tianjin, 2012, World Scientific, Hackensack, 2013, pp. 399-404.

[8] P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer, New York, 1997.

[9] P. Etingof, V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004) 627-654, 782-783.

[10] P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. Math. 162 (2005) 581-642.

[11] D.E. Evans, T. Gannon, The exoticness and realisability of twisted Haagerup-Izumi modular data, Comm. Math. Phys. 307 (2011) 463-512.

[12] D.E. Evans, T. Gannon, Near-group categories and their doubles, Adv. Math. 255 (2014) 586-640.

[13] D.E. Evans, T. Gannon, Generalised Haagerup and Ee subfactors and their modular data, April 2012.

[14] D.E. Evans, T. Gannon, The search for the exotic - subfactors and conformal field theories, in: Burenkov, et al. (Eds.), Progress in Analysis: Proceedings of the 8th Congress of the ISAAC, Moscow, 2011, vol. 1, People's Friendship University, Moscow, 2012, pp. 8-25.

[15] T. Gannon, Comments on nonunitary conformal field theories, Nuclear Phys. B 670 (2003) 335-358.

[16] T. Gannon, The theory of vector-valued modular forms for the modular group, in: W. Kohnen, R. Weissauer (Eds.), Conformal Field Theory, Automorphic Forms and Related Topics, Springer Verlag, 2014, pp. 247-286.

[17] P. Grossman, N. Snyder, Quantum subgroups of the Haagerup fusion categories, Comm. Math. Phys. 311 (2012) 617-643.

[18] U. Haagerup, Principal graphs of subfactors in the index range 4 < [M : N] < 3 + V3, in: H. Araki, et al. (Eds.), Subfactors, World Scientific, River Edge, NJ, 1994, pp. 1-38.

[19] T. Hayashi, S. Yamagami, Amenable tensor categories and their realizations as AFD bimodules, J. Funct. Anal. 172 (2000) 19-75.

[20] M. Izumi, Subalgebras of infinite C*-algebras with finite Watatani indices, I. Cuntz algebras, Comm. Math. Phys. 155 (1993) 157-182.

[21] M. Izumi, The structure of sectors associated with Longo-Rehren inclusions, I. General theory, Comm. Math. Phys. 213 (2000) 127-179.

[22] M. Izumi, The structure of sectors associated with Longo-Rehren inclusions, II. Examples, Rev. Math. Phys. 13 (2001) 603-674.

[23] M. Izumi, Notes on the Haagerup series, January 2012.

[24] W.G. Leavitt, The module type of homomorphic images, Duke Math. J. 32 (1965) 305-311.

[25] V. Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995) 279-327.

[26] S. Morrison, E. Peters, N. Snyder, Categories generated by a trivalent vertex, arXiv:1501.0686.

[27] M. Müger, From subfactors to categories and topology II. The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra 180 (2003) 159-219.

[28] M. Müger, Tensor categories: a selective guided tour, Rev. Un. Mat. Argentina 51 (2010) 95-163.

[29] N.C. Phillips, Analogs of Cuntz algebras on Lp spaces, arXiv:1201.4196.

[30] N.C. Phillips, Private communication.

[31] K.-H. Rehren, Chiral observables and modular invariants, Comm. Math. Phys. 208 (2000) 689-712.

[32] F. Xu, Algebraic orbifold conformal field theories, Proc. Natl. Acad. Sci. USA 97 (26) (2000) 14069-14073.

[33] S. Yamagami, A categorical and diagrammatical approach to Temperley-Lieb algebras, arXiv:math/ 0405267.