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arXiv:1208.1500v1 [math.QA] 7 Aug 2012

Near-group fusion categories and their doubles

David E. Evans

School of Mathematics, Cardiff University,

Senghennydd Road, Cardiff CF24 4AG, Wales, U.K.

e-mail: EvansDE@cf.ac.uk

Terry Gannon

Department of Mathematics, University of Alberta,

Edmonton, Alberta, Canada T6G 2G1

e-mail: tgannon@math.ualberta.ca

August 8, 2012

Abstract

A near-group fusion category is a fusion category C where all but 1 sim-

ple objects are invertible. Examples of these include the Tambara-Yamagami

categories and the even sectors of the D(1)

are infinitely many others. We classify the near-group fusion categories, and

compute their doubles and the modular data relevant to conformal field theory.

Among other things, we explicitly construct over 40 new finite depth subfac-

tors, with Jones index ranging from around 6.85 to around 14.93. We expect

all of these doubles to be realised by rational conformal field theories.

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and E6 subfactors, though there

Contents

1 Introduction2

2 The near-group systems

2.1The numerical invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3First class: Near-group categories with n′= n − 1 . . . . . . . . . . . . . . . . . . . .

2.4The remaining class: n′a multiple of n . . . . . . . . . . . . . . . . . . . . . . . . . .

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7

11

16

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3Explicit classifications

3.1Which finite group module categories are near-group? . . . . . . . . . . . . . . . . .

3.2The type G + 0 classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3The near-group categories for the trivial group G . . . . . . . . . . . . . . . . . . . .

3.4 The type G + n − 1 C∗-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5At least as many T’s as S’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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25

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4Tube algebras and modular data

4.1The tube algebras of near-group systems . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 The first class near-group C∗-categories: n′= n − 1 . . . . . . . . . . . . . . . . . . .

4.3The tube algebra in the second class . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 The modular data for the double of G + n when n is odd

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33

36

44

46. . . . . . . . . . . . . . .

References50

1Introduction

Considerable effort in recent years has been directed at the classification of subfactors

of small index. Subfactors of index ≤ 5 are now all known (see e.g. [23]). The

classification for index ≤ 4 was established some time ago. The Haagerup subfactor

[13] with Jones index (5+√13)/2 ≈ 4.30278, the Asaeda-Haagerup subfactor [1] with

index (5 +√17)/2 ≈ 4.56155, and the extended Haagerup subfactor [2] with index

≈ 4.37720, arose in Haagerup’s classification [13] of irreducible finite depth subfactors

of index between 4 and 3 +√3 ≈ 4.73205. A Goodman-de la Harpe-Jones subfactor

[11], coming from the even sectors of the subfactor corresponding to the A1,10⊂ C2,1

conformal embedding, has index 3 +√3. Then comes the Izumi-Xu subfactor 2221

[19] with index (5+√21)/2 ≈ 4.79 and principal graph in Figure 1, coming from the

G2,3⊂ E6,1conformal embedding.

The punchline is that, at least for small index, there are unexpectedly few sub-

factors. Does this continue with higher index? Are the aforementioned subfactors

exotic, or can we put them into sequences? In [19], Izumi realised the Haagerup and

Izumi-Xu subfactors using endomorphisms in Cuntz algebras, and suggested that his

construction may generalise. More precisely, to any abelian group G of odd order,

Izumi wrote down a nonlinear system of equations; any solution to them corresponds

to a subfactor of index (|G| + 2 +?|G|2+ 4)/2. He showed the Haagerup subfactor

solutions for the next several G, explained that the number of these depends on the

prime decomposition of |G|2+4, and argued that the Haagerup subfactor belongs to

an infinite sequence of subfactors and so should not be regarded as exotic.

Izumi in [19] also associated a second nonlinear system of equations to each finite

abelian group; to any solution of this system he constructs a subfactor of index

(|G| + 2 +?|G|2+ 4|G|)/2 and with principal graph 2|G|1, i.e. a star with one edge

an example). Izumi then found solutions for G = Zn(n ≤ 5) and Z2× Z2. G = Z1

and Z2correspond to the index < 4 subfactors A4and E6, respectively; his solution

for Z3 provides his construction for Izumi-Xu. An alternate construction of 2221,

involving the conformal embedding G2,3⊂ E6,1, is due to Feng Xu as described in

the appendix to [5] (see also [14]). As we touch on later in the paper, there may be

a relation between the series containing the Haagerup subfactor, and that containing

the Izumi-Xu subfactor.

corresponds to G = Z3, and that there also is a solution for G = Z5. In [10] we found

of length 1 and |G| edges of length 2 radiating from the central vertex (Figure 1 is

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Figure 1. The 231 principal graph

One of our tasks in this paper is to construct several more solutions to Izumi’s

second family of equations, strongly suggesting that this family also contains infinitely

many subfactors. But more important, in this paper we study a broad class of systems

of endomorphisms, the near-group fusion categories, including the Izumi-Xu series as

a special case. We obtain a system of equations, generalising those of Izumi, providing

necessary and sufficient conditions for their existence. We identify the complete list

of solutions to the first several of these systems, which permits us the construction

of over 40 new finite-depth subfactors of index < 15,

A fusion category C [9] is a C-linear semisimple rigid monoidal category with

finitely many simple objects and finite-dimensional spaces of morphisms, such that

the endomorphism algebra of the neutral object is C. The Grothendieck ring of a

fusion category is called a fusion ring. Perhaps the simplest examples are associated

to a finite group G: the objects are G-graded vector spaces ⊕gVg, with monoidal

product Vg⊗ V′

examples group categories. The category Mod(G) of finite-dimensional G-modules is

also a fusion category.

We’re actually interested in certain concrete realisations of fusion categories, which

we call fusion C∗-categories: the objects are endomorphisms (or rather sectors, i.e.

equivalence classes of endomorphisms under the adjoint action of unitaries) on some

infinite factor M, the spaces Hom(ρ,σ) of morphisms are intertwiners, and the prod-

uct is composition. Two fusion C∗-categories are equivalent iff they are equivalent as

fusion categories — all that matters for us is that the factor M exists, not which one

it is. Every finite-depth subfactor N ⊂ M gives rise to two of these, one correspond-

ing to the principal, or N-N, sectors and the other to the dual principal, or M-M,

ones. For example, given an outer action α of a finite group G on an infinite factor N,

we get a subfactor N ⊂ N×G = M coming from the crossed product construction:

the N-N system realises the group category for G, while the M-M system realises

Mod(G). Not all fusion categories can be realised as fusion C∗-categories (e.g. the

modular tensor categories associated to the so-called nonunitary Virasoro minimal

models are not fusion C∗-categories).

Perhaps the simplest nontrivial example of the extension of a fusion category is

when the category C has precisely 1 more simple object than the subcategory C0, and

the latter corresponds to a finite abelian group. More precisely, simple objects [g] in C0

correspond to group elements g ∈ G, with tensor product [g][h] = [gh] corresponding

to group multiplication. The simple objects of C consist of the [g], together with

h= (V ⊗ V′)gh. Its fusion ring is the group ring ZG. We call such

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some object we’ll denote [ρ]. Then [ρ] must be self-conjugate, [g][ρ] = [ρ] = [ρ][g], and

[ρ]2= n′[ρ]+?

of [1] must be 1).

We call these near-group categories of type G + n′. In this paper we restrict to

abelian G, and we reserve n always for the order of G. Examples of these have been

studied in the literature:

g∈G[g] (the multiplicities n′′

gin the second term must be independent of

g because of equivariance [g][ρ] = [ρ]; because [ρ] is its own conjugate, the multiplicity

• the Ising model and the module category of the dihedral group D4, which are

of type Z2+ 0 and Z2× Z2+ 0, respectively;

• more generally, the Tambara-Yamagami systems are by definition those of type

G + 0;

• the A4,E6and Izumi-Xu subfactors are of type G + n for G = Z1,Z2,Z3re-

spectively;

• more generally, Izumi’s second hypothetical family would be of type G + n;

• the D(1)

Z2+ 1, Z2+ 1, and Z3+ 2, respectively;

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subfactor and the module category of groups S3and A4are of type

• more generally, the representation category of the affine group Aff1(Fq)∼=

Fq×F×

More precisely, Aff1(Fq) is the group of all affine maps x ?→ ax + b where a ∈ F∗

and b ∈ Fq. It has precisely q = pkconjugacy classes, with representatives (a,0)

and (1,1). It has precisely q − 1 1-dimensional representations, corresponding to the

characters of F∗

and is the nontrivial summand of the natural permutation representation of Aff1(Fq)

on Fqgiven by the affine maps: (a,b).x = ax + b.

In this paper we classify the near-group C∗-categories G+n′, in the sense that we

obtain polynomial equations in finitely many variables, whose solutions correspond

bijectively to equivalence classes of the near-group C∗-categories. Given any near-

group C∗-category C with n′> 0, we identify a natural subfactor ρ(M) ⊂ M whose

even systems are both identified with C. We also work out the principal graph of the

closely related subfactor ρ(M) ⊂ MG. By contrast, we can realise some but not all

C with n′= 0, as the even sectors of a subfactor.

There is a fundamental dichotomy here: n′either equals n−1, or is a multiple of

n, where as always n = |G|. When n′< n, we have a complete classification:

Fact. Let G be any abelian group of order n.

(a) There are precisely two C∗-categories of type G + 0.

(b) When n′is not a multiple of n = |G|, the only C∗-categories of type G + n′are

Mod(Aff1(Fn+1)), except for n = 1,2,3,7 which have 1,2,1,1 additional C∗-categories.

In all cases here, n + 1 is a prime power, n′= n − 1, and G = Zn+1.

qof a finite field Fqis of type Zq−1+ (q − 2).

q

q. The remaining irrep is thus of dimension?q(q − 1) − q − 1 = q−1,

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This is our Corollary 4 and Proposition 5 respectively, proven below. Type G+0

and type Zn+ n − 1 fusion categories were classified by Tambara-Yamagami [30]

and Etingof-Gelaki-Ostrik [8], respectively; we find that for these types, all fusion

categories can be realised as C∗-categories. Our proof of (a) is independent of and

much simpler than [30].

Conjecture 1.

inequivalent subfactors with principal graph 2n1whose principal even sectors satisfy

the near-group fusions of type G + n.

For every nontrivial cyclic group G = Zn, there are at least 2

We have verified this for n ≤ 13. For those n, the complete classification is given

in Table 2 below. In the process, we construct dozens of new finite depth subfactors of

small index with principal even sectors of near-group type. This classification for n =

3 yields a uniqueness proof (up to complex conjugation) for the principal even sectors

of the Izumi-Xu 2221 subfactor; this can be compared to Han’s uniqueness proof [14]

of the 2221 subfactor. Again, our proof is independent of and both considerably

shorter and simpler than the original one. We do not yet feel confident speculating

on systems with n′> n; the corresponding subfactors would have principal graph as

in Figure 3 below.

Two morals can be drawn from this paper together with our previous one [10]. One

is that there is surely a plethora of undiscovered finite-depth subfactors, of relatively

small index. This is in marked contrast to the observations of e.g. [24], who speak of

the ‘little desert’ in the interval 5 < [M : N] < 3+√5. The situation here is probably

very analogous to the classification of finite groups, which also is very tame for small

orders. The second moral is that, when the fusions are close to that of a group, a very

promising approach to the classification and construction of corresponding systems

of endomorphisms, equivalently C∗-categories, or the corresponding subfactors, is the

Cuntz algebra method developed in e.g. [19] and championed here. This approach

also makes the computation of the tube algebra and corresponding modular data

etc (to be discussed shortly) completely accessible. In contrast, the technique of

planar algebras is more robust, able to handle subfactors unrelated to groups, such

as Asaeda-Haagerup and the extended Haagerup. But planar algebra techniques

applied to e.g. the Haagerup fail to see that it (surely) lies in an infinite family. In

a few minutes the interested computer can construct several more subfactors of the

type described in Conjecture 1, using the Cuntz algebra method here, each of which

would be a serious challenge for the planar algebra method.

The underlying presence of groups here begs the question of K-theory realisations

of these fusion rings. For example, the fusion ring of the near-group C∗-categories

when n′= n − 1 can be expressed as KAff1(Fq)

in the other class, i.e. when n′∈ nZ?

An important class of fusion categories are the modular tensor categories [31],

which are among other things braided and carry a representation (called modular

data) of the modular group SL2(Z) of dimension equal to the rank of the category,

from which e.g. the fusion coefficients can be computed. These arise from braided

systems of endomorphisms on an infinite factor, from representations of completely

0

(1). Is there a K-theoretic expression

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