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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.
5
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 . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
11
16
21
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
26
26
26
28
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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
33
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;
5
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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rational conformal nets, or from the modules of a rational vertex operator algebra.
There is a standard construction, called the quantum or Drinfeld double, to go
from fusion categories (with mild additional properties) to modular tensor categories
[25]. We construct the doubles of our C∗-categories, following the tube algebra ap-
proach [18], and in particular explicitly compute its modular data. As with the
Haagerup series, our formulae are unexpectedly simple. This simplicity also chal-
lenges the perceived exoticness of these subfactors.
A natural question is, are these modular tensor categories realised by conformal
nets of factors, or by rational vertex operator algebras (VOAs)? Ostrik (see Appendix
A in [5]) shows that the double of Izumi-Xu 2221 has a VOA interpretation, in fact it
is the affine algebra VOA corresponding to G2,3⊕A2,1. No construction is known for
the large G. Curiously, this VOA conformally embeds into that of E6,1⊕A2,1(which
realises the fusions of the double of Z3), and this was where [10] suggests to look for
the VOA associated to the double of the Haagerup. Could there be a relation between
Izumi-Xu 2221 and the Haagerup? Other reasons suggest a relationship between 291
and the Haagerup. We discuss this latter possibility briefly in Subsections 3.5 and
4.4.
More generally we could consider quadratic extensions of a group category. More
precisely, let G be a finite group (not necessarily abelian) and suppose [ρ]∗= [ρ][gρ]
for some gρ∈ G. Let N be any subgroup of G: we require [g][ρ] = [ρ] iff g ∈ N.
Then [ρ][g] = [ρ] iff g ∈ gρNg−1
[g] for g ∈ G as well as [gi][ρ] for representatives giof cosets G/N. Let φ be any
isomorphism G/N → G/N′; we require [g][ρ] = [ρ][g′] iff g′∈ φ(gN). Then [ρ]2=
?
categories correspond to the choice N = G and gρ= 1; the Haagerup-Izumi series
[19, 10] corresponds to G = Z2n+1, N = 1, φ(g) = −g, n′
Haagerup subfactor at index (5 +√13)/2 corresponds to G = Z3. It would be very
interesting to extend the analysis in this paper to this larger class.
Here is a summary of our main results. Theorem 1 associates numerical invariants
to a near-group C∗-category, which according to Corollary 1 completely characterise
the category. Corollary 2 (and the end of Subsection 2.2) associate to each C∗-
category two subfactors and work out their principal graphs. Theorem 2 establishes
the fundamental dichotomy of near-group C∗-categories: either n′= n−1 or n′∈ nZ.
When n′= n− 1, Theorem 3 lists the identities necessarily obeyed by the numerical
invariants and shows they are also sufficient. Theorem 4 does the same when n|n′. In
Proposition 5 we find all near-group C-categories with n′= n−1; we see that almost
all of these are known. In Table 2 we list the first several with n′= n, and find that
almost none of these are known. In Theorem 5 and Corollary 6 we work out the tube
algebra and modular data for any near-group C∗-category with n′= n − 1. [19] had
found a very complicated expression for the modular data when n′= n; we notice in
Subsection 4.4 that it collapses to cosines.
ρ
=: N′. The simple objects in this category are
g∈N[g] +?
in′
i[gi][ρ]. We require φ to satisfy g−1
large class of examples should be accessible to a similar treatment. The near-group
ρφ(φ(g))gρ= g for all g ∈ G. This
i= 1; in particular, the
Note added in proof. After completing this manuscript, we received in July 2012
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[21] from Masaki Izumi, which overlaps somewhat the contents of our paper. In
particular, he also obtained necessary and sufficient conditions for the Cuntz algebra
construction to realise a near-group C∗-category of type G + n′. On the one hand,
unlike us, he does not require G to be abelian, and he allows the possibilities of
an H2-twist. On the other hand, unlike us, he does not address principal graphs
of associated subfactors, nor the tube algebra, nor the modular data (simplified or
otherwise) for the doubles, and he does not construct new solutions of the resulting
equations and hence does not construct new subfactors.
2 The near-group systems
2.1The numerical invariants
Let G be a finite abelian group (written additively) with order n = |G|, and as usual
write?G for its irreps. Let M be an infinite factor, ρ a self-conjugate irreducible
of G on M. Suppose the following fusion rules hold:
endomorphism on M with finite statistical dimension dρ< ∞, and α an outer action
[αgρ] = [ρ] = [ραg],
?
(2.1)
[ρ2] =
g
[αg] ⊕ n′[ρ], (2.2)
for some n′∈ Z≥0. Then the dρsatisfies d2
ρ= n′dρ+ n so
dρ=n′+√n′2+ 4n
2
=: δ.(2.3)
Let C(G,α,ρ) denote the fusion C∗-category generated by α,ρ. We call these, C∗-
categories of type G + n′.
Definition 1. By a pairing ?g,h? on G we mean a complex-valued function on G×G
such that for all fixed g ∈ G, both ?g,∗?,?∗,g? ∈?G. By a symmetric pairing we mean
for which the characters ?g,∗? are distinct for all g.
Note that a nondegenerate pairing is equivalent to a choice of group isomorphism
G →?G, g ?→ φg, by φg(h) = ?g,h?. The nondegenerate symmetric pairings for
Theorem 1. Let G,α,ρ be a C∗-category of type G + n′. Suppose in addition that
H2(G;T) = 1. Then there are n + n′isometries Sg,Tz(g ∈ G, z ∈ F) satisfying the
Cuntz relations, such that αgρ = ρ, ραg= Ad(Ug)ρ, for a unitary representation Ug
of G of the form
Ug=
?g,h?ShS∗
a pairing satisfying ?g,h? = ?h,g?. By a nondegenerate pairing we mean a pairing
G = Znare ?g,h? = e2πımgh/nfor some integer m coprime to n.
?
h
h+
?
z
uz,gTzT∗
gz,(2.4)
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where G permutes the z ∈ F and ?g,h? is a pairing on G. Moreover, αg(Sh) = Sg+h
and αg(Tz) = ? z(g)Tzfor some ? z ∈?G. Finally,
ρSg=
?
sδ−1?
?
h
?g,h?Sh+
?
h+
x,z
ux,gagx,zTxTz
?
U∗
g, (2.5)
ρ(Tz) =
?
h,x
? x(h)bz,xShT∗
x+
x,h
? x(h)b′
z,xTxShS∗
?
w,x,y
b′′
z;w,x,yTwTxT∗
y, (2.6)
for some sign s ∈ {±1} and complex parameters ay,z,bz,x,b′
F.
Proof. Our argument follows in part that of the first theorem of [20]. Because [αgρ] =
[ρ], there exists a unitary Wg∈ U(M) for each g ∈ G, satisfying αgρ = Ad(Wg)ρ.
But
Ad(Wg+h)ρ = αhαgρ = Ad(αh(Wg)Wh)ρ
z,x,b′′
z;w,x,y, for w,x,y,z ∈
(2.7)
for all g,h ∈ G, so αh(Wg)Wh = ξ(g,h)Wg+h for some 2-cocycle ξ ∈ Z2(G;T).
Because H2(G;T) = 1, we can require that ξ be identically 1, by tensoring Wgwith
the appropriate 1-coboundary. Since G is a finite group and α is outer, the α-cocycle
Wgis a coboundary, so there exists a unitary V ∈ U(M) so that Wg= αg(V∗)v for
all g ∈ G. This means Ad(αg(V ))αg(ρ) = Ad(V )ρ, i.e. αg(Ad(V )ρ) = Ad(V )ρ.
Thus if we replace ρ by Ad(V )ρ we obtain αgρ = ρ as endomorphisms, not just as
sectors.
This has exhausted most of the freedom in choosing ρ. The fusion [ραg] = [ρ]
means ραg = Ad(Ug)ρ for some unitaries Ug; because H2(G;T) = 1, we can in
addition insist that g ?→ Ugdefines a unitary representation of G. Note that we still
have a freedom in replacing Ugwith ψ(g)Ugfor any character ψ ∈?G.
ρ2(x) =
Sgαg(x)S∗
The fusion (2.2) means
?
g∈G
g+
?
z∈F
Tzρ(x)T∗
z, (2.8)
where Sgand {Tz}z∈Fare bases of isometries for the intertwiner spaces Hom(αg,ρ2)
and Hom(ρ,ρ2) respectively (so ρ2(x)Sg = Sgαg(x) etc). Then (2.8) implies Sg,Tz
obey the Cuntz relations. Since Ad(Ug)ρ2= ραgρ = ρ2, Uhmaps Hom(αg,ρ2) to
itself and Hom(ρ,ρ2) to itself, i.e. UhSg = µg(h)Sgand UhTz =?
and the matrices u define a unitary representation on Hom(ρ,ρ2). This gives us
?
wu(h)z,wTwfor
some µg(h),u(h)z,w∈ C. Since Uh+h′ = UhUh′, we have that µg∈?G for each g ∈ G,
Ug=µh(g)ShS∗
h
h+
?
z,y
u(g)z,yTzT∗
y.(2.9)
Define U′
of G. For this reason we may assume that µ0is identically 1.
g= µ0(g)Ug. Then ραg= Ad(U′
g)ρ and U′
gis still a unitary representation
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half-braidings, and precisely one each for σ = αg+ αh for each g ?= h. In this
Tambara-Yamagami case, as analysed in Section 3 of [19], Tube∆ is isomorphic as a
C∗-algebra to C4n⊕(M2×2)n(n−1)/2, and elementary expressions for the modular data
fall out directly.
Thanks to Proposition 6, the case n = n′reduces to that studied in [19], and so
its tube algebra is fully analysed in Section 6 of [19]. We find there is a unique half-
braiding and simple summand C in Tube∆ for each σ = αg, while each σ = ρ + αg
corresponds to a unique summand M2×2and half-braiding. σ = ρ + αg+ αh(g ?= h)
gives a unique braiding and summand M3×3. Finally, there are exactly n(n + 3)/2
half-braidings with σ = ρ, and each contributes a C to Tube∆. Thus
Tube∆n′=n∼= Cn(n+5)/2⊕ (M2×2)n⊕ (M3×3)n(n−1)/2.
The modular data for n′= n is described in [19] as follows. First, find all functions
ξ : G → T and ω ∈ T, τ ∈ G such that
(4.51)
?
g
ξ(g) =√nω2a(τ)c3− nδ−1,(4.52)
c
?
k
b(g + k)ξ(k) = ω2c3a(τ)ξ(g + τ) −√nδ−1, (4.53)
ξ(τ − g) = ω c4a(g)a(τ − g)ξ(g), (4.54)
?
k
ξ(k)b(k − g)b(k − h) = c−2b(g + h − τ)ξ(g)ξ(h)a(g − h) − c2δ−1.(4.55)
There will be a total of n(n + 3)/2 such triples (ωj,τj,ξj).
The n(n + 3) primaries fall into four classes:
1. n primaries, denoted ag, g ∈ G;
2. n primaries, denoted bhfor h ∈ G;
3. n(n − 1)/2 primaries, denoted ck,l= cl,kfor k,l ∈ G, k ?= l;
4. n(n + 3)/2 primaries, denoted dj, corresponding to the triples (ωj,τj,ξj).
We can write the S and T matrices in block form as
T = diag(?g,g?;?h,h?;?k,l?;ωj),
(δ + 1)?g,h′?−2
?h,h′?−2
(δ + 2)?k + l,h′?
−δ?τj,h′?
(4.56)
S =1
λ
?g,g′?−2
(δ + 1)?h,g′?−2
(δ + 2)?k + l,g′?
δ?τj,g′?
(δ + 2)?g,k′+ l′?
(δ + 2)?h,k′+ l′?
(δ + 2)(?k,k′??l,l′? + ?k,l′??l,k′?)
0
δ?g, τj′?
−δ?h,τj′?
0
Sj,j′
, (4.57)
where
Sj,j′ = ωjωj′
?
g
?τj+τj′+g,g?+δωjωj′c6a(τj)a(τj′)n−1?
g,h
ξj(g)ξj′(h)?τj− τj′ + h − g,h − g?.
(4.58)
This is all perfectly simple, except for the n(n + 3)/2 × n(n + 3)/2 block Sj,j′.
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4.4The modular data for the double of G+n when n is odd
The point of this subsection is to compute the mysterious part (4.58). We will show
that, rather unexpectedly, Sj,j′ is built up from a quadratic form on an abelian group
of order n + 4.
Definition 3. Let G be any finite abelian group. By a nondegenerate quadratic
form Q on G we mean a map Q : G → Q/Z such that Q(−g) ≡ Q(g) (mod 1) for
all g ∈ G, and ?,?Q : G × G → T defined by ?g,h?Q = e2πı(Q(g+h)−Q(g)−Q(h))is a
nondegenerate symmetric pairing in the sense of Definition 1.
For example, when G = Znfor n odd, these are precisely Q(g) = mg2/n for any
integer m coprime to n. More generally, for |G| odd, the nondegenerate quadratic
forms and nondegenerate symmetric pairings are in natural bijection.
case, we can always write G as Zn1× ··· × Znkwhere Q restricted to each Zniis
nondegenerate and ?Zni,Znj? = 1 for i ?= j. When |G| is even, things are more
complicated but G will have precisely |G/2G| nondegenerate quadratic forms for
each nondegenerate symmetric pairing ?,?. The map a : G → T of Corollary 5 is the
exponential of a nondegenerate quadratic form.
Given a nondegenerate quadratic form Q and a ∈ Z, define the Gauss sum
1
?|G|
For example, note from (3.2) that the quantity c3of Corollary 5 is a Gauss sum.
Provided aQ is nondegenerate, αQ(a) will be a root of unity (this is a consequence of
Proposition 7(a) below). All Gauss sums needed in this paper can be computed from
the classical Gauss sums, corresponding to G = Znand Q(g) = mg2/n, which equal
where
b
is the Jacobi symbol. For n even, these classical Gauss sums are not
modulus 1, because mg2/n is degenerate in Zn.
In such a
αQ(a) =
?
k∈G
exp(2πıaQ(k)).
?am
n
?
for n ≡41
for n ≡43
for n ≡42
ı?am
n
?
am
0
(1 ± ı)?n
?
when n ≡40 and a ≡4±m
,
?a
?
Proposition 7(a) Let Q be a nondegenerate quadratic form on any abelian group G.
Define matrices
SQ
g,h=
α
?|G|?g,h?Q,TQ
g,h= β δg,hexp(2πıQ(g)),(4.59)
for any α,β ∈ C. Then SQ,TQdefine modular data iff α = ±1 and β3= ααQ(1). In
this case, the identity is a0.
(b) Let G,G′be abelian groups of odd order n and n + 4 respectively. Choose any
nondegenerate quadratic forms Q and Q′on them, and write ?g,h? = ?g,h?(n+1)/2
Q
and
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?β,γ?′= ?β,γ?(n+5)/2
Let Φ consist of the following n(n+3) = n+n+n(n−1)/2+n(n+3)/2 elements: ag
∀g ∈ G; bh∀h ∈ G; ck,l= cl,k∀k,l ∈ G with k ?= l; and dm,γ= dm,−γ∀m ∈ G,γ ∈ G′,
γ ?= 0. Define
TQ,Q′= diag(?g,g?;?h,h?;?k,l?;?m,m??γ,γ?′),
where λ is given in (4.14). Then these define modular data iff αQ(1)αQ′(1) = −1.
The identity is a0.
Q′
for all g,h ∈ G, β,γ ∈ G′, so Q(g) = ?g,g? and Q′(γ) = ?γ,γ?′.
SQ,Q′=1
λ
?g,g′?2
(δ + 1)?g,h′?2
?h,h′?2
(δ + 2)?k + l,h′?
−δ?m, h′?2
(δ + 2)?g, k′+ l′?
(δ + 2)?h,k′+ l′?
?k,k′??l,l′? + ?k,l′??l,k′?
0
δ?g, m′?2
−δ?h,m′?2
0
?
(δ + 1)?h,g′?2
(δ + 2)?k + l,g′?
δ?m, g′?
(δ + 2)
??
2
−δ?m, m′??γ,γ′?′+ ?γ,γ′?′?
The straightforward proof is by direct calculation: S2= C, S∗= CS, ST∗S =
TS∗T, and Verlinde’s formula (4.11). In (a), α2= 1 arises from the requirement that
S2be a permutation matrix. The conditions α3β3= αQ(−1) and αQ(−1)αQ′(−1) =
−1 for (a) and (b) respectively both come from ST∗S = TS∗T. We find that the
fusion coefficients of part (a) are Nk
and charge-conjugation C acts by −1. In (b), charge-conjugation sends ag?→ a−g,
bh?→ b−h, ck,l?→ c−k,−l, dm,γ?→ d−m,γ. The nonzero fusion coefficients there are
Nag,ah,ak=Nag,bh,bk= Nbg,bh,bk= Nbg,bh,dk,γ= δ(g + h + k);
Nag,ch,k,ch′,k′=δ(g + h + h′)δ(g + k + k′) + δ(g + h + k′)δ(g + h′+ k) ∈ {0,1};
Nag,dh,β,dk,γ=δ(g + h + k)δβ,γ;
Nbg,bh,ck,k′=Nbg,ck,k′,dh,γ= δ(2g + 2h + k + k′);
Nbg,ch,k,ch′,k′=δ(2g + h + k + h′+ k′) + δ(g + h + h′)δ(g + k + k′)
+ δ(g + h + k′)δ(g + h′+ k) ∈ {0,1,2};
Nbg,dh,β,dk,γ=δ(g + h + k)(1 − δβ,γ);
Ncg,h,cg′,h′,dk,γ=δ(g + h + g′+ h′+ 2k);
Ncg,h,dk,β,dk′,γ=δ(g + h + 2k + 2k′);
Ncg,h,cg′,h′,cg′′,h′′=δ(g + h + g′+ h′+ g′′+ h′′)(1 + δ(g + g′+ g′′) + δ(g + h′+ g′′)
+ δ(g + g′+ h′′) + δ(g + h′+ h′′)) ∈ {0,1,2};
Ndg,γ,dg′,γ′,dg′′,γ′′=δ(g + g′+ g′′)(1 − δ(γ + γ′+ γ′′) − δ(γ − γ′+ γ′′) − δ(γ + γ′− γ′′)
− δ(γ − γ′− γ′′)) ∈ {0,1},
where we write δ(g) = 1 or 0 depending on whether or not g = 0.
We’ll let MDG,G′(Q,Q′) denote the modular data of (b). Of course, associating
SL2(Z) representations to quadratic forms is an old story. See for instance [26],
who study these in similar generality (though their G are p-groups, and they require
β = 1), and call these Weil representations. To our knowledge, Proposition 7(b) is
completely new, but what is more important is its relation to near-group doubles:
g,h= δk,g+h, every primary is a simple-current,
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Conjecture 2. When |G| = n is odd, the modular data for G+n is MDG,G′(Q,Q′),
where Q is the nondegenerate quadratic form on G corresponding to ?,?, and Q′is a
nondegenerate quadratic form on some abelian group G′of order n + 4.
This is true for all groups G of odd order ≤ 13 (recall in Table 2). The easi-
est way to verify modular datum are equivalent is to first identify their T matrices
(straightforward since they must have finite order) and then compare the floating
point values of the S matrices — see Section 4.1 of [10] for details. In all cases in
Table 2, G′ ∼= Zn+4, except for the first entry for G = Z5when G′= Z3× Z3. The
quadratic form Q′is then identified in Table 2 by the integer m′in the Q′column.
For the first entry of G = Z5, Q′(γ1,γ2) = (γ2
Even for n = 3, this is vastly simpler than the modular data as it appears in
Example A.1 of [19]. In fact we have no direct proof that they are equal for n = 3 —
our proof that they are the same is that they both yield nonnegative integer fusions,
they have the same T matrices, and their S matrices are numerically close. The
simplicity of this modular data MDQ,Q′(G,G′) supports our claim that the doubles
of these near-group categories G + n should not be regarded as exotic. We would
expect that these doubles are realised by rational conformal nets of factors, and by
rational vertex operator algebras.
The quantum-dimensions Sx,0/S0,0are 1,δ+1,δ+2,δ for primaries of type a,b,c,d
respectively. We see from the above that the agare simple-currents, and obey the
fusions ag∗ ah= ag+h, ag∗ bh= bg+h, ag∗ ch,k= cg+h,g+k, and ag∗ dh,γ = dg+h,γ.
They form a group isomorphic to G, and act without fixed-points. They supply the
ultimate explanation for the G-action of Proposition 6.7 of [19]. The phases ϕg(x)
defined by Sagx,y= ϕg(y)Sx,yare ?g,h? for ah,bh,dh,γ, and ?g,k + l? for ck,l.
The Galois symmetry is useful in understanding the modular invariants. For
ℓ ∈ Z×
equals 1. Similarly, bℓ
1+ γ2
2)/3.
n(n+4), aℓ
g= aℓgor bℓgdepending on whether or not the Jacobi symbol
g= bℓgor aℓgrespectively. Finally, cℓ
All parities ǫℓ(x) = +1 except for ǫℓ(dg,γ) =
Galois symmetry is what led us to the simplified modular data given above.
A modular invariant is a matrix Z with nonnegative integer entries (often formally
written as a generator function Z =?
There are exactly 3 type I modular invariants when both n and n+4 are prime (e.g.
for n = 3):
?
Z2=
g
g,h
?
ℓ
n(n+4)
?
g,h= cℓg,ℓhand dℓ
. The requirement of a coherent
g,γ= dℓg,ℓγ.
?
ℓ
n(n+4)
?
a,bZa,bchachb), with Z0,0= 1, which commutes
with the modular data S,T. It is called type I if Z can be written as a sum of squares.
Z1=
g
|ag|2+
?
g
|bg|2+
?
?
|cg,h|2,
g,h
|cg,h|2+
?
Z3= |a0+ b0+
g,γ
|dg,γ|2,
?
|ag+ bg|2+ 2
?
g?=0
cg,0|2.
The most important modular invariants are the monomial ones, of form Z =
|ΣZa,0cha|2, as explained in Section 1.3 of [10], as they give a canonical endomor-
phism θ as a sum of sectors, and can be used to recover the original system from
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its double. This is M¨ uger’s forgetful functor [25]. For example, there are exactly 3
monomial modular invariants for the modular data of the double of the Haagerup
subfactor; these should correspond bijectively to the three systems found in [12] which
are Morita equivalent to the principal even Haagerup system (see their Theorem 1.1),
as each of those must correspond to a monomial modular invariant.
We see however that for the MDG,G′(Q,Q′) modular data there is only one generic
monomial modular invariant, namely Z3. This suggests that Grossman-Snyder per-
haps isn’t as interesting here as it was for the Haagerup (at least not for general n).
On the other hand, recall our comments earlier that the type G + n systems with
n = n′= ν2may be related to the Haagerup-Izumi system for groups of order ν.
Consider first G = Znfor n = ν2a perfect square and write H = νG∼= Zν; then
MDG,G′(Q,Q′) has at least one other monomial invariant, namely
?
Alternatively, when G = H1× H2where each Hi∼= Zν, and ?H1,H2?Q= 1 (which
can always be arranged), another monomial invariant is
Z4= |
h∈H
ah+
?
h∈H
bh+ 2
?
h<h′∈H
ch,h′|2. (4.60)
Z′
4= |a0+ b0+
?
h∈H1,h′∈H2
c(h,0),(0,h′)|2.(4.61)
We would expect that systems of type Zν2 +ν2or Zν×Zν+ν2should have nontrivial
quantum subgroups in the sense of [12].
It isn’t difficult to see why n+4 arises here, i.e. why it can’t be replaced by some
other positive integer n′. In particular, after some work, the nonzero fusions of the
form Nb,b,bare found to be 4/(n′− n);,and the ST∗S = TS∗T calculation requires
the product of Gauss sums for G and G′to be −1, which forces 4|(n′− n).
When G has even order, the situation is similar but (as always with n even)
somewhat messier; we will provide its modular data elsewhere. Again we have n
simple-currents (the ag), but for each g ∈ G of order 2, agnow has n/2 fixed-points,
which complicates things. The T entries for the first several even G are provided by
the pairs (m′,m′′) in the Q′column of Table 2, and from this the S matrix follows
quickly from the equations of the last subsection. In particular,
?
n
n(n+4)
Tdg,γ,dg,γ=
ξ?g,g?
n
ξm′γ2
n+4
ξm′′(1+nγ)2
if γ + n/2 is odd
if γ + n/2 is evenξ?m,m−1?
where τγ= 0,1 for γ +n/2 odd respectively even. Here 1 ≤ γ ≤ (n+4)/2 and g ∈ G
except for γ = (n + 4)/2 when g ∈ G/2.
Recall our observation in Section 3 of [10] that the modular data of the double of
the Haagerup-Izumi series at G = Znresembles that of the affine algebra so(n2+4)(1)
at level 2. The analogous statement here is that the modular datum of the double of
type Zn+n near-group systems resemble that of the affine algebra so(n+4)(1)at level
2. In particular, for an appropriate choice of Q′(corresponding to m′= (n+3)/2), this
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recovers Td0,γ,d0,γand Sd0,γ,d0,γ′. This could hint at ways to construct the corresponding
vertex operator algebra.
Acknowledgement.
The authors thank the Erwin-Schr¨ odinger-Institute, Cardiff School of Mathematics, and
Swansea University Dept of Computer Science, for generous hospitality while researching this paper.
They thank Masaki Izumi for sharing with us an early draft of [20] and informing us of [8], and
Eric Rowell for informing us of [29]. Their research was supported in part by EPSRC, EU-NCG
Research Training Network: MRTN-CT-2006 031962, and NSERC.
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