CALT-TH-2021-006
Topological Field Theory with Haagerup Symmetry
Tzu-Chen Huang
a
, Ying-Hsuan Lin
b,a
a
Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125, USA
b
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
jimmy@caltech.edu, yhlin@fas.harvard.edu
Abstract
We construct a (1+1)
d
topological field theory (TFT) whose topological defect lines
(TDLs) realize the transparent Haagerup
H
3
fusion category. This TFT has six vacua,
and each of the three non-invertible simple TDLs hosts three defect operators, giving
rise to a total of 15 point-like operators. The TFT data, including the three-point
coefficients and lasso diagrams, are determined by solving all the sphere four-point
crossing equations and torus one-point modular invariance equations. We further verify
that the Cardy states furnish a non-negative integer matrix representation under TDL
fusion. While many of the constraints we derive are not limited to the this particular
TFT with six vacua, we leave open the construction of TFTs with two or four vacua.
Finally, TFTs realizing the Haagerup
H
1
and
H
2
fusion categories can be obtained by
gauging algebra objects. This note makes a modest offering in our pursuit of exotica
and the quest for their eventual conformity.
arXiv:2102.05664v2 [hep-th] 28 Apr 2021
Contents
1 Introduction
2
2 Topological field theory extended by defects
6
2.1 Fusion category of topological defect lines . . . . . . . . . . . . . . . . . . .
6
2.2 Local operators and commutative Frobenius algebra . . . . . . . . . . . . . .
8
2.3 Defect operators, defect operator algebra, and lassos . . . . . . . . . . . . . .
8
2.4 General observables, crossing symmetry and modular invariance . . . . . . .
10
3 Spectral constraints by Haagerup symmetry
12
3.1 The Haagerup fusion ring with six simple objects . . . . . . . . . . . . . . .
13
3.2 Action on local operators and representation theory . . . . . . . . . . . . . .
13
3.3 Modular invariance and vacuum degeneracy . . . . . . . . . . . . . . . . . .
15
4 Transparency and
Z
3
symmetry
17
4.1
Z
3
relations for lassos and dumbbells . . . . . . . . . . . . . . . . . . . . . .
17
4.2
Z
3
action on defect operators . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5 Bootstrap constraints
20
5.1 Local operator algebra and associativity . . . . . . . . . . . . . . . . . . . .
21
5.2 Mixed local and
ρ
defect operators . . . . . . . . . . . . . . . . . . . . . . .
22
5.3
ρ
action on local operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.4 Torus one-point modular invariance . . . . . . . . . . . . . . . . . . . . . . .
26
6 Topological field theory with Haagerup
H
3
symmetry
28
6.1 Local operator algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6.2 Topological field theory with six vacua . . . . . . . . . . . . . . . . . . . . .
31
7 Boundary conditions and NIM-reps
35
8 Realization of Haagerup
H
1
and
H
2
via gauging
39
1
9 Prospective questions
40
A The
F
-symbols for the Haagerup
H
3
fusion category
41
B Crossing symmetry of
ρ
defect operators
43
1 Introduction
The best cultivated terrains in the landscape of (1+1)
d
conformal field theories (CFTs)
are rational conformal field theories (RCFTs) [1], free theories, and orbifolds [2, 3] thereof.
Exactly marginal deformations of orbifold twist fields bring us into more interesting realms,
and when roamed far enough provide candidates with weakly coupled holographic duals.
But the full landscape is believed to be vaster. The conformal bootstrap bounds on various
quantities such as the twist gap [4–6] are not saturated by known CFTs, and numerical
studies of certain renormalization group flows, such as that from the three-coupled three-
state Potts model [7], indicate the existence of fixed points with irrational central charges.
However, such fixed points are evasive of current analytic methods. Even for RCFTs, a full
classification has not been achieved.
The full set of interesting observables in a (1+1)
d
CFT is not limited to the correlation
functions of local operators. There are boundaries and defects that interact with the local
operators in nontrivial ways, and are together subject to stringent consistency conditions.
Some of the data, like the fusion category [8, 9] furnished by the topological defect lines
(TDLs) [10–12], are mathematically rigid structures that exist independently of quantum
field theory. A simple example of a fusion category is a group-like category, which consists of
the specification of a discrete symmetry group together with its anomaly. Fusion categories
generalize symmetries and anomalies, and constrain the deformation space of quantum field
theory. The preceding remarks beg the following question:
Q1:
Given a fusion category, is there a (1+1)
d
CFT whose TDLs (or a subset thereof )
realize the said category?
The (2+1)
d
Turaev-Viro theory [13] or Levin-Wen string-net model [14] constructed out
of a fusion category
C
is a bulk phase whose anyons are described by the Drinfeld center
Z
(
C
). By placing the bulk phase on a slab between a gapped boundary and another boundary
condition
B
, and further compactifying on a circle, the resulting theory
T
[
B
] would be a CFT
with TDLs described by
C
.
Q1
is thus interpreted as the existence/classification problem
of boundary conditions for the bulk phase.
1
From a purely (1+1)
d
perspective, statistical
1
The authors thank Shu-Heng Shao and Yifan Wang for discussions.
2
height models which take
C
(and the choice of a distinguished object) as the microscopic
input have recently been shown by Aasen, Fendley, and Mong [15] to host macroscopic TDLs
described by
C
. One can explore the phases of such models in search of a CFT.
A closely related question is the following:
Q2:
Given a modular tensor category (MTC), is there a vertex operator algebra (VOA)
whose representations realize the said category?
The phrase VOA could be replaced by diagonal RCFT, in which the fusion ring of Ver-
linde lines (TDLs commuting with the VOA) is isomorphic to the fusion ring of the VOA
representations. The correspondence between MTC and (1+1)
d
RCFT traces its origin to a
seminal series of papers by Moore and Seiberg [16–20], and is conjectured to be one-to-one
thought a construction or proof is lacking. Note that an affirmative answer to
Q2
implies
an affirmative answer to
Q1
: Given a fusion category
C
, if one can find a VOA that realizes
the Drinfeld center
D
(
C
), then gauging the diagonal RCFT by an algebra object gives a
non-diagonal RCFT whose TDLs realize
C
C
op
.
2
The explicit realization of many categories in CFT is not known. A famous example
is the Haagerup fusion category, which has a special place in the history of category and
subfactor theory. Subfactors have inherent categorical structure, and serve as a major source
of fusion categories. While Ocneanu [21] and Popa [22] classified subfactors with Jones
indices less than or equal to 4, Haagerup and Asaeda [23] constructed one—the Haagerup
subfactor—with Jones index
5+
√
13
2
, the smallest above 4 [24]. As the title of [23] suggests,
the Haagerup subfactor was deemed
exotic
since its construction at the time did not fit into
any infinite family. Later work by Izumi [25], Evans and Gannon [26] postulated that the
Haagerup subfactor does in fact fit into an infinite family, and furthermore constructed the
first few members. This development suggested that the Haagerup may not be exotic after
all. Nonetheless, for various categorical conjectures, the explicit demonstration in the case
of Haagerup is viewed as a key test of a conjecture’s legitimacy and generality.
There are actually three inequivalent unitary Haagerup fusion categories, commonly de-
noted by
H
1
,
H
2
,
H
3
. Most of this note concerns the Haagerup
H
3
fusion category, which did
not descend directly from the Haagerup subfactor of [23, 24], but was instead constructed
by Grossman and Snyder [27]. Because the fusion ring (reviewed in Section 3.1) is non-
commutative, the Haagerup
H
3
fusion category cannot possibly be realized by Verlinde
lines [28–31] in a diagonal RCFT. To our knowledge, its realization as general TDLs (need
not commute with the full VOA) is not known in any CFT. To connect to Verlinde lines, one
must consider the MTC that is Drinfeld center of Haagerup. In fact, Evans and Gannon [26]
constructed
c
= 8 characters for the Haagerup modular data, and used it to surmise possible
2
The authors thank Sahand Seifnashri for a discussion.
3
constructions of the VOA through the Goddard-Kent-Olive coset construction [32] and its
generalizations [33–35], or through the generalized orbifold construction (gauging an algebra
object) of Carqueville, Fr ̈ohlich, Fuchs, Runkel, and Schweigert [36–38] (see [10] for a re-
cent discussion). Recently, Wolf [39] considered the Haagerup anyon chain and numerically
searched for CFT phases, but with inconclusive results.
3
To date, a
bona fide
construction
remains an important open question. By trying to construct CFTs that realize more exotic
fusion categories, one hope is that light would be shed beyond the current borders of known
(R)CFTs.
Concerning the gapped phases of (1+1)
d
quantum field theory, described by (1+1)
d
topological field theories (TFTs) extended by defects [41, 42], a related but simpler question
can be asked:
4
Q3:
Given a fusion category, is there a (1+1)
d
TFT whose TDLs (or a subset thereof )
realize the said category?
Thorngren and Wang [51] has proven that
C
-symmetric defect TFTs are in bijection
with
C
-module categories, and since the regular module category always exists,
Q3
has been
affirmatively answered. However, their proof utilizes the (1+1)
d
Turaev-Viro state-sum [13]
or Levin-Wen string-net [14] construction, and it is generally unclear how the axiomatic TFT
data can be extracted. We are thus led to the next question:
Q4:
Given a fusion category, can one construct the axiomatic data of a (1+1)
d
TFT
whose TDLs (or a subset thereof ) realize the said category?
This question has been answered for group-like categories by Wang, Wen, and Witten [52]
and by Tachikawa [11], and for categories with fiber functors (the resulting TFT has a
unique vacuum) by Thorngren and Wang [51]. In [12], it was shown that for a variety of
CFTs, the TFT data can be solved solely from the input of the fusion categorical data, by
bootstrapping the consistency conditions. For general categories, a construction of the bulk
Frobenius algebra was given by Komargodski, Ohmori, Roumpedakis, and Seifnashri [53],
but the full defect TFT data remains unsolved.
5
3
Anyon chains generalize the golden chain of Feiguin et al [40], and arise in a limit of the statistical height
models of Aasen, Fendley, and Mong [15].
4
There are various notions of TFT with different amounts of structure, the most common being closed
TFT [43–46] and open/closed TFT [47–50]. The defect TFT of [41, 42] is an overarching formalism that
can incorporate multiple closed TFTs and their boundaries and interfaces. The minimal structure that
incorporates the data of TDLs is a defect TFT containing a single closed TFT; mathematically speaking, it
is a bicategory with a single object, whose 1-morphisms are the TDLs, and whose 2-morphisms are the local
and defect operators. The full enrichment by boundaries and interfaces with other closed TFTs is beyond
the scope of this note.
5
After publication of the first version of this note, Kantaro Ohmori and Sahand Seifnashri suggested to
the authors that a construction of the full defect TFT data may be possible through a generalization of [53].
4
The preceding questions are ultimately connected. A CFT realizing a certain fusion
category is connected to a TFT realizing the same category under TDL-preserving renor-
malization group (RG) flows, and this principle strongly constrains the infrared fate of CFTs.
In the space of TDL-preserving RG flows and without fine-tuning, they must either flow to
gapped phases described by TFTs, or to “dead-end” CFTs [54], which correspond to gapless
phases protected by fusion categories [51].
6
As desirable as fully universal answers to the preceding questions are, a more pragmatic
approach may be to first examine fusion categories for which the answers are not known.
This note makes a modest offering in this approach of pursuing exotica in the quest for
their eventual conformity: the construction of a TFT realizing the Haagerup
H
3
fusion
category with fully explicit axiomatic TFT data. The construction is of bootstrap nature,
by solving the full cutting and sewing consistency conditions. A prerequisite in this approach
is the explicit knowledge of the
F
-symbols. They were implicit in the work of Grossman and
Snyder [27] (using a generalization of the approach by Izumi [25]), and also explicitly obtained
by Titsworth [57], Osborne, Stiegemann, and Wolf [58]. In [59], the present authors recast
the
F
-symbols in a gauge that manifests the transparent property, which greatly simplify
our present computational endeavor.
The remaining sections are organized into steps of the construction and discussions of
further ramifications. Section 2 reviews the generalities of topological field theory extended
by defects, and formulates the defining data and consistency conditions. Section 3 intro-
duces the Haagerup fusion ring with six simple objects/TDLs, studies its representation
theory, and constrains the vacuum degeneracy using modular invariance. Section 4 studies
the relations among dynamical data implied by transparency and
Z
3
symmetry. Section 5
delineates the constraints of associativity and torus one-point modular invariance. Section 6
solves the constraints to construct a topological field theory with Haagerup symmetry. Sec-
tion 7 examines the expectation that the boundary conditions furnish a non-negative integer
matrix representation (NIM-rep) of the fusion ring. Section 8 discusses the relations among
topological field theories by gauging algebra objects. Section 9 ends with some prospec-
tive questions. Appendix A contains the
F
-symbols for the Haagerup
H
3
fusion category.
Appendix B analyzes the general crossing symmetry of defect operators.
6
Such phases generalize the notion of (group-like) symmetry-protected gapless phases [55] and perfect
metals [56].
5
2 Topological field theory extended by defects
This section introduces the axiomatic data of a topological field theory (TFT) extended by
defects, and the consistency conditions they must satisfy.
2.1 Fusion category of topological defect lines
The nontrivial splitting and joining relations of a finite set of topological defect lines (TDLs)
are captured by a fusion category. A classic introduction to fusion categories can be found
in [8, 9], and expositions in the physics context can be found in [10, 12]. Here we follow the
latter and present a lightening review of the key properties of TDLs.
Topological defect lines are (generally oriented) defect lines whose isotopic transforma-
tions leave physical observables invariant. We restrict ourselves to considering sets of TDLs
with finitely many
simple
TDLs
{L
i
}
; the others, the
non-simple
TDLs, are direct sums of
the simple ones.
7
Among the simple TDLs there is a trivial TDL
I
representing nothingness.
Furthermore, every TDL
L
has an orientation reversal
L
, as depicted by the equivalence
L
=
L
.
(2.1)
Whenever a TDL is isomorphic to its own orientation reversal,
L
=
L
, we omit the arrows
on the lines.
8
A general configuration of TDLs involves junctions built out of trivalent vertices. The
allowed trivalent vertices are specified by the fusion ring
L
i
L
j
=
N
k
ij
L
k
,
(2.2)
where
N
k
ij
∈
Z
≥
0
are the fusion coefficients. To simplify the discussion, it is assumed that
(1) the fusion coefficients (dimensions of junction vector spaces) are zero or one, and (2) the
trivalent vertices are cyclic-permutation invariant.
9
In conformity with [12,59], we adopt the
7
See [60] for progress in incorporating “non-compact” topological defect lines.
8
The orientation cannot be completely ignored if the TDL has an orientation-reversal anomaly (nontrivial
Frobenius-Schur indicator) [12]. This subtlety does not arise for the Haagerup and is therefore neglected.
9
Both assumptions are satisfied by the transparent Haagerup
H
3
fusion category. The reader is referred
to [12] for a general discussion without these assumptions.
6
counter-clockwise convention for trivalent vertices, such that
L
3
L
1
L
2
(2.3)
is allowed when
I ∈ L
1
L
2
L
3
. To completely specify a trivalent vertex, a junction vector
must be chosen from the junction vector space
V
L
1
,
L
2
,
L
3
.
10
The collection of choices for all
trivalent vertices formed by all simple TDLs constitutes a gauge.
The fusion product of a simple TDL
L
with its orientation reversal contains the trivial
TDL,
L
L
=
I
+
···
,
(2.4)
because clearly
I
L
L
=
L
(2.5)
is allowed. Another important notion is
invertibility
. A TDL
L
is invertible if
L
L
=
I
, and
non-invertible otherwise. Invertible TDLs are equivalent to background gauge bundles for
finite symmetry groups [61, 10].
The splitting and joining of TDLs can be decomposed into basic
F
-moves that are charac-
terized by the
F
-symbols. In a given gauge, the
F
-symbols are
C
×
-numbers, and an
F
-move
is the equivalence between the two configurations
L
5
L
1
L
2
L
3
L
4
=
∑
L
6
(
F
L
1
,
L
2
,
L
3
L
4
)
L
5
,
L
6
L
6
L
3
L
2
L
1
L
4
.
(2.6)
The
F
-symbols must satisfy the pentagon identity, which can only have finitely many solu-
tions (up to gauge equivalence) for a given fusion ring due to Ocneanu rigidity [62, 8].
10
In the path integral language, a junction vector specifies the boundary conditions of quantum fields at
a trivalent vertex.
7
2.2 Local operators and commutative Frobenius algebra
Topological defect lines act on local operators by circling and shrinking. In conformity
with [12, 59], we adopt the clockwise convention for action on local operators,
O
L
=
̂
L
(
O
)
.
(2.7)
For instance, if
O
q
is a local operator with
Z
3
-charge
q
, and if
α
is the TDL corresponding
to the generator of
Z
3
, then
O
q
α
=
ω
q
O
q
.
(2.8)
The data of local operators is captured by a commutative Frobenius algebra [45, 46].
Commutativity guarantees that a projector basis exists:
{
π
a
, a
= 1
,...,n
V
|
π
a
π
b
=
δ
ab
π
a
}
,
(2.9)
where
n
V
denotes the number of vacua. In this basis, the nontrivial data is captured in the
overlap of the projectors with the identity,
i.e.
the one-point functions
〈
π
a
〉
. Most of this
note does not work in the projector basis, because for us it is more convenient to work in a
basis that simplifies the TDL actions as much as possible. However, the projector basis will
figure in the discussion of boundary states in Section 7.
2.3 Defect operators, defect operator algebra, and lassos
Associated to every topological defect line
L
is a defect Hilbert space
H
L
, which contains
states quantized on a spatial circle with twisted periodic boundary conditions. Via the
state-operator map,
|O〉
L
7→
O
(
x
)
L
(2.10)
H
L
is also the Hilbert space of point-like
defect operators
on which
L
can end. Defect Hilbert
spaces are equipped with a norm
H
L
⊗H
L
→
C
,
(2.11)
8
which defines a hermitian structure. The hermitian conjugate of
O
will be denoted by
O
.
The spectral data of a topological field theory extended by defects consists of the set of
local operators, their representations under the fusion ring, and the set of defect operators.
The dynamical data consists of the operator product
O
1
⊗O
2
∈H
L
1
⊗H
L
2
7→
L
3
L
1
O
1
L
2
O
2
∈H
L
3
(2.12)
and the lasso action
O
4
∈H
L
4
7→
L
1
L
2
L
3
L
4
O
4
∈H
L
1
.
(2.13)
When
L
1
=
L
4
=
I
and
L
2
=
L
3
, the above diagram becomes (2.7), and the lasso action
reduces to the TDL action
̂
L
2
on local operators that maps
H
to
H
. The lasso action is
a generalization that maps a defect Hilbert space
H
4
to another defect Hilbert space
H
1
.
In the following, for TDLs ending on defect operators, the labeling of the former will be
suppressed as it is implied by that of the latter.
The closest analog of charge conservation for a non-invertible TDL
L
is to circle a pair of
local operators by
L
, and impose the commutativity of (1) taking the local operator product
and (2) performing an
F
-move and studying the defect operator product, as illustrated
below:
“
charge
”
conservation
O
1
O
2
O
1
×O
2
O
1
L
′
O
2
∑
L
′
(
F
L
,
̄
L
,
̄
L
̄
L
)
I,
L
′
L
L
L
L
∑
O∈O
1
×O
2
̂
L
(
O
)
(2.14)
9
By the use of the norm, the operator product is equivalently encoded in the three-point
coefficients
c
(
O
1
,
O
2
,
O
3
) =
O
1
O
2
O
3
∈
C
,
(2.15)
and the lasso action is encoded in the lasso coefficients
O
1
L
2
L
3
O
4
∈
C
.
(2.16)
In the above, vacuum expectation values are implicitly taken. The three-point coefficients
are invariant under cyclic permutations
c
(
O
1
,
O
2
,
O
3
) =
c
(
O
2
,
O
3
,
O
1
) =
c
(
O
3
,
O
1
,
O
2
)
,
(2.17)
and complex conjugate under reflections
c
(
O
1
,
O
2
,
O
3
) =
c
(
O
1
,
O
3
,
O
2
)
∗
.
(2.18)
The lasso coefficients also enjoy the symmetries
O
1
L
2
L
3
O
4
=
O
4
L
3
L
2
O
1
=
O
1
L
2
L
3
O
4
∗
.
(2.19)
2.4 General observables, crossing symmetry and modular invari-
ance
A general observable in a topological field theory extended by defects is the vacuum ex-
pectation value of a graph—a configuration of topological defect lines with junctions and
endpoints—on a Riemann surface.
11
On the sphere, any graph can be expanded into a sum
of local operators, and taking the vacuum expectation value amounts to computing the over-
lap with the identity. The basic building blocks for this computation are the three-point
11
Each observable can be interpreted as a transition amplitude over some time function, with nontrivial
topology changes and defect dressing. See [10] for an exposition from this perspective.
10
and lasso coefficients introduced earlier, and the computation also involves basic manipula-
tions of TDLs such as
F
-moves. Observables on general Riemann surfaces can be reduced
to those on the sphere by a pair-of-pants decomposition. The equivalence of the various
ways of building the same observable on a general Riemann surface is guaranteed by the
four-point crossing symmetry and torus one-point modular invariance [12], generalizing the
situation without defects argued by Sonoda [63, 64] and by Moore and Seiberg [16, 19]. In
the following, all local and defect operators are taken to be canonically normalized under
the hermitian structure,
〈
O
L
O
〉
= 1
.
(2.20)
On the sphere, the four-point correlator of local and defect operators
O
i
∈ H
L
i
bridged
by an internal
L∈L
1
L
2
∩
L
4
L
3
can be decomposed into three-point coefficients by cutting
across
L
(with the cut shown by the dotted lines),
L
O
1
O
2
O
3
O
4
=
∑
O∈H
L
c
(
O
1
,
O
2
,
O
)
c
(
O
3
,
O
4
,
O
)
.
(2.21)
Under an
F
-move,
L
O
1
O
2
O
3
O
4
=
∑
L
′
L
′
O
3
O
2
O
1
O
4
(
F
L
1
,
L
2
,
L
3
L
4
)
L
,
L
′
,
(2.22)
where each graph appearing on the right can be decomposed into three-point coefficients by
cutting across
L
′
. Crossing symmetry is the equivalence of
∑
O∈H
L
c
(
O
1
,
O
2
,
O
)
c
(
O
3
,
O
4
,
O
) =
∑
L
′
(
F
L
1
,
L
2
,
L
3
L
4
)
L
,
L
′
∑
O
′
∈H
L
′
c
(
O
2
,
O
3
,
O
′
)
c
(
O
4
,
O
1
,
O
′
)
.
(2.23)
The modular invariance of the torus one-point function begins with performing
F
-moves
11
on the configuration
L
1
L
2
L
1
L
2
L
3
L
4
O
=
∑
L
′
∈L
O
L
1
(
F
L
4
,
L
O
,
L
1
L
2
)
L
3
,
L
′
L
1
L
2
L
1
L
2
L
′
L
4
O
=
∑
L
′
∈
L
2
L
O
(
F
L
1
,
L
2
,
L
O
L
3
)
L
4
,
L
′
L
1
L
2
L
1
L
2
L
′
L
3
O
(2.24)
and demanding the equivalence of the two cuts shown by the dotted lines:
∑
L
′
∈L
O
L
1
∑
O
1
∈H
L
1
∑
O
′
∈H
L
′
(
F
L
4
,
L
O
,
L
1
L
2
)
L
3
,
L
′
c
(
O
,
O
1
,
O
′
)
O
1
L
2
L
4
O
′
=
∑
L
′
∈
L
2
L
O
∑
O
2
∈H
L
2
∑
O
′
∈H
L
′
(
F
L
1
,
L
2
,
L
O
L
3
)
L
4
,
L
′
c
(
O
,
O
2
,
O
′
)
O
2
L
1
L
3
O
′
.
(2.25)
3 Spectral constraints by Haagerup symmetry
This section studies the modular constraints on the spectral data—the set of local opera-
tors, their representations under the fusion ring, and the set of defect operators—when the
theory is known to contain topological defect lines (TDLs) realizing the Haagerup
H
3
fusion
category.
12
3.1 The Haagerup fusion ring with six simple objects
The Haagerup
H
3
fusion category was constructed by Grossman and Snyder [27] as a variant
(Grothendieck equivalent) of the
H
2
fusion category that directly came from the Haagerup
subfactor [23, 24]. There are six simple objects/TDLs, which we denote by
I
, α, α
2
, ρ, αρ, α
2
ρ.
(3.1)
The fusion ring is fully specified by the relations
α
3
= 1
, αρ
=
ρα
2
, ρ
2
=
I
+
Z
,
Z ≡
2
∑
i
=0
α
i
ρ.
(3.2)
For shorthand,
ρ
i
≡
α
i
ρ.
(3.3)
In the rest of this note, we use unoriented solid lines to denote the non-invertible self-dual
simple TDLs
ρ
i
, and oriented dashed lines to denote the invertible ones:
=
α
,
=
̄
α
,
ρ
i
.
(3.4)
There are two gauge-inequivalent unitary fusion categories realizing the above fusion ring,
denoted
H
2
and
H
3
by Grossman and Snyder [27]. Whereas the Haagerup
H
2
fusion category
descended directly from the Haagerup subfactor [23, 24], the Haagerup
H
3
fusion category
was constructed by Grossman and Snyder [27] based on
H
2
. It turns out to be easier to
work with
H
3
, but the analysis in this section applies to both
H
2
and
H
3
. The
F
-symbols
for
H
3
were implicit in the work of Grossman and Snyder [27] (using a generalization of the
approach by Izumi [25] for
H
2
), and also explicitly obtained by Titsworth [57], Osborne,
Stiegemann, and Wolf [58]. In [59], the present authors recast the
F
-symbols in a gauge that
manifests the transparent property, a notion we introduce in Section 4. The transparent
F
-symbols are given in Appendix A.
3.2 Action on local operators and representation theory
To describe how topological defect lines forming the Haagerup
H
3
fusion category act on
local operators, we should first study the complex representation theory of its fusion ring.
Since the fusion ring is non-commutative, the action of TDLs cannot be simultaneously
diagonalized. We work in a basis in which the action of
Z
3
is diagonal.
13
•
For a state
|
φ
〉
neutral under
Z
3
,
ρ
|
φ
〉
=
αρ
|
φ
〉
=
α
2
ρ
|
φ
〉
,
Z|
φ
〉
= 3
ρ
|
φ
〉
,
(3.5)
hence there are two one-dimensional representations,
ρ
|
φ
〉
=
3
±
√
13
2
|
φ
〉
.
(3.6)
•
For a state
|
φ
〉
with unit
Z
3
-charge,
α
|
φ
〉
=
ω
|
φ
〉
, αρ
|
φ
〉
=
ρα
2
|
φ
〉
=
ω
2
ρ
|
φ
〉
, α
2
ρ
|
φ
〉
=
ρα
|
φ
〉
=
ωρ
|
φ
〉
.
(3.7)
It follows that
Z|
φ
〉
= 0, and hence
ρ
2
|
φ
〉
=
|
φ
〉
.
(3.8)
If
ρ
|
φ
〉
and
|
φ
〉
were equal up to a phase, then there would be two possible one-
dimensional representations with
ρ
|
φ
〉
=
±|
φ
〉
,
(3.9)
which is in conflict with
αρ
=
ρα
2
. Hence
ρ
|
φ
〉
and
|
φ
〉
must be independent, and the
representation is two-dimensional. In the (
|
φ
〉
, ρ
|
φ
〉
) basis,
α
=
(
ω
0
0
ω
2
)
, ρ
=
(
0 1
1 0
)
.
(3.10)
The above classification of irreducible representations is summarized in Table 1. In a
reflection-positive quantum field theory, the identity operator transforms in a one-dimensional
representation with positive charges. Here, under the reflection-positive assumption, the
identity operator must transform in the + representation.
r
α
ρ
+
1
3+
√
13
2
−
1
3
−
√
13
2
2
(
ω
0
0
ω
2
) (
0 1
1 0
)
Table 1: Irreducible representations of the Haagerup fusion ring with six simple ob-
jects/TDLs.
14
3.3 Modular invariance and vacuum degeneracy
Let
n
V
denote the number of vacua (local operators), and
n
±
and
n
2
be their multiplicities
of representations (in the notation of Table 1). Clearly,
n
V
=
n
+
+
n
−
+ 2
n
2
.
Consider the modular invariance of the torus partition function with the non-invertible
TDL
ρ
wrapped around a one-cycle
ρ
(3.11)
The horizontal cut computes the trace over the action of
̂
ρ
in the Hilbert space
H
of local
operators, and the vertical cut simply counts the dimensionality of the defect Hilbert space
H
ρ
. Modular invariance requires
Tr
H
̂
ρ
= Tr
H
ρ
1
∈
Z
≥
0
.
(3.12)
Given the representation content of the Haagerup fusion ring, summarized in Table 1, we
immediately conclude that
n
+
=
n
−
, and the number of vacua must be even. Let us write
n
1
≡
n
+
=
n
−
(3.13)
to denote the multiplicity of each one-dimensional representation. Using the U(
n
2
) freedom,
we can choose a basis of local operators to represent
̂
ρ
in block diagonal form
̂
ρ
=
n
+
⊕
p
=1
(
3 +
√
13
2
)
⊕
n
−
⊕
q
=1
(
3
−
√
13
2
)
⊕
(
0 1
1 0
)
⊕···⊕
(
0 1
1 0
)
.
(3.14)
Modular invariance (3.12) also implies that the defect Hilbert space
H
ρ
is 3
n
1
-dimensional,
i.e.
the TDL
ρ
can end on
n
ρ
= 3
n
1
(3.15)
independent defect operators. And similarly for each of the other
ρ
i
.
Consider the modular invariance of the torus partition function with the invertible TDL
15
n
V
n
1
=
n
+
=
n
−
n
2
n
α
=
n
̄
α
n
ρ
=
n
αρ
=
n
α
2
ρ
n
P
2
1
0
2
3
15
4
1
1
1
3
15
4
2
0
4
6
30
6
1
2
0
3
15
6
2
1
3
6
30
6
3
0
6
9
45
Table 2: Possible numbers of point-like operators that satisfy the torus one-point modular
invariance (3.11) and (3.16). Here
n
V
denotes the total number of vacua (local operators),
comprised of
n
r
copies of representation
r
, where
r
= +
,
−
,
2
;
n
L
denotes the number of
defect operators in
each
L
, for
L
=
α,
̄
α, ρ, αρ, α
2
ρ
;
n
P
denotes the total number of point-
like (local and defect) operators. Only the highlighted cases with
n
1
= 1
, n
ρ
= 3
, n
P
= 15
are considered in this note.
α
wrapped around a one-cycle
α
(3.16)
Modular invariance requires
Tr
H
̂
α
= Tr
H
α
1
∈
Z
≥
0
.
(3.17)
Hence the
α
TDL hosts
n
α
= 2
n
1
−
n
2
(3.18)
defect operators. The total number of point-like operators is
n
P
≡
g
+ 2
n
α
+ 3
n
ρ
= (2
n
1
+ 2
n
2
) + 2(2
n
1
−
n
2
) + 9
n
1
= 15
n
1
.
(3.19)
The first few possibilities are listed in Table 2 in the order of increasing
n
V
. Whenever
n
2
= 0, the
Z
3
symmetry is not faithfully realized on the vacua. In the following, we consider
the three minimal cases totaling
n
P
= 15 point-like operators, highlighted in Table 2; each
case has
n
1
= 1 and
n
ρ
= 3. Eventually we will succeed in constructing a TFT realizing
n
V
= 6, but along the way we also derive various constraints on
n
V
= 2
,
4.
16
4 Transparency and
Z
3
symmetry
This note works in a gauge of the
H
3
fusion category that manifests its “transparent” prop-
erty [59]—the associator involving any invertible topological defect line (TDL) is the identity
morphism. In terms of the
F
-symbols, it means that every
F
-symbol with an external in-
vertible TDL takes value one. Hence invertible TDLs can be attached or detached “freely”,
changing the isomorphism classes of other involved TDLs but without generating extra
F
-
symbols. Several diagrammatic identities are illustrated below:
(
a
)
ρ
i
ρ
i
+1
ρ
i
=
ρ
i
(
b
)
ρ
i
ρ
i
+1
ρ
i
=
ρ
i
(
c
)
ρ
i
ρ
i
+1
ρ
i
+2
=
ρ
i
ρ
i
+2
(
d
)
ρ
i
ρ
k
ρ
k
+1
ρ
j
ρ
j
+1
=
ρ
i
ρ
k
+1
ρ
j
+1
(
e
)
ρ
i
ρ
i
+1
=
ρ
i
ρ
i
−
1
ρ
i
+1
=
ρ
i
ρ
i
−
1
ρ
i
+1
(
f
)
ρ
i
ρ
i
=
ρ
i
ρ
i
+1
ρ
i
=
ρ
i
ρ
i
−
1
ρ
i
(4.1)
Importantly, the four-way junctions in (
e
) and (
f
) are unambiguously defined.
In [59], transparency and the
Z
3
symmetry were exploited to reduce the pentagon identity
so that the
F
-symbols could be efficiently solved. Below, in attempting to construct a
topological field theory, the utilization of the
Z
3
symmetry is also essential in reducing the
amount of independent data.
4.1
Z
3
relations for lassos and dumbbells
Let
O
q
be a local operator with
Z
3
-charge
q
∈{
0
,
±
1
}
, and consider the lasso
ρ
i
O
q
ρ
j
.
(4.2)
17
The
Z
3
symmetry relates lassos with different triples (
q,i,j
) as follows: replace
O
q
using the
equalities
O
q
=
ω
q
O
q
=
ω
−
q
O
q
(4.3)
and fuse the
Z
3
symmetry line with
ρ
i
(apply (4.1)(
b
) and then (
d
)) to obtain the relations
ρ
i
O
q
ρ
j
=
ω
q
ρ
i
−
1
O
q
ρ
j
=
ω
−
q
ρ
i
+1
O
q
ρ
j
.
(4.4)
Next consider the dumbbell
ρ
i
ρ
k
ρ
j
,
(4.5)
where each empty dot denotes an arbitrary local operator insertion. The
Z
3
action on the
dumbbell (circling it with a clockwise
Z
3
loop) gives (see (4.1)(
e
) for the meaning of the
four-way junction)
ρ
i
ρ
k
ρ
j
=
ρ
i
ρ
k
ρ
j
=
ρ
i
−
1
ρ
k
−
1
ρ
j
+1
(4.6)
Combining (4.4) and (4.6), we obtain an identity that leaves the side loops intact and only
changes the handle
ρ
i
ρ
k
O
q
1
O
q
2
ρ
j
=
ω
−
q
1
−
q
2
ρ
i
ρ
k
O
q
1
O
q
2
ρ
j
+1
,
(4.7)
which will prove useful in Section 5.3. A mnemonic is that the
Z
3
symmetry line measures
the
opposite
Z
3
-charge of the local operators
O
q
1
and
O
q
2
placed inside a dumbbell, because
the
Z
3
symmetry line changes orientation when it crosses a
ρ
i
TDL, as illustrated in (4.1)(
f
).
18
4.2
Z
3
action on defect operators
Recall that each
ρ
i
TDL hosts three independent defect operators. We work in an orthonor-
mal basis and denote them by
o
iA
, i
= 0
,
1
,
2
, A
= 1
,
2
,
3
,
with
〈
o
iA
o
jB
〉
=
δ
ij
δ
AB
.
(4.8)
Note that there is still an O(3)
3
basis freedom. The
Z
3
action on a defect operator
o
i
∈H
ρ
i
is defined by the lasso (see (4.1)(
e
) for the meaning of the four-way junction)
ρ
i
+1
o
iA
=
|
o
iA
〉
ρ
i
ρ
i
+1
=
|
o
iA
〉
ρ
i
ρ
i
+1
,
(4.9)
where in the last diagram, the left and right edges of the square are identified to represent a
cylinder. Performing the
Z
3
action three times on
H
ρ
i
becomes a trivial action, as illustrated
by the sequence of
F
-moves
=
=
.
(4.10)
We make use of the O(3)
2
⊂
O(3)
3
basis freedom such that the lasso (4.9) representing the
Z
3
action takes
Z
3
:
o
1
A
→
o
2
A
→
o
3
A
→
o
1
A
.
(4.11)
The
Z
3
action also gives rise to relations among the dynamical data. For instance,
consider the
Z
3
action on the operator product of
o
iA
and
o
iB
o
iB
o
iA
=
o
iA
o
iB
.
(4.12)
If the vacuum expectation value is taken, possibly in the presence of other local operators,
the
Z
3
symmetry line can be deformed to shrink in some other patch while picking up the
19
Z
3
-charges of other local operators. This process gives rise to identities among correlators.
Similarly, the
Z
3
action
o
iA
o
jB
o
kC
=
o
iA
o
jB
o
kC
=
o
iA
o
jB
o
kC
(4.13)
implies identities among different three-point coefficients, when the sphere vacuum expecta-
tion value is taken.
We can nucleate
Z
3
loops inside or outside a lasso to change the species of the
ρ
i
TDLs,
resulting in the relations
ρ
i
O
q
o
jB
=
ρ
i
−
1
O
q
o
j
+1
,B
=
ω
q
ρ
i
−
1
O
q
o
jB
,
ρ
k
ρ
`
o
jB
o
iA
=
ρ
k
−
1
ρ
`
−
1
o
j
+1
,B
o
iA
=
ρ
k
−
1
ρ
`
−
1
o
jB
o
i
+1
,A
.
(4.14)
5 Bootstrap constraints
Given the spectral constraints derived in Section 3, our goal now is to solve for a minimal
defect topological field theory (TFT) with a total of
n
P
= 15 point-like operators, and the
number of vacua (local operators) can be
n
V
= 2
,
4
,
6. For each case, there is one nontrivial
Z
3
-neutral local operator
v
and three defect operators
o
iA
on each
ρ
i
line. The remaining
four point-like operators can be two pairs of
Z
3
-charged local operators
u
a
,
̄
u
a
, two pairs of
Z
3
defect operators
w
a
∈H
α
,
̄
w
a
∈H
α
2
, or a pair of each.
In this section, we delineate constraints of crossing symmetry and modular invariance
that were formulated in generality in Section 2.4. For simplicity, we ignore the constraints
involving
Z
3
defect operators
w
a
,
̄
w
a
, and only consider the part of crossing symmetry that
20
is equivalent to the associativity involving at least one local operator. More general crossing
symmetry is deferred to Appendix B.
We reserve the
i
= 0
,
1
,
2 index for the species of the
ρ
i
line, the
A
= 1
,
2
,
3 index for the
species of the defect operators of each
ρ
i
line, and the
a
= 1
,...,n
2
index for the species of
Z
3
-charged local operators. Note that the
Z
3
-charged operators
u
a
,
̄
u
a
have a U(
n
2
) basis
freedom.
5.1 Local operator algebra and associativity
The most general local operator algebra consistent with the
Z
3
symmetry is
v
×
v
= 1 +
βv , v
×
u
a
=
∑
b
ξ
ab
u
b
,
u
a
×
̄
u
b
=
δ
ab
+
ξ
ab
v , u
a
×
u
b
=
∑
c
σ
abc
̄
u
c
.
(5.1)
The following are the constraints from associativity.
•
u
a
u
b
u
c
σ
abc
=
σ
bca
,
∑
d
σ
abd
ξ
ed
=
∑
d
σ
bcd
ξ
ad
,
∑
e
σ
abe
σ
cde
=
∑
e
σ
ade
σ
bce
=
∑
e
σ
ace
σ
bde
.
(5.2)
Hence
σ
abc
is totally symmetric.
•
u
a
̄
u
b
v
ξ
ab
=
̄
ξ
ba
, δ
ab
+
βξ
ab
=
∑
c
ξ
ac
̄
ξ
bc
=
∑
c
̄
ξ
bc
ξ
ac
,
(5.3)
The first condition says that
ξ
ab
is Hermitian, which allows us to use the U(
n
2
) basis
freedom to diagonalize
ξ
ab
. Then the second condition, which also encompasses the
associativity of
u
a
vv
, is solved by
ξ
ab
=
ξ
a
δ
ab
, ξ
a
=
β
±
√
β
2
+ 4
2
.
(5.4)
•
u
a
u
b
v
∑
c
σ
abc
̄
ξ
cd
=
∑
c
ξ
bc
σ
acd
=
∑
c
ξ
ac
σ
bcd
.
(5.5)
21
•
u
a
u
b
̄
u
c
∑
d
σ
abd
̄
σ
dce
=
δ
bc
δ
ae
+
ξ
bc
ξ
ae
.
(5.6)
In the special case of
a
=
e
and
b
=
c
,
∑
d
σ
abd
̄
σ
dba
= 1 +
ξ
a
ξ
b
.
(5.7)
5.2 Mixed local and
ρ
defect operators
The most general operator algebra involving mixed local and
ρ
defect operators is
o
iA
ρ
i
o
iB
=
δ
AB
+
κ
i
AB
v
+
∑
a
(
̄
λ
i
AB
;
a
u
a
+
λ
i
AB
;
a
̄
u
a
)
,
ρ
i
o
iA
v
=
∑
B
κ
i
AB
ρ
i
o
iB
,
ρ
i
o
iA
u
a
=
∑
B
λ
i
AB
;
a
ρ
i
o
iB
,
(5.8)
where
κ
i
AB
and
λ
i
AB
;
a
are both symmetric in
A,B
, and the
Z
3
action (4.13) implies that
κ
i
+1
AB
=
κ
i
AB
, λ
i
+1
AB
;
a
=
ω
−
1
λ
i
AB
;
a
.
(5.9)
The following are the constraints from associativity.
•
o
iA
o
iB
v
o
iA
ρ
i
o
iB
v
=
δ
AB
v
+
κ
i
AB
(1 +
βv
) +
∑
a,b
(
̄
λ
i
AB
;
b
ξ
ba
u
a
+
λ
i
AB
;
b
̄
ξ
ba
̄
u
a
)
=
κ
i
AB
+
(
∑
C
κ
i
AC
κ
i
BC
)
v
+
∑
C
κ
i
AC
∑
a
(
̄
λ
i
BC
;
a
u
a
+
λ
i
BC
;
a
̄
u
a
)
.
(5.10)
Hence,
∑
C
κ
i
AC
κ
i
BC
=
δ
AB
+
βκ
i
AB
,
(5.11)
which also encompasses the associativity of
o
iA
vv
, and
∑
C
κ
i
AC
λ
i
BC
;
a
=
∑
b
λ
i
AB
;
b
̄
ξ
ba
.
(5.12)
22
•
o
iA
o
iB
u
a
o
iA
ρ
i
o
iB
u
a
=
δ
AB
u
a
+
∑
b
κ
i
AB
ξ
ab
u
b
+
∑
b,c
̄
λ
i
AB
;
b
σ
abc
̄
u
c
+
λ
i
AB
;
a
+
∑
b
λ
i
AB
;
b
ξ
ab
v
=
λ
i
AB
;
a
+
(
∑
C
λ
i
AC
;
a
κ
i
BC
)
v
+
∑
C
λ
i
AC
;
a
∑
b
(
̄
λ
i
BC
;
b
u
b
+
λ
i
BC
;
b
̄
u
b
)
.
(5.13)
Hence,
∑
C
λ
i
AC
;
a
̄
λ
i
BC
;
b
=
δ
AB
δ
ab
+
κ
i
AB
ξ
ab
,
∑
C
λ
i
AC
;
a
λ
i
BC
;
b
=
∑
c
̄
λ
i
AB
;
c
σ
abc
.
(5.14)
5.3
ρ
action on local operators
Let us study the analog of charge conservation (2.14) for the non-invertible TDLs
ρ
i
. We
will constrain the
ρ
i
action on local operators,
ρ
i
1
=
ζ ,
ρ
i
v
=
−
ζ
−
1
v ,
ρ
i
u
a
=
ω
i
∑
b
R
ab
̄
u
b
,
(5.15)
and the lassos on local operators,
ε
i
A
≡
ρ
i
v
o
iA
, γ
i
aA
≡
ρ
i
u
a
o
iA
.
(5.16)
The
Z
3
action relations (4.14) imply that
ε
i
+1
A
=
ω
−
i
ε
i
A
, γ
i
+1
aA
=
ω
−
i
γ
i
aA
.
(5.17)
Note that in writing (5.4), we already used the U(
n
2
) freedom to diagonalize the operator
product
u
a
̄
u
b
, so we can no longer use it to simplify
R
ab
. In the following, we make frequent
use of the explicit values of the
F
-symbols
(
F
ρ
i
,ρ
i
,ρ
i
ρ
i
)
I
,
I
=
ζ
−
1
,
(
F
ρ
i
,ρ
i
,ρ
i
ρ
i
)
I
,ρ
j
=
ζ
−
1
, ζ
≡
3 +
√
13
2
(5.18)
from (A.7).
23
First, let us revisit the requirement that
u
a
transforms as a representation of the fusion
ring.
12
The consideration of
ρ
ρ
u
a
=
u
a
+
∑
i
ρ
i
u
a
(5.19)
leads to a constraint
∑
c
R
ac
R
cb
=
δ
ab
+
∑
i
ω
i
R
ab
=
δ
ab
,
(5.20)
where the left side comes from shrinking the inner and outer
ρ
loops consecutively, and the
right side from fusing them before shrinking.
13
Now, following the
↓
direction in (2.14), we circle
ρ
i
on the operator product of local
operators, and apply the
F
-move to obtain
ρ
i
O
q
1
O
q
2
I
=
∑
j
′
ρ
i
ρ
i
O
q
1
O
q
2
ρ
j
′
.
(5.21)
Using the
Z
3
action (4.7), we can simplify the sum of dumbbells to
3
ρ
i
ρ
i
O
q
1
O
q
2
ρ
j
∣
∣
∣
∣
∣
∣
∣
Z
3
-charge
−
(
q
1
+
q
2
)
,
(5.22)
where
j
is arbitrary. We might as well set
j
=
i
. In the following, we equate the above to
the
→↓
direction of (2.14) where the local operator product is taken first.
•
v
×
v
= 1 +
βv
ζ
−
3
(1 +
βv
) + 3
ζ
−
1
ρ
i
ρ
i
v
v
ρ
i
∣
∣
∣
∣
∣
∣
∣
Z
3
-neutral
=
ζ
−
ζ
−
1
βv .
(5.23)
12
The representation given in (3.10) was specialized to a particular basis for
u
a
. Here we prioritize the
use of the U(
n
2
) basis freedom to diagonalize
ξ
ab
in (5.4), so the requirement that
u
a
transforms as a
representation needs to be rewritten in a basis-independent fashion.
13
The fusion of the two
ρ
TDLs can be understood as an
F
-move followed by the shrinking of the
ρ
loop.
24
Hence,
∑
A
(
ε
i
A
)
2
=
√
13
,
∑
A,B
ε
i
A
κ
i
AB
ε
i
B
=
−
√
13
3
ζ
−
1
β .
(5.24)
•
u
a
×
̄
u
b
=
δ
ab
+
ξ
ab
v
ζ
−
1
(
δ
ab
+
∑
c,d
R
ac
R
bd
ξ
dc
v
)
+ 3
ζ
−
1
ρ
i
ρ
i
u
a
̄
u
b
ρ
i
∣
∣
∣
∣
∣
∣
∣
Z
3
-neutral
=
ζδ
ab
−
ζ
−
1
ξ
ab
v .
(5.25)
Hence,
∑
A
γ
i
aA
̄
γ
i
bA
=
ζδ
ab
,
(5.26)
∑
A,B
γ
i
aA
̄
γ
i
bB
κ
i
AB
=
−
1
3
(
ξ
ab
+
∑
c,d
R
ac
R
bd
ξ
dc
)
.
(5.27)
•
u
a
×
u
b
=
∑
c
σ
abc
̄
u
c
ζ
−
1
ω
−
i
∑
d,e,f
R
ad
R
be
̄
σ
def
u
f
+ 3
ζ
−
1
ρ
i
ρ
i
u
a
u
b
ρ
i
∣
∣
∣
∣
∣
∣
∣
Z
3
-charge 1
=
ω
−
i
∑
c,f
σ
abc
R
cf
u
f
.
(5.28)
Hence,
∑
A,B
γ
i
aA
γ
i
bB
̄
λ
i
AB
;
f
=
1
3
ω
−
i
(
ζ
∑
c
σ
abc
R
cf
−
∑
d,e
R
ad
ρ
be
̄
σ
def
)
.
(5.29)
•
u
a
×
v
=
∑
b
ξ
ab
u
b
−
ζ
−
2
ω
i
∑
b,c
R
ab
̄
ξ
bc
̄
u
c
+ 3
ζ
−
1
ρ
i
ρ
i
u
a
v
ρ
i
∣
∣
∣
∣
∣
∣
∣
Z
3
-charge
−
1
=
ω
i
∑
b,c
ξ
ab
R
bc
̄
u
c
.
(5.30)
Hence,
∑
A
γ
i
aA
λ
i
AB
;
c
ε
i
B
=
1
3
ω
i
∑
b
(
ζξ
ab
R
bc
+
ζ
−
1
R
ab
̄
ξ
bc
)
.
(5.31)
25
5.4 Torus one-point modular invariance
Consider the torus one-point modular invariance (2.25) in the special case of
L
2
=
L
O
=
I
,
L
3
=
L
4
=
L
1
.
(5.32)
•
v
with
Z
3
symmetry line
v
(5.33)
Let us denote the three-point coefficient of
v
with
Z
3
defect operators by
̃
ξ
a
=
c
(
v,w
a
,
̄
w
a
)
.
(5.34)
Let us write down
vertical cut = horizontal cut
(5.35)
for different numbers of vacua.
(a)
n
V
= 6
0 =
c
(
v,v,v
) +
ω
∑
a
c
(
v,u
a
,
̄
u
a
) +
ω
2
∑
a
c
(
v,
̄
u
a
,u
a
)
=
c
(
v,v,v
)
−
∑
a
c
(
v,u
a
,
̄
u
a
)
=
β
−
ξ
1
−
ξ
2
.
(5.36)
(b)
n
V
= 4
̃
ξ
=
β
−
ξ .
(5.37)
(c)
n
V
= 2
̃
ξ
1
+
̃
ξ
2
=
β .
(5.38)
26