Topological field theory with Haagerup symmetry

J. Math. Phys. 

63

, 042306 (2022); 

https://doi.org/10.1063/5.0079062

63

, 042306

© 2022 Author(s).

Topological field theory with Haagerup

symmetry

Cite as: J. Math. Phys. 

63

, 042306 (2022); 

https://doi.org/10.1063/5.0079062

Submitted: 17 November 2021 • Accepted: 27 March 2022 • Published Online: 21 April 2022

Tzu-Chen Huang

and

Ying-Hsuan Lin

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Topological field theory with Haagerup symmetry

Cite as: J. Math. Phys.

, 042306 (2022); doi: 10.1063/5.0079062

Submitted: 17 November 2021

•

Accepted: 27 March 2022

•

Published Online: 21 April 2022

Tzu-Chen Huang

1, a)

and Ying-Hsuan Lin

2, b)

AFFILIATIONS

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, USA

jimmy@caltech.edu

Author to whom correspondence should be addressed:

yhlin@fas.harvard.edu.

Also at:

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA.

ABSTRACT

We construct a (1

topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup

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 this particular TFT with

six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup

and

fusion categories

can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual

conformity.

Published under an exclusive license by AIP Publishing.

https://doi.org/10.1063/5.0079062

I. INTRODUCTION

The best cultivated terrains in the landscape of (1

conformal field theories (CFTs) are rational conformal field theories (RCFTs),

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. However, 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,

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

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, such as 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

Turaev–Viro theory

or Levin–Wen string-net model

constructed out of a fusion category

is a bulk phase whose anyons

are described by the Drinfeld center

(

)

. By placing the bulk phase on a slab between a gapped boundary and another boundary condition

and further compactifying on a circle, the resulting theory

[

]

would be a CFT with TDLs described by

is thus interpreted as the

existence/classification problem of boundary conditions for the bulk phase.

From a purely (1

perspective, statistical height models that

take

(and the choice of a distinguished object) as the microscopic input have recently been shown by Aasen, Fendley, and Mong

to host

macroscopic TDLs described by

. One can explore the phases of such models in search of a CFT.

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A closely related question is as follows:

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 Verlinde lines (TDLs commuting with the VOA)

is isomorphic to the fusion ring of the VOA representations. The correspondence between MTC and (1

RCFT traces its origin to a

seminal series of papers by Moore and Seiberg

17–21

and is conjectured to be one-to-one although a construction or proof is lacking. Note that

an affirmative answer to

implies an affirmative answer to

: Given a fusion category

, if one can find a VOA that realizes the Drinfeld

center

(

)

, then gauging the diagonal RCFT by an algebra object gives a non-diagonal RCFT whose TDLs realize

⊠

, where

is the

opposite

category of

, physically corresponding to orientation-reversing all TDLs.

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

and Popa

classified subfactors with Jones indices less than or equal to 4, Asaeda and Haagerup

constructed

one—the Haagerup subfactor—with Jones index

√

, the smallest above 4.

As the title of Ref. 25 suggests, the Haagerup subfactor was

deemed

exotic

since its construction at the time did not fit into any infinite family. Later work by Izumi,

Evans, and Gannon

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 denoted by

. Most of this article con-

cerns the Haagerup

fusion category, which did not descend directly from the Haagerup subfactor of Refs. 25 and 26, but was instead

constructed by Grossman and Snyder.

Because the fusion ring (reviewed in Sec. III A) is non-commutative, the Haagerup

fusion cat-

egory cannot possibly be realized by Verlinde lines

30–33

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 the Drinfeld center

of Haagerup. In fact, Evans and Gannon

constructed

8 characters for the Haagerup modular data and used it to surmise possible con-

structions of the VOA through the Goddard–Kent–Olive coset construction

and its generalizations

35–37

or through the generalized orbifold

construction (gauging an algebra object) of Carqueville, Fröhlich, Fuchs, Runkel, and Schweigert

38–40

(see Ref. 10 for a recent discussion).

Recently, Wolf

considered the Haagerup anyon chain and numerically searched for CFT phases, but with inconclusive results.

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

quantum field theory, described by (1

topological field theories (TFTs) extended by

defects,

43,44

a related but simpler question can be asked.

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

argued that

-symmetric defect TFTs are in bijection with

-module categories, and since the regular module

category always exists,

has been affirmatively answered. However, their construction utilizes the Turaev–Viro state-sum

or Levin–Wen

string-net

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

and by Tachikawa

and for categories with fiber

functors (the resulting TFT has a unique vacuum) by Thorngren and Wang.

In Ref. 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

et al.

but the full defect TFT data remains unsolved.

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 renormalization group (RG) flows (if a TDL-preserving relevant operator exists), and this principle strongly

constrains the infrared fate of CFTs. In the space of TDL-preserving RG flows and without fine-tuning, they must flow either to gapped phases

described by TFTs or to “dead-end” CFTs,

which correspond to gapless phases protected by fusion categories.

46,51

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 article 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

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

-symbols. They were implicit in the work of Grossman and Snyder

(using a generalization of the approach by Izumi

) and also

explicitly obtained by Titsworth

and Osborne, Stiegemann, and Wolf.

In Ref. 54, the present authors recasted the

-symbols in a gauge

that manifests the transparent property, which greatly simplify our present computational endeavor.

Sections II-IX are organized into steps of the construction and discussions of further ramifications. Section II reviews the generalities of

topological field theory extended by defects and formulates the defining data and consistency conditions. Section III introduces the Haagerup

fusion ring with six simple objects/TDLs, studies its representation theory, and constrains the vacuum degeneracy using modular invariance.

Section IV studies the relations among dynamical data implied by transparency and

symmetry. Section V delineates the constraints of

associativity and torus one-point modular invariance. Section VI solves the constraints to construct a topological field theory with Haagerup

symmetry. Section VII examines the expectation that the boundary conditions furnish a non-negative integer matrix representation (NIM-

rep) of the fusion ring. Section VIII discusses the relations among topological field theories by gauging algebra objects. Section IX ends with

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some prospective questions. Appendix A contains the

-symbols for the Haagerup

fusion category. Appendix B analyzes the general

crossing symmetry of defect operators.

II. 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.

A. 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 Refs. 8 and 9, and expositions in the physics context can be found in Refs. 10 and 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 transformations leave physical observables invariant. We

restrict ourselves to considering sets of TDLs with finitely many

simple

TDLs

{

}

; the others, the

non-simple

TDLs, are direct sums of the

simple ones.

Among the simple TDLs, there is a trivial TDL

representing nothingness. Furthermore, every TDL

has an orientation

reversal

, as depicted by the equivalence

(1)

Whenever a TDL is isomorphic to its own orientation reversal,

, we omit the arrows on the lines.

A general configuration of TDLs involves junctions built out of trivalent vertices. The allowed trivalent vertices are specified by the fusion

ring

(2)

where

∈

≥

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.

In conformity with Refs. 12 and 54, we adopt

the counter-clockwise convention for trivalent vertices such that

(3)

is allowed when

∈

. To completely specify a trivalent vertex, a junction vector must be chosen from the junction vector space

The collection of choices for all trivalent vertices formed by all simple TDLs constitutes a gauge.

The fusion product of a simple TDL

with its orientation reversal contains the trivial TDL,

+ ⋅ ⋅ ⋅

(4)

because clearly

(5)

is allowed. Another important notion is

invertibility

. A TDL

is invertible if

and non-invertible otherwise. Invertible TDLs are

equivalent to background gauge bundles for finite symmetry groups.

10,59

The splitting and joining of TDLs can be decomposed into basic

-moves that are characterized by the

-symbols. In a given gauge, the

-symbols are

-numbers, and an

-move is the equivalence between the two configurations

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(6)

The

-symbols must satisfy the pentagon identity, which can only have finitely many solutions (up to gauge equivalence) for a given fusion

ring due to Ocneanu rigidity.

8,60

B. Local operators and commutative Frobenius algebra

Topological defect lines act on local operators by circling and shrinking. In conformity with Refs. 12 and 54, we adopt the clockwise

convention for action on local operators,

(7)

For instance, if

is a local operator with

-charge

and if

is the TDL corresponding to the generator of

, then

(8)

The data of local operators are captured by a commutative Frobenius algebra.

61,62

Commutativity guarantees that a projector basis exists,

{

. . .

∣

}

(9)

where

denotes the number of vacua. In this basis, the nontrivial data are captured in the overlap of the projectors with the identity, i.e., the

one-point functions

⟨

⟩

. Most of this article 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 Sec. VII.

C. Defect operators, defect operator algebra, and lassos

Associated with every topological defect line

is a defect Hilbert space

, which contains states quantized on a spatial circle with

twisted periodic boundary conditions. Through the state-operator map,

(10)

is also the Hilbert space of point-like

defect operators

on which

can end. Defect Hilbert spaces are equipped with a norm

⊗

→

(11)

which defines a Hermitian structure. The Hermitian conjugate of

will be denoted by

The spectral data of a topological field theory extended by defects consist of the set of local operators, their representations under the

fusion ring, and the set of defect operators. The dynamical data consist of the operator product

(12)

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and the lasso action

(13)

When

and

, the above diagram becomes (7), and the lasso action reduces to the TDL action

on local operators that

maps

. The lasso action is a generalization that maps a defect Hilbert space

to another defect Hilbert space

. 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

is to circle a pair of local operators by

and impose the com-

mutativity of (1) taking the local operator product and (2) performing an

-move and studying the defect operator product, as illustrated as

follows:

(14)

By the use of the norm, the operator product is equivalently encoded in the three-point coefficients

(15)

and the lasso action is encoded in the lasso coefficients

(16)

In the above, vacuum expectation values are implicitly taken. The three-point coefficients are invariant under cyclic permutations

(

)

(

)

(

)

(17)

and complex conjugate under reflections

(

)

(

)

∗

(18)

The lasso coefficients also enjoy the symmetries

(19)

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D. General observables, crossing symmetry, and modular invariance

A general observable in a topological field theory extended by defects is the vacuum expectation value of a graph—a configuration of

topological defect lines with junctions and endpoints—on a Riemann surface.

On the sphere, any graph can be expanded into a sum of

local operators, and taking the vacuum expectation value amounts to computing the overlap with the identity. The basic building blocks

for this computation are the three-point and lasso coefficients introduced earlier, and the computation also involves basic manipulations of

TDLs, such as

-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,

generalizing the situation without defects argued by Sonoda

64,65

and by Moore and

Seiberg.

17,20

In the following, all local and defect operators are taken to be canonically normalized under the Hermitian structure,

(20)

On the sphere, the four-point correlator of local and defect operators

∈

bridged by an internal

∈

∩

can be

decomposed into three-point coefficients by cutting across

(with the cut shown by the dotted lines),

(21)

Under an

-move,

(22)

where each graph appearing on the right can be decomposed into three-point coefficients by cutting across

′

. Crossing symmetry is the

equivalence of

∑

∈

(

)

(

)

∑

′

(

)

′

∑

′

∈

′

(

′

)

(

′

)

(23)

The modular invariance of the torus one-point function begins with performing

-moves on the configuration

(24)

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and demanding the equivalence of the two cuts shown by the dotted lines,

(25)

III. SPECTRAL CONSTRAINTS BY HAAGERUP SYMMETRY

This section studies the modular constraints on the spectral data—the set of local operators, 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

fusion

category.

A. The Haagerup fusion ring with six simple objects

The Haagerup

fusion category was constructed by Grossman and Snyder

as a variant (Grothendieck equivalent) of the

fusion

category that directly came from the Haagerup subfactor.

25,26

There are six simple objects/TDLs, which we denote by

αρ

(26)

The fusion ring is fully specified by the following relations:

αρ

ρα

≡

∑

(27)

For shorthand,

≡

(28)

In the rest of this article, we use unoriented solid lines to denote the non-invertible self-dual simple TDLs

and oriented dashed lines to

denote the invertible ones,

(29)

There are two gauge-inequivalent unitary fusion categories realizing the above fusion ring, denoted as

and

by Grossman and

Snyder.

Whereas the Haagerup

fusion category descended directly from the Haagerup subfactor,

25,26

the Haagerup

fusion category

was constructed by Grossman and Snyder

based on

. It turns out to be easier to work with

, but the analysis in this section applies

to both

and

. The

-symbols for

were implicit in the work of Grossman and Snyder

(using a generalization of the approach by

Izumi

for

) and also explicitly obtained by Titsworth

and Osborne, Stiegemann, and Wolf.

In Ref. 54, the present authors recasted

the

-symbols in a gauge that manifests the transparent property, a notion we introduce in Sec. IV. The transparent

-symbols are given in

Appendix A.

B. Action on local operators and representation theory

To describe how topological defect lines forming the Haagerup

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

is diagonal.

●

For a state

∣

⟩

neutral under

∣

⟩

αρ

∣

⟩

∣

⟩

∣

⟩

∣

⟩

(30)

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TABLE I.

Irreducible representations of the Haagerup fusion ring with six simple

objects/TDLs.

√

−

√

(

)

(

)

Hence, there are two one-dimensional representations,

∣

⟩

√

∣

⟩

(31)

●

For a state

∣

⟩

with unit

-charge,

∣

⟩

∣

⟩

αρ

∣

⟩

ρα

∣

⟩

∣

⟩

∣

⟩

ρα

∣

⟩

ωρ

∣

⟩

(32)

It follows that

∣

⟩

0, and, hence,

∣

⟩

∣

⟩

(33)

∣

⟩

and

∣

⟩

were equal up to a phase, then there would be two possible one-dimensional representations with

∣

⟩

∣

⟩

(34)

which is in conflict with

αρ

ρα

. Hence,

∣

⟩

and

∣

⟩

must be independent, and the representation is two-dimensional. In the

(∣

⟩

∣

⟩)

basis,

⎛

⎜

⎝

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

(35)

The above classification of irreducible representations is summarized in Table I. 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.

C. Modular invariance and vacuum degeneracy

Let

denote the number of vacua (local operators) and

and

be their multiplicities of representations (in the notation of Table I).

Clearly,

−

Consider the modular invariance of the torus partition function with the non-invertible TDL

wrapped around a one-cycle

(36)

The horizontal cut computes the trace over the action of

in the Hilbert space

of local operators, and the vertical cut simply counts the

dimensionality of the defect Hilbert space

. Modular invariance requires

∈

≥

(37)

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Given the representation content of the Haagerup fusion ring, summarized in Table I, we immediately conclude that

−

, and the number

of vacua must be even. Let us write

≡

−

(38)

to denote the multiplicity of each one-dimensional representation. Using the U

(

)

freedom, we can choose a basis of local operators to

represent

in the block diagonal form

⊕

(

√

)

⊕

−

⊕

(

−

√

)

⊕

⎛

⎜

⎝

⎞

⎟

⎠

⊕ ⋅ ⋅ ⋅ ⊕

⎛

⎜

⎝

⎞

⎟

⎠

(39)

Modular invariance (37) also implies that the defect Hilbert space

is 3

-dimensional, i.e., the TDL

can end on

(40)

independent defect operators and similarly for each of the other

Consider the modular invariance of the torus partition function with the invertible TDL

wrapped around a one-cycle

(41)

Modular invariance requires

∈

≥

(42)

Hence, the

TDL hosts

−

(43)

defect operators. The total number of point-like operators is

≡

(

)

(

−

)

(44)

The first few possibilities are listed in Table II in the order of increasing

. Whenever

0, the

symmetry is not faithfully realized

on the vacua. In the following, we consider the three minimal cases totaling

15 point-like operators, highlighted in Table II; each case has

1 and

3. Eventually, we will succeed in constructing a TFT realizing

6, but along the way, we also derive various constraints on

2, 4.

TABLE II.

Possible numbers of point-like operators that satisfy the torus one-point

modular invariance (36) and (41). Here,

denotes the total number of vacua

(local operators), comprised of

copies of representation

, where

=

−

;

denotes the number of defect operators in

each

for

=

αρ

;

and

denotes the total number of point-like (local and defect) operators. Only the

highlighted cases with

=

=

=

15 are considered in this article.

−

αρ

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IV. TRANSPARENCY AND

SYMMETRY

This article works in a gauge of the

fusion category that manifests its “transparent” property

—the associator involving any invertible

topological defect line (TDL) is the identity morphism. In terms of the

-symbols, it means that every

-symbol with an external invertible

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

-symbols. Several diagrammatic identities are illustrated as follows:

(45)

Importantly, the four-way junctions in

(

)

and

(

)

are unambiguously defined.

In Ref. 54, transparency and the

symmetry were exploited to reduce the pentagon identity so that the

-symbols could be efficiently

solved. Below, in attempting to construct a topological field theory, the utilization of the

symmetry is also essential in reducing the amount

of independent data.

relations for lassos and dumbbells

Let

be a local operator with

-charge

∈

{

}

, and consider the lasso

(46)

The

symmetry relates lassos with different triples

(

)

as follows: replace

using the equalities

(47)

and fuse the

symmetry line with

[apply (45)

(

)

and then

(

)

] to obtain the following relations:

(48)

Next, consider the dumbbell

(49)

J. Math. Phys.

, 042306 (2022); doi: 10.1063/5.0079062

, 042306-10

Published under an exclusive license by AIP Publishing