SciPost Phys. 13, 067 (2022)

Higher central charges and topological boundaries
in 2+1-dimensional TQFTs
Justin Kaidi1? , Zohar Komargodski1† , Kantaro Ohmori1,2‡ ,Sahand Seifnashri1§ and Shu-Heng Shao3,4◦
1 Simons Center for Geometry and Physics, Stony Brook University,
Stony Brook, NY 11794-3636, USA
2 Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
3 School of Natural Sciences, Institute for Advanced Study, Princeton NJ, USA
4 C. N. Yang Institute for Theoretical Physics, Stony Brook University,
Stony Brook, NY 11794-3840, USA? jkaidi@scgp.stonybrook.edu † zkomargodski@scgp.stonybrook.edu‡ kant.ohmori@gmail.com § sahand.seifnashri@stonybrook.edu ◦ shuhengshao@gmail.com

Abstract
A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition
for the existence of a topological boundary condition is that the chiral central chargec− has to vanish. In this paper, we consider conditions associated with “higher" central
charges, which have been introduced recently in the math literature. In terms of these
new obstructions, we identify necessary and sufficient conditions for the existence ofa topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative
to the identification of a Lagrangian subgroup. Our proof relies on general aspects ofgauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric
way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious
duality in the partition functions of Abelian TQFTs, which begs for an explanation via the
3d-3d correspondence.
Copyright J. Kaidi et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 05-04-2022
Accepted 16-08-2022
Check forupdatesPublished 26-09-2022
doi:10.21468/SciPostPhys.13.3.067

Contents
1 Introduction
1.1 The Chiral Central Charge
1.2 Obstructions Beyond Anomalies
1.3 Gapped Boundaries and Lagrangian Algebras

2
3
4
5

2 Abelian TQFTs
2.1 Lagrangian Subgroups
2.2 Obstructions from 3-Manifold Invariants

7
8
10

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SciPost Phys. 13, 067 (2022)

2.3
2.4

Complete Obstructions to Gapped Boundaries
Another Point of View

13
15

3 Non-Abelian TQFTs
3.1 Lagrangian Algebra Anyons
3.2 Gauging the Lagrangian Algebra
3.3 Topological Boundaries and the Turaev-Viro TQFT
3.4 The Galois Action and the Higher Central Charges

18
19
24
25
26

A Review of 2+1d TQFTs

30

B Surgery Calculation of Partition Functions

35

C Partition Function of Abelian CS Theories

38

References

41

1

Introduction

The boundary physics of a Quantum Field Theory is essential in many applications. A curious
phenomenon is that the boundary must sometimes support gapless modes, even though thebulk has an energy gap above its vacuum. In such cases, the bulk theory is well-described by aTopological Quantum Field Theory (TQFT), but the boundary does not become topological in
the deep IR. In other words, a TQFT does not always admit a topological boundary condition,and it is natural to ask under what conditions such a boundary does or does not exist.
Consider a QFT with global symmetry G, which could be continuous or discrete. If thesymmetry G has an ’t Hooft anomaly, then no boundary condition respecting G can exist.1
Therefore, let us focus on the case in which the symmetry G has no ’t Hooft anomalies. Morespecifically, we assume the bulk to be a nontrivial, gapped, G-Symmetry Protected Topological(SPT) phase. The continuum description of an SPT phase is in terms of a classical field theoryof the G background gauge fields. This is also known as an invertible field theory [2]. In
this case G-preserving boundary conditions can exist. But if we require that we can couple
the theory and its boundary to background G-gauge fields in a gauge invariant way, thenthe boundary cannot be trivially gapped because the nontrivial SPT phase results in anomaly
inflow into the boundary.2 In particular, when the SPT corresponds to a perturbative anomaly
for a continuous G, then the boundary has to be gapless.3
Another way to think about this situation is in the language of interfaces. Indeed, using
For instance, take G = U(1) in 3+1 dimensions. To prove that there does not exist a U(1)-preserving boundary
condition, we assume by contradiction that such a boundary condition does exist. Then weRcan couple the theory
to a background gauge field A and write the gauge variation of the partition function as M αF ∧ F where α isthe gauge variation parameter and F is the field strength. The manifold M now has a boundary and hence thevariation of the partition function does
R not make sense since F ∧ F no longer has quantized integrals. If one tries
to fix this issue by adding a term c ∂ M αA ∧ dA to the anomalous gauge transformation (with some coefficientRc), then the Wess-Zumino consistency condition is no longer obeyed since [δα , δβ ] log Z = c ∂ M (αdβ − β dα)F .
See [1] for a recent exposition of this topic.
2
Here we fix the scheme so that the trivial theory on the other side of the boundary has vanishing SPT phase.
Then a nontrivial SPT phase in the theory of interest is enough to guarantee a nontrivial boundary.
3
This is because the anomalous Ward identity ensures that the separate (d − 1)-point function of the currents
is nontrivial.
1

2

SciPost Phys. 13, 067 (2022)

the folding trick one can reinterpret an interface between the theories T and T 0 as a boundary
condition for the theory T × T 0 (T 0 stands for the orientation reversal of T 0 ). What we said
above therefore implies that a symmetry-preserving interface between T and T 0 exists only if
T and T 0 have the same anomaly for the symmetry under discussion. Furthermore, if T and T 0
have no anomaly for the symmetry G but are both in some SPT phase, a symmetry-preserving
interface can be trivially gapped only if the two SPT phases agree.
In this paper, we consider a more general case in 2+1-dimensions, where the bulk theoryis gapped and has nontrivial anyon excitations. The continuum description is in terms of a
nontrivial TQFT, rather than a classical field theory. The main objective of this paper is tounderstand when topological boundary conditions for a general 2+1-dimensional TQFT canexist. For simplicity, we will only focus on bosonic (non-spin) theories in this paper. We willuse “gapped boundary" as a synonym for topological boundary, and these two terms will be
used interchangeably.
The problem of finding topological boundary conditions of a 2+1d TQFT has a long history
in the high energy physics, the condensed matter physics, and the mathematical literature. In
the case of abelian TQFTs, this has been discussed in [3] from a field theory point of view.
Mathematicians have studied this problem in the context of modular tensor category [4–6]
and of fully extended field theories [7]. In the condensed matter literature, it has been studied
extensively in the context of gapped boundaries of topological order, e.g. in [8–18]. In thispaper, we will derive new results and provide a geometric interpretation for some of the known
results.

1.1

The Chiral Central Charge

We start with a discussion of a non-spin, invertible field theories in 2+1 dimensions without
any symmetry. Consider the following classical field theory on a three-manifold M :
Z


k
2
L=
Tr ω ∧ dω + ω ∧ ω ∧ ω , k ∈ Z .
π
(1)
12
3
M
Here ω is the spin connection of M . This invertible field theory depends on the metric as well
as a choice of the framing of M .
To get rid of the framing dependence, we can consider a closely related invertible field
theory given by the η-invariant with coefficient k (normalized appropriately). It is an invertible
field theory that is independent of the framing. In the condensed matter literature, this is
known as Kitaev’s E8 phase [19]. Finally, the ratio of Kitaev’s E8 phase and (1) is the (E8 )1Chern-Simons (CS) theory, in the quantization scheme of [20]. The latter is topological but
depends on the choice of the framing.
On general grounds, we could always add massive degrees of freedom in the UV. This
amounts to changing the IR field theory by an invertible field theory, such as the (E8 )1 CS
theory. This has the effect of shifting the coefficient k by 1.
Next, we would like to discuss boundary conditions of a 2+1-dimensional theory, subject
to the freedom of stacking an invertible field theory.
Suppose the 2+1-dimensional theory is in a gapless phase described by a CFT. The boundary of a CFT must always support a displacement operator, which has a nontrivial scaling
dimension and thus renders the boundary non-topological.
The case of primary interest in this paper is when the bulk 2+1-dimensional theory is a
TQFT. Such a TQFT describes the low energy limit of a gapped phase with long-range entanglement. There are nontrivial vacuum degeneracies on different spatial two-manifolds. In
this case, one defines the effective coefficient of the gravitational Chern-Simons term in theinfrared (1) via a contact term in the two-point function of the energy-momentum tensor. It
3

SciPost Phys. 13, 067 (2022)

does not have to be quantized with k ∈ Z and it is physically measurable through the so-called
thermal Hall conductance. Equivalently, the quantization of a 2+1-dimensional TQFT (suchas the Chern-Simons theory) generally leads to a term (1) with non-integral k [20]. The chiral
central charge of the bulk TQFT is then defined byc− = 8k .

(2)

Since we can stack an invertible field theory on top of our TQFT, c− is only determined by the
anyon data mod 8Z.
A famous statement is that if c− ∈/ 8Z, then there is no topological boundary condition.
In this case, the boundary theory is not a well-defined 1+1-dimensional theory on its own
right. Rather, it is a “relative” theory, which is not invariant under diffeomorphisms. Thishas to be captured by the gapless modes living on the boundary. Furthermore, these gaplessmodes cannot be removed by adding ordinary 1+1-dimensional theories on the boundary,which always have properly quantized gravitational anomaly c− ∈ 8Z.
An interesting fact is that there are TQFTs with c− = 0 mod 8 and yet no topologicalboundary conditions. Given that the only presently known obstruction due to symmetries andanomalies is the chiral central charge c− , it is puzzling that theories with c− = 0 mod 8 can still
have protected gapless edge modes. One such example is the U(1)2 × U(1)−4 Chern-Simons
theory, which has c− = 0, and yet its boundary has to be gapless.

1.2

Obstructions Beyond Anomalies

We now give a summary of our results. We start with the Abelian TQFTs. The basic data of an
Abelian TQFT includes a one-form symmetry group G, which is generated by the |G| anyonsof the theory.
It is known that an Abelian TQFT admits a gapped boundary if and only if there is a sufficiently large non-anomalous subgroup of G, known as a Lagrangian subgroup L [3, 6, 11, 12].
Gauging the subgroup L of the Abelian TQFT then results in the trivial theory. The anyons
generating the Lagrangian subgroup can terminate on this topological boundary. This is often
referred to as the Abelian anyon condensation [21] (see [22] for a review). While this in principle solves the problem of topological boundary conditions for Abelian TQFTs, in practice, theidentification of a Lagrangian subgroup of anyons is difficult. Our main goal in this paper will
be to provide a more readily computable alternative.
We begin by discussing higher obstructions to the existence of topological boundaries,
generalizing the chiral central charge c− . Recall that the chiral central charge c− , in a preferredchoice of scheme, appears as the phase of the partition function of the TQFT on S 3 . One ofour main results is to show that a nontrivial phaseξ M :=Z[M ],|Z[M ]|(3)of the partition function Z[M ] of an Abelian TQFT on a three-manifold M withgcd(|H1 (M )|, |G|) = 1 ,(4)

is also an obstruction to the existence of a topological boundary condition. The proof of thistheorem relies on general aspects of gauging a one-form global symmetry.
When the three-manifold M is the lens space L(n, 1), this phase is known as the higher
central charge [23, 24], which we denote by ξn . The higher central charge admits a simple
expression in terms of the spins θ (a) of the anyons a:Pna θ (a).
(5)ξn := P| a θ (a)n |4

SciPost Phys. 13, 067 (2022)

Importantly, we have restricted to the case of gcd(n, |G|) = 1 so far.
Our next result is that there is an extension of the higher central charges ξn such thatgcd(n, |G|) = 1 is not required, but the more general condition
2|G|
gcd n,= 1,(6)
gcd(n, 2|G|)is satisfied. We will prove that an Abelian TQFT admits a topological boundary condition if andonly if all these extended higher central charges ξn satisfying (6) are trivial. Since ξn = ξn+2|G| ,there are only finitely many higher central charges to compute. These phases provide a highly
computable alternative to the known condition in terms of Lagrangian subgroups.
Along the way, we will prove that an Abelian TQFT admits a topological boundary if andonly if it is an Abelian Dijkgraaf-Witten theory [25].4 This gives a complete classification of all
possible Abelian topological field theories with gapped boundaries.
As an aside, we report a curious relation for the partition function of Abelian TQFT. Any
Abelian TQFT can be described by an Abelian Chern-Simons theory with a K-matrix K [26](see also [27–30]). On the other hand, any three-manifold ML can be obtained by removing
a link from S 3 and gluing in a union of solid tori specified by a linking matrix L.
Let ZK [L] be the partition function of the Abelian TQFT labeled by K on the three-manifold
ML labeled by L. The partition function of an Abelian TQFT does not depend on the detail of thelink other than L. We review the basic ingredients in the surgery construction of 3-manifolds
in Appendix B.
In Appendix C we will show that the partition function on ML satisfies the following identity
2πi

ZK [L] = e 8 sgn(K)sgn(L)

det L
det K

1/2

ZL [K] .

(7)

This identity holds as long as the matrices are coprime (see Appendix C for the definitionof coprimality of matrices). This symmetry between the Chern-Simons K-matrix K and thesurgery linking matrix L calls for a geometric interpretation, perhaps in terms of the 3d-3d
correspondence [31].5

1.3

Gapped Boundaries and Lagrangian Algebras

Next we consider the problem of topological boundaries in non-Abelian TQFTs. For nonAbelian theories, there generically does not exist a one-form symmetry whose gauging leads
to the desired topological boundary. In other words, not all gapped boundaries correspond to
Abelian anyon condensation.
As a simple example, take any TQFT T and consider T × T . This clearly admits a topological boundary since by the folding trick this is the same as the trivial interface between T
and itself, which exists tautologically. However, in general, T × T does not have any one-form
symmetry, and hence the existence of this gapped boundary cannot always be understood bygauging a Lagrangian subgroup.
Nonetheless, there is an extension of the notion of a Lagrangian subgroup to a Lagrangianalgebra. The Lagrangian algebra is a certain non-simple anyon with various special properties,
such as having quantum dimension coinciding with the total quantum dimension of the TQFT.6
4Note that a Dijkgraaf-Witten theory based on a finite Abelian group can be an Abelian or a non-Abelian TQFT.
Here by Abelian Dijkgraaf-Witten theories we mean those that are Abelian TQFTs.
5
On a related topic, [32] discusses constructions of 2+1d TQFTs from three-manifold compactifications of the
6d N = (2, 0) theory.
6
In general, gapless boundary conditions of a TQFT is not represented by any (possibly non-simple) anyon.
The fact that this is the case for gapped boundary conditions is reminiscent of the situation in 1+1 dimensions,

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SciPost Phys. 13, 067 (2022)

As we will review in Section 3.3, there is a one-to-one correspondence between Lagrangianalgebras and topological boundary conditions in unitary 2+1d TQFTs [4, 6].
We will provide a geometric picture and construction of the Lagrangian algebra from a
topological boundary. This description pictorially trivializes the defining properties of a Lagrangian algebra. Inserting a fine mesh of this algebra object allows us to formalize the notionof “condensing non-Abelian anyons." We will see that the condensation of these anyons again
leads to a gapped boundary.
For the particular case of T × T , if we label the anyons by ai and those of T by ãi (where ãi
is obtained by time reversal from ai ) then the Lagrangian algebra is the following (non-simple)
anyon:MA=ai ⊗ ãi .
(8)i

The non-Abelian anyon condensation in many ways appears as a natural generalization of“gauging" these non-Abelian anyons.7 However, while there is a well-defined mathematicalprocedure, there is no clear physical understanding of what it means to “gauge" such an algebra
anyon other than the insertion of a mesh. In particular, there is no clear understanding in termsof a sum over some gauge fields.8
The existence of Lagrangian algebras is studied in the context of modular tensor categories by mathematicians. By the correspondence between Lagrangian algebras and topological boundaries, this is closely related to the problems we are interested in this work. The
basic idea is that one can define an equivalence relation between 2+1d TQFTs whenever there
exist a topological interface between them. More precisely, we say topological theories T1 andT2 are Witt equivalent, if the theory T1 × T 2 has a gapped boundary (or equivalently, has a
Lagrangian algebra) [4, 5]. These equivalence classes form an Abelian group called the Wittgroup. The group structure is given by taking tensor product of two TQFTs. More precisely,
the group multiplication is defined by [T1 ] · [T2 ] := [T1 × T2 ], where [T ] denotes the Wittclass of the theory T . The group inverse is given by [T ]−1 = [T ], since T × T has a gappedboundary and hence is trivial in the Witt group.
Having introduced the Witt group, we will review the theorem [23, Theorem 4.4] thatthe phase of lens space partition functions ξn with gcd(n, NFS) = 1, are invariants of the Wittgroup. In other words, these phases are obstructions to the existence of a gapped boundaryfor non-Abelian theories. Here NFS is the Frobenius-Schur exponent, i.e. the smallest integer
such that θ (a)NFS = 1 for of all anyons a. This shows that even in the non-Abelian case there
are important obstructions beyond the chiral central charge.
Along the way we will uncover a few elementary but interesting properties of the lensspace partition functions in theories with a gapped boundary. These properties will be derived
by making use of Galois theory.
For the convenience of the reader, we summarize various equivalent properties of TQFTs
in Table 1. The details will be discussed in the main body of the paper. See, in particular,
Section 2.3 and 3.3 and references therein. In the lower right corner, although there is no
analogous condition like the vanishing of ξn at present, we will discuss necessary conditions
for the existence of a topological boundary.
where the conformal boundary conditions are represented by linear combinations of bulk local operators via the
Cardy conditions [33]. One can view the constraints that we derive on this non-simple anyon as 2+1 dimensionalcounterparts of the Cardy conditions.
7
We thank F. Burnell, T. Devakul, P. Gorantla, H. T. Lam, and N. Seiberg for many discussions on this point.
8
The generalized notion of gauging in terms of inserting a fine mesh of (non-invertible) topological lines in
1+1d QFTs was introduced by [34–36] and reviewed in [37]. See [38] for a discussion in the context of category
theory.

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SciPost Phys. 13, 067 (2022)

Table 1: We summarize various sufficient and necessary conditions for the existenceof topological boundary conditions, both in the case of Abelian and non-Abelian
TQFTs. The conditions in each column are equivalent to each other. In the first column, |G| denotes the number of simple anyons, and ξn is the higher central charge(i.e., the phase of the L(n, 1) lens space partition function).
Abelian TQFTs

Non-Abelian TQFTs∃ Topological boundary condition∃ Topological boundary condition∃ Lagrangian subgroup∃ Lagrangian algebra

Abelian Dijkgraaf-Witten theoryPθ (a)n2|G|ξn = | Pa θ (a)n | = 1 for gcd n, gcd(n,2|G|) = 1

Turaev-Viro theory / Drinfeld center?

aOrganization
The rest of this paper will be organized as follows. In Section 2 we will focus on Abelian theories, beginning with a review of the known results about Lagrangian subgroups in Section 2.1.
Then in Section 2.2 will discuss the new class of obstructions ξ M labelled by 3-manifolds M ,including the higher central charges. We then extend the higher central charges to a complete
set of obstructions in Section 2.3, giving an alternative set of necessary and sufficient conditions beyond the usual Lagrangian subgroups. We end the Abelian discussion with Section
2.4, which gives an alternative viewpoint on these obstructions. This alternative viewpoint
partly generalizes to the non-Abelian case and allows to establish several general result about
the properties of ξ M in Abelian theories. For instance, the phases ξ M are always 8th roots ofunity.
We then move on to a discussion of the non-Abelian case in Section 3, beginning witha detailed geometric definition of Lagrangian algebra anyons in Section 3.1. Utilizing various geometric considerations, we find many properties that Lagrangian algebra anyons mustsatisfy.
In Section 3.2 we discuss a generalized notion of gauging of the Lagrangian algebra. Section 3.3 reviews various sufficient and necessary conditions for the existence of a topologicalboundary conditions for general non-Abelian TQFTs. Section 3.4 discusses the non-Abeliananalogs of the higher central charges. Using Galois theory we find several general propertiesof the higher central charges.
Finally, some background information is collected in the Appendices. Appendix A givesa brief review of 2+1d TQFT. Appendix B reviews the surgery construction of TQFT partition functions. Appendix C obtains various results regarding the partition function of Abelian
Chern-Simons theories labelled by the matrix K on manifolds with linking matrix L, including
the identity (7) given above.

2

Abelian TQFTs

In this section we discuss the question of topological boundary conditions in Abelian TQFTs.
Along the way, we will phrase some of the results in the literature in terms of gauging higherform symmetries. Throughout this section, we will make use of various basic properties of
2+1d TQFT reviewed in Appendix A.

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SciPost Phys. 13, 067 (2022)

2.1

Lagrangian Subgroups

A Lagrangian subgroup L [39] of an Abelian TQFT is a subgroup of the bosons (i.e. anyons
with spin θ (a) = 1) such that |L|2 = |G|, where G is the Abelian group obtained from thefusion of anyons. It follows from this definition that9θ (a×b)1. Every two lines in L have trivial braiding, i.e. B(a, b) = θ (a)θ (b) = 1 for all a, b ∈ L.
2. Any line that is not in L has nontrivial braiding with at least one line in L.
Note that the choice of the Lagrangian subgroup for a given Abelian TQFT is generically notunique.
In [3, 6, 11, 12] the following result was shown,
Theorem 2.1 A c− = 0 Abelian bosonic TQFT has a gapped boundary if and only if there existsa Lagrangian subgroup L.
We now provide an interpretation of this result in terms of gauging one-form symmetries.
Gauging One-Form Symmetries and Dijkgraaf-Witten Theories
The anyons of an Abelian TQFT generate a one-form symmetry group G, and the ’t Hooft
anomalies of G are captured by the spins θ (a) of the anyons [40–42]. In particular, one-form
symmetries generated by bosonic lines are non-anomalous and can be gauged. (Note howeverthat the set of the bosons in an Abelian TQFT is generally not closed under fusion, and hence
does not form a group). In this language, Theorem 2.1 can be restated as:• A c− = 0 Abelian bosonic TQFT has a gapped boundary if and only if there is a nonanomalous subgroup L of the one-form symmetry group G such that |L|2 = |G|.
Another statement, more within the framework of the low-energy TQFT which allows us todefine c− only mod 8, is• An Abelian bosonic TQFT has a gapped boundary after stacking with appropriate copiesof the (E8 )1 Chern-Simons theory if and only if there is a non-anomalous subgroup L of
the one-form symmetry group G such that |L|2 = |G|.
As explained in the introduction, stacking copies of the (E8 )1 Chern-Simons theory does notchange anyon date of the TQFT. All it does is to shift the gravitational Chern-Simons coefficientk from (1) by an integer. Equivalently, stacking copies of the (E8 )1 Chern-Simons theory adds
edge modes with c− ∈ 8Z.
Note that in 2+1d, every time we gauge a non-anomalous discrete one-form symmetry L,
we gain a quantum zero-form symmetry bL in the gauged theory, where bL = Hom(L, R/Z) is
the Pontryagin dual of L [40, 43]. This is the 2+1d version of the quantum symmetry of theorbifold theory in 1+1d [44].
Let us explain this point further. First recall that coupling the theory to a background gauge
field is equivalent to the insertion of a certain network of L symmetry lines into the spacetime
manifold M . For a discrete and anomaly-free L, the insertions only depend on the homology
9
The first property is obvious from the expression of the braiding phase in terms of the spins. The second property can be shown as follows. Since elements in L braid trivially with each other, the braiding phase B(a, b) inducesÔa homomorphism from L × G/L to U(1), which can be alternatively viewed as a homomorphism h : L → G/L,
Ò = Hom(H, U(1)) is the Pontryagin dual of H. Since the braiding is non-degenerate, h must be injective.
where H
Moreover, the Lagrangian condition |L|2 = |G| implies that h must be surjective as well. The homomorphism hbeing surjective then implies that the homomorphism h0 : G/L → bL is injective, which proves the second property.

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SciPost Phys. 13, 067 (2022)

class of the network, and hence by Poincaré duality the network is characterized by a cocyclea ∈ H 2 (M , L). Gauging the symmetry is equivalent to summing over gauge fields and hence
making a dynamical. Now we can introduce a background gauge field â ∈ H 1 (M , bL ), and add
the term 〈â, a〉 to the action. The gauge field â ∈ H 1 (M , bL ) is a gauge field for the quantum bL
zero-form symmetry. The partition function of the gauged theory becomes
Z 0 [M , â] =|H 0 (M , L)||H 1 (M , L)|X

e2πi 〈â,a〉 Z[M , a] .

(9)a∈H 2 (M ,L)

Here 〈., .〉 : H 1 (M , bL ) × H 2 (M , L) → R/Z is the intersection pairing on cohomology. Z[M , a]stands for the partition function on manifold M coupled to a background gauge field a. An
important property of the quantum bL zero-form symmetry is that it can be gauged to retrieve
the original theory, since1

X|H 0 (M , bL )|Z 0 [M , â] = Z[M , 0] .

(10)â∈H 1 (M ,bL )Now let us consider an Abelian bosonic TQFT with a Lagrangian subgroup L. We cangauge this non-anomalous one-form symmetry subgroup L to obtain another TQFT. Following
the three-step process of gauging a one-form symmetry outlined in [42, 45, 46], and using theproperties of L described at the beginning of this section, one finds that the gauged TQFT istrivial. But if this is the case, then as was just reviewed the original TQFT can be retrieved
from the trivial TQFT by gauging an bL zero-form symmetry, possibly with a nontrivial action
for the bL gauge fields. In other words, the TQFT is a finite group gauge theory bL , possibly witha nontrivial Dijkgraaf-Witten twist [25].10
Since conversely every Dijkgraaf-Witten gauge theory also admits a gapped boundary condition (i.e. the Dirichlet boundary condition) we are led the following result:11
Corollary 2.2 An Abelian bosonic TQFT has a gapped boundary if and only if it is an Abelian
Dijkgraaf-Witten gauge theory.
This statement also follows from the discussion in [3, 6].
Example: Z2 Gauge Theory
Let us consider Z2 gauge theory as an example. The Lagrangian is given by [48–50],

L=

2 (1) (1)a db ,
2π(11)

and the theory admits four anyons,1,e ≡ eiHa(1),m ≡ eiHb(1),f ≡ eiHa(1) +i

H

b(1)

.

(12)

Their spins are θ (1) = θ (e) = θ (m) = 1 and θ ( f ) = −1. They obey the Z2 × Z2 fusion rules,e × e = m × m = f × f = 1,

e×m= m×e = f ,m× f = f ×m = e,

f × e = e × f = m.

(13)

We can find the particular element of H 3 (bL , U(1)) specifying the Dijkgraaf-Witten twist as follows. By restricting
the braiding phase B : G × G → U(1) to G/L × L, we obtain a non-degenerate pairing between G/L and L, which
provides an isomorphism between G/L and bL [39]. Since G can be viewed an extension of G/L ' bL by L, thisdetermines a class in H 2 (bL , L). This class can be further mapped to a 3-cocycle in H 3 (bL , U(1)) which is the DijkgraafWitten twist [3].
11
See, for example, [47] for a recent discussions on boundary conditions of the Dijkgraaf-Witten gauge theory.
10

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SciPost Phys. 13, 067 (2022)

The subset of bosonic anyons is {1, e, m}, but this is not closed under fusion and hence
does not form a subgroup. Instead, we have the following two bosonic subgroups,(e)

Z2 = {1, e} ,

(m)

Z2

= {1, m} .

(14)

Since 22 = 4, each of these is a Lagrangian subgroup, and there are consequently two corresponding choices for gapped boundaries conditions. Indeed, gauging either one of the two(e)Lagrangian subgroups (but not both) gives the trivial theory. For example, gauging Z2 gives
2 (1)2 (2) (0)a (d b(1) − c (2) ) +c dφ ,
2π
2π(15)where c (2) is a two-form gauge field and φ (0) is a 2π-periodic compact scalar. The latter
enforces the condition that c (2) is a Z2 -valued two-form gauge field. The gauge transformationsarea(1) ∼ a(1) + dα(0) , b(1) ∼ b(1) + dβ (0) + γ(1) ,(16)
φ (0) ∼ φ (0) + α(0) ,c (2) ∼ c (2) + dγ(1) .
In terms of the gauge-invariant combination ā(1) := a(1) − dφ (0) , we can rewrite the gauged
2 (1)Lagrangian as 2πā (d b(1) − c (2) ) modulo total derivatives. After integrating out ā(1) , thisindeed becomes a trivial TQFT.
The gapped boundary separates this trivial phase from the Z2 gauge theory phase. To
demonstrate that the boundary is empty let us analyze the theory (11) on a half-plane y ≥ 0.
Choosing the gauge a t = b t = 0, we are restricted by the Gauss law to satisfy that the x ycomponents of both field strengths vanish, leaving us with flat connections. If we impose at
the boundary a|| = 0 (resp. b|| = 0) then we do not have to restrict the b (resp. a) gaugetransformations at the boundary, and hence the b (resp. a) flat connections can be removed
everywhere on the disk leaving us with an empty theory.
Each of these boundary conditions breaks the Z2 × Z2 one-form symmetry to Z2 . For(m)instance, if we impose a|| = 0 then the Z2 one-form symmetry is broken and the Wilson line
e from (12) can end (condense) on the boundary.

2.2

Obstructions from 3-Manifold Invariants

We have seen that the existence of a gapped boundary in an Abelian TQFT is tantamount to
the existence of a Lagrangian subgroup. In practice though, it is not always straightforward tocheck if such a subgroup exists. In this subsection, we will provide a new set of obstructions
which have the virtue of being highly computable, though with the disadvantage that they
will provide only necessary, not sufficient, conditions to the existence of a gapped boundary.
In the following subsection, we will generalize these obstructions to ones yielding necessary
and sufficient conditions.
To begin, let ZT [M ] be the partition function of a c− = 0 Abelian bosonic TQFT T on
a closed, oriented, connected three-manifold M . Since c− = 0, we can choose a scheme inwhich the partition function is topological and independent of the choice of the framing. In
this scheme ZT [M ] coincides with the Reshetikhin-Turaev invariant of three-manifolds, whichis reviewed in Appendix B.
As we have seen, the Abelian TQFT T has a gapped boundary if and only if there exists a
non-anomalous one-form symmetry subgroup L with |L|2 = |G|, where G is the Abelian groupof anyons. Furthermore, the TQFT obtained by gauging L is trivial. In equations, this meansthat|H 0 (M , L)| X
1= 1
ZT [M , a] ,(17)
|H (M , L)|2a∈H (M ,L)

10

SciPost Phys. 13, 067 (2022)

where a is a two-form gauge field of the one-form symmetry L, and ZT [M , a] is the partition
function coupled to the gauge field a.
Consider a three-manifold M with the following property:
gcd(|H1 (M )|, |G|) = 1 .

(18)

With this condition we also demand that H1 (M ) = H1 (M , Z) has a finite order. For any Lagrangian subgroup L, this condition is equivalent togcd(|H1 (M )|, |L|) = 1 ,(19)since |L|2 = |G|. Next, (19) is equivalent to
H1 (M , L) = 0 ,(20)

as follows from H1 (M , L) = H1 (M )⊗ L and the fact that Zn ⊗Zm = Zgcd(n,m) .12 Poincare duality
further implies that
H 2 (M , L) ∼(21)
= H1 (M , L) = 0 ,and using the universal coefficient theorem, we also have H 1 (M , L) ∼= Hom(H1 (M ), L) = 0.
Altogether, we conclude that on a manifold satisfying (18), one has H 2 (M , L) = 0 andhence there are no nontrivial two-form gauge fields of the one-form symmetry L. It follows
that gauging L is completely trivial on such manifolds, and so (17) reduces to
ZT [M ] =1,
|L|if gcd(|H1 (M )|, |G|) = 1 ,(22)

where we have used H 0 (M , L) = L and H 2 (M , L) = 0. This gives rise to a new set of obstructions: namely, the phase of ZT [M ] on manifolds satisfying (18) is an obstruction to theexistence of a gapped boundary. We summarize this by means of the following theorem,
Theorem 2.3 A c− = 0 Abelian bosonic TQFT T has a gapped boundary only if ZT [M ] > 0 on
every closed oriented three-manifold M with gcd(|H1 (M )|, |G|) = 1.
Let M = L(n, 1) be the orientation reversal of the n-th lens space.13 In this case
H1 (M ) = Zn . Using the surgery presentation of lens spaces (reviewed in Appendix B), thepartition function of a c− = 0 Abelian bosonic TQFT T on L(n, 1) is given by
ZT [L(n, 1)] =1 Xθ (a)n .
|G| a

The phases of these partition functions are then given byPna θ (a)ξn := P.
| a θ (a)n |

(23)

(24)

These quantities have made a previous appearance in the math literature, where they went
under the name of higher central charges [23, 24]. By using various techniques from Galois
theory, the authors of those references were able to show that the higher central charges with
gcd(n, |G|) = 1 are indeed obstructions to having a Lagrangian subgroup. Now we have seenthat the higher central charges are just special cases of the more general obstructions of Theorem 2.3, which can be obtained from any three-manifold with gcd(|H1 (M )|, |G|) = 1.
Here ⊗ is the tensor product of Abelian groups, not to be confused with the direct product ×. See [51] for adefinition.
13
We choose to work with L(n, 1) as opposed to L(n, 1) in order to avoid an inconvenient sign in n below.
12

11

SciPost Phys. 13, 067 (2022)

Example: U(1)2N1 × U(1)−2N2
One of the simplest bosonic Abelian TQFTs is U(1)±2N with N ∈ N. The spins of the anyons
are


s2
θ (s) = exp 2πi,s = 0, 1, · · · , 2N − 1 .
(25)
4NThese anyons generate a one-form symmetry group G = Z2N . The order of the T -matrix
Tss0 = δss0 θ (s) is 4N .
Because this theory has chiral central charge c− = ±1 mod 8, it cannot admit a gappedboundary. However, we can instead consider U(1)2N1 × U(1)−2N2 with N1 , N2 ∈ N, which has
vanishing chiral central charge c− = 0 mod 8. In this case the total one-form symmetry is
G = Z2N1 × Z2N2 and the total number of anyons is |G| = 4N1 N2 .
As reviewed above, a bosonic Abelian TQFT admits a gapped boundary if and only if it hasa Lagrangian subgroup. The K-matrix of U(1)2N1 × U(1)−2N2 is


2N1
0
,
(26)
K=
0
−2N2
and in this case it turns out that the existence of a Lagrangian subgroup is equivalent to finding
a two-dimensional integer vector Λ such that Λ T KΛ = 0 (see, for example, [11]). We thus
conclude that U(1)2N1 × U(1)−2N2 has a gapped boundary if and only ifp(27)
N1 N2 ∈ N .
Note that the only if direction in (27) is trivial. There is a Lagrangian subgroup of anyons only
if the total number of anyons is a square, but 4N1 N2 is a square if and only if N1 N2 is a square.
Below we will rephrase this condition in terms of the higher central charges. Let us compute the higher central charges of U(1)2N1 × U(1)−2N2 using (24). We begin with the generalized Gauss sum,




2N
−1
2N
−1
X
X
s2
1 + i −1 p
4Nn
θ (s) =
exp 2πin="n 4N, when gcd(n, 4N ) = 1 , (28)
4N
2
n
s=0
s=0
where

¨
"n =

1,

if

n = 1 mod 4i,if

n = 3 mod 4(29)


and ac is the Jacobi symbol. (See, for example, Appendix B of [42] for the definition.) The
higher central charges of U(1)2N1 × U(1)−2N2 can then be easily obtained:


N1 N2
ξn (U(1)2N1 × U(1)−2N2 ) =
,
(30)
n

 
where we have used anb = na nb . Recall that the higher central charges ξn are defined only Nfor gcd(n, |G|) = gcd(n, 4N1 N2 ) = 1, for which the Jacobi symbol ni is always ±1.
To show equivalence with the Lagrangian subgroup condition, we use the following resultin number theory14
Proposition 2.4 N is a perfect square if and only if N= 1 , ∀ odd prime n such that gcd(n, N ) = 1 .
n(31)

It then follows that the condition (27) is equivalent to the triviality of all the higher central
charges (30). We have therefore shown that the Abelian TQFT U(1)2N1 × U(1)−2N2 admits agapped boundary if and only if all of its higher central charges ξn are trivial.
14

This proposition was proved, for example, in [52].

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SciPost Phys. 13, 067 (2022)

Abelian TQFTs with Vanishing Higher Central Charges
With this initial success, one might be tempted to suppose that all Abelian TQFTs with trivial
higher central charges admit a gapped boundary. But as we will now see, this assumption isincorrect. There are Abelian TQFTs with trivial ξn for all gcd(n, |G|) = 1, and yet they do not
admit any gapped boundary [24].
Let us begin by noting that for every prime number p, there is a unique TQFT A p with thefollowing properties:• Fusion rules G = Z p × Z p .
• No non-anomalous one-form symmetry.
• ξn = −1 for all n with gcd(n, p) = 1.
For example, for p = 2 the TQFT in question is A2 = S pin(8)1 CS theory. For p odd,
 thetheory can be decomposed as A p = A p,2 × A p,−2m , where m is an integer such that mp = −1.
Here we are using the notation introduced in [42], where A p,m are defined to be the minimal
2
TQFTs with Z p fusion rule and spins θ (a) = exp(2πi ma2p ) for a ∈ Z p .
Since A p has no non-anomalous one-form symmetry, it follows that A p × Aq for p 6= q also
has no non-anomalous one-form symmetry. The theory thus cannot admit a gapped boundary.
But because the higher central charges of each factor theory are −1, the higher central chargesof the product theory are trivial (in particular, the chiral central charge vanishes mod 8). Hencethe theories A p × Aq for p 6= q provide examples of theories which do not admit a gappedboundary, but have trivial higher central charges with gcd(n, |G|) = 1.15

2.3

Complete Obstructions to Gapped Boundaries

For an Abelian TQFT with one-form symmetry of order |G|, we have seen that a necessary
condition for the presence of a gapped boundary is the triviality of the higher central charges
ξn , as defined in (24), for all n such that gcd(n, |G|) = 1. We now show that expanding therange of n, we can also obtain a necessary and sufficient condition. In particular, we will show
the following:
Theorem 2.5 An Abelian TQFT T with Frobenius-Schur
exponent
 NFS admits a gapped boundNFS
ary if and only if ξn (T ) = 1 for all n such that gcd n, gcd(n,N ) = 1.
FS

Recall that the Frobenius-Schur exponent NFS is defined as the smallest integer such that
θ (a)NFS = 1 for of all anyons a. Three remarks are in order:• Note that the condition on n can be restated as follows. Take the prime factorization
NFS = N1 N2 . . . Nk of NFS, where Ni = pi αi for distinct prime numbers p1 , . . . , pk . Thenfor each i, either Ni divides n or gcd(n, Ni ) = 1.16
• Since ξn = ξn+NFS , it suffices to check different n modulo NFS.
• For any positive integer k, we can replace NFS by kNFS in Theorem 2.5. In other words,
if we scan over all 0 < n < kNFS such that gcd (n, kNFS/gcd(n, kNFS)) = 1, then n mod
NFS scans over exactly all solutions of gcd (n, NFS/gcd(n, NFS)) = 1. In particular, we canreplace NFS by 2|G| since NFS divides 2|G| (which follows from (37) below).
15

In fact these theories and their products generate all the elements in the Witt group of Abelian theories withtrivial higher central charges [24].
16
In other words, for any common prime factor p of n and NFS , the exponent of p in n must be greater than orequal to that in NFS .

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SciPost Phys. 13, 067 (2022)

In order to prove Theorem 2.5, we will use some simple facts about the factorization of
Abelian TQFTs. We begin with the following,
Lemma 2.6 An Abelian TQFT T with Frobenius-Schur exponent NFS admits a factorization

T = T p1 × T p2 × · · · × T pk ,(32)

where T pi are TQFTs labelled by distinct primes pi , such that the number of anyons in T pi isa positive integer power of pi . If we denote the Frobenius-Schur exponents of T pi by Ni , then
NFS = N1 N2 . . . Nk .
At the level of fusion rules, the existence of such a factorization is obvious since, by the Chinese remainder theorem, finite Abelian groups admit such a factorization. But to establish thefactorization at the level of TQFTs, it is also necessary to show that anyons in the different
factors braid trivially. To show this, first note that for any anyon ai in T pi , the order mi of ai
must divide the total number of anyons in T pi by Lagrange’s theorem. This means that mi is a
power of pi , and thus gcd(mi , m j ) = 1 for i 6= j. Given anyons a1 and a2 in T p1 and T p2 withrespective orders m1 and m2 , the braiding must satisfy
B(a1 , a2 )m1 = B(a1 , a2 )m2 = 1 ,(33)

by the multiplicative property of the braiding phase. But since m1 and m2 are coprime, theonly solution to this is the trivial phase. This justifies the decomposition.
To prove the factorization of the Frobenius-Schur exponent, we note that the spins satisfy22θ (a m ) = θ (a)m . Considering an anyon ai of order mi in T pi , we conclude that θ (ai )mi = 1
and thus, since mi is a power of pi , Ni is as well. This tells us that gcd(Ni , N j ) = 1 for i 6= j,from which NFS = N1 N2 . . . Nk follows.

Having shown that any Abelian TQFT T admits a factorization as in (32), we now show
the following,
Lemma 2.7 T admits a gapped boundary if and only if all the factors T pi do.
Denote the total number of anyons in T by |T |. Recall that T admits a gappedboundaryp
if and only if there exists a Lagrangian subgroup L, i.e., a subgroup of |L| = |T | lines withtrivial spins. This in particular requires that |T | is a perfect square, in which case the orders of
the theories T pi appearing in the prime factor decomposition are also perfect squares. Clearlysuch an L, if exists, admits a decomposition as L = L p1 × · · · × L pk by the exact same reasoningÆ
as for T . Furthermore, we see that |L pi | = |T pi | and, since the lines in L have trivial spin,
those in L pi do as well. Thus each L pi serves as a Lagrangian subgroup for T pi , proving theforward direction of the theorem. Conversely, given a Lagrangian subgroup for each T pi , it iseasy to see that the product of these subgroups gives a Lagrangian subgroup for T , since lines
in different factors have trivial braiding.

We are now in a position to prove Theorem 2.5. We begin by proving the forward direction, namelyof a gapped boundary for T implies ξn (T ) = 1 for all n that the existenceNFSsuch that gcd n, gcd(n,N=1.
By
Lemma 2.6, the theory T admits the factorization (32)
FS )with NFS = N1 . . . Nk . The condition on n is equivalent to requiring that for any i, either
gcd(n, Ni ) = 1 or Ni | n. By Lemma 2.7 the existence of a gapped boundary for T means that
all factors T pi have gapped boundaries as well. Now consider any factor T pi ; as we have reviewed in the previous subsection, for gcd(n, Ni ) = 1, the existence of a gapped boundary
implies ξn (T pi ) = 1 [23]. On the other hand, if Ni | n, then ξn (T pi ) = 1 trivially. Either way,
14

SciPost Phys. 13, 067 (2022)

when a gapped boundary exists we see that the relevant higher central charges for the prime
factors are trivial. Noting thatξn (T ) = ξn (T p1 )ξn (T p2 ) . . . ξn (T pk ) ,(34)then completes the proof of the forward direction.

NFS
For the converse direction, assume that ξn (T ) = 1 for all n such that gcd n, gcd(n,N
= 1.
FS )
In particular, we can consider n of the form n = N1 . . . Nr−1 Nr+1 . . . Nk ñ with gcd(ñ, Nr ) = 1.
Since all Ni except for Nr divide n, we haveξn (T ) = ξn (T pr ) .

(35)

If we now take ñ to scan over all totatives of Nr , then n also scans over all totatives of Nr . Thus
if ξn (T ) = 1 for all such n, we conclude that ξn (T pr ) = 1 for all gcd(n, Nr ) = 1. For theories
where the number of anyons is a prime power, as is the case for T pr , it is already known that
this implies the presence of a gapped boundary (see [53, Appendix A.7] for a proof). Repeating
this for all r and utilizing Lemma 2.7 then completes the proof.
Summary on Topological Boundary Conditions of Abelian TQFTs
We summarize the discussions up to this point by the following statement. If we work modulo
invertible field theories (such as the (E8 )1 CS theory), then the following conditions for abosonic, Abelian TQFT are equivalent:• It admits a topological boundary condition.
• It has a Lagrangian subgroup, i.e., a non-anomalous subgroup L of the one-form symmetry G satisfying |L|2 = |G|.
• It is an Abelian Dijkgraaf-Witten gauge theory.
Pθ (a)n
2|G|• ξn = | Pa θ (a)n | = 1 for all n such that gcd n, gcd(n,2|G|) = 1.
a2.4

Another Point of View

In Section 2.2 we established that if T has a gapped boundary, then ZT [M ] is positive on every
closed oriented three-manifold M with gcd(|H1 (M )|, |G|) = 1. (More precisely, this is true insome particular scheme which always exists if c− = 0. This is the scheme where the partition
functions are topological invariants and framing-independent.)
There is another point of view on this result which partly generalizes to non-Abelian theories. The idea is to think of ZT [M ] as the S 3 partition function of some different (Abelian)theory, with a possibly different number of anyons. Assuming the existence of a Lagrangian
subgroup in the original theory, one can prove the existence of a Lagrangian subgroup in theauxiliary theory, which guarantees that it has a vanishing chiral central charge mod 8 andhence that its S 3 partition function is positive. Translating back to the original theory proves
the positivity of ZT [M ]. This point of view, besides partly generalizing to non-Abelian theories,
also allows to establish various properties of the partition functions ZT [M ]. For instance, thephase of ZT [M ] is always an 8th root of unity (whether or not c− = 0 in the original theory).

15

SciPost Phys. 13, 067 (2022)

K-matrix
To see how this works, first recall that any Abelian TQFT can be presented as a Chern-Simons
theory with some K-matrix K [26] (see also [27–30]). These theories have Lagrangians

L=

1
K I J a I ∧ daJ ,
4π

(36)

where the a I are U(1) gauge fields and I = 1, ..., |K|. The matrix K is symmetric and integral.
For the theory to be bosonic, we furthermore require that the diagonal entries of K are even.
The anyons in these theories are labeled by |K|-dimensional integer-valued vectors m ∈ Z|K| ,and fusion of two anyons m and n corresponds to addition of vectors m + n. The associatedtopological spins are given by
T −1
(37)
θ (m) = eπi m K m .
Of course, not all integer-valued vectors m describe independent anyons. There are only
finitely many anyons which are independent and furnish a non-degenerate braiding matrix. Indeed, the braiding phase is calculated from the topological spins as usual for Abelian theories,
T −1

B(m, n) = e2πi m K

n

,

(38)

so if we shift m → m + K · m̃ for arbitrary m̃ ∈ Z|K| , the braiding is left unchanged. The
topological spin is also invariant under such a shift,17 and therefore the space of anyon labels,
or more precisely the Abelian group of anyons, is
G = Z|K| /K · Z|K| .

(39)

In particular, there are |G| = | det K| independent anyons.
Let us evaluate the L(1, 1) ' S 3 partition function in such a K-matrix theory. To do so, weneed to perform the sum (23)
X


T −1
1
Z L(1, 1) =eπi m K m .
(40)| det K|m∈Z|K| /K·Z|K|

By referring to [54, Theorem 1] (see also [26]), one finds
X
2πi
T −1
1
1
eπi m K m = p
e 8 sgn(K) ,| det K|
| det K|m∈Z|K| /K·Z|K|(41)

where sgn(K) is the signature of the symmetric matrix K. This is the familiar statement thatthe chiral central charge is the signature of the matrix K.
We would now like to consider the partition function of the theory on a general 3-manifold
obtained by surgery on a link with linking matrix L (a brief review of surgery is given in Appendix B). Like K, the linking matrix L is an integral-valued symmetric matrix, though now
the diagonal does not have to consist of even numbers. For Abelian theories the partition
function depends only on the linking matrix, since for any two simple anyons the fusion channel is unique. The resulting manifold has H1 (M ) isomorphic to Z|L| /L · Z|L| , and in particular|H1 (M )| = | det L|.
Using the methods reviewed in Appendix B, one finds that the partition function of thetheory with K-matrix K on a 3-manifold with linking matrix L takes the form (cf. equation(123))
X
T−1
1
ZK [L] =eπi m (K ⊗L)m .
(42)|L|/2+1/2| det K|m∈Z|K||L| /(K⊗1)·Z|K||L|
17

This is why K has to have even integer entries on the diagonal. Otherwise, there would be transparent spin
1/2 anyons. Such theories with a transparent spin 1/2 anyon are called spin TQFTs.

16

SciPost Phys. 13, 067 (2022)

It is difficult to evaluate this in closed form. However, we now show that under certain conditions on the matrices K and L we can reinterpret (42) as the S 3 partition function of an
auxiliary theory, which can then be computed using (41). Similar calculations have been donein [55].
The Auxiliary Theory
The most general situation in which we can recast (42) as the S 3 partition function of an
auxiliary theory is when the matrices K ⊗ 1 and 1 ⊗ L are coprime. Two n × n integral matrices
C and D (both assumed to have non-zero determinant) are called coprime if there exist integral
matrices A and B such that [56]ADT − BC T = 1 .
(43)
When C DT is an even symmetric matrix (this is often referred to as a symmetric pair), there
exists an integral symplectic matrix whose lower row is (C, D). In other words, there exists a“preferred” choice of A and B in (43) such thatAT D − C T B = 1 ,AT C = C T A ,B T D = DT B .

(44)

In the context we are interested in, we take C = K ⊗ 1 and D = 1 ⊗ L. These matrices commute and each of them is symmetric, and hence they form a symmetric pair. This symmetricpair is even because the elements on the diagonal of K are even. Hence coprimality is equivalent to demanding the existence of integral matrices A and B satisfying (44). In Appendix C,
we prove that such A and B exist if and only if gcd(| det K|, | det L|) = 1. Since in the context
of surgery | det L| is the order of H1 , the condition gcd(| det K|, | det L|) = 1 is precisely the one
we are interested in for the question of gapped boundaries, namely gcd(|G|, |H1 (M )|) = 1.
Assuming comprimality, we now interpret (42) as the S 3 partition function of an auxiliary
Abelian theory. In particular, the auxiliary theory is one whose anyons generate the Abeliangroup Z|K||L| /(K ⊗ 1) · Z|K||L| , with spinsT−1

θ (m) = eπi m (K

⊗L)m

.

(45)

For this data to define a legitimate theory, we must check that the braiding is a bilinear map
with θ being its quadratic refinement, and that the resulting braiding matrix is non-degenerate.
θ (z y)θ (x y)θ (z x)
Regarding the first check, it is immediate that θ (z x y) = θ (x)θ ( y)θ (z) , and hence the braiding
is bilinear. As for non-degeneracy, we have to check that (117) holds. This requires that
X
T
−1
T
−1
e2πi a (K ⊗L)c + 2πi b (K ⊗L)c = | det K||L| δ (a + b) ,(46)
c∈Z|K||L| /(K⊗1)·Z|K||L|where¨δ (a) =

1

if a ∈ (K ⊗ 1) · Z|K||L|0

otherwise.

To show this, we can trivially rewrite the LHS asX
T−1
e2πi [1⊗L·(a+b)] (K ⊗1)c = | det K||L| δ ((1 ⊗ L) · (a + b)) ,c∈Z|K||L| /(K⊗1)·Z|K||L|

(47)

(48)

where the sum was evaluated by interpreting it in terms of the braiding matrix of a theory
consisting of |L| copies of the original theory defined by K. Since the original theory was
assumed to have a non-degenerate braiding, we were able to evaluate the sum using (117).
If we could now show that δ ((1 ⊗ L) · a) = δ (a), we would have successfully proven the
non-degeneracy of the braiding for the auxiliary theory.
To do so, suppose that17

SciPost Phys. 13, 067 (2022)(1 ⊗ L) · a = (K ⊗ 1) · m for some integer vector m. Setting C = K ⊗ 1 and D = 1 ⊗ L, by
the assumption of coprimality there exist integer matrices A and B such thatAT D − C B = 1 ,(49)

where we have used the fact that C is symmetric. Acting with both sides of the equation on
the vector a, we find AT C · m − C B · a = a. But we also know that AT C = CA (from (44) andthe fact that C is symmetric) and hence(K ⊗ 1) (A · m − B · a) = a ,(50)which means that δ ((1 ⊗ L) · a) = δ (a) as we wanted.
We have thus learned that the spins given in (45) give a consistent quadratic refinementof a bilinear map on Z|K||L| /(K ⊗ 1) · Z|K||L| . It follows that the auxiliary theory proposed is alegitimate Abelian TQFT, and that (42) can be interpreted as its S 3 partition function. We maythen use the general S 3 result (41) to conclude thatXÆT −1
eπi m K ⊗Lm = | det K||L| ξK,L ,(51)m∈Z|K||L| /(K⊗1)·Z|K||L|
for some 8th root of unity ξK,L . It follows that
ZK [L] =Æ11| det K||L| ξK,L =ξK,L .
|L|/2+1/2| det K|
| det K|1/2(52)

This proves that the phase of ZK [L] is always an 8th root of unity, consistent with the factthat 8 copies of any Abelian theory has a gapped boundary [4, Section 5.3]. We have also
determined the absolute value of these partition functions.
We now prove that if the original theory defined by K has a gapped boundary, then ξK,L = 1.
To do so, we assume that the original theory has a Lagrangian subgroup of anyons
A ⊂ Z|K| /K · Z|K| , where θ (a) = 1 for all a ∈ A (and in particular this means that B(a, b) = 1
for all a, b ∈ A). Then in the auxiliary theory, A⊗|L| is also a Lagrangian subgroup.
To prove A⊗|L| is Lagrangian, we have to show that it has the correct size and moreover the spin of allpanyons in this subgroup is trivial. The first property follows easily since|A⊗|L| | = |A||L| = | det K||L| . For the second property, note that an arbitrary anyon in thissubgroup can be represented by a vector m = m1 ⊕ · · · ⊕ m|L| ∈ Z|K||L| for some mi representing
an anyon in A. To show that the spin of m given by (45) is trivial, note thatX
1 T −11X
m (K ⊗ L)m =L i j mTi K−1 m j +L ii mTi K−1 mi ∈ Z .
2
2
i< j
i

(53)

Above we have used 21 mTi K−1 mi ∈ Z, which is true because A is Lagrangian. Thus the auxiliary
theory must have vanishing chiral central charge, and hence its S 3 partition function, namely(52), must have ξK,L = 1. This gives an alternative proof of Theorem 2.3.

3

Non-Abelian TQFTs

In this section we discuss the existence of topological boundary conditions for non-Abelian
unitary 2+1d TQFTs. As before we set c− = 0 (which means that the discussion applies totheories with c− ∈ 8Z after an appropriate stacking with copies of the invertible (E8 )1 CS
theory) and work with framing-independent topological theories.

18

SciPost Phys. 13, 067 (2022)AshrinkFigure 1: Anyon A from gapped boundary condition.

3.1

Lagrangian Algebra Anyons

For general non-Abelian TQFTs, the question of whether a gapped boundary exists cannot be
reduced to a question about a Lagrangian subgroup of anyons. However, there is a related
object known as a Lagrangian algebra anyon which will allow us to make statements about
gapped boundaries in the non-Abelian case.
Let us assume that a gapped boundary exists in a given non-Abelian TQFT. Then we can cut
out a small cylindrical tube and introduce the topological boundary condition on the surfaceof the tube. Since the boundary condition is topological we can change the radius of thecylindrical tube at will. Shrinking it must therefore define a line defect, which is equivalent toa direct sum of simple anyons (see Figure 1)MA=
Z0a a ,
(54)a∈I

where Z0a are some non-negative integers and I is the set of simple anyons.
The vector Z0a obeys many nice properties. Below we will show that Z0a is an eigenvectorof the S- and T -matrices of the TQFT with eigenvalue 1. A corollary is that the symmetricmatrix Za b := Z0a Z0b commutes with the S and T matrices.
We can compute the integers Z0a by considering the S 2 × S 1 partition function with theinsertion of parallel anyons A and ā along S 1 [20], as shown in Figure 2. On the one hand,
this partition function is equal to the dimension of the Hilbert space on S 2 punctured by A andā, which is equal to dim Hom(A ⊗ ā, 1) = dim Hom(A, a) = Z0a . On the other hand, viewing
A as the empty cylindrical tube, this configuration is topologically equivalent to the solid torus
D2 × S 1 with the gapped boundary condition on its boundary and the insertion of the anyon aalong S 1 . From this point of view the partition function is equal to the dimension of the disk
Hilbert space punctured by a, and equating the two results gives
Z0a = dim H(D2 ; a) .

(55)

By the state/operator correspondence, the Hilbert space H(D2 ; a) is the same as the space of
operators living at the intersection of the line a with the boundary. Therefore Z0a counts thenumber of distinct ways that a can end on the gapped boundary. In particular Z0a 6= 0, if andonly if the line a can end (condense) on the boundary.
The anyon A defined above has various special properties that we will explore in the remainder of this subsection. Any anyon with these properties is known as a Lagrangian algebra [4], so we see that the existence of a gapped boundary implies the existence of a Lagrangianalgebra anyon. By introducing the notion of gauging the algebra anyon, we will see how theoriginal TQFT turns into a trivial theory.
Z00 = 1First we show that the trivial anyon must be contained in A. To prove this, we must showthat there exists a non-zero morphism between A and the trivial anyon, meaning that the19

SciPost Phys. 13, 067 (2022)A
unshrink−−−−−→

−−−−→

a

a

a

Figure 2: The anyons a and A wrap an S 1 (not shown) and are located at points on
S 2 . Replacing the algebra anyon A with an empty tube turns the S 2 into a disk, andthe partition function of this configuration gives the dimension of the Hilbert spaceon D2 with the insertion of a.

Lagrangian algebra anyon A can end. This is most easily seen by replacing A with an empty
tube, and noting that we can cap off the empty tube. For instance, we can imagine putting the
gapped boundary condition on the boundary of a three-dimensional ball which is topologicallyequivalent to an elongated cigar. By shrinking the width of the cigar, the configuration canbe interpreted as the anyon A morphing into the identity line on both ends. This means that
Hom(A, 1) is non-empty, and thus Z00 6= 0.
One can further argue that Z00 = dim H(D2 ) = 1 for simple gapped boundary condition. Indeed, we will now show that if the disk Hilbert space is not one-dimensional, then theboundary condition is not simple and can be decomposed into the direct sum of Z00 simple
gapped boundary conditions. To show this, first recall that by the state/operator correspondence H(D2 ) can be identified with the algebra of boundary (point) operators. These boundary
operators are topological and live on a two-dimensional surface. Hence their “OPE" defines acommutative Frobenius algebra18 analogous to the algebra of point operators in 1+1d TQFTs(see for instance [57]). In other words, H(D2 ) can be identified with the S 1 Hilbert space of
a 1+1d TQFT constructed from the topological boundary condition. When we fix the threedimensional bulk to be a genus-g handlebody, we can take the topological boundary condition
to be this 1+1d TQFT on a genus-g surface.
Assuming unitarity (reflection-positivity), the Frobenius algebra H(D2 ) must be semisimple (separable) [57, 58]. Therefore it must contain a complete set of projection operators(idempotents)"1 , "2 , . . . , " Z00 ∈ H(D2 ) ,"i " j = δi j "i .
(56)
When inserted on the boundary, the topological boundary operator "i ∈ H(D2 ) will project
the gapped boundary condition onto its i-th simple component. In other words, smearing (or,condensing) "i over the boundary, which is equal to inserting one, defines a boundary condition
that is contained in the original boundary condition as a summand. Hence we have establishedthat in a unitary theory any topological boundary condition decomposes into simple boundary
conditions with one-dimensional disk Hilbert spaces.
TZ = Z
Next we show that Z0a must satisfy (T Z)0a = Z0a . (This also means that Zab := Z0a Z0b commutes with the T -matrix.) Since the boundary is gapped, Dehn twisting leaves the partition
function invariant. Hence for all a with Z0a 6= 0 we have θ (a) = 1. In other words, the anyons
18
For instance, the unit and the trace of this algebra is given by the 3-ball partition function with boundary
insertions of these local operators.

20

SciPost Phys. 13, 067 (2022)

unshrinkA−−−−−→aa
D2 × S 1

S3

Figure 3: The Hopf link between a and A can be evaluated to (S Z)0a . On the other
hand, replacing A by an empty tube gives a solid torus with a wrapping the noncontractible cycle, which can be evaluated to the dimension of the Hilbert space on
D2 with an insertion of a, i.e. Z0a .

that make up A must have zero spin, and henceX
Tab Z0b = Z0a .

(57)

b

SZ = Z
Furthermore, Z0a must satisfy (S Z)0a = Z0a . (As we remarked earlier, from this also follows
that Za b := Z0a Z0b commutes with the S matrix.) To see this, consider the S 3 partition function
with the insertion of the Hopf link between A and a, as shown in Figure 3. This is equal to
the Hopf link amplitude, times the S 3 partition function without any insertions (the latter ofwhich is equal to S00 in theories with a vanishing chiral central charge mod 8) [20]. Splitting
the algebra anyon A via (54) and then applying (104), we get X
Z S 3 ; Hopf(A, a) =Sab Z0b .
(58)
b

On the other hand, thinking of A as the empty tube this configuration is topologically equivalent to the solid torus with the gapped boundary condition on its boundary and the anyon ainserted along its longitude. In this case the partition function evaluates to the dimension of
the disk Hilbert space punctured by a, and using (55) we conclude thatX
Sab Z0b = Z0a .
(59)b

We have therefore proven that the matrices S and T have a common non-negative integer
eigenvector Z0a with eigenvalue 1. Furthermore, Z0a can be shown to obey [15]:XcZ0a Z0b ≤
Nab
Z0c .
(60)
c

The conditions (57), (59), and (60) are only necessary conditions for the Lagrangian algebra anyon, but not sufficient [59]. More generally, for the genus-g surface Σ g , there exist
a vector |A〉 ∈ H(Σ g ) that is preserved by all genus-g mapping class group (MCG) transformations. The state |A〉 is defined by the path integral on Σ g × [0, 1], where we put the
topological boundary condition on Σ g × {0}. The path integral prepares the boundary state
21

SciPost Phys. 13, 067 (2022)|A〉 on the Hilbert space on the other side of the interval Σ g × {1}. Since MCG transformations
do not change the topology of the boundary, they act trivially on the topological boundary
condition. This
P shows that indeed |A〉 is a singlet of the MCG representation. For g = 1 wehave |A〉 = a Z0a |a〉, where |a〉 is the state prepared by the path integral on the solid torus
with an insertion of anyon a wrapping its non-contractible cycle.
Quantum dimension dim(A)The existence of the eigenvector Z0a is very constraining. In particular, such an eigenvector
can only exist if c− = 0 mod 8. To see this, note that S and T satisfyc−(S T )3 = e2πi 8 S 2 .

(61)

Then we can act on both sides of this equation on Z, and find that this is only consistent if
c− = 0 mod 8.
Moreover,P setting a = 0 in (59), we find that the quantum dimension of A, denoted bydim(A) = a Z0a da , is equal to the total quantum dimension
dim(A) =

vX
t

da2 .

(62)

a

−1
Here we have made use of (106) (fixing b to the identity tells us that da = S0a S00), together
3with (105). These statements can be translated to facts about the S partition function. Indeed, the 3-sphere can be obtained by gluing two solid tori along their boundary with an
S-transformation, and thus
1
Z[S 3 ] = S00 =.
(63)
dim(A)

Another way to obtain (63) is to consider an unknot of A inside S 3 . On the one hand theresult is by definition Z[S 3 ]dim(A). On the other hand, we can blow up the unknot and obtaina disc with topological boundary condition and no insertion. The partition in this presentation
is manifestly 1 (note that Z00 = 1). We have therefore derived (63).
Note that all of these constraints nicely generalize facts that we have explained in great
detail about the Abelian theories. For instance, in the Abelian case A is nothing more than adirect sum of the lines in the Lagrangian subgroup, each appearing with multiplicity 1. Above
we have rederived the facts that the spins of the lines in the Lagrangian subgroup all vanish andthat the dimension of the Lagrangian subgroup must be the square root of the total number ofanyons.
AAA
FA
and RAA
A

Additional constraints on the Lagrangian algebra A arise if we consider carving out junctionsof cylinders (pairs of pants), such that the boundary is our gapped boundary condition. Clearly
since the boundary is topological, we can do arbitrary smooth transformations of these junctions and find “trivial" F and R moves for A, as shown in Figure 4. More precisely, let |µ〉 ∈ VAAA
be the junction vector/operator between three A specified by putting the gapped boundary
condition on a pair of pants. Then we have (cf. equation (97))
AAAFA· |µ〉 ⊗ |µ〉 = |µ〉 ⊗ |µ〉 , and RAA(64)
A · |µ〉 = |µ〉 ,AAwhich is consistent since A has zero spin.
Mathematically, |µ〉 ∈ Hom(A ⊗ A, A) satisfying the above conditions define an associative
commutative algebra structure on A and is referred to as the multiplication of the algebra [60].
22SciPost Phys. 13, 067 (2022)

µ

−→

←−

=

µ

µ
−→

µ

µ

←−

−→

=

µ

←−

µ

Figure 4: Replacing the A anyons with empty tubes makes it clear that fusion andbraiding are trivial. This gives a geometric interpretation of the defining axioms of a
Lagrangian algebra A.

A Lagrangian algebra object is an associative commutative algebra with a unique unit, that hastrivial spin and satisfies the Lagrangian property (62). For more details, see Definition 1.1 andFigure 2 of [60]. Here we have provided a geometric interpretation of these defining axioms
in Figure 4.
RCFT Interpretation
Finally, we close this subsection with a discussion of what the topological boundary conditionmeans for the boundary RCFTs.
For simplicity, we assume the bulk TQFT to be a Chern-Simons theory. The Chern-Simons
theory has a standard Dirichlet boundary condition that supports the chiral WZW model. Let
the space be a disk with the standard Dirichlet boundary condition, and insert an anyon a at
a point in the bulk of the disk. This corresponds to the character χa of the boundary chiralalgebra [46, 61].
Next, we make another hole in the disk with the conjugate boundary conditions. This thengives rise to the diagonal modular invariant partition function of the boundary WZW model:
X
Z[τ, τ̄] =χa (τ)χ̄a (τ̄) .
(65)aIn other words, this is the same as compactifying the Chern-Simons theory on an interval with
the Dirichlet boundary condition for the gauge fields on the two boundaries of the interval.
Now let us further assume that the Chern-Simons theory admits a topological boundary
condition. Then we can consider another configuration where on the second hole we impose

23

SciPost Phys. 13, 067 (2022)

the topological boundary condition. This leads to a new modular invariant partition function
X
Z[τ] =
Z0a χa (τ) ,(66)
awhich is purely holomorphic. Indeed, the new topological boundary can be collapsed to a
direct sum of anyons and hence introduces no new τ̄ dependence.
More generally, we can put such a Chern-Simons theory on a general Riemann surface
times an interval. We impose the standard Dirichlet boundary condition on one end, and the
topological boundary condition on the other end. The compactification of the interval thengives a holomorphic CFT. We conclude that the existence of a topological boundary condition
implies that the chiral algebra of the boundary RCFT can be extended to a single module.19

3.2

Gauging the Lagrangian Algebra

We have seen that the existence of a gapped boundary implies the existence of an anyon A
with various special structures and properties. As previously mentioned, such an anyon iscalled a Lagrangian algebra.
While A generally consists of non-Abelian anyons, it is possible to talk about the gauging/condensation of A and argue that gauging A leads to a trivial theory. One can thenrecover the topological boundary condition by means of “Dirichlet boundary conditions" after
gauging A.
Let us explain how this gauging is done geometrically. When we gauge a one-form symmetry (i.e. condense Abelian anyons) we are instructed to sum over all possible network of said
anyons [40]. However, in the non-Abelian case it is no longer true that the only fusion channel of A and A is inside A, and there is no subgroup (or more precisely, fusion subcategory)structure. Summing over general knots made out of A then generally leads to contradictions
because the crossing move is nontrivial.
Instead of summing over all possible configurations of anyons, by gauging a Lagrangianalgebra we will mean inserting a fine mesh of A. A fine mesh is defined as the graph that is dual
to a triangulation of the space-time manifold. More specifically, let M1 be the 1-skeleton of atriangulation of a 3-manifold M , i.e. the union of the edges and vertices of the triangulation
Ò1 be the dual 1-skeleton, i.e. the fine mesh. Now consider fattening M1 totetrahedra. Let M
get a regular neighborhood N of M1 which is topologically a handlebody. Moreover, deletingÒ = M − int(N ) which is also a handlebody, and which can be
the interior of N from M we get NÒ1 .20 Inserting the algebra anyon A on the fine mesh is
obtained by fattening the fine mesh MÒ from the space-time and putting the gapped boundary condition onequivalent to deleting N
the boundary of N .
Therefore after inserting A on the fine mesh, we are left with a handlebody N of genus
g = e − v + 1 (where e and v are the number of edges and vertices of M1 ) and with the gappedboundary condition on ∂ N . To compute this partition function we can use the standard cutting
and gluing similar to the one used in the context of 1+1d TQFTs [58]. Since we are assumingthat the gapped boundary is simple, the Hilbert space on the disk D2 is one-dimensional andthe partition function isZ[N ] = Z[D3 ]1−g ,(67)19Note a subtlety regarding the modular invariance of (66): the invariance under the S transformation is guarc−
anteed by construction, but the T -matrix in 1+1d differs by e2πi 24 from the T -matrix of the MTC and hence theinvariance of (66) under T transformations is only guaranteed if c− = 0 mod 24 and not just c− = 0 mod 8 (cf.
(110)). Alternatively, we could always add some copies of (E8 )1 to correct the issue.
20
Ò is known as a Heegaard splitting. See for
The decomposing of the 3-manifold M into handlebodies N and N
instance [62, Chapter 9].

24

SciPost Phys. 13, 067 (2022)

where D3 is the three-dimensional ball. Since Z[D3 ] is positive by unitarity, we can set
Z[D3 ] = 1 by adding the appropriate Euler counter-term on the boundary.21 Therefore, we
see that after inserting a fine mesh of A the partition function on any 3-manifold trivializes,
which means that gauging a Lagrangian algebra anyon trivializes the theory.22
We can now define a topological interface between the original theory and the one obtained
from gauging A. This interface is defined by putting the “Dirichlet boundary conditions" forA, meaning that the mesh is terminated on the interface. More precisely, consider the original
theory in region M of the space-time that is connected via a codimension-one interface toregion M 0 where the gauged theory lives. To define the topological boundary condition, we
first put the original theory on the whole space-time, i.e. on M ∪ M 0 . We triangulate thespace-time including the codimension-one interface. Now we can insert A on the fine meshin region M 0 , which is the graph dual to the tetrahedra of M 0 . Such a configuration defines a
topological boundary condition for the original theory, since as we argued above the gauged
theory is trivial.
The above discussion gives an unambiguous procedure to compute the partition functionsof the gauged theory in terms of those of the original theory decorated by the anyons. It is
however not entirely clear to what extent it can really be interpreted as gauging some generalizations of global symmetries. Among other things, there is no clear notion of a backgroundgauge field for A. We leave this point for future investigations.

3.3

Topological Boundaries and the Turaev-Viro TQFT

In this subsection, we make contact with some known facts about the relation between the
Lagrangian algebra, topological boundary conditions, and the Turaev-Viro TQFT.
For Abelian TQFT, Theorem 2.1 states that the existence of a topological boundary condition is equivalent to the existence of a Lagrangian subgroup. This theorem generalizes to
non-Abelian TQFTs [4, 6]: a bosonic TQFT has a topological boundary condition if and only if
it has a Lagrangian algebra A. In Section 3.1 we gave a geometric interpretation of the only
if part of the theorem.
What is the generalization of Corollary 2.2 that an Abelian TQFT admits a topologicalboundary if and only if it is an Abelian Dijkgraaf-Witten theory? It is natural to ask if every
non-Abelian TQFT with a topological boundary can be viewed as the pure gauge theory ofsomething. The answer is given by the Turaev-Viro TQFT, which we briefly review below.
For any unitary fusion category F , there is a state-sum 2+1d TQFT known as Turaev-Viro(Barret-Westbury) theory [64, 65]. This class of TQFTs can be realized as the low-energy limitof the Levin-Wen string-net lattice model [66, 67].
For instance, when the lines in the fusion category F are all invertible (F = VecωG for3some ω ∈ H (G, U(1))), the Turaev-Viro TQFT reduces to the discrete G-gauge theory with
Dijkgraaf-Witten twist ω [25].
For more general unitary fusion category F , the corresponding Turaev-Viro TQFT canbe thought of as F -gauge theory in the following sense. One begins with the trivial theory
equipped with trivially acting topological surface defects with the fusion rules of F , and onethen gauges these topological defects to obtain the Turaev-Viro TQFT [68].23
Now we are ready to state another sufficient and necessary condition for the existence of a
topological boundary condition: A TQFT admits a topological boundary condition if and only
21
In the algebra language, this is equivalent to normalizing the multiplication such that composing the unit andcounit give the quantum dimension of A, i.e. βA = 1 in the notation of [63].
22Note that any associative commutative algebra with a unique unit and trivial spin in a unitary braided fusion category B can be gauged. The Lagrangian property (62) only implies that after gauging all the nontrivial
topological lines in B disappear.
23
The operation of gauging topological surface defects is the inverse operation of gauging topological line defects.

25

SciPost Phys. 13, 067 (2022)

if it is a Turaev-Viro TQFT [6,7]. Given that the latter can be viewed as a pure gauge theory ofa unitary fusion category F , this statement is the non-Abelian generalization of Corollary 2.2.
Summary on the Topological Boundary Conditions for Non-Abelian TQFTs
Similar to the summary at the end of Section 2.3 for the Abelian TQFTs, we summarize the sufficient and necessary conditions for topological boundary conditions for general non-Abelian
TQFTs. If we work modulo invertible field theories (such as the (E8 )1 CS theory), then thefollowing conditions for a bosonic TQFT are equivalent:• It admits a topological boundary condition.
• It has a Lagrangian algebra.
• It is a Turaev-Viro TQFT.24
It would be interesting to find a complete list of obstructions similar to ξn for the Abelian
TQFTs .

3.4

The Galois Action and the Higher Central Charges

We have just seen that the existence of a Lagrangian algebra anyon A is necessary and sufficientcondition for the existence of a gapped boundary. But in practice it is not easy to check whether
such an A exists in a given theory, since this requires knowledge of e.g. the F -matrices.
We will now present some necessary conditions which do not require the F or R matrices,
but only depend on the modular data, i.e. the S- and T -matrices. In particular, we will discuss
the non-Abelian generalizations of the higher central charges ξn which have been proved tobe obstructions to topological boundary conditions [23].
One simple but very restrictive condition that we have already discussed is the existenceof an integer-valued vector Z0a satisfying
SZ = Z ,TZ = Z ,Z00 = 1 .

(68)

The rest of this subsection will be dedicated to deriving some more subtle conditions. To
introduce these conditions, some background information will be necessary.
c
Say that we fix a basis of simple anyons I = {a, b, c, . . . } with Nab∈ Z≥0 . The space of MTC
data consistent with this basis choice is given by the space of solutions to a set of polynomial
equations, such as the pentagon and hexagon identities. This means that the R, F, S, and T
matrices take values in a certain field extension of Q. Associated to this field extension is a
Galois group, and we can use Galois conjugation to map a given set of MTC data to a new setof MTC data [71, 72] (see also [73, 74]).
For the modular data in theories with vanishing chiral central charge c− = 0 mod 8, therelevant field extension and Galois group are [75, 76]
 2πi Q(T, S) = Q e NFS ,Gal(T, S) = Z∗N ,(69)
FS

where Z∗N is the multiplicative group consisting of all elements n ∈ ZN such thatgcd(n, N ) = 1.25 Here NFS is the Frobenius-Schur exponent, i.e. the smallest integer suchthat θ (a)NFS = 1 for of all anyons a.
24

A Witten-Reshetikhin-Turaev type TQFT whose MTC is the Drinfeld center of a unitary fusion category isequivalent to a Turaev-Viro TQFT based on the same fusion category [69, 70].
25
If the chiral central charge does not vanish, the S matrix may contain elements which are not in the above
field extension.

26

SciPost Phys. 13, 067 (2022)

Let σ be an element of Gal(T, S). By abuse of notation, we will identify it with the corresponding integer number σ mod NFS. Anytime we encounter an NFS-th root of unity ζ, we
simply raise it to the corresponding power to obtain its Galois conjugate,
σ

N
FS
X
n=1


qn ζ

n

=

NFS
X

qn ζσn ,where qn ∈ Q .

(70)

n=1

The T -matrix therefore transforms in an obvious way
σ(T ) = T σ ,(71)since σ(θa ) = θaσ . The S-matrix transforms in a more complicated way [72]. There is a grouphomomorphism from the field extension into signed permutations of the labels I , such thatthe element σ ∈ Z∗N maps to some permutation σ : I → I (by further abuse of notation)
FSobeyingσ(Sa b ) = εσ (a)Sσ(a)b = εσ (b)Saσ(b) ,(72)
with εσ (a) = ±1. The fact that the Galois action on the S-matrix simply induces a (signed)
permutation of the anyons is not obvious. Note that if we were to consider the ratios Sab /Sa0
then the action of the Galois group would reduce to just a permutation of the a anyon.
Another important result is the congruence subgroup property of the modular representation defined by the S- and T -matrices [75, 76]. Note that in general the S- and T -matricesc−define a projective representation of the modular group because of the factor of e2πi 8 in (109).
However, when c− = 0 mod 8 this factor is trivial and we get an ordinary representation ρ ofSL(2, Z) given by




1 1
0 −1
7→ T .
(73)
7→ S ,
ρ:
ρ:
0 1
1 0
The congruence property then says that the kernel of this modular representation contains theprincipal congruence subgroup Γ (NFS), defined by


 



1 0
a b
a b
(mod N ) .
(74)=∈ SL(2, Z) :
Γ (N ) =
c d
0 1
c d
Note that Γ (N ) is just the kernel of the linear map πN : SL(2, Z) → SL(2, ZN ) defined by taking
the mod N reduction of the elements of SL(2, Z). Therefore the modular representation ρ
factors through πNFS , and hence can be thought of as a representation of SL(2, ZNFS ).26 This
allows us to compute the Galois conjugate modular representation ρσ in terms of ρ, without
knowing the signed permutation:




a b
a σb
ρσ
=ρ
,
(75)
c d
σ̄c d
where σ̄ ∈ Z is a multiplicative inverse of σ modulo NFS, i.e. σσ̄ = 1 (mod NFS).
We can write the Galois action on the S- and T -matrices as σ(S) = Gσ S = SGσ−1 andσ(T ) = T σ , where (Gσ )a b = εσ (a)δσ(a)b . Using (75), we find an alternative expression forσ(S) that doesn’t involve εσ (a):σ(S) = T σ S T σ̄ S T σ .
It can further be shown that σ2 (T ) = Gσ T Gσ−1 [76, Theorem II].
26

More precisely, there exist a representation ρNFS of SL(2, ZNFS ) such that ρ = ρNFS ◦ πNFS .

27

(76)

SciPost Phys. 13, 067 (2022)

Galois Action on Lagrangian Algebras
If the original theory has a gapped boundary, then as we have discussed there exists a LaL
grangian algebra anyon A = a Z0a a, and hence a non-negative integer eigenvector Z satisfying S Z = Z. We will now show that upon Galois conjugation, the algebra anyon is left
unchanged!
Indeed, we may begin by doing a Galois conjugation to both sides of the equation S Z = Z,givingσ(Sab )Z0b = Z0a .
(77)
Here we have used the fact that the Z0a are integers, and thus are left invariant by Galois
conjugation. Note that (77) is equivalent to εσ (a)Sσ(a)b Z0b = Z0a . Using S Z = Z once again,
we can then derive a constraint
Z0σ(a) = εσ (a)Z0a .
(78)
L
Since the algebra anyon is given by A = a Z0a a, this means that the permutation can only
send anyons in A to other anyons in A with the same multiplicity. In addition, since Z is
positive, εσ (a) = 1 when a is contained in A, and hence we conclude that
Z0σ(a) = Z0a⇒σ(A) = A .

(79)

In other words, the permutation induced by the Galois group is constrained in such a way thatit preserves the algebra anyon!
The various nice properties of A are also preserved under Galois conjugation. For example,
since under Galois conjugation the spins are simply multiplied by σ, and since the original
spins vanished for A, the same is also true after the conjugation, i.e. σ(T )Z = Z. We havealso seen in (77) that σ(S)Z = Z. Finally, since for the algebra anyon the F and R matrices
are “trivial", the same will remain true after conjugation. More precisely, we must show thatfor the Galois conjugate theory there exists a |µ〉 ∈ VAAA satisfying (64). To show this, firstnote that since the fusion coefficients do not change under Galois conjugation, we can identify
the fusion vector spaces Vca b before and after Galois conjugation. The multiplication of thealgebra is just a particular vector|µ〉 ∈ VAAA =

Z0c
Z0a M
Z0b M
M M
a,b,c∈I α=1 β=1 γ=1

In this fixed basis (64) is equivalent toAAA〈ν ⊗ ρ| FA|µ ⊗ µ〉 = δνµ δρµ ,AA

andVcab .

〈ν|RAAA |µ〉 = δνµ ,

(80)

(81)

for all |ν〉, |ρ〉 ∈ VAAA . Since the numbers on the RHS of equations in (81) are integer, they do
not change under Galois conjugation. Hence the algebra multiplication |µ〉 is preserved under
Galois conjugation.
Obstructions from Galois Conjugation
Having shown that Galois conjugation maps a theory with a Lagrangian algebra to anothertheory with such an algebra, we are ready to draw some conclusions regarding gapped boundaries.
We start from a TQFT that admits a topological boundary condition. Since its (possibly nonunitary) Galois conjugate theory also admits a topological boundary, we expect that similar to
the unitary case its S 3 partition function
Zσ [S 3 ] = σ(S00 ) ,
28

(82)

SciPost Phys. 13, 067 (2022)

is positive. Here Zσ [S 3 ] stands for the S 3 partition function of the Galois conjugate theory.
Indeed, σ(S00 ) is positive. The proof is given as follows: noting that σ(S00 ) = εσ (0)Sσ(0)0 andusing Z00 6= 0 as well as the constraint (78) to find εσ (0) = 1, we findZσ [S 3 ] = Sσ(0)0 ,(83)

which is positive.
We can now obtain some necessary conditions for the existence of gapped boundaries inour original theory by computing the S 3 partition function of its Galois conjugate using a
different surgery presentation. In particular, S 3 is homeomorphic to the lens space L(1, 1),
and therefore (cf. equation (124))Zσ [S 3 ] = σ(S T S)00 .
Of course if the original theory has a gapped boundary then it has a vanishing chiral central
charge and hence (S T S)00 = S00 , which is an identical expression to what we wrote in (83).
Using (75) we find




−1 0
−1 0
⇒ σ(S T S) = ρ= S T σ̄ S .
STS = ρ1 −1
σ̄ −1Thus the S 3 partition function of the Galois conjugate theory is (S T σ̄ S)00 , which is just theL(σ̄, 1) partition function of the original theory. This gives us a prediction for the partition
functions on lens spaces:Z L(σ, 1) = Sσ̄(0)0 ≥ S00 ,(84)
which in particular is positive. Therefore we find that if a theory has a gapped boundary thenits lens space partition function, which admits the simple formula
1 X 2 σZ L(σ, 1) = 2d θ ,D a a a(85)

must be positive when gcd(σ, NFS) = 1.
To summarize, the higher central charges for general non-Abelian TQFTs are defined as
P 2
σa da θ (a)
P
(86)
ξσ :=
.
| a da2 θ (a)σ |A TQFT admits a topological boundary condition only if ξσ = 1 for all gcd(σ, NFS) = 1 [23].
Moreover, the inequality
qPon the RHS of (84) implies the following interesting number2
theoretic property of D =a da ,D ≥ σ(D) > 0 .

(87)This means that all roots of the minimal polynomial of D are positive and less than or equal
to D.
RCFT Interpretation via Galois Conjugation
We close by briefly mentioning the relation of these obstructions to RCFT. The Galois action
defined above on the modular data S and T extends in the straightforward way to 1+1 dimensions. At the level of the RCFT characters, it was shown in [74, 77] that the Galois transformation given by σ ∈ Z∗N induces the action of the Hecke operator Tσ̄ , which is defined asFSfollows. Consider an RCFT with d characters, which may be organized into a d-dimensional
29

SciPost Phys. 13, 067 (2022)

vector-valued modular function χ = (χ1 , . . . , χd ). Then the σ-th Hecke operator Tσ for anyeFS) = 1 is defined as27
prime σ such that gcd(σ, N(Tσ χ)a (τ) = εσ̄ (a)χσ̄(a) (στ) +

σ−1
X
j=0

χa



eFS
τ + jN
σ


.

(88)

eFS but not necHaving the expression of Tσ for prime σ, Hecke operators for σ coprime to N
essarily prime are constructed in appendix of [77].
The action of the Hecke operator on the characters gives a new set of characters, which
may be interpreted as those of the Galois conjugate RCFT. An illustrative example is to consider
the action of Tσ on the characters of the Lee-Yang minimal model; one finds that the actionof T7 , T13 , and T19 gives rise to the respective characters of (G2 )1 , (F4 )1 , and (E7 1 )1 , all of2
which are in the same Galois orbit.
With this in mind, we may reinterpret the phases of Z[L(σ, 1)] discussed above as theusual chiral central charges of the 1+1d RCFTs related by the Hecke transformation Tσ to theoriginal RCFT on the boundary. In other words, the higher central charges are just the usualchiral central charges for appropriate conjugate CFTs.
For Abelian TQFTs with a one-form symmetry group G, we had a larger set of obstructions in addition to the higher central charges. These additional invariants arise from threemanifolds with gcd(|H1 (M )|, |G|) = 1 (see Section 2.2). Furthermore, they also arisefrom lens
2|G|
spaces with an extended range of allowed n, i.e., those n such that gcd n, gcd(n,2|G|) = 1 (seeSection 2.3). We do not have a concise 1+1d interpretation for these higher obstructions.

Acknowledgements
We thank M. Barkeshli, N. Benjamin, F. Burnell, Y.-A. Chen, T. Devakul, D. Freed, P. Gorantla,
H. T. Lam, N. Seiberg, C. Teleman, J. Wang, and F. Yan for helpful discussions. We also thank
G. Moore for comments on the draft. ZK is supported in part by the Simons Foundation grant488657 (Simons Collaboration on the Non-Perturbative Bootstrap) and the BSF grant no.
2018204. SHS was also supported by the Simons Collaboration on Ultra-Quantum Matter,
which is a grant from the Simons Foundation (651440, NS).

A

Review of 2+1d TQFTs

In this appendix we review the classification of unitary 2+1d TQFTs. These theories are described by a pair
(C , c− ) ,(89)

where C is a unitary modular tensor category (UMTC) [78–82] and c− ∈ Q is the chiral central
charge28 of the boundary CFT. Note that C determines the chiral central charge c− modulo 8,
and hence C and c− are not independent. TQFTs defined by such data are known as WittenReshetikhin-Turaev theories [20, 83].
When c− 6= 0 the theory is not strictly speaking topological and the partition function has
a mild dependence on the geometry/metric.29 The metric dependence implies a perturbative27
eFS is the order of the RCFT T -matrix, which is related to the TQFT T -matrix used so far by
Note that Nc−−2πi 24eeFS | 12NFS .
T =e
T (cf. (110)). By [76, Theorem II] we have NFS | N
πk2

c− = c − c̄ is related to the thermal Hall conductance KH = c− 6ħhB T .
29
Note that the scheme that we are using here is different from Witten’s original quantization of Chern-Simons

28

30

SciPost Phys. 13, 067 (2022)

gravitational/diffeomorphism anomaly [84] on the boundary. So for c− 6= 0 the theory on theboundary cannot be gapped. Since in this paper we are interested in 2+1d theories with agapped boundary, from now on we only focus on theories with zero chiral central charge, i.e.
c− = 0.
In this work we are only interested in theories with a unique vacuum on S 2 . Note that vacuum degeneracy on S 2 leads to nontrivial topological point operators via the state/operator
correspondence. In a unitary theory the algebra of topological point operators is a semisimple
Frobenius algebra which has a complete set of idempotents (projection operators). These
projection operators correspond to different superselection sectors (universes) of the theory [85–87]. Hence any unitary 2+1d TQFT can be decomposed into theories with a onedimensional S 2 Hilbert space. So there is no loss of generality by restricting to theories with aunique vacuum on S 2 .
The line defects (a.k.a anyons) of a 2+1d TQFT are an important set of observables that
determine the theory uniquely. These line defects along with their various properties form theunitary modular tensor category C . Mathematically speaking, a category consists of a set ofobjects and a set of morphisms between those objects. The objects of the category C correspond to the line defects of the TQFT, while the morphisms between two objects correspondto junctions between the corresponding lines. A unitary modular tensor category has extra
structures besides the objects and morphisms. Below we review these structures in terms of
the line defects of the TQFT. For more details see Appendix E of [81] or Section 5 of [88].
1. Fusion:
a⊗b'

M

c
c.
Nab

(90)

c∈I

In a general fusion category, multiplication does not have to be commutative. However, in a UMTC, due to the existence of an invertible R-matrix (below) it is true thatc
Nac b = Nba. Nac b is interpreted as the dimension of the Hilbert space of the junction
cof three lines. There are two related vector spaces, Vcab and Vab, corresponding to theccsplitting and fusion vector spaces, respectively. They satisfy dim(Vcab ) = dim(Vab) = Nab.
Denoting the trivial anyon by 0 we can regard any anyon as itself being a vector spacea
Va0, seen by attaching the trivial anyon.
b ...bMore generally there are vector spaces Va11...anm corresponding to the fusion space ofn anyons into m anyons. These vector spaces have a natural basis in terms of tensor
cproducts of the elementary spaces Vcab and Vab. For instance, we may obtain any vector in
cde
Va b by first picking a vector in Vab for some e, and then another vector in Vecd . Therefore
M
cd
e
Vab
=
(Vecd ⊗ Vab
).
(91)
e

0
For every anyon a there is an anyon ā with the property that Vaāand V0aā are onedimensional vector spaces. In addition, there are canonical isomorphisms from e.g. Vcab
to Vācb , given by taking the a anyon in Vcab backwards in time. Requiring that these
isomorphisms are unitary gives a preferred normalization for the inner product on the
splitting and fusion spaces.

In this normalization, which is widely used in the literature, an unknot of a gives
da = a,(92)

theory in [20]. Witten adds a (not necessarily properly quantized) gravitational Chern-Simons term to cancel themetric dependent of the theory and instead introduces a framing dependence. In our scheme the partition function
does not depend on the framing of the 3-manifold.

31

SciPost Phys. 13, 067 (2022)

which is called the quantum dimension of a. By bending the lines and rotating the figure
cdby 90 degrees one finds an isomorphism between Vaband Vac̄b̄d , and further using thedecomposition to splitting and fusion spaces we find
X
X
f
f
e
e
Nba
Ncd
=
Nac̄ N .
(93)
b̄d

e

f

Using that Nac̄ = N fac = Ncaf and Nff= Nbdf , we rewrite the above equation as
b̄d

X

e
e
Nba
Ncd
=

e

X

Nbdf Ncaf .

(94)

f

Defining the matrices Na = (Na )cb , the above equation reads Nb Nc T = Nc T Nb , i.e.
[Nb , Nc T ] = 0 for all b, c. This guarantees that the matrices Nc are all mutually diagonalizable. An important fact is that the vector of quantum dimensions da satisfies
Xcda d b =
Nab
dc ,
(95)
c

i.e. d is an eigenvector of Na with eigenvalue da .
cd
Above we have used the fact that vector spaces Vaband Vac̄b̄d have the same dimension.
More generally, the vector space Vdabc admits distinct decompositions in terms of splitting
and fusion spaces:
M af
M
Vd ⊗ Vfbc .
Veab ⊗ Vdec ,
(96)
e

f

In addition to the requirement that these lead to a vector space of the same dimension,
L ab
L af
ec
bc
there exist fusion matrices Fda bc :which allow us toe Ve ⊗ Vd →f Vd ⊗ Vf
translate all vectors in one basis to the other. The F -matrices (6j-symbols)
af
Fda bc e f : Veab ⊗ Vdec → Vd ⊗ Vfbc ,(97)

are famously subject to the pentagon equation and also the triangle equation (the latter
simply ensures that no harm is done in attaching the trivial anyon at will).
2. R-matrices (Braiding):ccAbove we have already used the fact that Nab= Nba. In fact, there is an invertible linearabba
map between the vector spaces Vc and Vc :
ab
ba
Rab
c : Vc −→ Vc ,(98)which is pictorially represented as
a

a

b
µ

b

X
=
(Rab
c )µν
ν

c

ν

.

(99)

c

We can write the matrix in components as Rabc;µν . The R-matrix is subject to two hexagon
equations that ensure its compatibility with the F matrices. An important conceptual
point is that braiding an anyon a around an anyon b depends on their fusion channel

32

SciPost Phys. 13, 067 (2022)c, and not only on the type of the anyons a, b alone. Using the R-matrix it is possible todefine the topological spin as
1
1 X
dc Tr (Raa
c )=
da c
da

θ (a) =

a .

(100)

The topological spin is always a root of unity with θ (a) = θ (ā) [89]. We sometimes
write θ (a) = e2πisa .
For our purposes it is important to quote the following nontrivial property of the braiding
matrix R. Applying the matrix twice gives an automorphism of the splitting spaceabab
R cba Rabc : Vc −→ Vc ,(101)

and it turns out that this map is proportional to the identity map up to an overall phase
given in terms of the topological spins:
R cba Rab
c =

θ (c)
idV ab .
θ (a)θ (b) c(102)

The topological spins are constrained by the hexagon identities through their relation to
the R-matrix. It is possible to derive one general constraint that is independent of the
F -matrices and is given purely in terms of the the topological spins and the dimensions
Nac b of the fusion spaces. For arbitrary labels w, x, y, z this constraint is
Y

p

p̄

p

p̄

p

p̄

q
q̄
q N x y Nwz

Pθ (p)Nx y Nwz +Nxz Nw y +N yz Nwx = (θ (w)θ (x)θ ( y)θ (z))

.

(103)

p

3. Modular S and T Matrices:
From the above data, it is possible to construct S and T matrices which behave similarly
to the S and T matrices of two-dimensional conformal field theory. The components Sabof the S-matrix are defined by
Sa b =

1X c
θ (c)
1 a
N
d =
D c a b̄ θ (a)θ (b) c
D

where

D=

vX
t

b

−1
da2 = S00
,

,

(104)

(105)

a

is the total quantum dimension of C .30 We will also find the following result useful,
b

b

a

=

Sab
S0b

.

(106)

1
∗
S is a symmetric matrix and, in addition, Sā b̄ = Sab and Sāb = Sab. Note that S0c = D
dc
and in the same way that dc was an eigenvector of N, S x c is an eigenvector for every x

Sax S bx X c=
Nab S x c .
S0x
c
30

In the mathematical literature, dim(C ) =P

2a da is called the (global) dimension of C .

33

(107)

SciPost Phys. 13, 067 (2022)

A few additional definitions: C = δāb , Tab = θa δab , and very importantly,
e

2πic−
8

=

1X 2
d θ .
D a a a(108)

It requires a proof that the right hand side of (108) is a pure phase – we do not review thegeneral proof here, but we will prove the Abelian case later. The quantity c− is identified
with the chiral central charge of the edge modes. The UMTC only determines it mod 8,
which is an important fact we will come back to later.
These matrices satisfy31
S2 = C ,(S T )3 = e

2πic−
8

C2 = 1 .

C,

(109)

2πic

e = θa e− 24 − δab in terms of which we get
Note that it is possible to define T
S2 = C ,e )3 = C ,(S T

C2 = 1 .

(110)

e is the T -matrix in the RCFT.
Here the matrix T
For the classification of low-rank UMTCs see [82, 90].

The Abelian Case
An anyon a is called Abelian if da = 1. This is equivalent to the requirement that for any
other anyon x, Nxc a is non-vanishing for only one c.32 If all the anyons are Abelian the fusion
rules therefore lead to an Abelian group G, and the corresponding theories are called Abelian
TQFTs. The inverse of the anyon a is ā.
Next we must find θ (a) which is a map from the group to U(1),θ : G → U(1) .

(111)

The main constraint on θ comes from (103) if we plug w = x̄ and z = ȳ. In this case we find
θ x y θ x ȳ = θ x2 θ y2 .

(112)

A neat way to encapsulate the information in (112) is to define a map
B : G × G → U(1)such that
B(x, y) =θ (x y).
θ (x)θ ( y)

(113)

(114)

B is of course just the full braiding of anyon x around anyon y, c.f. (102). In the special
case of Abelian theories, the fusion channel of x, y is unique and hence the braiding phasecan be defined as a function of x, y only. From property (112) we immediately conclude that1
B(x̄, y) = B ∗ (x, y) = B(x,y) . In fact, B is a bilinear map on the Abelian group G as it satisfies
B(z x, y) = B(z, y)B(x, y) .
31

(115)

In a non-unitary theory there is an ambiguity in the sign of the S-matrix since there is no positivity constraint,
hence (109) only determines c− mod 4. More precisely if (S, T, c− ) is a solution, (−S, T, 4 + c− ) is also a solution.
32c
The proof involves using (95) along with the symmetries of the symbol Nab. First we assume by contradictionthat Nxca 6= 0 and Nxda 6= 0 for c 6= d. Then we find d x ≥ dc + dd . But on the other hand Nxca = Nācx , and thereforedc ≥ d x . These two are clearly incompatible if the anyon d exists.

34

SciPost Phys. 13, 067 (2022)

The condition B(x̄, y) = B ∗ (x, y) =1
B(x, y)follows from the bilinear property since1
1 = B(0, y) = B(x x̄, y) = B(x, y)B(x̄, y) and hence B(x̄, y) = B ∗ (x, y) = B(x,y) . The topological spins θ are called a quadratic refinement of the bilinear map.
θ (z y)θ (x y)θ (z x)
To derive the bilinear property (115), we must show that θ (z x y) = θ (x)θ ( y)θ (z) . Thisin turn follows from the fact that the S-matrix columns are eigenvectors of the fusion matrix,θ (x ȳ)θ (x)θ ( y)as per (107). Indeed, using (112) we find S x y = p1 θ (x)θ ( y) = p1 θ (x y) , and then (107)
|G|

reads

θ (a)θ 2 (x)θ (b) θ (x)θ (a b)=,θ (a x)θ (b x)θ (x a b)

|G|

(116)

which is exactly the required condition for (115) to hold.
Finally, the bilinear map B(x, y) is non-degenerate in theories with a unitary S-matrix.
That is, for every anyon, there is another anyon that braids with it nontrivially. From theunitarity of the S-matrix we haveX
|G|δa b̄ =B(a, c)B(b, c) .
(117)
cTherefore if such an anyon x 0 6= 0 that has trivial
P braiding with all anyons existed we could
have set a = x 0 , b = 0 and found |G|δ x 0 ,0 = c B(x 0 , c) which is a contradiction since theleft hand side vanishes for x 0 6= 0 while the right hand side is equal to |G| by assumption.
(This statement holds very generally: The S-matrix (104) is unitary when the braiding is nondegenerate.)PP θ (a)θ (c)A simple consequence of (117) is |G|δa0 = c B(a, c) = c θ (ac) . For a 6= 0 we getP θ (c)Pθ (c)0 = c θ (ac) and hence also 0 = c,a6=0 θ (ac) . The sum over a can be traded for a sum over
all elements b of G such that b = ac except the elements b = c which can be added by hand.
P θ (c)P
Hence, |G| = c,b θ (b) = c,b θ (c)θ (b)∗ . This is equivalent to the statement that the righthand side of (108) is a pure phase.

B

Surgery Calculation of Partition Functions

In this appendix we review the computation of 2+1d TQFT partition functions via the Dehn
surgery presentation of 3-manifolds. Given a UMTC C , there is a topological invariant associated to any 3-manifold known as the Reshetikhin-Turaev (RT) invariant [79,80,83]. For C withchiral central charge c− = 0 (mod 8), the RT-invariant coincides with the partition functionof metric-independent 2+1d TQFTs in question. Note that when C is the category of representations of the quantum group associated with the Lie group G at level k, the RT-invariant
is almost the same as Witten’s Gk CS invariant [20]. The only difference is that in Witten’squantization the partition function of the theory depends on a choice of 2-framing [91] of
the 3-manifold, arising from the addition of an improperly quantized gravitational CS term
to cancel the metric dependence of the theory. In [91], Atiyah showed that one can always
define a canonical 2-framing, and it turns out that the partition function of Witten’s CS theoryon 3-manifolds with this canonical 2-framing coincides with the RT invariant.

Dehn Surgery
In order to describe the computation of the RT invariant, we must first review the surgery
presentation of 3-manifolds; for more details see Chapter 9 of the book by Rolfsen [62].

35

SciPost Phys. 13, 067 (2022)

Framed Links (Ribbon Graphs): According to the Lickorish-Wallace theorem, any closedoriented 3-manifold can be obtained by performing Dehn surgery on a framed link (i.e. ribbon
graph) in the 3-sphere S 3 . A link L in S 3
L = K1 ∪ K2 ∪ · · · ∪ Km

(118)

is a collection of knots K1 , K2 , · · · , Km ⊂ S 3 that might link non-trivially with each other. Eachknot Ki is called a component of the link L. If we equip these knots with framing then we get aframed link. Framing of a knot K is a normal vector field along K. If we push the knot K along
this vector field (slightly, such that it does not intersects with K) we get another knot K 0 . Thisdefines a ribbon bounded by K and K 0 – see for instance Fig. 3 of [20]. So a framed knot is
nothing but an orientable ribbon. The integer number counting the number of right-handed
twists in the ribbon, or equivalently the linking number of K with K 0 , is called the framing
coefficient of the knot. For example, for K the unknot with framing coefficient n, we have

K

n

=


K

0

K

n times .

Consider a generic framed link, or equivalently ribbon graph, in S 3
n

n

nm
,
L = K1 1 ∪ K2 2 ∪ · · · ∪ Km

(119)

where Ki is the i-th component of the link with framing coefficient ni ∈ Z. Given such a framed
link in S 3 , the following integral surgery procedure gives a closed and orientable 3-manifold:
1. Drill out a tubular neighborhood Ni (homeomorphic to the solid torus) of the knot Ki .
2. Glue back Ni with a homemorphism hi : ∂ Ni → ∂ Ni ⊂ S 3 such thathi : µi 7→ Ki0 ,(120)

where µi is the contractible meridian curve of Ni and Ki0 ⊂ ∂ Ni is the knot specified by
the framing of Ki as explained above. In particular, the curve Ki0 becomes contractible
after the gluing.
All closed and orientable 3-manifolds can be obtained by such integral surgery procedure,
and the resulting 3-manifolds depend only on the data specified by the framed link in equation(119). However, there is a slightly more general procedure known as rational surgery if weallow the framing coefficients (a.k.a. surgery coefficients) to be rational numbers instead ofintegers.
For rational surgery, we allow hi in (120) to be a homeomorphism specified by an arbitrary
element of S L(2, Z) (the mapping class group of Ni ). To parameterize such elements, we picka canonical basis for H1 (∂ Ni ). Take µi to be the meridian (hence trivial in H1 (Ni )), and λi tobe the longitude curve on the boundary of Ni . Then for any hi we can writehi ([µi ]) = pi [µi ] + qi [λi ] ∈ H1 (∂ Ni ) ,(121)

where [µi ] and [λi ] are the homology classes of the meridian and longitude, and pi , qi ∈ Z.
The rational number ri = pi /qi ∈ Q is called the surgery coefficient of the knot Ki , and specifies
the homology class of the curve Ki0 up to a sign.33 This extends the surgery procedure explained
above to the case where the surgery coefficients ni are rational numbers.
33

Note that this does not determine the isotopy type of hi uniquely. However, the homeomorphism type of theresulting 3-manifold only depends on the link L and the surgery coefficients.

36

SciPost Phys. 13, 067 (2022)

Note that the procedure of constructing 3-manifolds starting from a framed link is notunique. A 3-manifold can have different surgery presentations. For the case of integral surgery,any two such surgery presentations are related by a Kirby move [92], and for rational surgery
there are extra moves such as Rolfsen’s twist to be able to go between different presentations[93] (see [94] for more details).

RT Invariant
Here we review the calculation of RT invariant on 3-manifolds using their surgery presentation;
for more details see [79, 80].
Colored Framed Links (Anyon Amplitudes): Given a unitary modular tensor category C
and a framed link L ⊂ S 3 , we can label/color each knot component of L with a simple objectof C to get a C -colored framed link. Moreover, if we choose an orientation on each knot, we
get an oriented colored link. Such an oriented colored link is equivalent to a configurationof anyons linking with each other. By choosing a time direction on S 3 , such a configuration
describes a process in which anyons-antianyons are created from the vacuum and annihilate
at a later time. Physically this process has an amplitude which we denote by 〈Lcol 〉 ∈ C. Since
this process happens in a topological theory, the amplitude is a topological invariant and only
depends on the isotopy type of the oriented colored link Lcol .
Using the braiding and modular matrices of C , the configuration of anyons described by
Lcol can be simplified to eventually calculate the amplitude 〈Lcol 〉 ∈ C (cf. equations (100) and(106)). Having defined the anyon amplitudes, we can now define the RT invariant associatedwith C . Take a 3-manifold M3 that can be obtained from integral surgery on the framed linknnnm
L = K1 1 ∪ K2 2 ∪ · · · ∪ K m
.

(122)

The RT-invariant is then given by [79]
RT(C , M3 ) =X
e−2πiσ(L)c− /8
da1 da2 · · · dam L(a1 , . . . , am ) ,D m+1
a ,...,a ∈ I
1

(123)

m

where I is the set of simple anyons in C , L(a1 , . . . , am ) is the anyon configuration
described bypinserting anyon ai with framing ni on the knot component Ki of L, D = dim(C ), and σ(L)is the signature of the linking matrix of L.34
For the case of manifolds obtained by plumbing on trees, the RT invariant can be expressedusing only the modular S and T matrices [95]. In particular, for the lens space L(p, q) = S 3 /Z p
partition functions, we have
Z L(p, q) = (S T a1 S · · · S T an S)00 ,
(124)
where
p
q

1

= a1 −
a2 −

··· −
34

.

1

(125)

1
an

To evaluate (123) we need to choose an orientation on L, but it is clear that the final answer does not dependon the choice of orientation since we are summing over all anyons.

37

SciPost Phys. 13, 067 (2022)

C

Partition Function of Abelian CS Theories

The (E8 )1 Theory
The (E8 )1 theory is a trivial massive theory with no nontrivial anyons which explicitly allows
to shift the infrared value of c− by 0 mod 8. The (E8 )±1 theory is therefore an invertible field
theory with c− = ±8. Stacking this theory, we can therefore generate all the possible invertible
field theories with c− = 0 mod 8.
The (E8 )1 CS theory can be constructed from the following K-matrix

2 −1 0
0
0
0
0
0
0
0
0
0
−1 2 −1 0


0
0
0
 0 −1 2 −1 0
0 −1 2 −1 0
0
0
0
K=
 ,
0
0 −1 2 −1 0 −1
0


0
0
0 −1 2 −1 0 
0
0
0
0
0
0 −1 2000
0
0 −1 0
0
2


(126)

which is the Cartan matrix of E8 Lie algebra. It is straightforward to verify that there is only atrivial anyon since det K = 1, and also that sgn(K) = 8.

Coprime Matrices
In Section 2.4 and the remainder of this appendix, we will make use of various properties of
coprime matrices that we now introduce. Two n × n integral matrices C and D (both assumed
to have non-zero determinant) are said to be coprime if there exist integral matrices A and B
such that−BC T + ADT = 1 .
(127)(The particular signs and transposition above are purely for later convenience.) We will findthe following theorem useful [96],2nTheorem C.1 Consider the n n× n minors of the matrix (C D). Then C and D are left coprime
if and only if the gcd of all these minors is equal to unity.
In particular, this theorem implies that if gcd(det C, det D) = 1, then the matrices are coprime.
An additional useful criterion is that if DC T is an even symmetric matrix, i.e. DC T = C DT
and the integers on the diagonal of DC T are all even, then one 
can make a “preferred” choiceA
B
of A and B in (127) such that we also have AB T = BAT and thus
is symplectic [56]. In
C D
this case the inverse matrices are symplectic and integral as well, and hence we have additional
relations
AT D − C T B = 1 , AT C = C T A , B T D = DT B .
(128)
In the context we will be interested in, we will have D = K ⊗ 1 and C = 1 ⊗ L. Thesematrices commute and each of them is symmetric, and thus they form a symmetric pair. Thissymmetric pair is even because the elements on the diagonal of K are even. If we assume
gcd(det K, det L) = 1 then we immediately infer that gcd(det(K ⊗ 1), det(1 ⊗ L)) = 1 and hence
these C and D are coprime. We now ask if the converse also holds: namely, does coprimalityof K ⊗ 1 and 1 ⊗ L imply that gcd(| det K|, | det L|) = 1?
The answer is in fact yes, as was quoted in Section 2.4. To show this, assume to the contrary
that gcd(| det K|, | det L|) 6= 1 . Then there should exist a common prime factor p that divides
38

SciPost Phys. 13, 067 (2022)

both det K and det L. By Cauchy’s theorem, there then exist elements (anyons) in Abeliangroups Z|K| /K · Z|K| and Z|L| /L · Z|L| whose order are equal to p. If we represent such elements
by integer vectors v and w respectively, we have
pv = K · x,

pw = L · y,(129)

for integer vectors x and y such that the vectors 1p x and 1p y are not integer. Consider now thevector 1p x ⊗ y. The action of both K ⊗ 1 and 1 ⊗ L on this vector produces an integral vector,
1
1 ⊗ L · x ⊗ y = x ⊗ w.
p

1
x ⊗ y = v ⊗ y,
K⊗1·
p





(130)

But if the matrices K ⊗ 1 and 1 ⊗ L were coprime, then by (127) we would have concluded
that the vector 1p x ⊗ y must be integer. But this is impossible because if 1p x ⊗ y is integer,p should divide x i y j for all i and j. So either all x i or all y j are integer, which contradicts
our assumption. Therefore if gcd(| det K|, | det L|) 6= 1 the matrices K ⊗ 1 and 1 ⊗ L cannot be
coprime.

Symmetry Between the Chern-Simons and Surgery Matrix
In (7) of the Introduction, we quoted an identity relating the partition function of the Abelian
TQFT with K-matrix K on a 3-manifold with linking matrix L to the same quantity with K and
L exchanged. In this final subsection we derive this identity.
To begin, recall from Appendix B that in the surgery presentation the partition function
can be obtained by summing over all anyons running on all knot components of the link:
ZK [L] =1| det K||L|/2+1/2

X

T

−1

eπi m (K

⊗L)m

.

(131)

m∈Z|K||L| /(K⊗1)·Z|K||L|

For coprime matrices C and D such that DC T is even and symmetric, there is an identity
generalizing Gaussian reciprocity [54, Theorem 1]:
1
pX| det D| m∈Z|D| /DT ·Z|D|T

eπi m (D

−1

C)m

=p

1

2πi

| det C|Te 8 sgn(DC )X

Te−πi m (C

−1

D)m

.

m∈Z|C| /C T ·Z|C|(132)
The reciprocity law says that we can exchange C and D up to an 8th root of unity and complex
conjugation.
p Choosing D = K ⊗ 1 and C = 1 ⊗ L, the LHS of (132) reduces to (131) up to a factor of| det K|. This immediately tells us that the theory with CS matrix K on a 3-manifold withlinking matrix L gives, up to some normalization factors, the same result as the theory on
linking matrix K with CS matrix L.
The theory with CS matrix L may be spin however. So for the proper physical interpretationof the statement that L and K are interchangeable, we need to discuss the partition functionsof spin TQFTs. The basic issue here is that in a spin theory the topological spin
T −1

θ (m) = eπi m K

m

(133)

is not a well-defined function on the quotient m ' m+K·p for integer vectors p. This is because
in a spin theory, there exists a transparent fermion which has spin 12 and braids trivially withall other anyons.35
35

For a discussion of the spin of line defects in spin vs. non-spin theories see for instance [97, 98].

39

SciPost Phys. 13, 067 (2022)

However, the braiding of the anyons given by
T −1

B(m, n) = e2πi m K

n

(134)

,

is still well-defined. Moreover by choosing a so-called integral Wu class W we can define aquadratic refinement of the braiding bilinear form as [26]1T −1

q(m) = eπi (m− 2 W) K

(m− 21 W)

,

(135)

which satisfies
q(m + n) − q(m) − q(n) + q(0) = B(m, n) .

(136)

The defining property of W is that WT · p = pT · K · p mod 2 for all integral vectors p. Thismeans in particular that Wi = Kii mod 2 for all i = 1, . . . , |K|. With this we find that q is awell-defined function on the quotient Z|K| /K · Z|K| defining the anyons.
Note that the defining property of W only fixes it up to shifts by 2p where p is any integral
vector. We must thus understand how this affects the quadratic function q. For this we note
that

1 
(137)
q(m) = θ m − W .
2
From this we see that shifting W by 2p amounts to a permutation of the anyons labels via theredefinition m 7→ m − p.
As pointed out in [26], the crucial difference between the quadratic refinement q in spin
theories and θ in non-spin theories is the fact that θ (0) = 1, whereas in general q(0) 6= 1. For
this reason we cannot think of q as the spin of anyons. However, when there is a choice ofW such that q(0) = 1, the theory can be viewed as a non-spin theory. This is because in thatcase the transparent fermion decouples from the detectable anyons, and the theory becomes
equivalent to the tensor product of a bosonic theory with θ = q and the trivial spin theory
containing the transparent fermion.
Similar to the bosonic case, we can compute the chiral central charge, or equivalently theL(1, 1) ' S 3 partition function, as [26]1c−e2πi 8 = p

X| det K| m ∈ Z|K| /K·Z|K|

1

T −1

eπi (m− 2 W) K

(m− 12 W)

.

(138)

Therefore W enables us to compute the S 3 partition function which admits a unique spin
structure. We now turn to the situation that there is a linking matrix L. In this case we must
guess what the appropriate quantity to sum over is. The natural guess is
ZK [L, x] =1
| det K||L|/2+1/2

1Xm∈Z|K||L| /(K⊗T

−1eπi (m− 2 W⊗x) (K⊗L)(m− 12 W⊗x)

.

(139)

1

)·Z|K||L|

But for this to be well-defined under the usual shifts m → m + K · p for arbitrary integralp ∈ Z|K||L| , we need that
(W ⊗ x)T (1 ⊗ L) · p = pT · (K ⊗ L) · p (mod 2) ,(140)

holds for all p. This forces us to choose LAB Wi x B = Kii LAA mod 2 for all i = 1, . . . , |K| andA, B = 1, . . . , |L|. Since Wi = Kii mod 2, this is equivalent to LAB x B = LAA mod 2 (assuming K
is odd). One obvious freedom in this equation is to shift x → x + 2s for any integral vectors ∈ Z|L| . As before, this can be reabsorbed by redefining the anyons. Therefore the space of
inequivalent solutions is¦©
|L|x ∈ Z2 : L · x = x mod 2 .
(141)
40

SciPost Phys. 13, 067 (2022)

These solutions correspond to what are known as characteristic sublinks of L, and are shown
to be in one-to-one correspondence with the spin structures on the 3-manifold associated withL [99, Appendix C] (see also [100]). This shows that spin theories require a choice of spin
structure in order for the partition function to be well defined, as one would expect.
However, when L is even (or more generally when K ⊗ L is even), W ⊗ x = 0 is always
an allowed solution of (140). In other words when the surgery linking matrix is even, there
is a preferred spin structure x = 0 given by the particular surgery presentation L. Hence we
can now interpret of the sum (131) for when K is odd but L is even as the partition functionof the Abelian spin CS theory K on 3-manifold L with some specific spin structure. Using thereciprocity formula (132), and following the normalization of partition functions we find theaforementioned symmetry:Æ

2πi| det K| ZK [L] = e 8 sign(K)sign(L)Æ| det L| ZL [K] .

(142)

One could imagine a tentative explanation of this symmetry in terms of the 3d-3d correspondence [31] as follows. Both sides of (142) can be perhaps interpreted as the partition
function of the 6d N = (2, 0) theory of U(1)-type considered in [101, equations (2.23)-(2.25)]
on the 6-manifold given by the tensor product of 3-manifolds with linking matrices K and L.
We can compactify the 6d theory on either of the two 3-manifolds. In each case, after thecompactification we find a 3d N = 2 pure CS theory with K-matrix K or L. In such supersymmetric theories, the fermions decouple and after integrating them out contribute some extra
normalization factor. It would be interesting to confirm that integrating out these fermions
gives the same normalization factor in (142).

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