On finite symmetries and their gauging in two dimensions
Accepted: March
On nite symmetries and their gauging in two
Lakshya Bhardwaj 0 1 3
Yuji Tachikawa 0 1 2
0 Kashiwa , Chiba 2778583 , Japan
1 Waterloo , Ontario, N2L 2Y5 Canada
2 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
3 Perimeter Institute for Theoretical Physics
It is wellknown that if we gauge a Zn symmetry in two dimensions, a dual Zn symmetry appears, such that regauging this dual Zn symmetry leads back to the original theory. We describe how this can be generalized to nonAbelian groups, by enlarging the concept of symmetries from those de ned by groups to those de ned by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a nonanomalous subgroup of an anomalous nite group is gauged: for example, the gauged theory can have nonAbelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of twodimensional topological quan
Anyons; Discrete Symmetries; Global Symmetries; Topological Field Theories

tum
eld theories whose symmetry is given by a category. We see explicitly that the gauged
version is a topological quantum eld theory with a new symmetry given by a dual category.
3 Symmetries as categories in two dimensions
HJEP03(218)9
Basic notions of symmetry categories
Comments
More notions of symmetry categories
Groups and representations of groups as symmetry categories
4 Gaugings and symmetry categories
Module and bimodule categories
Duality of C(G) and Rep(G)
Gauging by an algebra object
4.4 Symmetries of the gauged theory from bimodules for the algebra object
Gauging of C(G) to get Rep(G) and vice versa
Gaugings and module categories
4.7 (Re)gauging and its e ect on the symmetry category
4.8 The e ect of the gauging on Hilbert space on S1
1 Introduction 2 Regauging of nite group gauge theories
Abelian case
NonAbelian case
{ i {
5
More examples of symmetry categories and their gauging
Symmetry category with two simple lines
Symmetry category of SU(2) WZW models and other RCFTs
Gauging a subgroup of a possiblyanomalous group
Integral symmetry categories of total dimension 6
Integral symmetry categories of total dimension 8
TambaraYamagami categories
6 2d TFT with C symmetry and their gauging
2d TFTs without symmetry
TFT with C symmetry on a cylinder
TFT with C symmetry on a general geometry
6.4 Gauged TFT with the dual symmetry
7 Conclusions
A Group cohomology
Introduction
Let us start by considering a twodimensional theory T with Zn symmetry. We can gauge it
to get the gauged theory T =Zn. This gauged theory is known to have a new Zn symmetry,
and regauging it gives us back the original theory: T =Zn=Zn = T [1]. It is also wellknown
that this phenomenon generalizes to any arbitrary nite Abelian group G. That is, gauging
a theory T with G symmetry results in a theory T =G with a new
nite Abelian group
symmetry G^ such that gauging the new theory by the new symmetry takes us back to the
original theory: T =G=G^ = T . One natural question arises: can it be generalized to higher
dimensions? Yes, according to [2], where the generalized concept of pform symmetries
the Hamiltonian of T =G. When G is nonAbelian, we will argue that the information of G
in T =G is captured by operators Ui which still commute with the Hamiltonian but these
operators are now in general nonunitary. These operators can be constructed by wrapping
a Wilson line for G along the spatial circle. Hence, Wilson lines should be thought of as
generalized symmetries for the theory T =G. In fact, we will also argue that there is a
natural notion of gauging this symmetry formed by Wilson lines such that gauging T =G
results back in the original theory T .
This raises the following question: how do we specify a generalized symmetry that
a theory can admit? In this paper, we give an answer to this question: a general
nite symmetry of a twodimensional theory is speci ed by a structure which is known to
mathematicians in the name of unitary fusion categories. We prefer to call it symmetry
categories.1 For the gauged theory T =G for possibly nonAbelian group G, the Wilson line
operators form Rep(G), which is a symmetry category formed by the representations of
G. Similarly, a general symmetry category C physically corresponds to more general line
operators of T .
We also discuss how a symmetry category C can be gauged. It turns out that there
is no canonical way of gauging a generic symmetry category. Pick one way M of gauging
the symmetry C of a theory T . Denote the gauged theory by T =M and its symmetry
category by C0. It then turns out that there exists a dual way M0 of gauging C0 such that
1We do not claim that this is the ultimate concept for the 0form
nite symmetry in two dimensions;
of all, we will see that symmetry categories C capture symmetries together with their
anomalies. Then, the machinery we spell out allows us to compute what is the symmetry
of the gauge theory T =H when we gauge a subgroup H of a possibly anomalous
avor
symmetry G. For example, if we gauge a nonanomalous Z2 subgroup of Z2
Z2
Z2 with
a suitable choice of the anomaly, we can get nonanomalous nonAbelian symmetry D8 and
Q8, the dihedral group and the quaternion group of order 8.2 In general, the symmetry
C of the gauged theory is neither a group nor Rep(G) for a nite group, and we need the
concept of symmetry categories to describe it.
There are also vast number of symmetry categories not related to nite groups, formed
by topological line operators of twodimensional rational conformal eld theories (RCFTs).
In particular, any unitary modular tensor category, or equivalently any MooreSeiberg data,
can be thought of as a symmetry category, by forgetting the braiding.
We should emphasize here that this generalization of the concept of symmetry from
that de ned by groups to that de ned by categories was already done long ago by other
authors, belonging to three somewhat independent lines of studies, namely in the study of
the subfactors and operator algebraic quantum
eld theories, in the study of representation
theory, and in the study of RCFTs. Each of the communities produced vast number of
papers, and not all of them can be cited here. We recommend to the readers textbooks by
Bischo , Longo, Kawahigashi and Rehren [6] and Etingof, Gelaki, Nikshych and Ostrik [7]
from the
rst two communities and the articles by Carqueville and Runkel [8] and by
Brunner, Carqueville and Plencner [9] from the third community as the starting points.
Our rst aim in this paper is then to summarize the content of these past works in
a way hopefully more accessible to other researchers of quantum
eld theory, including
the authors of this paper themselves, emphasizing the point of view related to the modern
study of symmetry protected topological phases. What we explain in this rst part of the
paper is not new, except possibly the way of the presentation, and can all be found in the
literature in a scattered form.
Our second aim is to axiomatize twodimensional topological quantum
eld theories
(TFTs) whose symmetry is given by a symmetry category C. This is a generalization of the
2Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the
phase structure of thermal 4d su(2) YangMills theory. One important step in the analysis is the symmetry
structure of the thermal system, which is essentially threedimensional. As a dimensional reduction from
4d, the system has a Z2
Z2 0form symmetry and a Z2 1form symmetry, with a mixed anomaly. Then the
authors gauged the Z2 1form symmetry, and found that the total 0form symmetry is now D8. This D8 was
then used very e ectively to study the phase diagram, but that part of their paper does not directly concern
us here. Their analysis of turning an anomalous Abelian symmetry by gauging a nonanomalous subgroup
into a nonAbelian symmetry is a 3d analogue of what we explain in 2d. See their section 4.2, appendix B
and appendix C. Clearly an important direction to pursue is to generalize their and our constructions to
arbitrary combinations of possiblyhigherform symmetries in arbitrary spacetime dimensions, but that is
outside of the scope of this paper.
{ 2 {
circles together with arbitrary network of line operators from C
.
The rest of the paper is organized as follows. First in section 2, as a preliminary, we
recall how gauging of a nite Abelian symmetry G can be undone by gauging the new
nite
Abelian symmetry G^, and then brie y discuss how this can be generalized to nonAbelian
symmetries G, by regarding Rep(G) as a symmetry. This e ort of generalizing the story to
a nonAbelian group makes the possibility and the necessity of a further generalization to
symmetry categories manifest. We exploit this possibility and describe the generalization
in detail in subsequent sections.
Second, we have two sections that form the core of the paper. In section 3, we introduce
the notion of symmetry categories, and discuss how we can regard as symmetry categories
both a
nite group G with an anomaly
and the collection Rep(G) of representations of
G. We then explain in section 4 that physically distinct gaugings of a given symmetry
category C correspond to indecomposable module categories M of C, and we describe how
to obtain the new symmetry C0 of the theory T =M for a given theory T with a symmetry C.
Third, in section 5, we give various examples illustrating the notions introduced up to
this point. Examples include the form of new symmetry categories C0 when we gauge a
nonanomalous subgroup H of an anomalous nite group G, and the symmetry categories
of RCFTs.
Fourth, in section 6, we move on to the discussion of the axioms of twodimensional
TFTs whose symmetry is given by a symmetry category C. We also construct the gauged
TFTs T =M given an original TFT T with a symmetry category C and a gauging speci ed
by its module category M. Sections 5 and 6 can be read independently.
Finally, we conclude with a brief discussion of what remains to be done in section 7. We
have an appendix A where we review basic notions of group cohomology used in the paper.
Before proceeding, we note that we assume that the spacetime is oriented throughout
the paper. We also emphasize that all the arguments we give, except in section 6, apply
to nontopological nonconformal 2d theories.
2
2.1
Regauging of nite group gauge theories
Abelian case
Let us start by reminding ourselves the following wellknown fact [1]:
Let T be a 2d theory with avor symmetry given by an Abelian group G. Let us
assume that G is nonanomalous and can be gauged, and denote the resulting
theory by T =G. Then this theory has the
Pontrjagin dual of G, such that T =G=G^ = T .
avor symmetry G^, which is the
Recall the de nition of the Pontrjagin dual G^ of an Abelian group G. As a set, it is given by
G^ = f : G ! U(1) j
is an irreducible representation g:
(2.1)
{ 3 {
Note that
is automatically onedimensional. Therefore the product of two irreducible
representations is again an irreducible representation, which makes G^ into a group. G and
G^ are isomorphic as a group but it is useful to keep the distinction because there is no
canonical isomorphism between them.
In the literature on 2d theories, gauging of a nite group G theory is more commonly
called as orbifolding by G, and the fact above is often stated as follows: a Gorbifold has a
dual G^ symmetry assigning charges to twisted sectors, and orbifolding again by this dual
G^ symmetry we get the original theory back. This dual G^ symmetry is also known as the
quantum symmetry in the literature.
This fact can be easily shown as follows. Let ZT [M; A] denote the partition function
HJEP03(218)9
of T on M with the external background G gauge eld A. Here A can be thought of as
taking values in H1(M; G). Then the partition function of the gauged theory T =G on M
is given by ZT=G[M ] / P
A ZT [M; A]. Here and in the following we would be cavalier on
the overall normalization of the partition functions. More generally, with the background
gauge eld B for the dual G^ symmetry, the partition function is given by
ZT=G[M; B] /
X ei(B;A)ZT [M; A]
A
(2.2)
(2.3)
(2.4)
where B 2 H1(M; G^) and ei(B;A) is obtained by the intersection pairing
ei( ; ) : H1(M; G^)
H1(M; G) ! H2(M; U(1)) ' U(1):
The equation (2.2) says that the partition function of T =G is essentially the discrete Fourier
transform of that of T , and therefore we dually have T = T =G=G^:
ZT [M; A] /
X ei(A;B)ZT=G[M; B]:
B
This statement was generalized to higher dimensions in e.g. [2]:
Let T be a ddimensional theory with pform
avor symmetry given by an
Abelian group G. Let us assume that G is nonanomalous and can be gauged,
and denote the resulting theory by T =G. Then this theory has the dual (d 2
p)form
avor symmetry G^, such that T =G=G^ = T .
B 2 Hd 1 p(M; G^).
2.2
NonAbelian case
The derivation is entirely analogous to the 2d case, except that now A 2 Hp+1(M; G) and
The facts reviewed above means that the nite Abelian gauge theory T =G still has the full
information of the original theory T , which can be extracted by gauging the dual symmetry
G^. It is natural to ask if this is also possible when we have a nonAbelian symmetry G,
which we assume to be an ordinary 0form symmetry.
{ 4 {
This is indeed possible3 by suitably restating the derivation above, but we will see
that we need to extend the concept of what we mean by symmetry. To show this, we
rst massage (2.4) in a suitable form which admits a straightforward generalization. Let
us consider the case of ZT [M; A = 0] for illustration. By Poincare duality, B can also be
represented as an element of H1(M; G^). Then, (2.4) can be rewritten as
ZT [M ] /
X
g^1; ;g^n
ZT=G[M; g^1;
; g^n]
where i 2 f1;
; ng labels generators of H1(M ) and g^i is an element of G^ associated to the
cycle labeled by i. Each summand on the right hand side, ZT=G[M; g^1;
; g^n], is then the
expectation value of Wilson loops in representations labeled by g^i placed along the cycle i.
Now, we can sum the G^ elements for each i separately to obtain
where W reg denotes the insertion of a Wilson line in the regular representation along the
cycle i. This is because, for an abelian G, the regular representation is just the sum of
representations corresponding to elements g^ of G^.
The relation (2.6) says that by inserting all possible Wilson lines on all possible cycles,
we are putting the delta function for the original gauge
eld A. We now note that the
relation (2.6) holds for a nonAbelian G as well, if we insert W reg not only for the generators
of H1(M ) but for the generators of 1(M ). This can be seen by the fact that tr g in the
i
regular representation is nonzero if and only if g is the identity.
i
The identity (2.6) means that the ungauged theory T can be recovered from the gauged
theory T =G by inserting line operators W reg in an appropriate manner. This is analogous
to the construction of the gauged theory T from the ungauged theory T by inserting line
operators representing the G symmetry in an appropriate manner. Given the importance
of Wilson lines in recovering the information of the ungauged theory, we assign the status
of dual symmetry to Wilson lines.
Let us phrase it another way. When G is Abelian, the dual (d
2)form G^ symmetries
can be represented by 1cycles labeled by elements of G^, forming a group. When G is
nonAbelian, the dual symmetry can still be represented by 1cycles labeled by representations
Rep(G) of G.
We can still multiply lines, corresponding to the tensor product of the
representations, and this operation reduces to group multiplication of G^ in the abelian
case.
But Rep(G) is not a group if G is nonAbelian.
Therefore this is not a
avor
symmetry group, it is rather a avor symmetry something. We summarize this observation
as follows:
Let T be a ddimensional theory with 0form avor symmetry given by a possibly
nonAbelian group G. Let us assume that G is nonanomalous and can be
gauged, and denote the resulting theory by T =G. Then this theory has Rep(G) as
the dual (d 2)form
avor symmetry `something', such that T =G=Rep(G) = T .
3That this is possible was already shown for twodimensional theories in [3], as an example of a much
more general story, which we will also review in the forthcoming sections. Here we describe the construction
in an elementary language.
{ 5 {
Symmetries as categories in two dimensions
Any nite group, possibly nonAbelian, can be the 0form symmetry group of a theory.
In addition to this, we saw in the last section that Rep(G), the representations of G, can
also be the (d
2)form symmetry something of a ddimensional theory. We do not yet
have a general understanding of what should be this something in general d dimensions for
general combinations of various pform symmetries. However, at least for d=2 and p=0,
we already have a clear concept for this something in the literature, which includes both
groups G and representations of groups Rep(G) and much more. In this section we explain
what it is.
3.1
Basic notions of symmetry categories
In two dimensions, a 0form
nite symmetry element can be represented by a line operator
with a label a. Inserting this line operator on a spacelike slice S corresponds to acting
on the Hilbert space associated to S by a possibly nonunitary operator Ua corresponding
to the symmetry element. Moreover, Ua commutes with the Hamiltonian H associated
to any foliation of the twodimensional manifold. In addition, Ua cannot change under
a continuous deformation of its path. Therefore the line operator under consideration is
automatically topological.
Topological line operators, in general, form a structure which mathematicians call a
tensor category. We want to restrict our attention to topological line operators describing
a nite symmetry. Such topological line operators form a structure which mathematicians
call a unitary fusion category. We call it instead as a symmetry category, to emphasize its
role as the nite symmetry of unitary twodimensional quantum
eld theory. We start by
stating the slogan, and then ll in the details:
Finite avor symmetries of 2d theories are characterized by symmetry categories.
3.1.1
Objects
The objects in a symmetry category C correspond to topological line operators generating
the symmetry. More precisely, a theory T admitting the symmetry C will admit topological
line operators labeled by the objects of C. Henceforth, we will drop the adjective topological
in front of line operators.
For any line operator labeled by an object a, we have a partition function of T with
the line inserted along an oriented path C, which we can denote as
h
a(C)
i
(3.1)
where the dots stand for additional operators inserted away from C.
3.1.2
Morphisms
The morphisms in a symmetry category C correspond to topological local point operators
which can be inserted between two lines. More precisely, consider two labels a and b, and
a path C such that up to a point p 2 C we have the label a and from the point p we have
{ 6 {
HJEP03(218)9
3.1.3
Existence of a trivial line
C contains an object 1, which labels the trivial line of T . We have
h
1(C)
i = h
i
:
3.1.4
Additive structure
Given two objects a and b, there is a new object a
We abbreviate a
a by 2a, a
a
Hilbert space obtained by wrapping a
3.1.5
Tensor structure
h
(a
b)(C)
i = h
a(C)
i + h
b(C)
i
:
a by 3a, etc. The linear operator Ua b acting on the
b on a circle is then given by Ua + Ub.
b in C. In terms of lines in T , we have
(3.2)
(3.3)
(3.4)
a
m
b
=
a
a⊗b
m
b
the label b. Then, we can insert a possiblylinechanging topological operator labeled by m
at p. We call such m a morphism from a to b, and denote this statement interchangeably
as either
HJEP03(218)9
m : a ! b
or
m 2 Hom(a; b):
The set Hom(a; b) is taken to be a complex vector space and it labels (a subspace of)
topological local operators between line operators corresponding to a and b in T . From
now on, we would drop the adjective topological in front of local operators.
Given two objects a and b, we have an object a
b in C. This corresponds to considering
two parallelrunning line operators a and b as one line operator. The linear operator Ua b
acting on the Hilbert space obtained by wrapping a
b on a circle is then given by UaUb.
The trivial object 1 acts as an identity for this tensor operation. That is, there exist
canonical isomorphisms a
1 ' a and 1
a ' a for each object a. We can always nd
an equivalent category in which these isomorphisms are trivial, that is a
1 = 1
a = a.
Hence, we can assume that these isomorphisms have been made trivial in C. Henceforth,
the unit object will also be referred to as the identity object.
Consider three lines C1;2;3 meeting at a point p, with C1;2 incoming and C3 outgoing.
We can put the label a, b, c on C1;2;3, respectively. We demand that the operators we can
put at the junction point p is given by m 2 Hom(a
b; c). This label a
b corresponds to
a composite line as can be seen by the following topological deformation shown in gure 1.
{ 7 {
The de nition of a
b here includes a choice of the implicit junction operator where
the lines labeled by a, b and a
b meet. In this paper, whenever we draw
gures with
such implicit junction operators, we always choose the operator to be the one labeled by
the identity morphism id : a
b ! a
Simplicity of the identity, semisimplicity, and
niteness
The simple objects a 2 C are objects for which Hom(a; a) is onedimensional. In general,
for any object x, there is always a canonical identity morphism from x to x which labels
the identity operator on the line labeled by x. For a simple object a, the existence of
the identity morphism implies that there is a natural isomorphism Hom(a; a) ' C as an
algebra. We assume for simplicity that the identity object 1 is simple.
We also assume that every object x has a decomposition as a nite sum
x =
M Naa
a
(3.5)
(3.6)
where Na is a nonnegative integer and a is simple. In other words, every object x is
semisimple.
Finally, we assume
niteness, that is the number of isomorphism classes of simple
objects is
nite. Below, we will be somewhat cavalier on the distinction between simple
objects and isomorphism classes of simple objects.
3.1.7
Associativity structure
The data in a symmetry category C includes certain isomorphisms implementing
associativity of objects
a;b;c 2 Hom((a
b)
c; a
(b
c))
which we call associators.4 Fusion matrices F for the MooreSeiberg data, and the
(quantum) 6j symbols for the (quantum) groups are used in the literature to capture the data
of associators.
The associator
a;b;c corresponds to a local operator which implements the process of
exchanging line b from the vicinity of a to the vicinity of c. See gure 2. They satisfy the
pentagon identity which states the equality of following two morphisms
((a
b)
c)
d ! (a
= ((a
(b
b)
c))
c)
d ! a
d ! (a
((b
b)
c)
(c
d) ! a
d) ! a
(b
(b
(c
(c
d))
d))
(3.7)
where each side of the equation stands for the composition of the corresponding associators.
The pentagon identity ensures that exchanging two middle lines b, c between two outer
lines a, d in two di erent ways is the same, see gure 3.
4Since we can and do choose the identity morphisms 1 a ! a and a 1 ! a to be trivial, the associator
a;b;c is also trivial when any of a, b, c is trivial.
{ 8 {
α
a
b
c
a
b
c
HJEP03(218)9
a
a b
the local operator labeled by the associator morphism is obtained by squeezing the region between
(a
b)
c and a
(b
c) shown above to a point.
tensoring of four lines lead to the same result.
R
a
=
=
R
a
=
L
a
a
a a
a
a
1
1
HJEP03(218)9
For every object a, C contains a dual object a . The line labeled by the dual object has
the property that
h
a(C)
i = h
a (C~)
i
:
Here, C~ denotes the same path C but with a reverse orientation, and the morphisms
attached at the junctions on C~ need to be changed appropriately as we explain below at
the end of this subsubsection.
We require that the dual of the dual is naturally isomorphic to the original object:
(a ) ' a. The dual operation also changes the order of the tensoring:
L
a
a
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(a
b) = b
a :
We demand that there are evaluation morphisms
and coevaluation morphisms5
aR : a
a
! 1;
aL : a
a ! 1
aR : 1 ! a
a;
aL : 1 ! a
a :
aR = aL
(pa
1) = aL
(1
pa 1)
aR = (1
pa 1)
aL = (pa
1)
L
a
gure 4.
we require
These label local operators corresponding to the process of folding a line operator a. See
We note that aR and aL are not necessarily equal. However, we require as part of
de nition of dual structure that they are related as follows
where pa is an isomorphism from a to a and pa is an isomorphism from a to a . Similarly,
5We use the convention that when something is denoted by x, cosomething is denoted by x. This usage
is unconventional, in particular for the case of coproduct for which
is de nitely the standard notation,
but it reduces the amount of notations that one has to remember.
=
a
a
HJEP03(218)9
category C.
condition with the associator
The data of pa and pa is referred to in the literature as a pivotal structure on the fusion
The evaluation and coevaluation morphisms have to satisfy the following consistency
( aR
1)
a;a ;a 1 (1
R
a ) = 1
as morphisms from a to a. This ensures that a line with two opposite folds in the right
direction can be unfolded as shown in gure 5. A similar identity is satis ed by aL and
L
a
which ensures that two opposite folds in the left direction can be unfolded.
Using evaluation and coevaluation morphisms of di erent parity we can construct
loops of lines
(dimCC a)id : 1
(dimC a)id : 1
R
!a a
L
!a a
a
a
L
!a 1;
R
!a 1:
These are morphisms from 1 to 1 and hence they are proportional to the identity morphism.
The proportionality factors de ne two numbers: the counterclockwise dimension dimCC a
of a and the clockwise dimension dimC a of a. See gure 6.
Since we can replace the label a by a at the cost of ipping the orientation of line, we
must have
Indeed, this follows from (3.12) and (3.13).
In fact, it turns out that we can further argue that
dimCC a = dimC a ;
dimCC a = dimC a:
dimCC a = dimC a
dim a:
To see this, place a small counterclockwise loop of line a around the \north pole" on
the sphere. Let there be no other insertions anywhere on the sphere. This evaluates to
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
The loop, if it contains no other operators in it, can then be shrunk and the partition function with
the loop is equal to dimCC a = dim a times the partition function without the loop, with all other
insertions unchanged.
dimCC a
ZS2 where ZS2 is the partition function on sphere. Now we can move the line
such that it looks like a small clockwise loop around the \south pole" on the sphere. This
evaluates to dimC a
ZS2 . Equating the two expressions we nd (3.19).6 This is a further
constraint on C. If the fusion category C satis es (3.19), then C is called a spherical fusion
category in the literature.
Since aR and aL are not necessarily equal, we have to specify whether a folding of line
a to the right should be read as the morphism
aR or the morphism
aL . Similarly there
is a speci cation of
aR vs.
aL . This issue can be dealt with in two ways, which are
technically equivalent but have a rather di erent avor.
One method.
One perspective is to regard that a line is always labeled by the pair (the
local orientation, an object in C). Then, a pair ("; a) and (#; a ) are isomorphic but not
actually the same. We note that this distinction needs to be made even when a ' a . Then
we make the rule that when a vertical line is labeled by ("; a) up to some point and then
labeled by (#; a ) from that point, we insert the pivotal structure pa 2 Hom(a; a) at that
point. This approach would be preferred by those who have no trouble with adding local
orientations as a new datum to a topological line operator.
Another method.
and between
aR vs.
Another perspective is to think that the change between aR and aL
aL is canceled by changing the nearby morphisms. This method
might be preferred by those who do not want to add local orientation as a new datum to
a topological line operator.
We emphasize that, in this approach, the operation of exchanging a by a with a
reversed orientation does not change the local operators at the junctions. Instead it changes
the way the local operators at the junctions are read as morphisms in the associated
symmetry category C.
6The authors thank ShuHeng Shao for discussion related to this point.
a
b
HJEP03(218)9
speci ed as aL . This changes the morphism from m on the left side to (pa 1
side. These two diagrams provide two di erent categorical representations of the same physical
con guration.
The following moves are su cient to specify what happens in any situation:
R
1. Consider a morphism a
equal to aL
1
a;a ;c (1
2. Consider a morphism aR
equal to aR
c;a ;a (n
1) : b
1
a;a ;c (1
n) : a
b ! c where n = (pa 1
1)
m : b ! a
b ! c where m : b ! a
c. This is
c;a ;a (m
1) : b
a ! c where m : b ! c
a . This is
a ! c where n = (pa
1)
m : b ! c
a .
3. Consider a morphism (1
m)
is equal to (1
is equal to (n
n)
1)
a ;a;b
1
b;a;a
4. Consider a morphism (m
1)
a ;a;b
aL : b ! a
1
b;a;a
aL : b ! c
aR : b ! a
aR : b ! c
c where m : a
a where m : b
c where n = m
(pa 1
1) : a
a where n = m
(1
pa) : b
b ! c. This
b ! c.
a ! c. This
a ! c.
These moves follow from (3.12) and (3.13). We draw a picture of the rst move in gure 7.
The other three moves are also described by similar pictures.
3.1.9
Unitary structure
The unitary structure requires an existence of a conjugatelinear involution sending
m 2 Hom(a; b) to my 2 Hom(b; a), generalizing the Hermitian conjugate in the standard
We require that the evaluation and the coevaluation morphisms are related by this
linear algebra.
conjugate operation:
aR = ( aL)y;
aL = ( aR)y:
(3.20)
We further require my
m 2 Hom(a; a) to be positive semide nite in the following
sense: since we assumed the semisimplicity and the
niteness of the number of simple
objects, Hom(a; a) can naturally be identi ed with a direct sum of a matrix algebra. Then
we require my
m to have nonnegative eigenvalues.
The above positivity condition requires dim a > 0 for all a. As we will see in section 3.3.4
below, the unitarity implies sphericity.
3.2
We have several comments:
Note that category theorists do not like unitary structures, since it is speci c to
the base eld C while they would like to keep everything usable for arbitrary base
eld. For this reason they often distinguish various concepts of
operations and
various structures satis ed by them, such as rigid structure, pivotal structure, spherical
structure and pseudounitary structure.7 If we consider unitary 2d quantum
eld
theories (or more precisely its Wickrotated versions which are re ectionpositive),
the unitary structure is the most natural one.
Operator algebraic quantum
eld theorists in fact work in this setting, since for
them the existence of the positivede nite inner product on the Hilbert space is
paramount. Unfortunately their papers often phrase purely categorical results in
the operator algebra theoretic language, which makes them somewhat harder for
outsiders to digest. From this point of view their review article [6] is very helpful,
where a concise translation between terminologies of two di erent schools is given.
Every property given above, except the simplicity of identity, semisimplicity and
niteness, is a straightforward expression of how topological lines and the junction
operators associated to symmetries should behave. We impose the simplicity of
identity, semisimplicity and niteness to make the situation tractable. When the
semisimplicity is dropped, the category is called a
nite tensor category; when the simplicity
of identity is dropped, the category is called a nite multifusion category; when both
are dropped, it is called a
nite multitensor category. When
niteness is dropped,
we simply drop the adjective \ nite".
Indeed, if we consider all topological lines in a given 2d theory and all topological
operators on topological lines, they might not in general form a unitary fusion category.
Rather, our point of view is that we take a subset of topological lines and subspaces
of topological operators on the lines so that they form a unitary fusion category, and
then it can be thought of as a symmetry of the 2d theory.
A very similar categorical structure was introduced by Moore and Seiberg [11] in
the analysis of 2d RCFTs and 3d TFTs. In the category theory they are now called
unitary modular tensor categories. In fact, the unitary modular tensor categories are
also unitary fusion categories, where the latter description is obtained by forgetting
the braiding.
7The rigid structure posits the existence of the left dual a and the right dual a, satisfying various
conditions. It can be shown that
isomorphisms a
' a. In our description, the pivotal structure relates aL and aR . A pivotal structure is
a ' a , and a
' a. The pivotal structure is a collection of
largest eigenvalue of (Na)bc for all simple a. For the de nition of Nacb, see section 3.3.3.
called spherical if dim a = dim a for all a. A spherical structure is called pseudounitary if jdim aj is the
3.3
Before discussing examples, it is useful to set up a few more notions:
`Homomorphisms' between symmetry categories
In the case of two groups G1 and G2, we have the concept of homomorphisms ' : G1 ! G2,
preserving the group multiplication. Similarly, we can talk about symmetry functors ' :
C1 ! C2 between two symmetry categories, together with the data specifying how the
structures listed above are mapped. Among them are isomorphisms
a;b 2 Hom('(a)
'(b); '(a
b))
which tell us how the tensor structure of C1 is mapped into the tensor structure of C2. For
example, the morphisms a;b map the associator of C1 to the associator of C2.
Two symmetry functors '; '0 : C1 ! C2 are considered equivalent when there is a set
of isomorphisms
a 2 Hom('(a); '0(a))
a b a;b = 0a;b( a
b):
such that
categories.
When a symmetry functor has an inverse, it is called an equivalence between symmetry
(3.21)
HJEP03(218)9
(3.22)
(3.23)
3.3.2
Products of symmetry categories
In the case of two groups G1 and G2, their product G1
G2 is also a group. Similarly,
given two symmetry categories C1 and C2, we denote their product as C1
C2, whose simple
objects are given by a1
a2 where a1;2 are simple objects of C1;2, respectively. This product
is called Deligne's tensor product of categories.
3.3.3
Fusion rule of unitary fusion categories
A symmetry category comes with a lot of structures. Sometimes it is useful to forget
about most of them as follows. For each isomorphism class of simple objects a, introduce
a symbol [a], and de ne their multiplication by [a][b] := P
c Nacb[c] when a
b = L
c Nacbc.
This makes nonnegative integral linear combinations of [a]'s into an algebra over Z+ with
a speci c given basis. We call this algebra R(C) the fusion ring of the symmetry category
C. In the case of modular tensor categories, this algebra is also called the Verlinde algebra.
We would often call this algebra as just the fusion rule of C.
Let n be the number of isomorphism classes of simple objects. Then we can regard
(Na)bc as n
n matrices and [a] 7! Na is the adjoint representation of the fusion ring.
3.3.4
Determination of dimensions of objects
The dimensions are xed by Nacb. To see this, consider the ndimensional vector v := (dim a)a
where a runs over the isomorphism classes of simple objects. Its entries are positive real
numbers thanks to the unitarity. Furthermore, v is the simultaneous eigenvector of all
Na's with eigenvalues dim a. Then by the PerronFrobenius theorem, dim a is the largest
eigenvalue of the matrix Na, which is guaranteed to be positive. The argument above
applies both to dimC and dimCC, and therefore the sphericity is implied by the unitarity.
Unitarity also guarantees dim a = dim a .
We de ne the total dimension of the symmetry category C by the following formula:
dim C =
X(dim a)2:
a
Here the sum runs over the isomorphism classes of simple objects.
(3.24)
(3.25)
HJEP03(218)9
Groups and representations of groups as symmetry categories
Symmetry categories C(G; )
As an example, let us recast an ordinary group G as a symmetry category. We rst regard
each element g 2 G as a simple object denoted by the same letter in the category. We
g
g0 := gg0;
g := g 1:
Taking g1;g2;g3 to be the identity maps, they clearly form a unitary fusion category, which
we denote by C(G).
More generally, the pentagon identity among
g1;g2;g3 says that
is a 3cocycle on G
valued in U(1).8 Denote the resulting fusion category by C(G; ). In the literature it is
often denoted VecG. We clearly have dim g = 1. Thus the total dimension of this symmetry
category is the order of the group.
When
1 and
2 di er by a coboundary of a 2cochain , we can construct an
equivalence of categories between C(G; 1) and C(G; 2) using the functor speci ed using the same
in (3.21). This means that in the de nition of C(G; ), one can regard
2 H3(G; U(1)).
It is also clear that any unitary fusion category whose simple objects are all invertible
can be made to be of this form. Summarizing,
A symmetry category C whose simple lines are all invertible is equivalent to
C(G; ) where G is a nite group and
is an element in H3(G; U(1)).
3.4.2
C(G; ) and the anomaly
As is by now familiar, this cohomology class
avor symmetry in two dimensions [12, 13]. One way to see it is as follows [2]: insert a
network of lines with trivalent junctions between them on the spacetime manifold
. Let
the lines be labeled by simple objects of C(G; ), that is by group elements. And let every
junction of the form g
g0 ! gg0 be labeled by the identity morphism gg0 ! gg0. Such a
con guration can also be thought of as reproducing the e ects of a background connection
on
which has holonomies given by g on crossing transversely a line labeled by g. Now,
consider a local region looking like the left hand side of gure 8. Move the lines such
8As already noted in footnote 4, in our convention
g1;g2;g3 is trivial whenever any of g1;2;3 is the identity.
Such a cocycle is called normalized. It is a wellknown fact in group theory that group cohomology can be
computed by restricting every cochains involved to be normalized.
2 H3(G; U(1)) speci es the anomaly of G
HJEP03(218)9
topological line operators. If the symmetry is anomalous, they lead to di erent partition functions.
that now it looks like the right hand side of gure 8. This changes the partition function
by
(g; g0; g00). The new background connection is just a gauge transform of the original
background connection. Hence, we see that
precisely captures the anomaly in the avor
symmetry. Morally, this means the following:
A symmetry category C includes the speci cation of its anomaly.
Fixing a group G, the set of its anomalies forms an Abelian group. Notice that, in our
language, C(G; ) for di erent
have the same fusion ring R = Z+G. Thus, we can ask
the following more general question: does the set of symmetry categories C with the same
fusion ring R form an Abelian group? The answer is that we need a coproduct on R. To
see this, let us recall why the anomaly of a avor symmetry forms an Abelian group from
the perspective of quantum
eld theory.
In general, given two theories T1;2, we can consider the product theory T1
T2 which
is just two decoupled theories considered as one. When Ti has avor symmetry group Gi,
the product T1
T2 has avor symmetry group G1
G2. When G1 = G2 = G, we can take
the diagonal G subgroup of G
G and regard T1
T2 to have avor symmetry G. Now,
when Ti has the anomaly i, we de ne the anomaly of T1
T2 to be the sum 1 + 2. This
abstractly de nes the addition operation on the anomaly.
The crucial step that does not directly generalize to symmetry categories is the
existence of the diagonal subgroup G
G
G. In order to de ne the addition operation on
the set of fusion categories sharing the same fusion rule R, similarly we need a coproduct
R ! R
3.4.3
R.
C(G; ) and the GSPT phases
Next, xing a 3cocycle , let us ask what is the autoequivalence of C(G; ), that is, the
self equivalence that preserves the structure as a symmetry category.
Pick an autoequivalence ' : g ! '(g) with the associated g;h 2 Hom('(g)'(h); '(gh)).
Clearly ' is an automorphism of G. Fixing ' to be the identity,
needs to be a 2cocycle
so that it does not change . Furthermore, two such 's are considered equivalent when
h
h
gh
g
they di er by a 2coboundary, due to (3.23). Therefore can be thought of as taking values
in H2(G; U(1)). Summarizing,
Autoequivalence of C(G; 0) is the semidirect product Aut(G) n H2(G; U(1)).
The Aut(G) part is clear: it just amounts to renaming the topological lines associated
to the group operation. How should we think of H2(G; U(1))? It is telling us that instead
of choosing the identity operator as the implicit junction operator for g
g0 ! gg0 as done
in
gure 1, we can choose g;g0 times the identity. This will not change the associator
but will change the partition function associated to a background connection on
. This
corresponds to coupling a twodimensional theory with C(G; ) symmetry with a
twodimensional bosonic symmetry protected topological (SPT) phase, which is speci ed by
the 2cocycle
protected by the avor symmetry G. The 2cocycle
is also known as a
discrete torsion of G.
to two trivalent junctions.
As an example, consider a torus with holonomies g; h around two 1cycles. They can
be represented using the topological lines as in gure 9. There, we resolved the intersection
We now change the operators at the two junctions to g;h and g 1;h 1 given by the
values of the 2cocycle. In total the phase of the partition function changes by
cg;h = g;h= h;g
(3.26)
which is the standard relation between the discrete torsion phase c on the torus and the
2cocycle [14]. We can thus generalize as follows:
Autoequivalences of a symmetry category C generalize the notion of renaming
and multiplying by SPTphases for a group symmetry.
We need to keep in mind however that the phases introduced by
in the general case do
not have an interpretation of multiplying a SPT phase protected by C, since the product of
two theories with symmetry C has symmetry C
C but is not guaranteed to have symmetry
C, as already discussed above.
3.4.4
Rep(G) as symmetry category
Next, let us discuss Rep(G) for a
nite group G. Its structure as a symmetry category
is straightforward: the objects are representations of G, the morphism space Hom(R; S)
(X ; ; X 0; 0 ; X 00; 00 ) = 00( ( ; 0));
(X ; ; X 0; 0 ; Y 00; 00 ) = 00( ( ; 0));
(X ; ; Y 0; 0 ; X 00; 00 ) = ( ; 00);
(Y ; ; X 0; 0 ; X 00; 00 ) = ( 0 00
1)( ( 0; 00));
(X ; ; Y 0; 0 ; Y 00; 00 ) = 00( ( ; ( 0) 1 00));
(Y ; ; X 0; 0 ; Y 00; 00 ) = ( 0; ( 0) 1( 00) 1);
(Y ; ; Y 0; 0 ; X 00; 00 ) = ( 0( 00) 1)( ( ( 0) 1; 00));
(Y ; ; Y 0; 0 ; Y 00; 00 ) = sgn( ) ( ( 0) 1; 0( 00) 1):
As a check of the computation, we can directly con rm that these de ne a 3cocycle on G.
In the eight cases Rep(D8), Rep(Q8), KP, TY and S
we discussed above, we always
have H = Z2 and the resulting group G = (Z2
Z2) o Z2 is D8. To see this, regard Z2
Z2
as the group of ipping the coordinates x and y of R2 generated by
(5.36)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
(5.44)
(5.45)
(5.46)
(6.1)
can be computed using the projections given above and the
associators (5.22), (5.23) and (5.24) of the original category. We nd
respectively, and Z2 acting on Z2
Z2 to be the exchange of x and y given by
(x; y) 7! ( x; y);
(x; y) 7! (x; y)
(x; y) 7! (y; x):
Dually, with a suitably chosen
on D8 and gauging the Z2 subgroup ipping the x
coordinate, we get the four symmetry categories given above.
6
6.1
2d TFT with C symmetry and their gauging
2d TFTs without symmetry
As a warmup, let us recall the structure of 2d TFTs without any symmetry. We follow
the exposition in [10] closely, see in particular their appendix A.
We start with a vector space V of states on S1 and one wants to de ne a consistent
transition amplitude
Z : V
m
! V n
corresponding to a given topological surface
with m incoming circles and n outgoing
circles. We need four basic maps I, I, M , M corresponding to four basic geometries
given in gure 25.
First, we construct maps IM : V
V ! C and M I : C ! V
V as in gure 26. This
inner product must be nondegenerate because it just corresponds to a cylinder geometry
which pairs a state on one circle with a dual state on the other circle. Using it, we can
I : V ! C
I : C ! V
M : V
V ! V
M : V ! V
identify V and V . Then, I is an adjoint of I and
M is an adjoint of M . Therefore,
to every property involving M , we can write down a corresponding property involving
M , and similarly for statements about I and I. This allows us to reduce the number
of independent statements we need to write down roughly by half; we do not repeat these
adjoint statements below.
We consider M as giving a product on V . There is no order on the two incoming
circles of a pair of pants and hence the product is commutative, see
gure 27. We can
also see that M is associative from
gure 28 and that I is a unit of the multiplication M
from
gure 29. Also, by composing these inner products with the product, we see that the
product is invariant under permuting three legs, see gure 30.
After these preparations, let us associate a map Z : V
m
! V n to a surface
with
m incoming circles and n outgoing circles. We pick a time coordinate t :
that at t = 0 we start with m initial circles and at t = 1 we
nish with n
! [0; 1] such
nal circles.
As time goes from 0 to 1, the number of circles generically stay constant but can either
increase or decrease by one unit at speci c times 0 = t0 < t1 < t2 <
< tp = 1. Cut
once in each interval (ti; ti+1). This divides
into p pieces. The geometry of each piece
contains some cylinders, which correspond to trivial transition amplitude, and exactly one
nontrivial geometry out of the four nontrivial cases shown in gure 25. This gives us an
expression for Z in terms of the four maps I; I; M; M .
However, one could choose a di erent time t0 which starts with same m initial circles
and ends with same n nal circles. In general, this would lead to a di erent cutting of
and
a di erent compositions of four maps I; I; M; M . We need to make sure that they agree.
We can continuously deform the time function t to obtain the time function t0. The
critical points ti will move under this deformation and will cross each other. It is also
possible for two critical points to meet and annihilate each other or for two critical points
to pop out of nowhere. We therefore need to ensure that Z
remains invariant when ti and
ti+1 cross each other, and when two critical points are created or annihilated. For this, we
just need to ensure that the twostep composition from the cut between ti 1 and ti to the
cut between ti+1 and ti+2 remains invariant under these processes.
All possible types of the topology changes were enumerated carefully in appendix A
of [10]. The cases are the following and their adjoints:
1. The creation or the annihilation of two critical points as shown in gure 29, or
2a. The exchange of two critical points as shown in gure 28, which we already
encoun2b. the situation gure 31 where the number of intermediate circles changes from one to
tered, or
three, or
2c. the situation gure 32 where the Acycle and the Bcycle of a torus is exchanged. In
more detail, on one side, a circle consisting of segments a; b; c; d in this order splits
to two circles consisting of a; b and c; d, which are now along the Acycle. They then
HJEP03(218)9
b
c
c
a
d
=
b a
c d
d
b
c
a
d
HJEP03(218)9
on the right, the time ows from inside to the outside, and the parallel edges of the boundary need
to be identi ed to form a torus. On one side, the intermediate two circles are along the Acycle,
and on the other side, they are along the Bcycle.
merge into a circle consisting of four segments with the order b; a; d; c. On the other
side, the two circles in the intermediate stage consists of segments b; c and d; a, and
are along the Bcycle.
The invariance of Z under the change 1 is the unit property itself, and the invariance
under the change 2a is the associativity itself. The invariance under the change 2b can be
reduced to associativity by using the cyclic invariance of the product, shown in gure 30.
Finally, under the topology change 3b, the map Z is trivially invariant.
In total, we have shown that a 2d TFT with no symmetry is completely de ned by a
vector space V with the four maps I; I; M; M with the conditions described above. Such
a vector space is known as a commutative Frobenius algebra V .
6.2
TFT with C symmetry on a cylinder
Let us now move on to the discussion of TFTs with symmetry given by a symmetry category
C. In this subsection we start with the simplest geometry, namely cylinders. We already
discussed basics in section 4.8.1. As mentioned there, we choose a base point along each
constanttime cicle, and call its trajectory the auxiliary line.
Basic ingredients.
We rst associate the Hibert space Va for a circle with a single
insertion of a line labeled by a 2 C. We require Va b = Va
Vb. We now associate a Hilbert
space Va;b;c;::: for a circle with insertions of transverse lines a, b, c, . . . by fusing them in
a xed particular order, starting from the closest line on the right of the base point and
then toward the right:
Va;b;c;::: := V( ((a b) c) ):
(6.2)
The case with three lines is shown in gure 33.
a b
space of a circle with the fused line operator.
b
a
b
Z(m) : Va ! Vb
Xa;b : Va b ! Vb a
Yb;a : Vb a ! Va b
We have two basic operations we can perform on the cylinder, see gure 34. One is to
insert a morphism m : a ! b, which de nes an operator Z(m) : Va ! Vb. Another is to
move the base point to the right and to the left, which de nes morphisms Xa;b : Va b ! Vb a
and Yb;a : Vb a ! Va b.
Assignment of a map to a given network.
With these basic operations, we can
assign a map Va;b;::: ! Vc;d;::: for a cylinder equipped with an arbitrary network of lines
and morphisms from the symmetry category C, where an incoming circle have insertions
a, b, . . . and an outgoing circle have insertions c, d, . . . .
We choose a time function t on it, and we call any time ti a critical point when either of
the following happens: i) there is an insertion of a morphism on a line, ii) there is a fusion
of two lines a; b into one line a
b or vice versa, and iii) a line crosses an auxiliary line. Note
that we do not allow the auxiliary line to bend backward in time, as part of the de nition.
We order 0 = t0 < t1 <
< tp 1 < tp = 1 so that the incoming circle is at t = 0 and
the outgoing circle is at t = 1. Each critical point of type i) gives a factor of Z(m), that
of type ii) gives a factor of Z( ) where
is an appropriate associator, and that of type
iii) gives a factor of X or Y . Then we de ne the map Va;b;
! Vc;d;
associated with this
time function t to be the composition of factors corresponding to these critical points.
Consistency of the assignment.
We now need to show that this assignment is
consistent. There are three types of changes under which the assignment needs to be constant,
namely
the change of the time function t,
then
and
a
c
c
a
d
a
d
d
b
=
=
b
a
c
: a
b ! c
d
d
d
b
HJEP03(218)9
two subnetworks give the same morphism.
the change of the positions of the auxiliary line, and
the change of the network in a disk region that does not change the morphism
within it.
The third point might need some clari cation. In the symmetry category C, a topologically
di erent network can correspond to the same morphism. Then we need to ensure that if
we replace a subnetwork on a cylinder accordingly, the resulting map on the Hilbert space
should also be the same, see
gure 35 This is not just a change in the time function,
therefore we need to guarantee the invariance separately.
The auxiliary line might cut though the subdiagram, as also shown in
gure 35, but
this does not have to be treated separately, since we can rst move the auxiliary line outside
of the disk region, assuming that it is shown that the auxiliary lines can be moved.
Then this third type of change can be just taken care of by assuming that we can fuse
two local operators, leading to the following constraint, see gure 36:
Z(n)Z(m) = Z(n
m):
Next, let us take care of the second type of change, where we move the auxiliary lines
keeping the network and the time function
xed. First, moving the auxiliary line back and
forth in succession should not do anything, so we have
(6.3)
(6.4)
Xa;b = Yb;a1;
for m : a ! a0 and n : b ! b0. We can also fuse two lines before crossing the auxiliary line,
see gure 40. This leads to the constraint
Xb;c aZ( b;c;a)Xa;b cZ( a;b;c) = Z( c;a;b 1)Xa b;c:
Finally, on the cylinder, the change in the time function itself does not do much, and
possible changes are already all covered. Thus, we see that to de ne a consistent TFT
therefore we have
see gure 38.
see gure 37. Rotating the base point all the way around should not do anything either,
Then we should be able to move the morphisms across the auxiliary line, leading to
two relations, as illustrated in gure 39:
Xa;1 = id;
Xa0;bZ(m
Xa;b0 Z(1
1) = Z(1
n) = Z(n
m)Xa;b;
1)Xa;b
a
b n
a
b
a
=
b
a
a
(6.5)
(6.6)
(6.7)
(6.8)
HJEP03(218)9
aʹ
a
b
a b
a⊗b
b
c
;
a
a b
a⊗b
bʹ
n
c
a
n
bʹ
HJEP03(218)9
with C symmetry on a cylinder, we need the data of an additive functor Z : C ! Vec with
morphisms Xa;b : Va b ' Vb a satisfying (6.5), (6.6), (6.7) and (6.8).
Generalized associators on the cylinder.
The relations so far guarantees that we
can always move the base point and change the order of the tensoring of lines in a
consistent manner. For example, the relation (6.8) means that there is a single wellde ned
isomorphism between V(a b) c and V(c a) b. We introduce a notation
A(a b) c!(c a) b : V(a b) c ! V(c a) b
(6.9)
for it, and call it a generalized associator on the cylinder. We similarly introduce generalized
associators for an arbitrary motion of the base point and an arbitrary rearrangement of
parentheses. Each such generalized associator have multiple distinctlooking expressions
in terms of sequences of Z( ), X and X 1, but they give rise to the same isomorphism.
6.3
TFT with C symmetry on a general geometry
Basic data. Let us discuss now the TFT with C symmetry on a general geometry. The
four basic geometries are given in gure 41. For a pair of pants, we need to join the two
auxiliary lines coming from each leg into a single auxiliary line. We take the point where
this happens to coincide with the critical point where two circles join to form a single circle.
In what follows, we will refer to the initial two legs of a pair of pants as the initial legs and
the nal leg as the product leg.
We can now associate to any geometry
with m initial legs and n nal legs with an
arbitrarily complicated network of lines and morphisms from C a linear map as follows.
We rst choose a time function t :
! [0; 1]. We call a time value ti critical when any
a
I : V1 ! C
I : C ! V1
Ma;b : Va
Vb ! Va b
b
M a;b : Va b ! Va
Vb
of the following happens: i) the topology of the constant time slice change, ii) there is
a morphism, or iii) a line crosses a auxiliary line. We order the critical times so that
0 = t0 < t1 < t2 <
< tp = 1. We cut
once in each interval (ti; ti+1), and associate to
each critical time ti one of the basic linear maps. We then compose them. We now need
to guarantee that this assignment is consistent.
Basic consistency conditions. Let us rst enumerate basic consistency conditions.
First, we de ne the pairing of Va and Va as in gure 42:
IZ( aL)Ma ;a : Va
M a;a Z( aL)I : C ! Va
Va ! C;
Va :
Then we require that
this pairing is nondegenerate and can be used to identify (Va) ' Va .
Under this pairing, the product Ma;b and the coproduct
M b ;a are adjoint, etc. This
again allows us to reduce the number of cases need to be mentioned below roughly by half.
L
Before proceeding, we note that we used a
,
aR to de ne the pairing. We can also
use aR ,
aR to de ne a slightly di erent pairing. Exactly which pairing to be used in each
situation can be determined by fully assigning orientations to every line involved in the
diagram. Below, we assume that every line carries an upward orientation, unless otherwise
marked in the gure.
Second, a morphism can be moved across the product, see gure 43:
Third, the map I de ned by the bowl geometry gives the unit, see gure 44:
Ma;b(Z(m)
idVb ) = Z(m
idb)Ma;b:
Ma;1(v
I) = v;
v 2 Va:
a*
HJEP03(218)9
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
b
a
b
=
a
b
m
a
aʹ
b
=
(6.15)
(6.16)
(6.17)
a
b⊗c
up to assoc.
=
b
a
b
Fourth, it is twisted commutative:
Xa;bMa;b(v
w) = Mb;a(w
v 2 Va;
w 2 Vb
as illustrated in gure 45. Fifth, it is associative up to the associator:
Z( a;b;c)Ma b;c(Ma;b
idc) = Ma;b c(ida
Mb;c);
as shown in gure 46.
shown in gure 47:
Sixth, we want to formulate that the product is symmetric under the cyclic permutation
of three circles. To do this we rst introduce a slightly generalized form of the product
M(a c ;c b )!a b : Va c
Vc b ! Va b
a
c b*
a
b*
a
b*
time ows from inside to the outside.
M(a c ;c b )!a b = Z(ida
idb )A(a c ) (c b )!a (c c) b Ma c ;c b
(6.18)
where cL : c
introduced in the last subsection.
c ! 1 is the evaluation morphism and A is the generalized associator
The generalized product has an alternative de nition as given in gure 48, where the
line c crosses three auxiliary lines. This gives an alternative expression
M(a c ;c b )!a b =
Z(ida b
R
c )A(c a) (b c)!(a b ) (c c )Mc a;b c(Xa;c
Xc;b )
(6.19)
and we demand
the right hand sides of the equations (6.18) and (6.19) are the same.
We can now formulate the cyclic symmetry of the product:
M(a c ;c b )!a b and M(c b ;b a )!c a are related by the inner products,
(6.20)
(6.21)
see gure 49. We can in fact derive this relation for general a, b, c just from the subcase
when c = 1 and the relations already mentioned. We keep the general case for cosmetic
reasons, since it looks more symmetric.
Seventh, we need a consistency on the torus. An incoming circle consisting of four
segments with lines a, b, c, d can rst split into two circles with two segments a, b and
b
a
a c*
c b*
d
a c*
c b*
b a
c d
d
(6.22)
(6.23)
(6.24)
V(a b) (c d) ! V(b a) (d c)
V(b c) (d a) ! V(c b) (a d)
action of X and the associators.
c, d each and then rejoins to form a circle with four segments in the order b, a, d, c;
another way this happens is that the two intermediate circles have segments b, c and d,
a, see gure 50. They each determine maps (M
(M )b;c;d;a : V(b c) (d a) ! V(c b) (a d) given by
)a;b;c;d : V(a b) (c d) ! V(b a) (d c) and
(M )a;b;c;d := Mb a;d c(Xa;b
(M )b;c;d;a := Mc b;a d(Xb;c
Xc;d) a b;c d;
Xd;a) b c;d a:
We then demand that they are equal up to the generalized associators:
A(b a) (d c)!(c b) (a d)(M )a;b;c;d = (M )b;c;d;aA(a b) (c d)!(b c) (d a):
Consistency in the general case.
We nally
nished writing down basic moves. Now
we can analyze the general moves. We again have three cases:
the change of the time function t on the surface ,
the change of the positions of the auxiliary line, and
the change of the network in a disk region that does not change the morphism.
Let us start by discussing the third case. This is in fact automatic once the rst two
cases are taken care of, since any disk region can be put into a cylinder under a topological
a b
a b
change, and then the auxiliary line can be moved o away from it. Then all we have to
assume is that Z(m) fuses appropriately, (6.3).
The change in the position of the auxiliary line can happen in the following three ways:
The auxiliary line can move within a single cylinder. This was already discussed in
the last subsection.
When a circle with line a and a circle with line b join to form a circle, the order of
a, b and the base point x in the product leg can either be x, a, b or a, b, x. The
invariance under this is the twisted commutativity (6.15).
A line c 2 C can cross the trivalent vertex of the auxiliary line. This move changes
the number of the intersection of the line c with the three auxiliary lines in one of
the two ways, 0 $ 3 or 1 $ 2. One example of the move 0 $ 3 is the equality (6.20)
of the two de nitions (6.18) and (6.19) of the generalized product. The move 1 $ 2
can be deduced by combining the twisted commutativity.
Finally we need to take care of the changes in the time function t. One possible change
is that a morphism and a product can happen in two di erent orders. The invariance under
this move is (6.13). Then there are topological changes in the cutting of the surface, which
again comes in the following varieties:
1. The creation or the annihilation of two critical points does gure 44. The consistency
under this change is the unit property (6.14).
2a. The exchange of two critical points does gure 46. The consistency under this change
is the associativity (6.16).
of the generalized product (6.21).
relation (6.24).
2b. The number of intermediate circles changes from one to three. One example is drawn
in gure 51. The consistency under this change can be reduced to the cyclic symmetry
2c. How the torus is decomposed is changed as in gure 50, for which we assigned a basic
Summarizing, a TFT with C symmetry is captured by the data (V ; Z; X; I; I; M; M )
satisfying the various relations listed above. Namely, on the cylinder, we have (6.3), (6.5),
(6.6), (6.7), (6.8), and on the general geometry, we have in addition (6.12), (6.13), (6.14),
(6.15), (6.16), (6.20), (6.21), and (6.24), and nally, diagrams turned upside down
correspond to adjoint linear maps.
Gauged TFT with the dual symmetry
Now we would like to discuss the de nition of the TFT T =A gauged by an algebra object
A in terms of the ungauged TFT T . We start from the data (V ; Z; X; I; I; M; M ) for the
original theory T .
is given by
The Hilbert space of the gauged theory was introduced in section 4.8.2. See that section
for some necessary background. We use the action by A on Vp, depicted in gure 24. This
P := Up;A( xR; xL) := Z(xL)Xp;AZ( xR)
(6.25)
where xL : A
p ! p and xR : p ! p
A are the morphisms de ning the Abimodule
structure on p. As we already discussed, P turns out to be a projector, and we de ne
Wp to be the projection P Vp. Now, we de ne the data (W ; Z~; X~ ; I~; I~; M~ ; M~ ) for T =A in
terms of corresponding data for T .
The morphism map Z~.
We de ne the new Z~ to be the restriction of the old Z. We
need to check that if the initial state lay in Wp
Vp then the
nal state also necessarily
lies in Wq, for a bimodule morphism p ! q. This can be checked by gluing a cylinder
on top of the
nal state which corresponds to the action of A. The wrapped A line can
then be taken across any bimodule morphism until it wraps the p line at the start of the
cobordism. But then the wrapped A line has no e ect and can be removed because the
initial state we started with is invariant under the action of A.
The base pointchange map X~ .
We now want to de ne the new X~p;q : Wp Aq ! Wq Ap.
We use
and
(see section 4.4) to de ne X~p;q = Z( )Xp;qZ( ). This is well de ned
because a wrapped A line at the end of the cobordism can be moved to an A line propagating
between a and b at the start of the cobordism which can be removed because of the de nition
of A.
The unit and counit maps I~ and I~. Now the new unit morphism I~ would be a map
from C to WA
the de nition of A. Similarly, ~I = IZ(v) where v : A ! 1 is the counit morphism in the
de nition of A. These are wellde ned as can be shown by manipulations similar to those
VA. We de ne it as I~ = Z(u)I where u : 1 ! A is the unit morphism in
we are now going to perform for the de nition of M~ .
The product and coproduct maps M~ and
M~ . The new M~ is de ned analogously
as M~ p;q = Z( )Mp;q. This M~ can be shown to be well de ned as a map Wp
Wq ! Wp Aq
by a series of manipulations using a lot of properties of A. See gure 52. We represent a
pair of pants as a 3punctured plane for ease of illustration. The lower punctures
correspond to input legs and the upper puncture corresponds to the product leg of the pair of
pants. Unlabeled lines correspond to A. To explain various manipulations, let us refer to
manipulations involving =i as \step i".
=4
p
p
p
p A q
p A q
q
q
q
=1
=3
=5
p
p
p
p A q
p A q
q
q
q
main text.
p, and the right action on p
A q is de ned by the right action on q.
In step 2, we introduce an A line wrapping the leg carrying q. We can do that because
the input state in that leg is invariant under the action of A.
In step 3, we rst use the fact that q is an Abimodule and then use the fact that q
is a right Amodule.
In step 4, we use the fact that q is a left module for A.
Finally, in step 5, we rst use the fact that q is a bimodule, then the fact that the
tensor product is
A, and then the fact that p is a bimodule.
Ultimately, we can simply remove both the A lines because the input states are
invariant under the action of A.
We de ne M~ as the adjoint of M~ .
To complete the de nition of T =A, we have to check that the operations de ned above
satisfy the various conditions that we described in the last section. Most of them just
concern some trivial topological manipulations of lines and are manifestly satis ed. Some
others, such as (6.14), can be checked using manipulations similar to the ones we have been
doing in this subsection. Yet some others, like the complicated relation (6.20) and (6.24)),
require us to simplify a lot of
and
, but this can be done. This completes the de nition
of T =A.
7
Conclusions
In this paper we reviewed the notion of unitary fusion categories, or symmetry categories as
we prefer to call them, and how they formalize the generalized notion of nite symmetries
of a twodimensional system.
We studied various explicit examples of such symmetry
categories, some of which are related rather directly to
nite groups and some of which
are not. We then studied how a symmetry category can be gauged and be regauged back.
We also de ned 2d topological quantum
eld theories admitting a symmetry given by a
symmetry category. Many questions remain. Here we mention just two.
The rst is how to generalize the constructions discussed in this paper to higher
dimensions. In a sense, this is a merger of the generalized symmetry in two dimensions in
the sense of this paper, and of the generalized symmetry in the sense of [2]. That there
should be something that combines both is clear: even in general spacetime dimension d,
the 0form symmetries can be any nonAbelian group G, possibly with an anomaly
specied by Hd(G; U(1)) in the bosonic case and by subtler objects in the fermionic case. Then
the (d
2)form symmetry needs to be extended at least to allow Rep(G). When d = 3, it
seems that the notion of 1form symmetries needs to be extended at least to include
modular tensor categories, with an action of the 0form symmetry group G with an anomaly.
What should be the notion in d = 4 and higher?
The second is to actually construct twodimensional systems T for a given symmetry
category C. For any group G without an anomaly, there is the trivial theory where the
Hilbert space is always onedimensional. How about the other cases?
We can roughly
classify symmetry categories C as follows, depending on the simplest possible theories T
that have C as a symmetry:
1. The simplest Csymmetric theories have onedimensional Hilbert space. These would
be Csymmetry protected topological (SPT) phases.
2. The simplest Csymmetric theories have
nitedimensional Hilbert space. These
would be Csymmetry enriched topological (SET) phases.
3. The simplest Csymmetric theories have in nitedimensional Hilbert space. Taking
the low energy limit, these would be Csymmetric conformal eld theories (CFTs).
4. There is no Csymmetric theory.
Clearly this classi cation forms a hierarchy, and it would be nice if we have a uniform
construction that tells easily which stage of the above classi cation a given symmetry
category C belongs to. There is a recent paper in this general direction [36], where a
construction of 2d theory starting from any given symmetry category C was discussed. We
hope to see more developments in the future.
Actually, there are various symmetry categories C constructed in the subfactor theory,
e.g. what is called the Haagerup fusion category, for which no Csymmetric theory is known.
If a theory symmetric under the Haagerup fusion category could be constructed, it would
be considered as a huge breakthrough.
Acknowledgments
The authors would like to thank Mikhail Kapranov for bringing the crucial reference [7]
to the authors' attention. LB is also grateful to Kevin Costello, Davide Gaiotto and Theo
JohnsonFreyd for helpful discussions. The authors thank Nils Carqueville and Eric Sharpe
for comments on, and Marcel Bischo , Kentaro Hori and ShuHeng Shao for pointing out
erroneous claims in an earlier version of the manuscript. The work of LB is partially
supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter
Institute is supported by the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Economic Development and Innovation. The
work of YT is partially supported in part by JSPS KAKENHI GrantinAid (WakateA),
No.17H04837 and JSPS KAKENHI GrantinAid (KibanS), No.16H06335 and by WPI
Initiative, MEXT, Japan at IPMU, the University of Tokyo.
The di erential squares to zero: d2 = 0. Then we de ne the group cohomology Hi(G; A)
as the cohomology of this di erential. Explicitly, the rst few di erentials are given by
df (g; h) = gf (h) f (gh) + f (g);
df (a; b; c) = af (b; c)
f (ab; c) + f (a; bc) f (a; b);
df (x; y; z; w) = xf (y; z; w) f (xy; z; w) + f (x; yz; w) f (x; y; zw) + f (x; y; z):
Some points on notation. It does not lead to any loss of generality if we assume
that every cochain/cocycle/coboundary is normalized, i.e. it is zero whenever at least one
of gi = 1. See e.g. [13]. We have assumed throughout the paper that every cochain is
normalized. We are often interested in Hi(G; U(1)) for i = 2; 3 where the action of G on
U(1) is taken to be trivial. Henceforth, we will assume the trivial action whenever we write
U(1). It is also convenient to treat U(1) elements as phases and in this case the + sign in
above de nitions should be replaced by the usual multiplication of phases. For instance
we have,
ogy valued in U(1).
df (g; h) =
f (h)f (g)
f (gh)
:
We have used the product notation throughout the paper in the context of group
cohomolIn this appendix, we collect various standard facts about group cohomology.
De nition. Given a nite group G and its module A, we de ne ncochains Cn(G; A) as
functions Gn ! A. The di erential is given by
df (g1; : : : ; gn+1) = g1f (g2; : : : ; gn+1)
n
i=1
+ X( 1)if (g1; : : : ; gigi+1; : : : ; gn+1) + ( 1)n+1f (g1; : : : ; gn): (A.1)
(A.2)
(A.3)
(A.5)
(A.6)
(A.7)
Pullback. Recall that given a map M1 ! M2 between two manifolds, one can pullback
nforms on M2 to nforms on M1. The analogous statement in group cohomology is that
Hi(G; A). Explicitly, let h : G ! G0, then h~ : Hi(G0; A) ! Hi(G; A) is given by
given a group homomorphism G ! G0, we obtain a module homomorphism Hi(G0; A) !
h~( )(g1;
; gi) = (h(g1);
; h(gi)) :
Cup product.
when A is a ring. If
There is an operation called cup product Ci(G;A) Cj(G;A) ! Ci+j(G;A)
2 Hi(G; A) and
2 Hj(G; A), then the cup product is de ned as
( [ )(g1;
; gi+j) = (g1;
; gi) (gi+1;
; gi+j):
It can be easily checked that this product descends to a product on cohomologies.
Onedimensional representations of G. Let us ask what is the meaning of H1(G;U(1)).
The 1cochains are maps from G to U(1) and imposing the cocycle condition turns them
into group homomorphisms. Hence H1(G; U(1)) is the group formed by onedimensional
representations of G. In particular, when G is a nite Abelian group, then H1(G; U(1)) ' G^
Projective representations of G. Now, let us ask what is the meaning of 2 H2(G;U(1)).
We want to interpret the (g1; g2) as the phases de ning a projective representation of G.
The cocyle condition reads
(g1; g2) (g1g2; g3) = (g2; g3) (g1; g2g3)
which is the associativity condition such phases are required to satisfy. Such a cocycle can
be shifted by a coboundary of the form
(A.8)
(A.9)
(A.10)
(A.11)
d (g1; g2) =
(g1) (g2)
0 ! H ! G ! K ! 0
which corresponds to rephasing of the group action on the projective representation. Thus,
we see that H2(G; U(1)) classi es the phases encountered in projective representations upto
rephasing. The usual representations correspond to the trivial element of H2(G; U(1)).
Crossed products and extensions of groups.
Consider an Abelian group H and
a (possibly nonAbelian) group K. Consider an action of K on H and use it to de ne
H2(K; H). An element
2 H2(K; H) can be used to de ne a group extension G of K by
H, that is there is a short exact sequence
and G is called the cross product of K and H and it is written as G = H o K. Explicitly,
the group G as a set is the direct product H
K with the group multiplication given as
follows
(h1; k1)(h2; k2) = (h1 + (k1 . h2) + (k1; k2); k1k2):
Here, . denotes the action of K on H via inner automorphism in G. The reader can verify
that the associativity of the group multiplication is ensured by the cocycle condition on .
Shifting
by a coboundary changes G upto isomorphism. Hence, group extensions of K
by an Abelian group H are classi ed by a group action of K on H along with an element
in H2(K; H) de ned using the group action.
Bicharacters on G. Consider an Abelian group G and a trivial module A of G. Given
a cohomology element in H2(G; A) represented by a cocycle (g; h), one can form
(g; h) =
(g; h)
(h; g) which is an antisymmetric function on G. This is indeed well de ned because
adding a coboundary to
doesn't change . When A is U(1) this
is a bicharacter, and
there's a bijection between an antisymmetric bicharacter on G and H2(G; U(1)).
A useful isomorphism.
Recall the familiar statement that any closed nform is exact
locally. In group cohomology, the analogous statement is that
Hi(G; R) = 1;
that is Hi(G; R) is trivial. We can use this to obtain Hi+1(G; Z) ' Hi(G; U(1)). This
follows from the long exact sequence associated to the sequence 0 ! Z ! R ! U(1) ! 0.
Explicit group cohomologies.
H (Zn; Z) = Z[x2]=(nx2).
H2((Zn)k; U(1)) = (Zn)k(k 1)=2.
H3((Zn)k; U(1)) = (Zn)k+k(k 1)=2+k(k 1)(k 2)=6, with generators given by
particular, H3(Zn; U(1)) = Zn and its generator has the cocycle
where a; b; c = f0; 1; : : : ; n
1g and hai is the mod n function to f0; : : : ; n
(i)(a; b; c) = e2 iai(bi+ci hbi+cii)=n2 ;
(i;j)(a; b; c) = e2 iai(bj+cj hbj+cji)=n2 ;
(i;j;k)(a; b; c) = e2 iaibjck=n
(a; b; c) = e2 ia(b+c hb+ci)=n2 :
(A.12)
(A.13)
(A.14)
(A.15)
1g. In
(A.16)
For Dm the dihedral group with m elements we have [37],
H (Dm; Z) = Z[a2; b2; c3; d4]=(2a2; 2b2; 2c3; md4; (b2)2 + a2b2 + (m2=4)d4): (A.17)
In particular, H2(D2n+1; U(1)) = 0, H3(D2n+1; U(1)) = Z4n+2; H2(D2n; U(1)) =
Z2
Z2, H3(D2n; U(1)) = Z2
Z2
Z2n. The explicit generators can be found in [38].
For Q8, the quaternion group, we have [39]
H (Q8; Z) = Z[A2; B2; C4]=(2A2; 2B2; 8C4; A22; B22; A2B2
4C4):
(A.18)
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
[1] C. Vafa, Quantum Symmetries of String Vacua, Mod. Phys. Lett. A 4 (1989) 1615 [INSPIRE].
[2] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP
[3] I. Brunner, N. Carqueville and D. Plencner, Discrete torsion defects, Commun. Math. Phys.
[4] E. Sharpe, Notes on generalized global symmetries in QFT, Fortsch. Phys. 63 (2015) 659
[5] D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and
Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
[6] M. Bischo , R. Longo, Y. Kawahigashi and K.H. Rehren, Tensor categories of
endomorphisms and inclusions of von Neumann algebras, arXiv:1407.4793 [INSPIRE].
[7] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories, Mathematical Surveys
and Monographs, vol. 205, AMS, Providence, RI (2015).
[8] N. Carqueville and I. Runkel, Orbifold completion of defect bicategories, Quantum Topol. 7
[9] I. Brunner, N. Carqueville and D. Plencner, A quick guide to defect orbifolds, Proc. Symp.
(2016) 203 [arXiv:1210.6363] [INSPIRE].
Pure Math. 88 (2014) 231 [arXiv:1310.0062] [INSPIRE].
hepth/0609042 [INSPIRE].
[10] G.W. Moore and G. Segal, Dbranes and ktheory in 2D topological eld theory,
[11] G.W. Moore and N. Seiberg, Lectures on RCFT, in Strings '89, Proceedings of the Trieste
Spring School on Superstrings, World Scienti c (1990)
[http://www.physics.rutgers.edu/ gmoore/LecturesRCFT.pdf] [INSPIRE].
[12] R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun.
Math. Phys. 129 (1990) 393 [INSPIRE].
[13] X. Chen, Z.C. Gu, Z.X. Liu and X.G. Wen, Symmetry protected topological orders and the
group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114
[arXiv:1106.4772] [INSPIRE].
[14] M.R. Douglas, Dbranes and discrete torsion, hepth/9807235 [INSPIRE].
[15] D. Tambara and S. Yamagami, Tensor categories with fusion rules of selfduality for nite
abelian groups, J. Algebra 209 (1998) 692.
Phys. 14 (2002) 733.
[16] P. Etingof and S. Gelaki, Isocategorical groups, Int. Math. Res. Not. (2001) 59,
[17] M. Izumi and H. Kosaki, On a subfactor analogue of the second cohomology, Rev. Math.
[18] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators 1. Partition
functions, Nucl. Phys. B 646 (2002) 353 [hepth/0204148] [INSPIRE].
[19] G. Schaumann, Traces on module categories over fusion categories, J. Algebra 379 (2013)
[20] P. Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory, Quantum
[21] L. Bhardwaj, D. Gaiotto and A. Kapustin, State sum constructions of spinTFTs and string
net constructions of fermionic phases of matter, JHEP 04 (2017) 096 [arXiv:1605.01640]
[22] L. Kong, Anyon condensation and tensor categories, Nucl. Phys. B 886 (2014) 436
382 [arXiv:1206.5716].
Topol. 1 (2010) 209 [arXiv:0909.3140].
8 (2003) 177 [math/0111139].
Math. Phys. 332 (2014) 669 [arXiv:1307.3141] [INSPIRE].
Math. Phys. 123 (1989) 177 [INSPIRE].
orbifolds, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09),
Prague, Czech Republic, August 3{8, 2009 [DOI:10.1142/9789814304634 0056]
University of Amsterdam (1995) [hepth/9511195] [INSPIRE].
[25] V. Ostrik , Fusion categories of rank 2, Math. Res. Lett . 10 ( 2003 ) 177 [math/0203255]. [26] V. Ostrik , Pivotal fusion categories of rank 3, Mosc . Math. J. 15 ( 2015 ) 373 [27] J. Fr ohlich , J. Fuchs, I. Runkel and C. Schweigert , Defect lines, dualities and generalised