SciPost Phys. 8, 015 (2020)

On gauging finite subgroups
Yuji Tachikawa
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Abstract
We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup A of a Γ -symmetric theory. Depending on how anomalous Γ is, we find that the symmetry of the gauged theory can be i) a
direct product of G = Γ /A and a higher-form symmetry Â with a mixed anomaly, whereÂ is the Pontryagin dual of A; ii) an extension of the ordinary symmetry group G by thehigher-form symmetry Â; iii) or even more esoteric types of symmetries which are nolonger groups. We also discuss the relations to the effect called the H 3 (G, Â) symmetry
localization obstruction in the condensed-matter theory and to some of the constructionsin the works of Kapustin-Thorngren and Wang-Wen-Witten.
Copyright Y. Tachikawa.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 07-06-2018
Accepted 09-01-2020
Check forupdatesPublished 31-01-2020
doi:10.21468/SciPostPhys.8.1.015

Contents
1

Introduction and summary22

Extension, anomaly and gauging
2.1 Preliminaries: anomaly, topological defects and group cohomology
2.2 From a nontrivial extension with a trivial anomaly
2.3 From a trivial extension with a nontrivial anomaly
2.4 Miscellaneous remarks
2.5 From a nontrivial extension with a nontrivial anomaly
2.6 Relation to other constructions
2.6.1 Relation to Witten [12] and Wang-Wen-Witten [13]2.6.2 Relation to Kapustin-Thorngren [10, 11]
2.7 Existence of finite Abelian extension where the anomaly trivializes

4
4
6
8
8
10
11
11
11
13

3

Incorporating ’t Hooft defects
3.1 An example
3.2 Extension of G[n] by A[m]3.3 A curious phenomenon

14
14
15
16

4

Subtler effects
4.1 D = 2
4.2 D > 2
4.3 Toward even subtler effects

17
17
19
19
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SciPost Phys. 8, 015 (2020)

A

Some algebraic topology
A.1 Eilenberg-Mac Lane spaces
A.2 Lyndon-Hochschild-Serre spectral sequence
A.3 Classifying space for the mixed symmetry

References

1

20
20
20
21
21

Introduction and summary

In this paper we start from a quantum field theory T in general spacetime dimension D with an
ordinary symmetry group Γ possibly with an anomaly. We pick a non-anomalous finite Abeliannormal subgroup A ⊂ Γ . The aim of this note is to answer the following naive question: whatis the symmetry group of the gauged theory T /A?
We know that the group G = Γ /A is part of the story. We also know that the gauged theory
has the (D − 2)-form symmetry Â[D−2] based on the Pontryagin dual group Â of A [1, 2]. Herethe notation G[n] stands for the group G regarded as an n-form symmetry.
The question is how G and Â[D−2] are mixed together. We will find that the result depends
on the anomaly of Γ :1. When Γ is anomaly-free, the symmetry of T /A is the direct product of G and Â[D−2] , witha mixed anomaly determined by the extension 0 → A → Γ → G → 0.
2. When Γ is anomalous but in a tame way as defined later in this paper, the symmetry of
T /A is an extension of G by Â[D−2] . In particular, G is no longer a subgroup of the total
symmetry. This extension is an example of a (D −1)-group, a group-like higher category.
3. When Γ is anomalous in a more wild manner, the symmetry of T /A is still composed of
G and Â[D−2] , but the higher-category describing the total symmetry is no longer grouplike.
Theories whose symmetry is given by higher categorical structures have also been discussed
in the past e.g. [3–5]. Our point here is that such theories arise naturally if we perform evena rather innocuous operation of gauging a finite normal1 Abelian group of a larger group.2
Our analysis can be thought of as a small extension of a few previous studies:• The first case above generalizes and streamlines the technique of Gaiotto, Kapustin, Komargodski and Seiberg [9] where a combination of 0-form symmetry and 1-form symmetry with a mixed anomaly was turned into a non-Abelian 0-form symmetry. Startingfrom SU(2) gauge theory they found the dihedral group (Z2 × Z2 ) o Z2 as the symmetry
on a thermal circle. With our formula it is trivial to generalize it to SU(N ) gauge theory,
for which we find (ZN × ZN ) o Z2 .
When we gauge a non-normal subgroup A, the story is more intricate. When D = 2, the resulting generalized
symmetry is known [6] to be given by a fusion category, whose simple object is labeled by a pair ([g], ρ) where[g] ∈ A \Γ /A is a double coset and ρ is an irreducible representation of A ∩ gAg −1 , providing examples of nongroup-like symmetry structures. It would be interesting to work out the higher-dimensional generalizations, wewill not pursue this direction further in this note.
2
The appearance of two-group symmetry when one gauges a continuous symmetry with a suitable mixed
anomaly is also discussed in a recent paper [7]. See also a recent paper [8] for the discussions of finite gauge
theories living on the boundary of gauged and ungauged Dijkgraaf-Witten theories.
1

2

SciPost Phys. 8, 015 (2020)• The first case above also generalizes the construction of Kapustin and Thorngren [10,11]
to general spacetime dimensions and sheds a somewhat new light on a finite gauge
theory construction in Witten [12] and Wang, Wen and Witten [13].
• The second case for D = 3 exhibits the effect called the H 3 (G, Â) “symmetry localization
anomaly” in the condensed-matter literature, see e.g. [14, 15]. We find that this effectdescribes the extension of an ordinary group G by a one-form symmetry Â.
• The second case also generalizes another work of Kapustin and Thorngren [16] fromD = 3 to D > 3.
• Finally we note that for D = 2, the same question posed above was asked and answeredin terms of the language of fusion categories long time ago, see e.g. [17–21]. For a recent
review, see e.g. [22] by the author and Bhardwaj. In this case the analysis is completely
general and allows non-Abelian or non-normal subgroup A.
In this note the discussions are phrased in terms of the anomaly of a D-dimensional theory
on a spacetime X D . That said, the general principle that the anomaly of a D-dimensional theoryis captured by the corresponding (D + 1)-dimensional invertible topological phase [23–25]
means that it is useful and essential to regard X D as a component of the boundary of a (D + 1)dimensional space YD+1 .
Organization of the note: The rest of the note is organized as follows. In Sec. , we start byanalyzing what happens when we gauge an Abelian normal subgroup A of an anomaly freesymmetry Γ . We then study the case of the direct product Γ = A × G with a mixed anomaly.
These two analyses turns out to be almost symmetric, and will be then be combined. We
will also connect our analysis to the previous works of Kapustin and Thorngren [10, 11, 16],
Witten [12] and Wang-Wen-Witten [13]. In Sec. 2.7, we discuss the generalization where both
G and A are higher-form symmetries. We find that this generalization allows us to include ’t
Hooft defects in a natural manner. Finally in Sec. 3.3, we discuss subtler effects where thetopological defects of the gauged theory contains objects which are no longer invertible. We
have an appendix 4.3 where we review the required algebraic topology very briefly.
Notations and conventions: We assume all groups to be finite for simplicity, although thenon-gauged part G can be readily made to be continuous. We write G[n] to mean that G is
an n-form symmetry, in the sense of [2]. We abbreviate G[0] as G. We use the corresponding
lower case letter in bold, g , for the flavor symmetry background for G. When n = 0, this issimply a G-bundle on the spacetime M , and when n > 0, this is an element of H n+1 (M , G).
Uniformly, one can say that the background g is a map from M to the Eilenberg-Mac Lanespace K(G, n + 1); for more on this, see Appendix 4.3.
We reserve the letter D to denote the spacetime dimension. We typically denote the spacetime as X D and consider it as a component of the boundary of a (D + 1)-dimensional space
YD+1 . The background fields such as g are also regarded as given on YD+1 , not just on X D .
We work in the convention that we only count Wilson lines as the topological operators we
obtain for free in a G-gauge theory. Namely, ’t Hooft defects require the coupling of the original
theory to a singular G-gauge field, and in this paper we consider this to be an additional datum.
In Sec. 2.7 we see that ’t Hooft defects can be included in the framework as well.

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SciPost Phys. 8, 015 (2020)

g3

g3
g1 g2

g1 g2 g3

g1

g2g3

= ω( g1, g2 , g3 ) ×g2

g1g2 g3

g2

g1

Figure 1: The anomalous phase is associated to the change in the topology of thedomain walls on X D implementing the G action.

2

Extension, anomaly and gauging

2.1

Preliminaries: anomaly, topological defects and group cohomology

Let us begin by recalling briefly how the anomaly of a finite group symmetry G in spacetime
dimension D is given by an element ω ∈ H D+1 (G, U(1)), and why it can be accounted for bycoupling it to a (D + 1)-dimensional bulk whose action is given by ω. We illustrate it in thecase D = 2.
We represent the background field for the G symmetry in terms of topological domain
walls. A domain wall is labeled by an element g ∈ G such that crossing the wall implements
the group action by g. Now, consider a situation where three domain walls g1 , g2 , g3 merge
to form a wall g1 g2 g3 . This can be done in two ways, see Fig. 1. Although they both describe
the same background G connection, a nontrivial phase ω(g1 , g2 , g3 ) ∈ U(1) can be produced
in an anomalous theory. In other words ω is a U(1)-valued 3-chain ∈ C 3 (G, U(1)).
When we consider four domain walls g1,2,3,4 joining to form a wall g1 g2 g3 g4 , we canchange how we join the walls either as
g1 (g2 (g3 g4 )) → g1 ((g2 g3 )g4 ) → (g1 (g2 g3 ))g4 → ((g1 g2 )g3 )g4(2.1)

g1 (g2 (g3 g4 )) → (g1 g2 )(g3 g4 ) → ((g1 g2 )g3 )g4 .

(2.2)

or as
The consistency of these two procedures requires the condition∂ ω(g1 , g2 , g3 , g4 ) :=ω(g2 , g3 , g4 ) − ω(g1 g2 , g3 , g4 ) + ω(g1 , g2 g3 , g4 ) − ω(g1 , g2 , g3 g4 ) + ω(g1 , g2 , g3 ) = 0, (2.3)
where we use an additive notation for U(1) = R/Z. This means that ω is a U(1)-valued 3cocycle ω ∈ Z 3 (G, U(1)). In general dimensions, ω is a U(1)-valued (D + 1)-cocycleω ∈ Z D+1 (G, U(1)).
To see that this anomaly can be accounted for by coupling it to a bulk (D + 1)-dimensional
theory, let us use the dual triangulation instead of the domain walls to visualize the setup.
Then two domain wall junctions on X D shown in Fig. 1 and the change between them canbe depicted as in Fig. 2, i.e. due to an attachment of a (D + 1)-simplex. Namely, we regard
the domain walls and the junctions not to be defined only on X D but also to extend into the(D + 1)-dimensional bulk YD+1 . The introduction of an additional simplex to YD+1 , which does
not change the G-bundle, does change how the domain walls in the bulk join on the boundary.
Then the phase ω(g1 , g2 , . . . , g D+1 ) can be thought of as the Boltzmann weight associated4

SciPost Phys. 8, 015 (2020)

g3
g1 g2 g3

g2

g3

g3
g1

g1 g2 g3

g1 g2

g2

attach a 3-simplex

g1

g1 g2 g3

g2 g3

g2

g1

Figure 2: The anomalous phase can be thought of as due to attaching a (D + 1)simplex of the bulk theory on YD+1 .

to this simplex of the (D + 1)-dimensional symmetry-protected-topological (SPT) phase withG symmetry on YD+1 . From this viewpoint, the constraint (2.3) and its generalizations toarbitrary dimensions D + 1 guarantee that the partition function assigned to a manifold with atriangulation does not depend on the choice of the triangulation; see e.g. [26,27] for a carefulcombinatorial treatment for low dimensional cases. The bulk theory is essentially the theory
introduced by Dijkgraaf-Witten [28] but G is now considered as a background field here, not
as a dynamical field to be summed over.
We also note the following: suppose we are given a D-chain L ∈ C D (G, U(1)). Then wecan add to the action a term3Z
L(g ) ∈ U(1),

(2.4)

XD

where g is the background G-bundle represented by the domain walls. This operation corresponds to the addition of contact and/or counter terms in the case of continuous flavor
symmetry.
For example, for the figure on the left hand side of Fig. 2, the total additional couplingis L(g1 , g2 ) + L(g1 g2 , g3 ). Similarly, on the right hand side, the total additional coupling isL(g1 , g2 g3 ) + L(g2 , g3 ). This means that the anomaly (D + 1)-chain ω is shifted toω → ω0 = ω + ∂ L,

(2.5)

∂ L(g1 , g2 , g3 ) := L(g2 , g3 ) − L(g1 g2 , g3 ) + L(g1 , g2 g3 ) − L(g1 , g2 ).

(2.6)

where
This means that only ω ∈ H D+1 (G, U(1)) matters up to the addition of the counter terms.
That said, in an actual computation using this approach we are required to pick a particularcocycle representing ω. For example, in Sec. 2.6, we will need to treat a trivial theory with
G-symmetry with the anomaly cocycle ω = 0 as a theory with the anomaly cocycle ω0 = ∂ L
for a nontrivial L, by adding this contact term L.
Now, suppose we are given a D-dimensional G-symmetric theory T with the anomaly ω,and we want to gauge a subgroup H ⊂ G. What is the condition imposed? We represent the3
Or to be more precise, we should have said “we can multiply the exponentiated action by the term (2.4)” since
this is a U(1)-valued phase.

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SciPost Phys. 8, 015 (2020)

H-bundle again using domain walls and their junctions. At each junction where two walls h1
and h2 meet and form the wall h1 h2 , we assign the Boltzmann weight µ(h1 , h2 ). In order forthe total action to be consistent under the change in the topology of the domain wall, we need
the condition∂ µ = ω|H .
(2.7)
This means that the element ω ∈ H D+1 (G, U(1)) when pulled back to H D+1 (H, U(1)) shouldbe trivial. In other words, H should be anomaly free.
We note here that the difference of two possible solutions to (2.7) is an element of H D (H, U(1)),
but that there is no canonical solution in general, unless ω|H itself happens to be zero. In thislatter case, we can take µ = 0. In the analyses we perform below, we almost always have thissimplifying situation, so we take µ = 0 unless otherwise mentioned.

2.2

From a nontrivial extension with a trivial anomaly

Suppose a theory T has a 0-form finite group symmetry Γ of the form
0 → A → Γ → G = Γ /A → 0.

(2.8)

We assume the whole group Γ is anomaly free. Then we can gauge the subgroup A with atrivial action. Denote the resulting gauged theory by T /A. What is the symmetry of T /A?
We assume A is Abelian. Then the extension (2.8) is specified by an elemente ∈ H 2 (G, A).

(2.9)

Our first result is the following:
When the anomaly of Γ is zero, T /A has the symmetry Â[D−2] × G whose total anomaly isgiven byZâ ∪ e(g ) ∈ U(1),(2.10)

YD+1

where â ∈ H D−1 (Y, Â), g is the background G connection, and e(g ) ∈ H 2 (Y, A) is the element
defined by the extension class e ∈ H 2 (G, A) and g .
Mathematically, e(g ) is determined as follows. g determines a map Y → BG which we denoteby the same letter. e ∈ H 2 (G, A) = H 2 (BG, A) can then be pulled back and gives an elementg ∗ (e) ∈ H 2 (Y, A). We denote the resulting element by e(g ) as a function of g determined bye.
When D = 2 this result was known in the mathematical literature [29, 30] for some time.
When D = 3 this was derived essentially in [16] and also in [9] (see their Sec. 4.1 and Appendix
B and C in particular) and an amazing dynamical application was given in the latter. Here we
stick to pure formalities and give a derivation, essentially the one given in Appendix B of [9].
Let us first recall how e determines the extension. We identify Γ with A × G as a set. Thenthe product of (0, g) and (0, h) is given by (e(g, h), gh). This e(g, h) is the two-cocycle defining
the extension. In terms of the domain wall operators implementing the action of Γ , this grouplaw can be drawn as the figure on the left of Fig. 3. We can represent these domain walls also
as on the right of Fig. 3. Namely, we regard the walls to implement the action of A × G, but
now the codimension-2 junction locus of two domain walls g, h ∈ G forming gh ∈ G serves as
a boundary for the domain wall labeled by e(g, h) ∈ A.
This can be also phrased as follows. When e = 0, the background field a for the symmetry
A is an element in H 1 (X , A). When e 6= 0, a is not quite a cocycle, since the domain walls can
now have boundaries. Instead, a is a cochain in C 1 (X , A) such that∂ a = e(g ),6(2.11)

SciPost Phys. 8, 015 (2020)(e(g, h), gh)ghe(g,h)(0, h)gh(0, g)

Figure 3: Domain walls for Γ as domain walls for A × G such that domain walls of A
can have boundaries as determined by the extension class e.

aa= χ(a) ×χχ

Figure 4: We have an additional phase when the Wilson line χ ∈ Â is moved acrossthe boundary of the domain wall labeled by an element of A.

e(g,h)e(g,h)ghχg= χ(e(g,h)) ×

gh
g

h

χ
h

Figure 5: Combining the two facts, we find that moving the χ wall across the junctionof g and h forming gh produces a phase.

gh
g

χ

= χ(e(g,h)) ×

gh
g

h

χ
h

Figure 6: Summing over the A-walls, we now have a mixed anomaly between Â andG.

where g is the background G bundle on X , and e is now thought of as a degree-2 characteristicclass valued in A. Note that the latter is an analogue of the standard Green-Schwarz coupling:d H = tr F ∧ F . In the absence of the Green-Schwarz effect, H is a closed 3-form. However, inthe presence of the Green-Schwarz coupling, H is not quite closed, but its failure is given by acharacteristic class constructed from another field.
Now consider a Wilson line operator labeled by χ ∈ Â. When we move a line labeled by χ
across a codimension-2 locus where the domain wall a ∈ A starts, we get an extra phase χ(a),see Fig. 4. Recalling that the junction of two walls labeled by g and h forming a wall labeled

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SciPost Phys. 8, 015 (2020)

by gh is a boundary from which the domain wall e(g, h) ∈ A emanates, we find that moving awall labeled by χ across the junction point produces a phase χ(e(g, h)), see Fig. 5. Gauging
A means to sum over all possible domain walls implementing the symmetry A, and therefore
A-walls are not considered backgrounds. We end up having the situation as shown in Fig. 6.
Namely, we now have a mixed anomaly between the (D − 2)-form symmetry Â and the 0-form
symmetry G, whose anomaly is given by (2.10).

2.3

From a trivial extension with a nontrivial anomaly

Next, let us consider the case when Γ = A × G, or equivalently the extension class is zero,e = 0 ∈ H 2 (G, A). Let the total anomaly of Γ = A × G be given byZa ∪ ẽ(g ),(2.12)

YD+1

where a ∈ H 1 (Y, A) and ẽ ∈ H D (G, Â). Our next claim is the following:
The symmetry of T /A is an “extension of G by Â[D−2] ”
0 → Â[D−2] → Γ̃ → G → 0,

(2.13)

whose extension class is given byẽ ∈ H D (G, Â).

(2.14)

Here we put an underline to Γ̃ to emphasize that this is a mixture of an ordinary 0-form
symmetry and a higher form symmetry. It should be possible to give a mathematical definition of this object Γ̃ itself in terms of the concept of (D − 1)-group4 , but for us it suffices to
describe what are the background fields for this extension5 . Namely, a background field forthis symmetry on a spacetime X is a pair (a, g ), where• g is a G bundle on X , and• a is a (D − 1)-cochain valued in Â such that ∂ a = ẽ(g ).
With this understanding, the derivation is entirely analogous to the first case discussed above
and is not repeated here. We also note again that the equation ∂ a = ẽ(g ) can be thought ofas a finite analogue of the Green-Schwarz effect.

2.4

Miscellaneous remarks• In the language of the topological defects, the extension can be described as follows.
Take a point p in the spacetime X D where D domain walls implementing elements
g1 ,. . . , g D of G meet and form the wall for g1 g2 · · · g D . Then, from this point, a defect line operator implementing an element â of Â[D−2] comes out, where â is given byâ = ẽ(g1 , . . . , g D ).
4
Here an n-group refers loosely to an n-category with a single object where every k-morphism is invertible in asuitable sense. As there are multiple subtly different notions of n-categories for n ≥ 2, it is unwieldy to be specific.
Note also that in group theory a p-group refers to a finite group whose order is a power of a prime p, but we are
not particularly interested in those in this note.
5
The situation is analogous to that of a non-commutative space M : We define the non-commutative algebra A
first, and then say that M is something whose algebra of functions is A, defining M indirectly.

8

SciPost Phys. 8, 015 (2020)• At a purely formal level, the ’t Hooft anomaly itself can be thought of as an extension, as
follows. Note that every QFT has a U(1)[D−1] symmetry, whose corresponding topological defect operator is simply a point operator given by an insertion of a complex numberof absolute value 1. Then the ’t Hooft anomaly ∈ H D+1 (G, U(1)) specifies an extension
0 → U(1)[D−1] → G̃ → G → 0.

(2.15)

This consideration, however, does not buy us much, since the U(1)[D−1] symmetry acts
in an entirely straightforward manner on the theory.6• When D = 3, the resulting mixed symmetry is a 2-group, and the corresponding 4d TQFT
was studied in [16]. An appearance of higher groups in physics was also noted in [3].
• The appearance of this extension in D = 3 in a theory with Â 1-form symmetry and G
domain walls is sometimes called as the “H 3 (G, Â) symmetry localization anomaly” inthe cond-mat literature, see e.g. [14, 15]. But this is somewhat of a misnomer, since
the element ẽ is a part of the group law in the theory T /A, and not the anomaly in theordinary sense. We prefer to call it as the H 3 (G, Â) phenomenon.
• In [15] it was pointed out that a system with a nonzero ẽ ∈ H 3 (G, Â) can be realized on aboundary of an A-gauge theory. This can be naturally understood from the point of viewof this note: a system with nonzero ẽ phenomenon can be realized as an A-gauge theoryof a theory with Γ symmetry with an anomaly ẽ. This latter theory can be realized ona boundary of a Γ -SPT. We gauge A of the entire setup. Then the original theory with anonzero ẽ phenomenon is realized on a boundary of an A-gauge theory.
• Another related point is as follows. When D = 3, there are many studies of TQFTs with a
G action. The mathematics is captured by a G-crossed modular tensor category C = ⊕ g C gwhere objects in C g are the boundary lines of a domain wall implementing an action ofg ∈ G. In this context people speak of the H 3 (G, Â) anomaly/phenomenon/obstruction[31]: we start from an action of G on a category Cid and asks if it is possible to extend
this data to a genuine G-crossed category C = ⊕ g C g . It is known that there is an obstruction which appears as the failure of the isomorphism of tensor products of three objects
x 1,2,3 ∈ C g1,2,3 . In a genuine category, we would like to havex 1 ⊗ (x 2 ⊗ x 3 ) ' (x 1 ⊗ x 2 ) ⊗ x 3(2.16)

but in general we are forced to havex 1 ⊗ (x 2 ⊗ x 3 ) ' ((x 1 ⊗ x 2 ) ⊗ x 3 ) ⊗ ẽ(x 1 , x 2 , x 3 ),(2.17)

where ẽ(x 1 , x 2 , x e ) is an invertible object in Cid . We define Â to be the subcategory of
the invertible objects in Cid . Then ẽ defines a three-cocycle representing an element in
H 3 (G, Â), and this is an obstruction to the existence of the G-crossed tensor category
C = ⊕g Cg .
A higher-categorical framework for arbitrary topological defects in D = 3 was given in[4] and their gauging was studied in [5]. The point is that when this H 3 (G, Â) obstruction
is nonzero, the resulting topological defects are described by a higher-category moregeneral than G-crossed modular tensor categories, and that such an example can bereadily obtained by gauging a normal Abelian subgroup of a group with an anomaly.
It should also be noted that even when the H 3 obstruction vanishes, there is the next
obstruction which takes value in H 4 (G, U(1)) [31]. From the physics point of view, it
6

The author thanks C. Córdova and N. Seiberg for discussions on this point.

9

SciPost Phys. 8, 015 (2020)is the anomaly of the G symmetry of the 3d TQFT under consideration. These issues
are treated in detail from the physics point of view in a paper by Benini, Córdova andHsin [32].

2.5

From a nontrivial extension with a nontrivial anomaly

The discussions in Sec. 2.2 and Sec. 2.3 are symmetric; indeed they are the same when D = 2.
We can in fact mix them, which is well known for D = 2 [29, 30, 33, 34].
Suppose a theory T has the symmetry Γ given by a group extension as above (2.8), withthe extension specified by e ∈ H 2 (G, A). As in the discussions above, a background Γ bundle γ
can be decomposed into a pair (a, g ) where a is a 1-cochain valued in A and g is a backgroundG bundle so that ∂ a = e(g ).
Given an elementẽ ∈ H D (G, H 1 (A, U(1))) = H D (G, Â),
(2.18)
we try to use the same formula a ∪ ẽ(g ) (2.12) to define an element in H D+1 (Γ , U(1)). This isin general inconsistent, since∂ (a ∪ ẽ(g )) = (e ∪ ẽ)(g ).

(2.19)e ∪ ẽ = 0 ∈ H D+2 (G, U(1)).

(2.20)

We need to require that
Assuming this, we can pick an element ω in C D+1 (G, U(1)) such that∂ ω = e ∪ ẽ ∈ C D+2 (G, U(1)).

(2.21)

Then we can define an element of H D+1 (Γ , U(1)) by the formulaZα(ẽ, ω) :=(a ∪ ẽ(g ) − ω(g )) .

(2.22)

YD+1

Note that ω is defined up to additions of elements in H D+1 (G, U(1)).
Now, let us gauge A. The gauged theory T /A has the symmetry Γ̃[D−2,0] (2.13) determinedby ẽ. The anomaly is given byZ
α̃(e, ω) :=(â ∪ e(g ) − ω(g )) .

(2.23)

YD+1

Summarizing, we have the following:
Given e ∈ H 2 (G, A), ẽ ∈ H D (G, Â) satisfying e ∪ ẽ = 0 and an element ω ∈ C D+1 (G, U(1))
such that ∂ ω = e ∪ ẽ, a theory with an ordinary symmetry
0→A→Γ → G →0

(2.24)

specified by e with an anomaly α(ẽ, ω) as in (2.22) and a theory with the mixed symmetry
0 → Â[D−2] → Γ̃ → G → 0(2.25)

specified by ẽ with an anomaly α̃(e, ω) as in (2.23) are exchanged by the gauging of A andÂ.
In terms of the Lyndon-Hochschild-Serre spectral sequence briefly reviewed in Appendix A.2,
we can study the anomaly H p+q (Γ , U(1)) in terms of H p (G, H q (A, U(1))). In this language, thecondition (2.20) is the vanishing d2 (ẽ) = 0.
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2.6

Relation to other constructions

To actually realize a theory T /A with a nontrivial extension of a group by a higher-form symmetry such as (2.13), we need to have a theory T with a finite group symmetry Γ with a
specified nontrivial anomaly. It is not obvious that such a theory T actually exists, but luckily
a construction for D = 2, 3 in terms of finite gauge theory was given by Kapustin and Thorngren [10,11] which was further generalized to arbitrary dimensions in Sec. 3.3 of Witten [12]and Wang-Wen-Witten [13]. In fact their construction itself can be thought of as an exampleof the discussions so far. We would like to elaborate on this point.
2.6.1

Relation to Witten [12] and Wang-Wen-Witten [13]We would like to reproduce the following statement from a paper by Witten [12] and anotherby Wang-Wen-Witten [13]7 :Given a group G and an anomaly ω ∈ H D+1 (G, U(1)), suppose we are given an Abelian
extension
0→A→Γ → G →0
(2.26)
such that ω trivializes in H D+1 (Γ , U(1)). Denote by T an almost trivial theory with Γ symmetry as described below. Then the gauged theory T /A has the symmetry G with an anomaly
ω.
In Sec. 5 of [13] an argument was given to find such an extension Γ by a suitable finite
Abelian A for any G and ω ∈ H D+1 (G, U(1)) such that the pullback of ω to Γ trivializes. We
will give a mathematical proof in Sec. 2.7. For a while, we assume that such an extension is
given.
We now specify the almost trivial theory T to be used. We take a specific cocycleω ∈ Z D+1 (G, U(1)), and pull it back to Γ . We denote the resulting cocycle by the same symbol
ω. As we assumed that this trivializes in the cohomology, there is a µ ∈ C D (Γ , U(1)) such thatω = ∂ µ. We now consider a completely trivial theory with Γ symmetry, and use µ as the action,
as explained in (2.4). This is the theory T used in the statement above. Note that this theory T
is not completely trivial, in the sense that it has a nonzero anomaly cocycle ω ∈ Z D+1 (Γ , U(1)),
although ω is trivial in H D+1 (Γ , U(1)). However, the anomaly cocycle restricted to A is trivial,
since ∂ µ|A = ω|A = 0. Therefore one can gauge A.
It is almost tautological from the construction that the gauged theory T /A has the anomaly
cocycle ω ∈ Z D+1 (Γ , U(1)). In fact this is a special case of our discussion in Sec. 2.5. Indeed,
the condition (2.21) is trivially satisfied since ẽ = 0. Then we just restrict (2.23) to G by settingâ = 0. We find that the anomaly is simply given by ω itself.
2.6.2

Relation to Kapustin-Thorngren [10, 11]

The argument above was mostly tautological, so let us re-analyze it in slightly more detail. This
brings us closer to the point of view of the two papers by Kapustin and Thorngren [10, 11].
We use the Lyndon-Hochschild-Serre spectral sequence, briefly reviewed in Appendix A.2.
As an assumption, we have a natural pullback H D+1 (G, U(1)) → H D+1 (Γ , U(1)) under which
ω is sent to zero. This pullback map is decomposed into steps in the LHS spectral sequence:
D+1,0

H D+1 (G, U(1)) = E2

D+1,0

 E3

D+1,0

D+1,0
 · · ·  E D+2 = E∞,→ H D+1 (Γ , U(1)),(2.27)In their papers they used the symbols (K, H, G) instead of our (A, Γ , G). Also, their papers contain many more
results, and here we are only scratching a tiny part of them.
7

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SciPost Phys. 8, 015 (2020)

where
D+1,0

E r+1

= E rD+1,0 / Im d r .

(2.28)

Note that every arrow except the last is a surjection and the last is an inclusion.
When ω is in the image of d2 : Now, suppose that ω is sent to zero in the first step, i.e. ω isin the image of d2 . Since d2 is a cup product with the extension class e, this means that there
D−1,1= H D−1 (G, Â) such that b ∪ e = ω. With this assumption, ω should be
is an element b ∈ E2
a coboundary. Indeed, we see that ∂ (b(g ) ∪ a) = ω(g ), where a as always satisfies ∂ a = e(e).
In the previous sections, we were always gauging A of a theory T with a trivial action but
in a presence of an anomaly cochain ω. When the anomaly cochain is zero and the action of
the A gauge theory is also zero, the setup is exactly as in Sec. 2.2, the symmetry is Â[D−2] × G
and the anomaly of the gauged theory is purely mixed and was given in (2.10). We do nothave an anomaly purely of G yet.
Now, let us “embed” the group G into the mixed group Â[D−2] × G by the formulag → (b(g ), g ),(2.29)

where the left hand side is a G bundle, and the right hand side is a background of Â[D−2] × G,
i.e. a pair of an element of H D−1 (−, Â) and a G-bundle. Note that when D = 2, an elementb ∈ H 1 (G, Â) actually gives a homomorphism b : G → Â, and the operation (2.29) indeed
comes from a homomorphism G → Â × G. Therefore an element b ∈ H D−1 (G, Â) is a natural
higher version of this homomorphism. Plugging this in to (2.10), we see that the anomaly isnowZZb(g ) ∪ e(g ) =YD+1

ω(g ).

(2.30)

YD+1

This embedding (2.29), in the language of the A gauge theory, can be realized as thecoupling in the action of the formZb(g ) ∪ a.

(2.31)XD

Stated differently, if we gauge the original theory T with the action (2.31), we get the gauged
theory T /A with an anomaly ω for the symmetry G.
When ω is in the image of d3 : Let us next consider the case when ω is killed at the second
step, i.e. when it is in the image of d3 . For simplicity of the presentation, we restrict to the caseD = 2. Then we have an element β ∈ H 0 (G, H 2 (A, U(1))) = H 2 (A, U(1)) such that d2 β = 0
and d3 β = ω.
For explicitness, we set A = (Zn )k , and we can now match the data with the analysis ofSec. 3.1 of Kapustin and Thorngren [11]. In our notation, this goes as follows.
From the data β ∈ H 2 (A, U(1)) we try the following expression as the action of the (Zn )kgauge field:
Z X
βi j a i ∪ a j ,(2.32)
X 2 i< jwhere i, j = 1, . . . , k labels the components of (Zn )k and βi j is now thought of as a Zn -valued
skew-symmetric matrix. This is the standard form of the discrete torsion for A = (Zn )k . With
the extension ∂ a i = e i (g ), however, the coboundary of the integrand of (2.32) isX
XX∂(
βi j a i ∪ a j ) = (βi j e i ∪ a j ) +βi j (−∂ (a i ∪1 e j ) + e i ∪1 e j ),(2.33)
i< j

i, j

i< j

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SciPost Phys. 8, 015 (2020)

which is a-dependent in general and therefore inconsistent.
To be able to rectify it, we need to require that the object βi j e i is a coboundary, βi j e i = ∂ b j .
Note that βi j e i naturally is an element of H 2 (G, Â) = H 2 (G, H 1 (A, U(1))), and can be identified
with d2 β in the LHS spectral sequence. Therefore such bi exists if and only if d2 β = 0.
Assuming the existence of bi , we then consider the action
Z X
X
βi j (a i ∪ a j + a i ∪1 e j (g )) +(X 2 i< jbi (g ) ∪ a i ).

The coboundary of the integrand is nowXXω=
β i j e i ∪1 e j +bi ∪ e i ∈ H 3 (G, U(1)),
i< j(2.34)i(2.35)iwhich can be identified with d3 β and describe the anomaly of the symmetry G of the gauge
theory.
Kapustin and Thorngren also gave the analysis of the case D = 3, A = Zn and described d4
explicitly. It would be interesting to actually construct the LHS spectral sequence using theirapproach.

2.7

Existence of finite Abelian extension where the anomaly trivializes

Let us now prove8 that for any ω ∈ H D+1 (G, U(1)) for D + 1 ≥ 2, there is a finite Abelian
extension
0 → A → Γ → G → 0,
(2.36)
such that ω trivializes in Γ .
It is well-known [35, Sec. 6] that in the algebraic definition of group cohomology, we canrestrict attention to normalized cochains, i.e. those cochains c such that c(g1 , . . . , g n ) = 0 if
any of g i = 1, where 1 ∈ G is the identity element. We always use normalized cochains below.
From the discussion in the sub-case ‘when ω is in the image of d2 ’ in the last subsection,
it suffices to construct a finite Abelian group A and cocycles b ∈ Z D−1 (G, Â), e ∈ Z 2 (G, A) suchthat b ∪ e = ω. As we will see, this can be done almost tautologically.
Consider a free Z-module A generated by the symbols x g1 ,g2 for g1,2 ∈ G, with the understanding that x 1,g = x g,1 = 0. One can easily check that the following makes A a left
G-module:
g1 x g2 ,g3 = x g1 g2 ,g3 − x g1 ,g2 g3 + x g1 ,g2 .
(2.37)
Then there is a universal 2-cocycle e ∈ Z 2 (G, A) given bye(g1 , g2 ) = x g1 ,g2 .

(2.38)

Then we can define a cochain b ∈ C D−1 (G, Â) by〈b(g1 , g2 , . . . , g D−1 ), x g D ,g D+1 〉 := ω(g1 , . . . , g D−1 , g D , g D+1 ),(2.39)

where 〈·, ·〉 is the pairing between Â and A. By construction, we have
ω=b∪e(2.40)

and also almost by construction, b is in fact a cocycle ∈ Z D−1 (G, Â) since〈(∂ b)(g1 , . . . , g D−1 , h), x g D+1 ,g D+2 〉 = (∂ ω)(g1 , g2 , . . . , g D−1 , h, g D , g D+1 ) = 0.
8

The author thanks Juven C. Wang for explaining the content of Sec. 5 of [13] as a physics proof.

13(2.41)

SciPost Phys. 8, 015 (2020)

2This almost does the job, but A ' Z(|G|−1) is an infinite Abelian group. This small issue
can be easily remedied: we take Ω ⊂ U(1) be the finite subgroup generated by the value ofthe cocycle ω(g1 , . . . , g D+1 ). Then we take A to be the finite group
2

A = A ⊗ Ω̂ ' Ω̂(|G|−1) .

(2.42)The tautological cocycle e ∈ Z 2 (G, A) and the cocycle b ∈ Z D−1 (G, Â) can be defined by thesame formulas (2.38) and (2.39), and we achieved ω = e ∪ b.

3

Incorporating ’t Hooft defects

3.1

An example

Take a 3d theory T with one-form symmetry A[1] × G[1] where A = Ĝ is an Abelian group.
Suppose the anomaly is given byZa ∪ g.

(3.43)Y4

For concreteness, let A = Zn . Then, in terms of topological defect operators, this just meansthat the loop operator of charge k for A and the nontrivial loop operator of charge l for G has
a nontrivial braiding phase exp(2πikl/n).
Of course, one example of such theory T is a pure G = Zn gauge theory with ’t Hooft loops;
the line operators associated to A[1] are Wilson lines of G, and the line operators associated toG[1] are ’t Hooft lines of G.
We can consider this situation as a generalization of our discussions so far: we have atrivial extension
0 → A[1] → A[1] × G[1] → G[1] → 0(3.44)
specified by ẽ = 0, with an anomaly specified by e = id : H 2 (X , G) → H 2 (X , A).
Now we can gauge A[1] . Our previous discussions applied to this case imply that the resulting gauged theory T /A has a symmetry
0 → Â[0] → Γ → G[1] → 0,

(3.45)

whose extension is specified by the same e, with the zero anomaly ẽ = 0. What does thisabstract nonsense mean, in concrete terms?
We first note that we took Â = G. Then, a background field for Γ is a pair (a, g ) whereg ∈ H 2 (X , G) = H1 (X , G) while a ∈ C 1 (X , Â) is such that∂ a = e(g ) = g .

(3.46)

This just means that a is a surface in X whose boundary is g .
When T is the pure Zn gauge theory with ’t Hooft loops, T /Â is a trivial theory with a Zn
symmetry with ’t Hooft loops. a is the domain wall representing the Zn action; g specifies the’t Hooft loops around which we have nontrivial Zn holonomy; and the relation (3.46) meansthat the ’t Hooft loops are boundaries of the domain walls.
Summarizing, the relation between ’t Hooft loops of a flavor symmetry and domain walls
representing the flavor symmetry can be thought of as an extension of a 1-form symmetry by
a 0-form symmetry. It is not clear what this reinterpretation buys us at present, but in Sec. 3.3
we at least point out a somewhat curious phenomenon.

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3.2

Extension of G[n] by A[m]Before getting there, let us state the general situation of an extension of n-form G symmetryby an m-form A symmetry. Here G is also assumed to be Abelian when n > 0. We use thebasics of Eilenberg-Mac Lane spaces below, which is briefly reviewed in Appendix A.1.
We pick two elementse ∈ H m+2 (K(G, n + 1), A),ẽ ∈ H D−m (K(G, n + 1), Â)(3.47)

satisfyinge ∪ ẽ = 0 ∈ H D+2 (K(G, n + 1), U(1)).

(3.48)

ω ∈ C D+1 (K(G, n + 1), U(1))(3.49)∂ ω = e ∪ ẽ.

(3.50)

0 → A[m] → Γ → G[n] → 0,

(3.51)We choose a cochain
such thatTo this, we associate an extension

whose extension class is specified by e. Concretely, this means that a background field for Γ isa pair (a, g ) where a ∈ C m+1 (X , A), g ∈ H n+1 (X , G) such that∂ a = e(g ),(3.52)

where e is now thought of as a cohomology operatione : H n+1 (−, G) → H m+2 (−, A).

(3.53)

Similarly, we have another extension
0 → Â[D−m−2] → Γ̃ → G[n] → 0(3.54)

for which we think of ẽ as a cohomology operationẽ : H n+1 (−, G) → H D−m (−, Â)(3.55)

so that the background fields satisfy∂ â = ẽ(g ).

(3.56)We define an anomaly for Γ by the formulaα(ẽ, ω) :=Z(a ∪ ẽ(g ) − ω(g ))(3.57)(â ∪ e(g ) − ω(g )) .

(3.58)

YD+1

and an anomaly for Γ̃ by the formulaα̃(e, ω) :=Z
YD+1

We then have:

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SciPost Phys. 8, 015 (2020)= (−1) ×

Figure 7: A curious skein relation in a 3d theory. Here a thik line represents a Z2 line
which is a boundary of a Z2 wall. When it protrudes across another Z2 wall, there isa minus sign.

Given e, ẽ and ω as above, a theory with the symmetry
0 → A[m] → Γ → G[n] → 0,

(3.59)

whose extension is e with an anomaly α(ẽ, ω) and a theory with the symmetry
0 → Â[D−m−2] → Γ̃ → G[n] → 0,

(3.60)

whose extension is ẽ with an anomaly α̃(e, ω) are exchanged by the gauging of A and Â.
The derivation is again entirely analogous to the previous ones given. In Appendix A.3, we
discuss the classifying space for Γ and gives an interpretation of the condition (3.48) in that
language.

3.3

A curious phenomenon

In Sec. 3.1 we considered the defects in a 3d Zn gauge theory, which has loop operators labeled
by Zn × Zn . We saw that when Zn is ungauged, we found that a Zn domain wall can end on a
Zn ’t Hooft loop.
Here we consider a 3d theory with a Z4 1-form symmetry instead, with the anomaly givenby the requirement that the topological line operator labeled by the generator of Z4 has a
braiding phase −1 with itself. An example is to take two copies of U(1)4 Chern-Simons theory
and identify the two Z4 1-form symmetries. Now, Z4 fits in an extension
0 → (Z2 )[1] → (Z4 )[1] → (Z2 )[1] → 0,

(3.61)

and we find that Z2 ⊂ Z4 is non-anomalous and can be gauged. This fits in the framework of
the last subsection.
The gauged theory then has a mixed symmetry
0 → (Z2 )[0] → Γ → (Z2 )[1] → 0,

(3.62)

with a nontrivial extension and a nontrivial anomaly. The extension is the same as we discussed
in (3.45) in Sec. 3.1, and it says that the Z2 domain wall can end on the Z2 line operator. The
anomaly appears as the fact that the intersection points of Z2 line operators on the Z2 domain
wall has a nontrivial “braiding” relation −1, see Fig. 7.
Again the author does not know how to make of this phase, but it might be useful toremember that there can be such a phase, between the symmetry-implementing domain walls
and the ’t Hooft loops.

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4

Subtler effects

Let us come back to the case when the symmetry Γ of a theory T is an ordinary group fitting
in an extension
0 → A → Γ → G → 0.
(4.63)
So far we considered what happens if we gauge A when the anomaly of Γ has a very specificform (2.22), essentially determined by a class ẽ ∈ H D (G, Â).
In general, an arbitrary anomaly of the group Γ , or equivalently an element of H D+1 (Γ , U(1)),
can be studied in terms of H p (G, H q (A, U(1))) with p + q = D + 1. The systematics is provided
by the LHS spectral sequence, briefly reviewed in Sec. A.2. So far, we are only utilizing two
components
H D+1 (G, H 0 (A, U(1))) = H D+1 (G, U(1))(4.64)andH D (G, H 1 (A, U(1))) = H D (G, Â).

(4.65)

We saw the latter includes the infamous H 3 (G, Â) obstruction which appears in the discussionof how a G symmetry would act on anyons in 3d TQFTs.
These are however only the first two steps in the long ladder. Other H p (G, H q (A, U(1)))
with q ≥ 2 would provide more esoteric behaviors among topological defect operators in
the gauged theory T /A. When D = 2, everything is understood and can be phrased in thelanguage of fusion categories, as reviewed e.g. [22]. There, the only remaining component is
H 1 (G, H 2 (A, U(1))), and it is known that a nonzero element in it would make the symmetry of
the gauged theory T /A to be non-group-like.
In this last section,9 we restrict our attention to the direct product Γ = A× G, and consider
the effect of an anomaly determined by an element t ∈ H D−1 (G, H 2 (A, U(1))). Namely, we
consider the case where the total anomaly of the original theory T is given byZa ∪ a ∪ t(g ),(4.66)

YD+1

where we first consider the integrand as the (D + 1)-cocycle valued in A ⊗Z A ⊗Z H 2 (A, U(1)),
then use the fact that elements of H 2 (A, U(1)) can naturally be thought of as elements of Â⊗Z Â
which is skew-symmetric to regard it as a U(1)-valued (D + 1)-cocycle.
Equivalently, the element t characterize the anomaly in the following manner. Put thetheory on M D−1 with a G-bundle g on it, reduce it to a one-dimensional theory, and consider
it as a theory with A symmetry. Then its A anomaly is given by the elementZt(g ) ∈ H 2 (A, U(1)).

(4.67)

M D−1

4.1

D=2

Let us first consider the case D = 2. We are going to gauge the symmetry A. When t is nonzero,
there is an S 1 compactification with nontrivial g to one dimension such that the gauge anomaly(4.67) is nonzero. In other words, there is a mixed flavor-gauge anomaly between the gauge
group A and the flavor group G. This usually means that G is not a symmetry anymore. If we
accept defeat in this manner, the symmetry of the gauged theory T /A is just Â, and we can nolonger recover the whole symmetry A × G by re-gauging Â of the theory T /A. We would not
accept this.
9

The author thanks Kantaro Ohmori for convincing me to write up the content of this section.

17

SciPost Phys. 8, 015 (2020)

Still, mathematicians have found a way to consider more topological defect operators in
the gauged theory T /A so that re-gauging Â recovers the whole symmetry A × G. They had
their own motivation in the construction. Here we give a physics interpretation.
When the spacetime is one dimensional, it is very easy to give a topological theory withsymmetry A with a given anomaly τ ∈ H 2 (A, U(1)): it is just a fancy name for a projective representation of A whose projective phase is characterized by τ. Therefore, we just add a projective
representation of A on top of a domain wall implementing g ∈ G such that t(g) ∈ H 2 (A, U(1))
is nonzero, to cancel the anomaly on the domain wall. This is a natural generalization of thefact that on a domain wall for id ∈ G, we consider a genuine representation of A, i.e. they are
ordinary Wilson lines labeled by elements of Â.
Summarizing, the topological line operators of the gauged theory T /A are labeled by thepairs of the form
(ρ, g) ∈ (Rep t(g) A, G),(4.68)
where Rep t(g) A denotes the set of projective representations of A whose phase is characterizedby t(g) ∈ H 2 (A, U(1)). These line operators fuse according to the following rule:(ρ, g) ⊗ (σ, h) = (ρ ⊗ σ, gh).

(4.69)

This rule is consistent, since ρ ⊗ σ has the projective phase characterized by t(gh) = t(g)t(h).
As an example, consider a theory T with the symmetry A × G = (Z2 × Z2 ) × Z2 with theanomalyZ
a1 ∪ a2 ∪ g .

(4.70)

Y3

We gauge A = Z2 × Z2 . Denote the elements of G = Z2 by 0 and 1. Then the labels for theirreducible lines are either• (â, 0) where â ∈ Â = Ẑ2 × Ẑ2 , or
• (ρ, 1) where ρ is the unique irreducible non-trivial projective representation of Z2 × Z2 ,which is two-dimensional.
Note that this set of labels and the resulting fusion rule are exactly the same as that of theirreducible representations of the dihedral group D8 with eight elements. This is as it shouldbe, since D8 is an extension
0 → Ĝ = Ẑ2 → D8 → A = Z2 × Z2 → 0,

(4.71)and therefore an irreducible representation of D8 can be thought of as possibly projective representations of Z2 × Z2 , whose projective phase is determined by how the first Ẑ2 is represented.
This observation has a natural explanation. Consider the same theory T but gauge G
instead. Here we can apply the result in Sec. 2.2; we see that T 0 = T /G has the symmetry D8 ,which contains Ĝ as a subgroup. Then the theory T /A we are considering is in fact
T /A = (T 0 /Ĝ)/A = T 0 /D8(4.72)

since the D8 symmetry of T 0 is composed of Ĝ and A. Therefore the theory T /A should have
all the Wilson lines of D8 .10
10This decomposition of an orbifold by a solvable non-Abelian group into repeated applications of Abelian orbifolds was utilized before in the context of the study of D-branes at orbifold singularities [36].

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SciPost Phys. 8, 015 (2020)

4.2

D>2

Now the extension to D > 2 is clear. We start from the theory T with the symmetry A × G,
with the anomaly specified by t ∈ H D−1 (G, H 2 (A, U(1))). Given a G-bundle g , we have onedimensional subloci specified by t(g ). Such a locus can be visualized as the place where (D−1)
domain walls g1 , . . . , g D−1 ∈ G meet to form a new wall labeled by g 0 = g1 · · · g D−1 , which
determine an elementt(g1 , . . . , g D−1 ) ∈ H 2 (A, U(1)).
(4.73)
We simply demand that such a one-dimensional locus should carry a projective representationof A whose projective phase is characterized by the element (4.73).
When D = 3, this construction should give rise to a 2-category as described in [4]. Since
the nontrivial projective representations of an Abelian groups are not one dimensional andtherefore not invertible, this gives an explicit example of such a 2-category which is not a
2-group.

4.3

Toward even subtler effects

So far we discussed the effect of elements of t ∈ H D−1 (G, H 2 (A, U(1))) in the gauged theory
T /A. We saw that each one-dimensional locus at the intersection of domain walls implementing the G action can naively carry a nonzero gauge anomaly determined by t, which is thencanceled by putting a projective representation of A.
The simpler effect from elements of ẽ ∈ H D (G, H 1 (A, U(1))), which we examined in moredetail in the earlier part of the note, can be phrased in a similar manner: each zero-dimensional
locus at the intersection of domain walls of G can naively carry a “nonzero gauge anomaly” ofa zero-dimensional A gauge theory, which is just a nonzero excess charge under A of the point
being considered. This is canceled by attaching a Wilson line of A.
From these considerations, we can describe, at least schematically, the effect of elementsof t ∈ H D−d (G, H d+1 (A, U(1))) would have on the structure of the topological defect operators.
Namely, on a dimension-d locus where D − d domain walls implementing g1 , . . . , g D−d meet
to form a domain wall labeled by g 0 = g1 · · · g D−d , there is naively a gauge anomaly given byt(g1 , . . . , g D−d ) ∈ H d+1 (A, U(1)).

(4.74)

We cancel this by introducing there some TQFT with symmetry A whose anomaly is given by
the expression (4.74) above. As we recalled in Sec. 2.6, it was argued that there is always sucha TQFT, so we can keep arbitrary walls implementing the G action in the theory, by puttingthese anomalous lower-dimensional TQFTs on the walls.
What complicates the cases d ≥ 2 as compared to the cases d = 0, 1 is the following.
For d = 0, 1, the full list of d-dimensional TQFTs with A symmetry with a given anomaly in
H d+1 (A, U(1)) is known, and therefore the structure of the topological defect operators of thegauged theory can be worked out concretely. For d ≥ 2, our understanding of the full list ofsuch TQFTs with A symmetry with a given anomaly is still rudimentary, thus precluding us
from a full description of the topological description of the gauged theory.

Acknowledgments
The author thanks Nati Seiberg for suggesting him to study the so-called H 3 (G, Â) “anomaly”
in the first place, during the Strings 2017 conference. It is also a pleasure for the author to
thank Francesco Benini, Clay Córdova, Kantaro Ohmori, Nati Seiberg, Po-Shen Hsin and Juven
C. Wang for inspiring discussions. The author also thanks Kantaro Ohmori for suggesting the19

SciPost Phys. 8, 015 (2020)

notation G[n] for a higher-form symmetry, Edward Witten for suggesting an improvement in
the presentation in an earlier version of the paper, and an anonymous referee for detailed
comments which improved the paper greatly. The author is partially supported in part byJSPS
KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grant-in-Aid (KibanS), No.16H06335, and also supported in part by WPI Initiative, MEXT, Japan at IPMU, theUniversity of Tokyo.

A

Some algebraic topology

In this subsection we very briefly review the basics of algebraic topology needed in the note.
For more details, please consult standard textbooks, e.g. [37].

A.1

Eilenberg-Mac Lane spaces

The Eilenberg-Mac Lane space K(G, n + 1) is a space (defined up to homotopy) such that the
only nontrivial homotopy group is πn+1 = G. When n > 0 this requires that G is Abelian.
The space K(G, n+1) serves as the classifying space for the background fields for the n-form
finite symmetry group G,11 which means the following:
When n = 0, the background field for a finite group symmetry G on a space X is a Gbundle. Distinct G-bundles on X are in bijective correspondence with the homotopy class ofmaps [X , K(G, 1)]. K(G, 1) is also denoted as BG, the classifying space of G. When n ≥ 0
and G Abelian, the background field for an n-form G symmetry on a space X is an element of
cohomology group H n+1 (X , G). Again, distinct elements are in bijective correspondence withthe homotopy class of maps [X , K(G, n + 1)]. Similarly, we can denote K(G, n + 1) as B n G.
This means that cohomology classes on K(G, n + 1) are characteristic classes for a background field for an n-form symmetry G. Indeed, given a background field on X , regard it
as a (homotopy class of a) map g ∈ [X , K(G, n + 1)]. Pick an M -valued cohomology classα ∈ H d (K(G, n + 1), M ). Then we can pull back M via g to X and find a cohomology classg ∗ (α) ∈ H d (X , M ). We denote it as α(g ), regarding α as an operation
α : [−, K(G, n + 1)] → H d (−, M ).

(A.75)

When G is Abelian, this means that α is a cohomology operation
α : H n+1 (−, G) → H d (−, M ).

(A.76)

The cohomology of the Eilenberg-Mac Lane space H d (K(G, 1), M ) is also known as thegroup cohomology and is also written as H d (G, M ). It has a purely algebraic definition which
we freely used in this note.

A.2

Lyndon-Hochschild-Serre spectral sequence

Given an Abelian extension of groups 0 → A → Γ → G → 0, the Lyndon-Hochschild-Serre(LHS) spectral sequence is a spectral sequence converging to H p+q (Γ , M ) whose second pagep,q
is given by E2 = H p (G, H q (A, M )). Using the classifying space, this is just the Leray-Serre
spectral sequence associated to the fibration
BA → BΓ → BG
11

(A.77)

This shift by 1 is somewhat regrettable but cannot be avoided now that there is already a standard terminology on the physics side. This is somewhat similar to the situation of a Dp brane having a (p + 1)-dimensional
worldvolume.

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SciPost Phys. 8, 015 (2020)

but there is also a purely algebraic description which is sometimes handier for concrete computations.
Both the topological and algebraic descriptions can be found in the original paper by
Hochschild and Serre [38],12 which contains the result that when the action of Γ on M istrivial, the differential
p,1
p+2,0
d2 : E2 → E2
(A.78)
or equivalently
d2 : H p (G, Hom(A, M )) → H p+2 (G, M )(A.79)

is given by the cup product by the extension class e ∈ H 2 (G, A).

A.3

Classifying space for the mixed symmetry

For an n-form symmetry G[n] , the classifying space is B(G[n] ) := B n+1 G := K(G, n + 1), theEilenberg-Mac Lane space. Given an extension
0 → A[m] → Γ → G[n] → 0(A.80)

K(A, m + 1) → BΓ → K(G, n + 1).

(A.81)

the classifying space is a fibration

Taking the homotopy cofiber, we have another fibration
BΓ → K(G, n + 1) → K(A, m + 2)

(A.82)

by which we have a class e ∈ H m+2 (K(G, n + 1), A) called the Postnikov class or the Postnikov
k-invariant. This class e in fact classifies the fibration (A.81).
The Leray-Serre spectral sequence for the fibration (A.81) is a spectral sequence whosep,q
second page is E2 = H p (K(G, n + 1), H q (K(A, m + 1), M )) converging to H p+q (BΓ , U(1)). A
routine modification of the argument in [38] which uses the multiplicative property of thespectral sequence says thatp,m+1
p+m+2,0dm+2 : Em+2 → Em+2
(A.83)
is given by the cup product by the extension class e.

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