Mixed Anomalies Two-groups Non-Invertible Symmetries and 3d Superconformal Indices

2025-05-02 0 0 740.04KB 46 页 10玖币

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Mixed Anomalies, Two-groups, Non-Invertible

Symmetries, and 3d Superconformal Indices

Noppadol Mekareeyaa,b and Matteo Sacchic

aINFN, sezione di Milano-Bicocca,

Piazza della Scienza 3, I-20126 Milano, Italy

bDepartment of Physics, Faculty of Science,

Chulalongkorn University, Phayathai Road,

Pathumwan, Bangkok 10330, Thailand

cMathematical Institute, University of Oxford,

Andrew-Wiles Building, Woodstock Road,

Oxford, OX2 6GG, United Kingdom

E-mail: n.mekareeya@gmail.com,matteo.sacchi@maths.ox.ac.uk

Abstract: Mixed anomalies, higher form symmetries, two-group symmetries and non-

invertible symmetries have proved to be useful in providing non-trivial constraints on

the dynamics of quantum ﬁeld theories. We study mixed anomalies involving discrete

zero-form global symmetries, and possibly a one-form symmetry, in 3d N ≥ 3 gauge

theories using the superconformal index. The eﬀectiveness of this method is demon-

strated via several classes of theories, including Chern-Simons-matter theories, such as

the U(1)kgauge theory with hypermultiplets of diverse charges, the T(SU(N)) theory

of Gaiotto-Witten, the theories with so(2N)2kgauge algebra and hypermultiplets in the

vector representation, and variants of the Aharony-Bergman-Jaﬀeris (ABJ) theory with

the orthosymplectic gauge algebra. Gauging appropriate global symmetries of some of

these models, we obtain various interesting theories with non-invertible symmetries or

two-group structures.

arXiv:2210.02466v3 [hep-th] 20 Jan 2023

Contents

1 Introduction 1

2N= 4 U(1) gauge theory with 2hypermultiplets of charge q5

2.1 The case of q= 1 6

2.2 The case of q= 2 9

3 Global symmetry group and anomalies of T(SU(N)) 11

4so(2N)kgauge algebra and Nfhypermultiplets 15

4.1 Compatibility with the duality 18

4.2 SO(2N)4K+2 gauge theory and open questions 20

4.3 U(1)kgauge theory with Nfhypermultiplets 21

5 Non-invertible symmetries in the ABJ-type theories 23

5.1 so(2N)2kgauge algebra with Nfadjoints 23

5.2 ABJ theories of the orthosymplectic type 27

6 Two-groups in the ABJ theories of the orthosymplectic type 29

7 Conclusions 32

A 3d supersymmetric index conventions 33

1 Introduction

Symmetry serves as a vital organising principle in the study of quantum ﬁeld theories.

It can provide highly non-trivial constraints in the theory, for example, via selection

rules and ’t Hooft anomalies. There have been vast recent developments in this line of

research. One of the main important ideas is that a number of properties of the sym-

metries can be formulated in terms of the associated topological defects. Speciﬁcally, if

the symmetry obeys the group law, it can be viewed as the fusion rule of the topological

defects as follows: the topological defects associated with the group elements gand h

can be merged to form a topological defect associated with the group element gh.

– 1 –

This point of view has led to a number of new concepts of generalised global

symmetries. This includes higher-form symmetries [1,2]1whose topological defects

have codimension greater than one and whose charged objects are extended operators,

and non-invertible symmetries whose topological defects do not have an inverse and so

do not form a group. Examples of the latter in 2d theories and in 3d TQFTs have been

known for some time, see for example [9–34], but only very recently they have been

studied in diﬀerent ﬁeld theories and especially in higher dimensions from many point

of views, see e.g. [35–64] and [65,66] for recent reviews. Yet another important idea

which is central to this paper is the coexistence of a zero-form and a one-form global

symmetry. This can happen in several ways, for example, they can form a direct or a

semi-direct product, there can be a mixed anomaly between them, or they can combine

to form a non-trivial extension, where the latter is known as a two-group symmetry

[17,67–70]. In this paper, we focus on the two-group symmetries that involve a discrete

one-form symmetry and a continuous zero-form symmetry. This type of symmetry has

been studied in a wide range of theories, see e.g. [69,71–82].

In this paper, we study mixed anomalies in three-dimensional superconformal ﬁeld

theories with N ≥ 3 supersymmetry using the superconformal index [83–88] as a main

tool. This provides a convenient and eﬃcient way to detect various mixed anomalies,

including those involving two discrete zero-form global symmetries and a continuous

zero-form ﬂavour symmetries. Speciﬁcally, motivated by [89], we calculate the index in

a particular way in order to study the monopole operators carrying fractional magnetic

ﬂuxes for both the gauge group and the Cartan subalgebra of the ﬂavour symmetry

group, whose existence might signal the presence of the anomaly. From the perspective

of the index, this is manifest in the fact that certain gaugings of the global symmetries

are not allowed.

We demonstrate these ideas in the context of several Chern-Simons-matter theories,

which include the U(1)kgauge theory2with hypermultiplets with arbitrary charge, the

theories with so(2N)kgauge algebra and hypermultiplets in the vector representation,

and several variants of the Aharony-Bergman-Jaﬀeris (ABJ) theories [90] with the

orthosymplectic gauge algebra. Among a number of these theories, we ﬁnd that gauging

a discrete zero-form global symmetry leads to a dual one-form symmetry that forms a

two-group structure with the zero-form ﬂavour symmetry [17]. We also study discrete

mixed anomalies for the T(SU(N)) theory of Gaiotto and Witten [91] and show as an

application of these how they can be used to recover some known facts about the global

form of the global symmetries of the 4d N= 2 theories of class S[92] from the 3d

1We remark that these have been worked out in many theories by means of defect groups, see

e.g. [3–8].

2In this paper, Gkdenotes gauge group or gauge algebra Gwith Chern-Simons coeﬃcient k.

– 2 –

mirror perspective.

Since it is going to play a crucial role in our discussion, let us brieﬂy review the

argument of [68] for why gauging a discrete symmetry with a suitable mixed anomaly

gives a theory with a two-group symmetry. Suppose that we have a d-dimensional

theory Ton a manifold Xdwith an anomaly encoded in the anomaly theory deﬁned

on Yd+1 such that ∂Yd+1 =Xd

exp 2πi

NZYd+1

Ap+1 ∪Θ!,(1.1)

where Ap+1 ∈Hp+1(Xd,Z[p]

N) is a background ﬁeld for a Z[p]

Np-form symmetry and Θ

is some class valued modulo Nconstructed from the background ﬁelds for some other

global symmetries. The full partition function of the theory including the anomaly

theory is schematically

ZT[Ap+1] = exp 2πi

NZYd+1

Ap+1 ∪Θ!ZDΦ exp (iS[Φ, Ap+1]) .(1.2)

When we gauge the symmetry Z[p]

Nwe promote Ap+1 to a dynamical ﬁeld ap+1 over

which we sum in the partition function and we introduce a (d−p−1)-cochain Bd−p−1

which is the background ﬁeld for the dual Z[d−p−2]

N(d−p−2)-form symmetry and which

couples to ap+1 in the partition function

ZT/Z[p]

[Bd−p−1] = X

ap+1∈Hp+1(Xd,Z[p]

exp 2πi

NZXd

ap+1 ∪Bd−p−1ZT[ap+1]

ap+1∈Hp+1(Xd,Z[p]

exp 2πi

N ZXd

ap+1 ∪Bd−p−1+ZYd+1

ap+1 ∪Θ!!

×ZDΦ exp (iS[Φ, ap+1]) .

(1.3)

We can now extend the coupling RXdap+1 ∪Bd−p−1to the bulk Yd+1 and exploit the

fact that ap+1 is a (p+ 1)-cocycle δap+1 = 0 to rewrite the partition function as

ZT/Z[p]

[Bd−p−1] = X

ap+1∈Hp+1(Xd,Z[p]

exp 2πi

NZYd+1

ap+1 ∪(δBd−p−1+ Θ)!

×ZDΦ exp (iS[Φ, ap+1]) .

(1.4)

– 3 –

The exponential factor is a gauge anomaly for the gauged Z[p]symmetry, so requiring

the partition function of the theory T/Z[p]

Nto be well-deﬁned we get δBd−p−1+ Θ =

0 mod N, or in other words in presence of non-trivial background ﬁelds for the other

symmetries which appear in Θ we have that Bd−p−1is not closed and instead

δBd−p−1= Θ .(1.5)

This means that after gauging a Z[p]

Np-form symmetry with the anomaly (1.1) we obtain

a dual Z[d−p−2]

N(d−p−2)-form symmetry which forms a two-group with Postnikov

class Θ with the other symmetries.

The superconformal index in three dimensions is also sensitive to one-form symme-

tries, a fact that was already exploited for example in [93,94]. In particular, it depends

on the global structure of the gauge group thorugh the summation over monopole sec-

tors, which can be changed when gauging a one-form symmetry. Thanks to this, the

index can also be used in an indirect way to detect anomalies involving a one-form

symmetry, and we will be particularly interested in the mixed anomalies between two

discrete zero-form symmetries and a one-form symmetry. By indirect, we mean that we

study the theory resulting from gauging the one-form symmetry which in three space-

time dimensions gives rise to a zero-form symmetry. In some cases we can see from the

index of the resulting theory that the additional gauging of a zero-form symmetry is

obstructed, thus indicating the presence of the mixed anomaly in the original theory.

As it was pointed out in [37,43,45], if the resulting anomaly takes a suitable form, then

it leads to interesting consequences after gauging. In particular, if we gauge the two

zero-form symmetries in the original theory, the remaining one-form symmetry is non-

invertible. On the other hand, if we gauge the one-form symmetry and one of the two

zero-form symmetries, the other zero-form symmetry is non-invertible. We investigate

this in the context of the 3d N= 3 theories with so(2N)2kgauge algebra with adjoint

hypermultiplets, and the ABJ theories with the orthosymplectic gauge algebra. In par-

ticular, for keven, we ﬁnd that the Pin(2N)2k×USp(2M)−kvariant3of the ABJ theory

has a non-invertible one-form symmetry, whereas the (Spin(2N)2k×USp(2M)−k)/Z2

and the (O(2N)2k×USp(2M)−k)/Z2variants of the ABJ theory have a non-invertible

zero-form symmetry.

The paper is organised as follows. In Section 2, we study the N= 4 U(1) gauge

theory with 2 hypermultiplets of charge q. This part serves as an introduction on

3In this paper, we use the same nomenclature of [95] and use Pin to denote the theory obtained

by straightforwardly gauging both the magnetic and the charge conjugation symmetry of the SO

theory. This symmetry is also called Pin+as opposed to another possible variant Pin−, especially in

non-supersymmetric set-ups, see for example [96].

– 4 –

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MixedAnomalies,Two-groups,Non-InvertibleSymmetries,and3dSuperconformalIndicesNoppadolMekareeyaa;bandMatteoSacchicaINFN,sezionediMilano-Bicocca,PiazzadellaScienza3,I-20126Milano,ItalybDepartmentofPhysics,FacultyofScience,ChulalongkornUniversity,PhayathaiRoad,Pathumwan,Bangkok10330,ThailandcMathematic...

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