11institutetext: Department of Physics, North Carolina State University,
Raleigh, NC 27607, USA22institutetext: Yukawa Institute for Theoretical
Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502,
Japan

# Study of gapped phases of 4d gauge theories using temporal gauging of the
$\mathbb{Z}_{N}$ 1-form symmetry

Mendel Nguyen<EMAIL_ADDRESS>2 Yuya Tanizaki
<EMAIL_ADDRESS>1 Mithat Ünsal<EMAIL_ADDRESS>
###### Abstract

To study gapped phases of $4$d gauge theories, we introduce the temporal
gauging of $\mathbb{Z}_{N}$ $1$-form symmetry in $4$d quantum field theories
(QFTs), thereby defining effective $3$d QFTs with
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ $1$-form symmetry. In this
way, spatial fundamental Wilson and ’t Hooft loops are simultaneously genuine
line operators. Assuming a mass gap and Lorentz invariant vacuum of the $4$d
QFT, the $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ symmetry must be
spontaneously broken to an order-$N$ subgroup $H$, and we can classify the
$4$d gapped phases by specifying $H$. This establishes the $1$-to-$1$
correspondence between the two classification schemes for gapped phases of
$4$d gauge theories: One is the conventional Wilson–’t Hooft classification,
and the other is the modern classification using the spontaneous breaking of
$4$d $1$-form symmetry enriched with symmetry-protected topological states.

††preprint: YITP-23-72

##  1 Introduction

Color confinement is one of the most remarkable phenomena in $4$d non-Abelian
gauge theories, and we are continuously developing various techniques to
understand its physical mechanism. Importantly, we are interested in the
vacuum structure in the space of gauge theories as well as its properties for
a specific theory. This motivation naturally leads us to classify the possible
vacua as states of quantum phases of matter.

The classification problem of possible gapped phases of $4$d $SU(N)$ gauge
theories (with adjoint matter) has a long history, and one of the key ideas is
to study the behavior of the interparticle potential for probe particles. We
can introduce the test quark as the Wilson loop operator, and the electric
charge of the test quark is characterized by the center elements of the gauge
group, $\mathbb{Z}_{N}\subset SU(N)$. Then, Wilson proposed that confinement
and Higgs phases are discriminated by studying whether the Wilson loop shows
the area law or the perimeter law Wilson:1974sk . Interestingly, we can also
consider magnetic particles as well as electric ones. The magnetic charges
belong to $\widetilde{\mathbb{Z}}_{N}=\pi_{1}(SU(N)/\mathbb{Z}_{N})$, whose
elements specify possible Dirac strings, and we can describe their worldlines
using ’t Hooft loops. The above observation leads to the Wilson–’t Hooft
classification, which says that the gapped phases are classified according to
the set of deconfined dyonic lines in
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ tHooft:1977nqb ;
tHooft:1979rtg ; tHooft:1981bkw .

In the modern perspective of generalized global symmetry in quantum field
theories (QFTs), this Wilson–’t Hooft classification is a bit mysterious. When
we consider a $4$d gauge theory with (generalized) locality, we need to
specify the global structure of the gauge group, such as $SU(N)$ vs.
$SU(N)/\mathbb{Z}_{N}$. Once the global structure is specified, we cannot have
both Wilson and ’t Hooft loops as genuine line operators Aharony:2013hda ;
Kapustin:2014gua . Only $N$ of $N^{2}$ dyonic lines are genuine line operators
and they have to be mutually local. The other lines are non-genuine and live
on the boundaries of topological surface operators, which explains the
Wilson–’t Hooft commutation relation kinematically. These mutually-local
dyonic lines specify an order-$N$ group
$G\,(\subset\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N})$, and the theory
has a $G$ $1$-form symmetry Gaiotto:2014kfa . It should be noted that so far
we have not discussed the dynamics of gauge theories at all in this paragraph;
everything is just about the definition of genuine line operators even though
the order-$N$ subgroup of $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$
appears similarly as in the Wilson–’t Hooft classification of gapped phases.

In this paper, let us always choose the global structure of the gauge group to
be $SU(N)$. Then the $1$-form symmetry is denoted by $\mathbb{Z}_{N}^{[1]}$,
which measures the $\mathbb{Z}_{N}$ electric charge of the Wilson loop. Since
the ’t Hooft lines are not genuine line operators, we do not have the magnetic
counterpart of the $1$-form symmetry that measures
$\widetilde{\mathbb{Z}}_{N}$. This situation would naturally raise the
question of why we need the whole set of dyonic lines to characterize gapped
phases in the Wilson–’t Hooft classification. Here, we wish to answer this
question and make a clear connection between the Wilson–’t Hooft
classification and the classification via the $1$-form symmetry.

To achieve this goal, we introduce “temporal gauging” of the $1$-form symmetry
and apply this technique to produce $3$d QFTs with
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ $1$-form symmetry out of $4$d
QFTs with $\mathbb{Z}_{N}$ $1$-form symmetry. We study the partition function
of these $3$d QFTs with
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$ in the presence
of the background gauge fields, which we call the ’t Hooft partition function
as it was first introduced by ’t Hooft in Ref. tHooft:1979rtg . Let us
emphasize that the temporal gauging is reversible, so the ’t Hooft partition
function carries the same amount of information as the $4$d partition
function. The ’t Hooft partition function turns out to be strongly constrained
by the $4$d Lorentz invariance of the original theory, and this is exactly the
setup that justifies the Wilson–’t Hooft classification. We show in Sec. 2
that the classification of the $4$d gapped phases according to the spontaneous
breaking of the $4$d $1$-form symmetry enriched with symmetry-protected
topological (SPT) states is in $1$-to-$1$ correspondence with the Wilson–’t
Hooft classification via the temporal gauging operation:

$\displaystyle(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}\xrightarrow{\mathrm{SSB}}(\mathbb{Z}_{n}^{[1]})_{4\mathrm{d}}\text{
enriched with the $\mathbb{Z}_{n}^{[1]}$ level-$k$ SPT state}$
$\displaystyle\xLeftrightarrow{1:1}\quad$
$\displaystyle\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}\xrightarrow{\mathrm{SSB}}H=\\{x(n,0)+y(k,-N/n)\in\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}\\}.$
(1)

The left-hand-side describes the characterization of the gapped phases in $4$d
QFT language and the right-hand-side describes it after performing the
temporal gauging, and these two are shown to be completely equivalent.

In Sec. 3, we discuss the situation where the $4$d $1$-form symmetry has a
mixed ’t Hooft anomaly. We shall see that the anomaly relation in $4$d is
translated into the higher-group structure of $3$d QFTs after the temporal
gauging. We can reproduce the anomaly matching constraint by combining the
higher-group structure with the symmetry breaking,
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}\xrightarrow{\mathrm{SSB}}H^{[1]}$,
while the higher-group structure itself is not sufficient to reach this
conclusion. We then introduce the $\mathsf{S}$ and $\mathsf{T}$ operations to
study the connection between different gapped phases in Sec. 4. These
operations generate an $SL(2,\mathbb{Z})$ action on the space of $4$d QFTs
with $\mathbb{Z}_{N}^{[1]}$ symmetry, and these operations give automorphisms
on $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ that relate different
order-$N$ subgroups
$H_{1}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}H_{2}$.
We apply it to the $\mathcal{N}=1^{*}$ supersymmetric Yang–Mills theory and
study its rich vacuum structure from this viewpoint.

##  2 Temporal gauging of $1$-form symmetry and ’t Hooft partition function

Throughout this paper, we will analyze the gapped phases of $4$d QFTs with
$\mathbb{Z}_{N}$ $1$-form symmetry, which is denoted by
$\mathbb{Z}_{N}^{[1]}$. For this purpose, we introduce the background gauge
field $B_{4\mathrm{d}}$ for the $1$-form symmetry and study properties of the
partition function

$\mathcal{Z}[B_{4\mathrm{d}}].$ (2)

This partition function is defined on any general $4$-dimensional Riemannian
manifold $M_{4}$. In the following, we especially pay attention to the case

$M_{4}=M_{3}\times S^{1}.$ (3)

We refer to this $S^{1}$ as the temporal direction, and denote its coordinate
by $\tau\sim\tau+L$. As we are still interested in the four-dimensional
dynamics, we basically assume that the size $L$ of $S^{1}$ is sufficiently
large and the phase is smoothly connected to the ground states. In the
following, we choose a spin structure for $M_{3}\times S^{1}$.

By regarding the size of $M_{3}$ to be much larger than that of $S^{1}$, we
can pretend that we are dealing with $3$d QFTs. Then, the $4$d
$\mathbb{Z}_{N}^{[1]}$ symmetry splits into Gaiotto:2014kfa

$(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}\Longrightarrow(\mathbb{Z}_{N}^{[1]})_{3\mathrm{d}}\times(\mathbb{Z}_{N}^{[0]})_{3\mathrm{d}}.$
(4)

Let $B_{m}$ and $A$ denote the background gauge fields for
$(\mathbb{Z}_{N}^{[1]})_{3\mathrm{d}}$ and
$(\mathbb{Z}_{N}^{[0]})_{3\mathrm{d}}$, respectively. Then they can be related
to the $4$d background gauge fields $B_{4\mathrm{d}}$ as Gaiotto:2017yup ;
Shimizu:2017asf ; Tanizaki:2017qhf ; Tanizaki:2017mtm

$B_{4\mathrm{d}}=B_{m}+A\wedge\frac{\mathrm{d}\tau}{L}.$ (5)

Here, $B_{m}$ does not have the temporal component, and we sometimes call it
the magnetic flux. The temporal-spatial component is expressed by the $1$-form
gauge field $A$.

We define the temporal gauging by the path integral in terms of $A$:111We
follow the convention that the background fields are denoted with upper case
and the dynamical ones are with lower case. When we promote the background
gauge fields to the dynamical ones, we change their letters to the
corresponding lower case letters, accordingly.

$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\int{\mathcal{D}}a\,\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}\cup
a\biggr{)}\,\mathcal{Z}[B_{m}+a\wedge(\mathrm{d}\tau/L)].$ (6)

Here, $\widetilde{B}_{e}$ is the background gauge field for the
$\mathbb{Z}_{N}$ $1$-form symmetry, $\widetilde{\mathbb{Z}}_{N}^{[1]}$, dual
to the original $\mathbb{Z}_{N}^{[0]}$ symmetry. Regarding
$\mathcal{Z}_{\text{tH}}$ as the partition function of the $3$d QFT defined on
$M_{3}$, it enjoys the
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$ symmetry and
$(\widetilde{B}_{e},B_{m})$ is the corresponding background $2$-form gauge
field. As this partition function (6) was first introduced by ’t Hooft in Ref.
tHooft:1979rtg  for the case of $M_{4}=T^{4}$, we shall refer to it as the ’t
Hooft partition function.

As this theory enjoys the $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$
$1$-form symmetry, we must have the corresponding line operators. Since the
$\mathbb{Z}_{N}^{[1]}$ symmetry is the $1$-form symmetry in the original $4$d
theory, let us refer to the corresponding operator as the Wilson loop,
$W_{(0,1)}(C)$. It is then natural to refer to the charged object of
$\widetilde{\mathbb{Z}}_{N}^{[1]}$ as the ’t Hooft loop, $W_{(1,0)}(C)$. In
general, the dyonic loop operator (with the magnetic charge
$m\in\mathbb{Z}_{N}$ and the electric charge $e\in\mathbb{Z}_{N}$) is denoted
as

$W_{(m,e)}(C)\,\mathrm{e}^{\mathrm{i}\int_{D}(m\widetilde{B}_{e}+eB_{m})}$ (7)

with some $\partial D=C$ in the presence of the background gauge fields.222We
note that these are genuine line operators in the effective $3$d QFT on
$M_{3}$ as the surface dependence appears only when we turn on background
$2$-form gauge fields.

Here, we need to emphasize that the ’t Hooft partition function (6) is
introduced to understand the possible phases of $4$d gauge theories, while
$\widetilde{B}_{e}$ and $B_{m}$ there are $3$d $\mathbb{Z}_{N}$ $2$-form gauge
fields. Even though $\mathcal{Z}[B_{4\mathrm{d}}]$ is covariant under Lorentz
transformations, the temporal-gauging procedure does not respect it, and thus
$\mathcal{Z}_{\text{tH}}$ may seem at first sight to be less useful compared
with the original one $\mathcal{Z}[B_{4\mathrm{d}}]$ for studying the $4$d
dynamics. Let us point out, however, that the temporal gauging is a reversible
operation, and thus $\mathcal{Z}[B_{4\mathrm{d}}]$ and
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ should carry the same
amount of information. Moreover, it turns out in the following that
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ provides a convenient tool
for the classification of gapped phases, and the physical meaning of each
phase also becomes quite transparent.

###  2.1 Positivity of the ’t Hooft partition function
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$

An important property of the ’t Hooft partition function is its semi-
positivity, and one can easily show it using reflection positivity.333We
assume that the $4$d theory is unitary and thus its path integral satisfies
reflection positivity.  Regard $S^{1}$ as the temporal direction. We pick the
antipodal points $\\{\tau=0,L/2\\}\subset S^{1}$ and choose
$M_{3}\times\\{0,L/2\\}\subset M_{4}$ as the reflection plane for the
Osterwalder–Schrader reflection.

Using the $1$-form gauge invariance, or the topological nature of the
codim-$2$ defects, we may set the specific alignment of the discrete gauge
fields. For the spatial part $B_{m}$, we require that $B_{m}$ does not depend
on $\tau$ at all so that $B_{m}$ is invariant under the Osterwalder–Schrader
reflection. To discuss the temporal gauge field $a\wedge(\mathrm{d}\tau/L)$,
we note the following trivial identity,

$\int{\mathcal{D}}a\,F[a]=\int{\mathcal{D}}a_{1}{\mathcal{D}}a_{2}\,F[a_{1}-a_{2}],$
(8)

which holds for any functionals $F[a]$. Using this identity, we find

$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\int{\mathcal{D}}a_{1}{\mathcal{D}}a_{2}\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}\cup(a_{1}-a_{2})}\mathcal{Z}\biggl{[}B_{m}+a_{1}\wedge\frac{\mathrm{d}\tau}{L}-a_{2}\wedge\frac{\mathrm{d}\tau}{L}\biggr{]}.$
(9)

Thus, the defects for the temporal directions can be doubled, and these two
defects can be put on arbitrary locations due to their topological nature. By
a suitable choice, they can be related by the Osterwalder–Schrader reflection.
Then, the reflection positivity ensures that

$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]\geq 0$ (10)

for any $\widetilde{B}_{e},B_{m}\in H^{2}(M_{3};\mathbb{Z}_{N})$.

We note that this positivity is achieved by the temporal gauging procedure.
Indeed, the ordinary partition function, $\mathcal{Z}[B_{4\mathrm{d}}]$, can
take complex values in general in the presence of the background gauge fields,
and it often provides us important information on the quantum phases of
matter. In general, temporal components of the gauge field flip their sign
under the Osterwalder–Schrader reflection, so this complex phase is consistent
with reflection positivity. In the case of the ’t Hooft partition function,
the positivity argument works as we sum up all the possible gauge fields
having temporal components (note that $\widetilde{B}_{e}$ only has purely
spatial components), and we find (10).

The physical meaning of the positivity (10) becomes more transparent if we
consider it in the operator formalism tHooft:1979rtg . Let $\mathcal{H}$ be
the Hilbert space when we quantize the theory on $M_{3}$, and let
$\hat{H}[B_{m}]$ be the Hamiltonian operator that contains the magnetic flux
$B_{m}$. We can further define the projection operator onto the electric flux
sector $\hat{P}[\widetilde{B}_{e}]$, which satisfies
$\hat{P}[\widetilde{B}_{e}]^{\dagger}=\hat{P}[\widetilde{B}_{e}]$,
$\hat{P}[\widetilde{B}_{e}]^{2}=\hat{P}[\widetilde{B}_{e}]$ and
$\sum_{\widetilde{B}_{e}}\hat{P}[\widetilde{B}_{e}]=\bm{1}_{\mathcal{H}}$.
Then the ’t Hooft partition function can be written as

$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\operatorname{Tr}_{\mathcal{H}}\Bigl{[}\hat{P}[\widetilde{B}_{e}]\,\mathrm{e}^{-L\hat{H}[B_{m}]}\Bigr{]},$
(11)

and then its positivity is quite manifest.

###  2.2 $\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ for gapped phases
and constraints from Lorentz invariance

In Ref. tHooft:1979rtg , ’t Hooft found the “duality equation” for
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ by considering a discrete
rotation of the torus $M_{4}=T^{4}$. By assuming that the system is gapped,
the duality equation implies that
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$ should be
spontaneously broken to an order-$N$ subgroup, i.e.,

$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}\xrightarrow{\mathrm{SSB}}H^{[1]},$
(12)

where the unbroken symmetry $H$ has order $N$, $|H|=N$, and satisfies444We
note that when $H$ is an order-$N$ subgroup of
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$, then the mutual locality
condition (13) is automatically satisfied. To see this, let us assume (to
derive a contradiction) that the mutual locality is violated, so that there
are $(x_{1},y_{1}),(x_{2},y_{2})\in H$ such that $M\equiv
x_{1}y_{2}-x_{2}y_{1}\not=0\bmod N$. Without loss of generality, this
$M\,(<N)$ can be taken to be a positive divisor of $N$. Then, the subgroup
$\\{a(x_{1},y_{1})+b(x_{2},y_{2})\\}\subset H$ has order $N^{2}/M>N$, and this
contradicts with $|H|=N$.

$\forall(x_{1},y_{1}),(x_{2},y_{2})\in
H,\quad\langle(x_{1},y_{1}),(x_{2},y_{2})\rangle\equiv
x_{1}y_{2}-x_{2}y_{1}=0\bmod N.$ (13)

This condition is referred to as the mutual locality condition.

In order to see how such a constraint on gapped phases arises, let us consider
what would happen if
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$ were not broken
at all. By gauging $\widetilde{B}_{e}$, we can undo the temporal gauging
procedure and this gives the delta-functional constraint on $A$ for
$(\mathbb{Z}_{N}^{[0]})_{3\mathrm{d}}$. This implies that
$(\mathbb{Z}_{N}^{[0]})_{3\mathrm{d}}$ is spontaneously broken, while
$(\mathbb{Z}_{N}^{[1]})_{3\mathrm{d}}$ is unbroken by assumption. Since both
of these symmetries arise from $(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}$, this
option obviously violates the $4$d Lorentz invariance. Similarly, if we assume
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$ were completely
broken, we find that $(\mathbb{Z}_{N}^{[0]})_{3\mathrm{d}}$ is unbroken while
$(\mathbb{Z}_{N}^{[1]})_{3\mathrm{d}}$ is broken, and again Lorentz invariance
is violated. These quick observations already tell us that Lorentz invariance
puts severe constraints and requires the correct amount of symmetry breaking
for $\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$, and we can
actually find that it is broken down to an exactly order-$N$ subgroup (with
mutual locality) when assuming a mass gap (and also one technical
assumption).555As this consequence is very similar to that of the anomaly-
matching constraint, one might wonder if this can be understood from the mixed
anomaly for $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ $1$-form
symmetry. However, this is not the case, and we emphasize that the $4$d
Lorentz invariance of the original theory plays a pivotal role here.

We shall give a review of the original argument by ’t Hooft in Appendix A to
be self-contained. Here, instead, let us perform explicit calculations of the
$4$d partition function $\mathcal{Z}[B_{4\mathrm{d}}]$ and
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ for gapped phases. We here
assume that the 4d $\mathbb{Z}_{N}$ $1$-form symmetry is spontaneously broken
to a subgroup,

$(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}\xrightarrow{\mathrm{SSB}}(\mathbb{Z}_{n}^{[1]})_{4\mathrm{d}},$
(14)

where $n$ is a positive divisor of $N$, and the vacuum state further acquires
a nontrivial SPT phase for the unbroken $(\mathbb{Z}_{n}^{[1]})_{4\mathrm{d}}$
symmetry. The low-energy theory becomes $\mathbb{Z}_{N/n}$ topological field
theory, and the partition function can be modeled as666We may consider more
general $2$-group gauge theories as possible models (see, e.g., Refs.
Gukov:2013zka ; Kapustin:2013uxa ; Kapustin:2013qsa ; Lan:2018vjb ; Johnson-
Freyd:2020usu ; Thorngren:2020aph ). Here, let us restrict our attention to
these simplest possibilities.

$\displaystyle\mathcal{Z}[B_{4\mathrm{d}}]$
$\displaystyle=\frac{|H^{0}(M_{4};\mathbb{Z}_{N/n})|}{|H^{1}(M_{4};\mathbb{Z}_{N/n})|}\sum_{b\in
H^{2}(M_{4};\mathbb{Z}_{N/n})}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N/n}\int
b\cup B_{4\mathrm{d}}\biggr{)}$
$\displaystyle\quad\times\exp\biggl{(}\frac{2\pi\mathrm{i}k}{n}\int\frac{1}{2}P_{2}\biggl{(}\frac{B_{4\mathrm{d}}}{N/n}\biggr{)}\biggr{)},$
(15)

where $P_{2}(B)=B\cup B+B\cup_{1}\mathrm{d}B$ is the Pontryagin square.777As
long as working on torsion-free $4$-manifolds, we can always take an integral
lift of the discrete gauge field $B$ and $P_{2}(B)$ can be simply thought of
as $B\cup B$ by identifying $B$ with one of the integral lifts. We shall use
this property throughout this paper to simplify computations. The $b$ field
refers to the discrete $\mathbb{Z}_{N/n}$ $2$-form gauge field (not a
$\mathbb{Z}_{N}$ gauge field) for the topological field theory, and its path
integral gives the delta-functional constraint on $B_{4\mathrm{d}}$ so that

$\int_{\Sigma}B_{4\mathrm{d}}\in\frac{N}{n}\mathbb{Z}$ (16)

for any closed $2$-cycle $\Sigma$. As $\int B_{4\mathrm{d}}$ is well-defined
$\bmod\,N$, we can regard $B_{4\mathrm{d}}/(N/n)$ as the $\mathbb{Z}_{n}$
$2$-form gauge field, and the second line on the right-hand-side of (15)
describes the level-$k$ SPT action for this unbroken $\mathbb{Z}_{n}^{[1]}$
symmetry with $k\sim k+n$ (given a spin structure).

Let us compute the ’t Hooft partition function for (15), which is given by

$\displaystyle\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$
$\displaystyle=\frac{1}{|H^{0}(M_{3};\mathbb{Z}_{N})|}\sum_{a\in
H^{1}(M_{3};\mathbb{Z}_{N})}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int\widetilde{B}_{e}\cup
a\biggr{)}$ $\displaystyle\quad\times\frac{|H^{0}(M_{3}\times
S^{1};\mathbb{Z}_{N/n})|}{|H^{1}(M_{3}\times
S^{1};\mathbb{Z}_{N/n})|}\sum_{b_{m}\in
H^{2}(M_{3};\mathbb{Z}_{N/n})}\sum_{a^{\prime}\in
H^{1}(M_{3};\mathbb{Z}_{N/n})}$
$\displaystyle\quad\times\exp\biggl{(}\frac{2\pi\mathrm{i}}{N/n}\int(b_{m}\cup
a+B_{m}\cup a^{\prime})+\frac{2\pi\mathrm{i}}{n}\frac{k}{(N/n)^{2}}\int
B_{m}\cup a\biggr{)}.$ (17)

We note that $|H^{1}(M_{3}\times
S^{1};\mathbb{Z}_{N/n})|=(N/n)^{\beta_{1}(M_{3})+1}$, so
$\frac{|H^{0}(M_{3}\times S^{1};\mathbb{Z}_{N/n})|}{|H^{1}(M_{3}\times
S^{1};\mathbb{Z}_{N/n})|}=\frac{1}{(N/n)^{\beta_{1}(M_{3})}}$. Here,
$\beta_{i}(M)=\mathrm{rank}\,H^{i}(M;\mathbb{Z})$ is the $i$-th Betti number.
The summation over $a^{\prime}$ gives the delta-functional constraint on
$B_{m}$, so we separate it from other path integrals:

$\displaystyle\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$
$\displaystyle=\frac{1}{(N/n)^{\beta_{1}(M_{3})}}\sum_{a^{\prime}\in
H^{1}(M_{3};\mathbb{Z}_{N/n})}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N/n}\int
a^{\prime}\cup B_{m}\biggr{)}$ $\displaystyle\quad\times\frac{1}{N}\sum_{a\in
H^{1}(M_{3};\mathbb{Z}_{N})}\sum_{b_{m}\in
H^{2}(M_{3};\mathbb{Z}_{N/n})}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int
a\cup\biggl{(}n\,b_{m}+\widetilde{B}_{e}+k\frac{n}{N}B_{m}\biggr{)}\biggr{)}$
$\displaystyle=\frac{1}{(N/n)^{\beta_{1}(M_{3})}}(N/n)^{\beta_{1}(M_{3})}\delta_{N}[nB_{m}]\times\frac{1}{N}N^{\beta_{1}(M_{3})}\delta_{N}\biggl{[}\frac{N}{n}\biggl{(}\widetilde{B}_{e}+k\frac{n}{N}B_{m}\biggr{)}\biggr{]}$
$\displaystyle=N^{\beta_{1}(M_{3})-1}\,\delta_{N}\biggl{[}nB_{m},\,\frac{N}{n}\widetilde{B}_{e}+kB_{m}\biggr{]}.$
(18)

Here, $\delta_{N}[B]$ is the delta functional that gives $1$ when
$\int_{\Sigma}B\in N\mathbb{Z}$ for every closed cycle $\Sigma$ and gives $0$
otherwise. This shows that the deconfined lines are generated by

$W_{(0,n)}(C)\,\mathrm{e}^{\mathrm{i}\int_{D}nB_{m}},\quad
W_{(N/n,k)}(C)\,\mathrm{e}^{\mathrm{i}\int_{D}((N/n)\widetilde{B}_{e}+kB_{m})},$
(19)

with $\partial D=C$, and thus the unbroken subgroup $H$ is given by

$H=\left\\{x\Big{(}n,0\Big{)}+y\Big{(}k,-\frac{N}{n}\Big{)}\right\\}\subset\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N},$
(20)

and we can readily confirm that the mutual locality condition (13) is
satisfied. We can also check that $|H|=N$ and moreover that every order $N$
subgroup of $\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ appears in this
way.888To see that every order $N$ subgroup of
$\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ appears, note that any such subgroup $K$
arises from an index $N$ sublattice $L$ of $\mathbb{Z}\times\mathbb{Z}$
containing $N\mathbb{Z}\times N\mathbb{Z}$ such that $K=L/(N\mathbb{Z}\times
N\mathbb{Z})$. Then, since $(N,0)$ and $(0,N)$ are linearly independent
vectors in $L$, a theorem on lattices implies that we can find a basis $u,v$
of $L$ of the form
$u\equiv\frac{1}{q}(N,0),v\equiv\frac{k}{N}(N,0)+\frac{1}{n}(0,N)$, with $q,n$
positive divisors of $N$ and $k$ an integer. The condition that $L$ have index
$N$ in $\mathbb{Z}\times\mathbb{Z}$ then implies that $N^{2}/qn=\langle
u,v\rangle=N$, i.e., that $N/q=n$. Hence, $K$ is precisely of the form (20).

In the above discussion, we start from the $4$d partition function (15) and
derive (18) by the temporal gauging, but we can reverse the logic to reproduce
(15) by performing the path integral of $\widetilde{B}_{e}$ of the ’t Hooft
partition function (18), which achieves the equivalence mentioned in (1), and
let us recapitulate it here:

$\displaystyle(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}\xrightarrow{\mathrm{SSB}}(\mathbb{Z}_{n}^{[1]})_{4\mathrm{d}}\text{
enriched with the $\mathbb{Z}_{n}^{[1]}$ level-$k$ SPT state}$
$\displaystyle\xLeftrightarrow{1:1}\quad$
$\displaystyle\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}\xrightarrow{\mathrm{SSB}}H=\\{x(n,0)+y(k,-N/n)\in\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}\\}.$
(21)

Therefore, the order-$N$ subgroup $H$ of
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ correctly characterizes the
gapped phases of $4$d QFTs with $\mathbb{Z}_{N}^{[1]}$ symmetry: Vacuum states
with different $H$ are distinguished as quantum phases.

###  2.3 Example: Lattice $SU(N)$ Yang–Mills theory at strong coupling

It would be useful to compute $\mathcal{Z}[B_{4\mathrm{d}}]$ and
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ in some microscopically
solvable model for concrete understanding of their behaviors. Here, let us
consider the strong-coupling expansion of the lattice $SU(N)$ gauge theory
with the Wilson action.

The Wilson action with the $\mathbb{Z}_{N}$ two-form gauge field is given by

$S_{W}[U_{\ell},B_{p}]=-\frac{1}{2g^{2}}\sum_{p}\operatorname{tr}\Bigl{(}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}+\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}^{\dagger}\Bigr{)},$
(22)

where $U_{\ell}$ denotes the $SU(N)$-valued link variable,
$U_{p}=\mathcal{P}\prod_{\ell\in p}U_{\ell}$ is the path-ordered products
along the plaquette $p$, and $B_{p}$ denotes the $\mathbb{Z}_{N}$-valued
plaquette variable, which is identified with $B_{4\mathrm{d}}$. The partition
function is given by

$\mathcal{Z}[B_{4\mathrm{d}}]=\int{\mathcal{D}}U_{\ell}\exp\Bigl{(}-S_{W}[U_{\ell},B_{p}]\Bigr{)}.$
(23)

We expand this partition function in terms of $1/g^{2}$ in the strong-coupling
expansion by using formulas of Haar integration, such as
$\int\mathrm{d}U(U)_{i_{1}i_{2}}(U^{\dagger})_{j_{1}j_{2}}=\frac{1}{N}\delta_{i_{1}j_{2}}\delta_{i_{2}j_{1}}$.
Let us then expand the path-integral weight up to the $O(1/g^{2})$ term for
each plaquette,

$\exp(-S_{W})\simeq\prod_{p}\left(1+\frac{1}{2g^{2}}\operatorname{tr}(\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}+\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}B_{p}}U_{p}^{\dagger})\right).$
(24)

Then the partition function can be represented as a sum over closed surfaces,

$\mathcal{Z}[B_{4\mathrm{d}}]\simeq\sum_{\Sigma\colon\text{closed
surface}}N^{\chi(\Sigma)}\left(\frac{1}{2Ng^{2}}\right)^{\mathrm{Area}(\Sigma)}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}\int_{\Sigma}B_{4\mathrm{d}}}.$
(25)

We can think of this expression as the sum over the worldsheets of confining
strings with the string tension $\sigma=\ln(2Ng^{2})$ in lattice units. When
$\Sigma$ is a contractible closed surface, we have
$\int_{\Sigma}B_{4\mathrm{d}}=0\bmod N$. Thus, the nontrivial
$B_{4\mathrm{d}}$ dependence appears only if the confining-string worldsheet
wraps around nontrivial $2$-cycles, and such processes are exponentially
suppressed:

$\mathcal{Z}[B_{4\mathrm{d}}]-\mathcal{Z}[0]\simeq O(\mathrm{e}^{-\sigma
L^{2}})\xrightarrow{L\to\infty}0,$ (26)

where $L$ is the length of $T^{4}$. This shows that, in the infinite-volume
limit, we can regard $\mathcal{Z}[B_{4\mathrm{d}}]\to 1$, which corresponds to
$n=N$ and $k=0$ in (15).

Now, let us perform the temporal gauging of (25) to find
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$. As we have found that the
$B_{4\mathrm{d}}$ dependence of $\mathcal{Z}[B_{4\mathrm{d}}]$ is
exponentially small, its Fourier transform localizes to $\widetilde{B}_{e}=0$
and we get
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]\propto\delta_{N}[\widetilde{B}_{e}]$.
More precisely, for $\widetilde{B}_{e}=0$,

$\displaystyle\mathcal{Z}_{\text{tH}}[0,B_{m}]$
$\displaystyle=\int{\mathcal{D}}a\,\mathcal{Z}[B_{m}+a\wedge\mathrm{d}\tau/L]$
$\displaystyle\simeq\mathcal{Z}[0]+O(\mathrm{e}^{-\sigma
L^{2}})\xrightarrow{L\to\infty}\mathcal{Z}[0].$ (27)

On the other hand, if we take $\int_{(T^{2})_{12}}\widetilde{B}_{e}=1$ as an
example of $\widetilde{B}_{e}\not=0$, then
$\exp(\mathrm{i}\int\widetilde{B}_{e}\cup
a)=\exp(\mathrm{i}\int_{S^{1}}a_{3}\mathrm{d}x^{3})$. To cancel this phase in
the summation of $a$, the confining-string worldsheet should wrap once around
the $3$-$4$ cycle, and we get

$\displaystyle\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e}(\not=0),B_{m}]$
$\displaystyle=\int{\mathcal{D}}a\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int
a_{3}\mathrm{d}x^{3}}\mathcal{Z}[B_{m}+a\wedge\mathrm{d}\tau/L]$
$\displaystyle\simeq O(\mathrm{e}^{-\sigma L^{2}})\xrightarrow{L\to\infty}0.$
(28)

We actually find
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\mathcal{Z}[0]\,\delta_{N}[\widetilde{B}_{e}]$
neglecting the exponentially small contributions as $L\to\infty$, and the
unbroken order-$N$ subgroup is
$H=\\{0\\}\times\mathbb{Z}_{N}\subset\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$.

##  3 Anomaly matching and the higher-group structure

In general, global symmetry in QFTs may have an ’t Hooft anomaly, which is an
obstruction to the promotion of the global symmetry to local gauge redundancy.
The ’t Hooft anomaly is invariant under any local and symmetric deformations
of QFTs, and thus the low-energy effective theory is strongly constrained as
it must reproduce the anomaly computed in ultraviolet. In this section, we
shall discuss the structure of the ’t Hooft partition function
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ when the original $4$d
$\mathbb{Z}_{N}^{[1]}$ symmetry has a mixed ’t Hooft anomaly.

Pure Yang–Mills theory: As an example, let us consider the generalized
anomaly, or global inconsistency, of pure $SU(N)$ Yang–Mills theory. The $4$d
Yang–Mills partition function $\mathcal{Z}^{\mathrm{YM}}_{\theta}$ has an ’t
Hooft anomaly involving the $\theta$ periodicity Gaiotto:2017yup ;
Tanizaki:2017bam ; Karasik:2019bxn ; Cordova:2019uob  and we can detect it by
introducing the background $\mathbb{Z}_{N}$ two-form gauge field
$B_{4\mathrm{d}}$:

$\mathcal{Z}^{\mathrm{YM}}_{\theta+2\pi}[B_{4\mathrm{d}}]=\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int\frac{1}{2}P_{2}(B_{4\mathrm{d}})\biggr{)}\mathcal{Z}^{\mathrm{YM}}_{\theta}[B_{4\mathrm{d}}].$
(29)

To satisfy the anomaly matching condition in the confined phase, the level
crossing of the ground state is mandatory, as the two confined states at
$\theta$ and $\theta+2\pi$ are distinct as $4$d SPT states with
$\mathbb{Z}_{N}^{[1]}$ symmetry.

Let us interpret this result using the ’t Hooft partition function:

$\mathcal{Z}_{\mathrm{tH},\theta}^{\mathrm{YM}}[\widetilde{B}_{e},B_{m}]=\int{\mathcal{D}}a\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int\widetilde{B}_{e}\cup
a\biggr{)}\mathcal{Z}_{\theta}^{\mathrm{YM}}[B_{m}+a\wedge\mathrm{d}\tau/L].$
(30)

By performing the temporal gauging on both sides of (29), we find that

$\displaystyle\mathcal{Z}_{\mathrm{tH},\theta+2\pi}^{\mathrm{YM}}[\widetilde{B}_{e},B_{m}]$
$\displaystyle=\int{\mathcal{D}}a\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int\widetilde{B}_{e}a}\mathcal{Z}^{\mathrm{YM}}_{\theta+2\pi}[B_{m}+a\wedge\mathrm{d}\tau/L]$
$\displaystyle=\int{\mathcal{D}}a\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int\widetilde{B}_{e}a}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int
B_{m}a}\mathcal{Z}^{\mathrm{YM}}_{\theta+2\pi}[B_{m}+a\wedge\mathrm{d}\tau/L]$
$\displaystyle=\mathcal{Z}_{\mathrm{tH},\theta}^{\mathrm{YM}}[\widetilde{B}_{e}+B_{m},B_{m}].$
(31)

This is nothing but the Witten effect Witten:1979ey , which claims that the
purely magnetic line at $\theta+2\pi$ is equivalent to the dyonic line at
$\theta$. Equivalently, the $(-1)$-form transformation,
$\theta\to\theta+2\pi$, induces the nontrivial action on the $1$-form
symmetries, $\widetilde{B}_{e}\to\widetilde{B}_{e}+B_{m}$, which is an example
of the higher-group structure Sharpe:2015mja ; Tachikawa:2017gyf ;
Cordova:2018cvg ; Tanizaki:2019rbk ; Hidaka:2020iaz ; Hidaka:2020izy .

We note that the Witten effect, or the higher-group structure, itself does not
give nontrivial constraints, in constrast to the ’t Hooft anomaly. The trivial
state,
$\mathcal{Z}_{\mathrm{tH},\theta}^{\mathrm{YM}}[\widetilde{B}_{e},B_{m}]=1$,
is consistent with the transformation,
$\widetilde{B}_{e}\to\widetilde{B}_{e}+B_{m}$, and this state is indeed
realized by the high-temperature Yang–Mills theory at any value of $\theta$.
The anomaly matching constraint is reproduced by considering the $4$d Lorentz
invariance. As discussed in Sec. 2, the gapped state with $4$d Lorentz
invariance must have the symmetry breaking
$\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}\xrightarrow{\mathrm{SSB}}H^{[1]}$,
and thus we can set the ’t Hooft partition function for a given $\theta$ to be

$\mathcal{Z}_{\mathrm{tH},\theta}^{\mathrm{YM}}[\widetilde{B}_{e},B_{m}]=\delta_{N}[nB_{m}]\delta_{N}[(N/n)\widetilde{B}_{e}+kB_{m}].$
(32)

Dialing the $\theta$ parameter, $\theta\to\theta+2\pi$, we should obtain

$\mathcal{Z}_{\mathrm{tH},\theta+2\pi}^{\mathrm{YM}}[\widetilde{B}_{e},B_{m}]=\delta_{N}[nB_{m}]\delta_{N}[(N/n)\widetilde{B}_{e}+(k+(N/n))B_{m}].$
(33)

As $k\sim k+n$, we should encounter a phase transition as a function of
$\theta$ if $N/n$ is not a multiple of $n$. This is always the case for
ordinary confinement phases, $n=N$, while the totally Higgs phase, $n=1$, does
not need the phase transition in $\theta$. This reproduces the consequence of
the $4$d ’t Hooft anomaly (29).

$\mathcal{N}=1$ supersymmetric Yang–Mills theory: The higher group structure
may be more evident in the $\mathcal{N}=1$ super Yang–Mills (SYM) case, where
the shift of the $\theta$ angle is related to the discrete chiral symmetry
$(\mathbb{Z}_{2N})_{\chi}$. Introducing the discrete chiral gauge field
$A_{\chi}$, we find the ’t Hooft anomaly,

$\mathcal{Z}^{\mathrm{SYM}}[A_{\chi}+\mathrm{d}\lambda_{\chi},B_{4\mathrm{d}}]=\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int\lambda_{\chi}\cup\frac{1}{2}P_{2}(B_{4\mathrm{d}})\biggr{)}\mathcal{Z}^{\mathrm{SYM}}[A_{\chi},B_{4\mathrm{d}}].$
(34)

By performing a similar computation as in (31), this relation is translated as

$\mathcal{Z}^{\mathrm{SYM}}_{\mathrm{tH}}[A_{\chi}+\mathrm{d}\lambda_{\chi},\widetilde{B}_{e},B_{m}]=\mathcal{Z}^{\mathrm{SYM}}_{\mathrm{tH}}[A_{\chi},\widetilde{B}_{e}+\lambda_{\chi}\cup
B_{m},B_{m}].$ (35)

The $0$-form chiral symmetry causes the Witten effect and induces a nontrivial
action on the $1$-form symmetry,
$\widetilde{B}_{e}\to\widetilde{B}_{e}+\lambda_{\chi}\cup B_{m}$.

Again, the higher-group symmetry itself does not require the degeneracy of
ground states, but it gives a nontrivial consequence when we further impose
the $4$d Lorentz invariance. If we assume that the system is in a confined
phase (i.e. $n=N$), then the ’t Hooft argument shows that the partition
function of a given vacuum should be described by

$\delta_{N}[\widetilde{B}_{e}+kB_{m}],$ (36)

with some $k\sim k+N$. Then, the higher-group structure discussed above
indicates that the discrete chiral transformation interchanges the vacuum with
label $k$ to the vacuum with label $k+1$:

(37)

The ’t Hooft partition function of $\mathcal{N}=1$ SYM theory is then given by

$\mathcal{Z}_{\text{tH}}^{\mathrm{SYM}}[\widetilde{B}_{e},B_{m}]=\sum_{k=1}^{N}\delta_{N}[\widetilde{B}_{e}+kB_{m}].$
(38)

These $N$ vacua are understood as the chiral broken vacua,
$(\mathbb{Z}_{2N})_{\chi}\xrightarrow{\mathrm{SSB}}\mathbb{Z}_{2}$, and the
label $k$ specifies the phase of the gluino condensate,
$\langle\lambda^{2}\rangle=\Lambda^{3}\mathrm{e}^{2\pi\mathrm{i}k/N}$. This is
exactly the vacuum structure expected from the anomaly matching condition
obtained before the temporal gauging, and the same information is found via
the higher-group structure combined with the constraint from $4$d Lorentz
invariance.

##  4 $\mathsf{S}$ and $\mathsf{T}$ operations on $4$d QFTs with the
$\mathbb{Z}_{N}$ $1$-form symmetry

Let us introduce the formal operations, $\mathsf{S}$ and $\mathsf{T}$, that
act on $4$d QFTs with the $\mathbb{Z}_{N}$ $1$-form symmetry:

$\displaystyle\mathsf{S}$
$\displaystyle\colon\mathcal{Z}[B_{4\mathrm{d}}]\mapsto\mathsf{S}\mathcal{Z}[B_{4\mathrm{d}}]\equiv\int{\mathcal{D}}b_{4\mathrm{d}}\,\mathcal{Z}[b_{4\mathrm{d}}]\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int
B_{4\mathrm{d}}\cup b_{4\mathrm{d}}\biggr{)},$ (39) $\displaystyle\mathsf{T}$
$\displaystyle\colon\mathcal{Z}[B_{4\mathrm{d}}]\mapsto\mathsf{T}\mathcal{Z}[B_{4\mathrm{d}}]\equiv\mathcal{Z}[B_{4\mathrm{d}}]\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int\frac{1}{2}P_{2}(B_{4\mathrm{d}})\biggr{)}.$
(40)

The $\mathsf{S}$ operation dynamically gauges the $\mathbb{Z}_{N}^{[1]}$ in
$4$d spacetime, and thus the original background gauge field is promoted to
the dynamical field $b_{4\mathrm{d}}$. The gauged theory acquires the dual
$\mathbb{Z}_{N}^{[1]}$ symmetry, and we introduce the background gauge field
$B_{4\mathrm{d}}$ that couples to it. The $\mathsf{T}$ operation just shifts
the local counterterm for the background gauge field $B_{4\mathrm{d}}$.

Here, we would like to emphasize that these $\mathsf{S}$ and $\mathsf{T}$
operations do not necessarily imply the duality/symmetry of a given $4$d QFT.
We can always apply these operations as long as the $4$d QFTs have a
$\mathbb{Z}_{N}$ $1$-form symmetry, and generically these operations generate
different QFTs. One may say that $\mathsf{S}$ and $\mathsf{T}$ are morphisms
in the category of $4$d QFTs with the $\mathbb{Z}_{N}^{[1]}$ symmetry (see
Refs. Kapustin:1999ha ; Witten:2003ya  for the case of $3$d $U(1)$ symmetry).
When the generated QFT is accidentally the same as the original one, these
operations may be regarded as self-duality operations. The $SU(N)$ Yang–Mills
theory with adjoint matter always has the self $\mathsf{T}$ duality associated
with $\theta\to\theta+2\pi$ as we see in (29). Examples with the full
$SL(2,\mathbb{Z})$ self-duality are the Cardy–Rabinovici model Cardy:1981qy ;
Cardy:1981fd ; Honda:2020txe ; Hayashi:2022fkw  and $\mathcal{N}=4$ SYM theory
Vafa:1994tf ; Donagi:1995cf ; Dorey:1999sj ; Polchinski:2000uf . In Sec. 4.2,
we shall discuss the $\mathcal{N}=1^{*}$ SYM theory in detail.

###  4.1 $\mathsf{S}$ and $\mathsf{T}$ operations on ’t Hooft partition
functions

Let us study how these operations act on the ’t Hooft partition function (6).
We define the $\mathsf{S}$ and $\mathsf{T}$ operations on
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$ by the temporal gauging of
the $\mathsf{S}$\- and $\mathsf{T}$-transformed partition functions,
respectively:

$\displaystyle\mathsf{S}\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]\equiv\int{\mathcal{D}}a\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}\cup
a\biggr{)}\mathsf{S}\mathcal{Z}\biggl{[}B_{m}+a\wedge\frac{\mathrm{d}\tau}{L}\biggr{]},$
(41)
$\displaystyle\mathsf{T}\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]\equiv\int{\mathcal{D}}a\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}\cup
a\biggr{)}\mathsf{T}\mathcal{Z}\biggl{[}B_{m}+a\wedge\frac{\mathrm{d}\tau}{L}\biggr{]}.$
(42)

We can compute these path integrals explicitly to express the right-hand-side
using $\mathcal{Z}_{\text{tH}}$. The $\mathsf{S}$ transformation is given by

$\displaystyle\mathsf{S}\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$
$\displaystyle=\int{\mathcal{D}}a\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}a}\int{\mathcal{D}}b_{m}{\mathcal{D}}a^{\prime}\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}(B_{m}a^{\prime}+b_{m}a)}\mathcal{Z}\biggl{[}b_{m}+a^{\prime}\wedge\frac{\mathrm{d}\tau}{L}\biggr{]}$
$\displaystyle=\int{\mathcal{D}}b_{m}{\mathcal{D}}a^{\prime}\,N^{b_{1}(M_{3})-1}\,\delta_{N}[\widetilde{B}_{e}+b_{m}]\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}B_{m}a^{\prime}}\mathcal{Z}\biggl{[}b_{m}+a^{\prime}\wedge\frac{\mathrm{d}\tau}{L}\biggr{]}$
$\displaystyle=\mathcal{Z}_{\text{tH}}[B_{m},-\widetilde{B}_{e}].$ (43)

Here, we decompose the dynamical $4$d $\mathbb{Z}_{N}$ gauge field as
$b=b_{m}+a^{\prime}\wedge\frac{\mathrm{d}\tau}{L}$. The $\mathsf{T}$
transformation is given by

$\displaystyle\mathsf{T}\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$
$\displaystyle=\int{\mathcal{D}}a\,\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}\widetilde{B}_{e}a}\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}\int_{M_{3}}B_{m}a}\mathcal{Z}\biggl{[}B_{m}+a\wedge\frac{\mathrm{d}\tau}{L}\biggr{]}$
$\displaystyle=\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e}+B_{m},B_{m}].$ (44)

Let us choose the generators $S,T\in SL(2,\mathbb{Z})$ to be

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\quad T=\begin{pmatrix}1&-1\\\
0&1\end{pmatrix},$ (45)

so that $SL(2,\mathbb{Z})=\langle S,T\,|\,S^{2}=(ST^{-1})^{3},S^{4}=1\rangle$
and $C\equiv S^{2}=(ST^{-1})^{3}$ corresponds to charge conjugation. If we
write $\vec{B}=(\widetilde{B}_{e},B_{m})^{t}$ as a column vector, then the
transformations (43) and (44) can be expressed as

$\displaystyle\mathsf{S}\mathcal{Z}_{\text{tH}}\Bigl{[}\vec{B}\Bigr{]}=\mathcal{Z}_{\text{tH}}\Bigl{[}S^{-1}\vec{B}\Bigr{]},\quad\mathsf{T}\mathcal{Z}_{\text{tH}}\Bigl{[}\vec{B}\Bigr{]}=\mathcal{Z}_{\text{tH}}\Bigl{[}T^{-1}\vec{B}\Bigr{]}.$
(46)

Thus, $\mathsf{S}$ and $\mathsf{T}$ transformations generate the
$SL(2,\mathbb{Z})$ action on the space of $4$d QFTs with the
$\mathbb{Z}_{N}^{[1]}$ symmetry. The explicit form of the action can be easily
identified when we use the ’t Hooft partition function
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]$, as we just need to perform
an $SL(2,\mathbb{Z})$ transformation on the background field
$\vec{B}=(\widetilde{B}_{e},B_{m})^{t}$. For example, we find

$(\mathsf{S}^{p_{1}}\mathsf{T}^{q_{1}}\mathsf{S}^{p_{2}}\mathsf{T}^{q_{2}}\cdots)\mathcal{Z}_{\text{tH}}[\vec{B}]=\mathcal{Z}_{\text{tH}}[(S^{p_{1}}T^{q_{2}}S^{p_{2}}T^{q_{2}}\cdots)^{-1}\vec{B}].$
(47)

As we can express

$(S^{p_{1}}T^{q_{2}}S^{p_{2}}T^{q_{2}}\cdots)=\begin{pmatrix}p&q\\\
r&s\end{pmatrix}\in SL(2,\mathbb{Z}),$ (48)

with some $p,q,r,s\in\mathbb{Z}$ with $ps-qr=1$, the above relation becomes

$(\mathsf{S}^{p_{1}}\mathsf{T}^{q_{1}}\mathsf{S}^{p_{2}}\mathsf{T}^{q_{2}}\cdots)\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\mathcal{Z}_{\text{tH}}[\,s\widetilde{B}_{e}-qB_{m},-r\widetilde{B}_{e}+pB_{m}].$
(49)

Let us now consider two different gapped phases specified by order-$N$
subgroups $H_{1}$ and $H_{2}$, and assume that there is an isomorphism
$H_{1}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}H_{2}$
induced by an automorphism of
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$. Then, there should be an
$SL(2,\mathbb{Z})$ operation that relates $H_{1}$ and $H_{2}$, and thus these
different phases are connected in the web of $\mathsf{S},\mathsf{T}$
operations.

###  4.2 Non-Abelian gapped phases in the $\mathcal{N}=1^{*}$ SYM theory

${\cal N}=1^{*}$ SYM theory is defined as the mass deformation of the $4$d
${\cal N}=4$ SYM theory. It is one of the most interesting theoretical
playgrounds for 4d theories as its vacuum structure is very rich Donagi:1995cf
; Dorey:1999sj ; Polchinski:2000uf . Our goal below is to express the ’t Hooft
partition functions for the massive vacua of ${\cal N}=1^{*}$ theory.

First, we provide a quick overview of ${\cal N}=1^{*}$ theory with $SU(N)$
gauge group starting with its ${\cal N}=4$ origin. In the ${\cal N}=1$
notation, the field content of ${\cal N}=4$ SYM theory is a vector multiplet
$V$ and three chiral multiplets $\Phi_{i}$ $(i=1,2,3)$ all in the adjoint
representation of $SU(N)$. The action of the theory consists of gauge
invariant kinetic terms of these fields plus a unique superpotential,

$\displaystyle
W=\frac{1}{g^{2}}\operatorname{tr}(\Phi_{1}[\Phi_{2},\Phi_{3}]).$ (50)

The ${\cal N}=1^{*}$ theory is obtained from ${\cal N}=4$ by adding a mass
term:

$\displaystyle\Delta
W=\frac{m}{2g^{2}}\left(\operatorname{tr}\Phi_{1}^{2}+\operatorname{tr}\Phi_{2}^{2}+\operatorname{tr}\Phi_{3}^{2}\right).$
(51)

In the $m\rightarrow\infty$ limit with arbitrarily small coupling constant at
the cut-off $m$, $i\tau(m)\rightarrow\infty$, with
$\Lambda^{3}=m^{3}\mathrm{e}^{\mathrm{i}2\pi\tau(m)/N}$ fixed, the theory
reduces to pure ${\cal N}=1$ theory, which is believed to be a confining gauge
theory. At finite $m$, the theory has an extremely rich classical and quantum
vacuum structure. Since we take the three masses non-zero, the moduli space of
the ${\cal N}=4$ theory is completely lifted and the theory has isolated
vacua. The classical vacua are determined by the solutions of the $F$-term
equations, $\partial W/\partial\Phi_{i}=0$, given by

$\displaystyle[\Phi_{i},\Phi_{j}]=-m\varepsilon_{ijk}\Phi_{k}.$ (52)

Therefore, the supersymmetric classical vacua can be expressed by three
$N\times N$ matrices which obey the standard commutation relations for the
$\mathfrak{su}(2)$ algebra. Up to gauge transformations, the classical vacua
are described as

$\Phi^{\prime}_{i}\equiv\frac{1}{\mathrm{i}m}\Phi_{i}=\underbrace{J^{(d_{1})}_{i}\oplus\cdots\oplus
J^{(d_{1})}_{i}}_{k_{d_{1}}}\oplus\underbrace{J^{(d_{2})}_{i}\oplus\cdots\oplus
J^{(d_{2})}_{i}}_{k_{d_{2}}}\oplus\cdots,$ (53)

where $J^{(d)}_{i}$ are the generators of $d$-dimensional irreducible
representation of $\mathfrak{su}(2)$, $k_{d}$ denotes its multiplicity, and
$N=d_{1}k_{d_{1}}+d_{2}k_{d_{2}}+\cdots$. Ignoring discrete factors
momentarily, the gauge structure at these vacua is reduced to
$[\otimes_{d}U(k_{d})]/U(1)$ at the classical level. If the classical vacuum
contains different $\mathfrak{su}(2)$ representations, i.e. $N\neq dk_{d}$,
then there will be unbroken $U(1)$’s in the infrared and it becomes a gapless
Coulomb vacuum.

Our primary interest in this paper is the ’t Hooft partition functions of
massive vacua, and they arise when the Higgs expectation values take the
following form:

$\displaystyle\Phi^{\prime}_{i}=\mathbf{1}_{N/d}\otimes
J_{i}^{(d)},\qquad(i=1,2,3),$ (54)

where $d$ is a divisor of $N$. As a result, the $SU(N)$ gauge group is Higgsed
as

$SU(N)\xrightarrow{\mathrm{Higgs}}\frac{SU(N/d)\times\mathbb{Z}_{N}}{\mathbb{Z}_{N/d}},$
(55)

where $SU(N/d)$ is the remaining continuous gauge group acting on the
$\bm{1}_{N/d}$ component, $\mathbb{Z}_{N}$ in the numerator is the center
subgroup of $SU(N)$, and they need to be divided by the common center
$\mathbb{Z}_{N/d}$. As the classically massless contents describe the
$\mathcal{N}=1$ $SU(N/d)$ SYM theory with emergent $\mathbb{Z}_{2(N/d)}$
discrete chiral symmetry, they are expected to be gapped by quantum
fluctuations with $N/d$ distinct vacua. In addition to these confinement
dynamics, the remnant discrete
$\mathbb{Z}_{N}/\mathbb{Z}_{N/d}\simeq\mathbb{Z}_{d}$ gauge field describes
the topological field theory for deconfined Wilson lines. As we have this
dynamics for each divisor of $N$, the total number of the massive vacua is
given by the divisor function,

$\displaystyle\sigma(N)=\sum_{d|N}d=\sum_{d|N}\left(\frac{N}{d}\right).$ (56)

As pointed out in Ref. Donagi:1995cf , every gapped state in the Wilson–’t
Hooft classification can be realized as one of these vacua in the
$\mathcal{N}=1^{*}$ SYM theory, and we will find their ’t Hooft partition
functions and study how these vacua behave under $\mathsf{S}$ and $\mathsf{T}$
transformations.999A recent paper Damia:2023ses  also studies the global
properties of the $\mathcal{N}=1^{*}$ gapped vacua using the analogous $S$ and
$T$ operations.

Depending on whether the prime factorization of $N$ is square free, the
$SL(2,\mathbb{Z})$ orbit of the massive vacua shows different features. In the
following, let us work on concrete examples, $N=6$ and $N=4$, as
demonstrations.

#### Massive vacua of $\mathcal{N}=1^{*}$ $SU(6)$ SYM theory (the case with
$N$ square-free)

Let us consider $SU(6)$ gauge theory as a pedagogical example. Positive
divisors of $N=6$ are $d=1,2,3$, and $6$, and we get the following table for
the Higgs expectation values to have massive vacua:

(62)

Let us discuss each of these vacuum structures to obtain the ’t Hooft
partition functions and their relation under $SL(2,\mathbb{Z})$.

Let us first discuss the Higgs phase, where $\Phi^{\prime}_{i}=J_{i}^{(6)}$
and the gauge group is Higgsed to the center $\mathbb{Z}_{6}$. We note that
this classical vacuum cannot be consistent with the nonzero magnetic flux
$B_{4\mathrm{d}}\not=0$, and a vortex must be created to satisfy the boundary
condition. Thus, the partition function with nonzero flux becomes
exponentially small,

$\mathcal{Z}[B_{4\mathrm{d}}]=\delta_{6}[B_{4\mathrm{d}}],$ (63)

and the corresponding ’t Hooft partition function is given by

${\cal Z}_{\rm tH}[\widetilde{B}_{e},B_{m}]=\delta_{6}[B_{m}].$ (64)

Indeed, in this Higgs phase, the electrically charged particles associated
with the triplet of adjoint scalars condense, and we naturally expect the
confinement of magnetic charges.

In the totally confined phase, the Higgs expectation values are zero,
$\Phi^{\prime}_{i}=\bm{1}\otimes J^{(1)}_{i}=0$, and the
$\mathbb{Z}_{6}^{[1]}$ symmetry is unbroken. At the classical level, gapless
gluons and gluinos are associated with the $SU(6)$ gauge group with
$\mathcal{N}=1$ supersymmetry, and quantum fluctuations should generate
confinement. In the $\mathcal{N}=1^{*}$ theory, the counterpart of discrete
chiral symmetry of $\mathcal{N}=1$ SYM does not exist as the potentially
anomalous $U(1)$ symmetry is broken at the classical level by the
superpotential (50), but the low-energy effective theory for the confinement
phase acquires an emergent $\mathbb{Z}_{2N}$ axial symmetry. As computed in
(37), the ’t Hooft partition functions for these gapped chiral-broken vacua
are given by

${\cal Z}_{\rm
tH}[\widetilde{B}_{e},B_{m}]=\delta_{6}[\widetilde{B}_{e}+kB_{m}],$ (65)

where $k=1,\ldots,N$ specifies the phase of the gluino condensate and they are
cyclically permuted by the $\mathsf{T}$ operation. Another important remark is
that the $k=0$ confining state is dual to the Higgs state by the $\mathsf{S}$
operation as expected from electromagnetic duality.

Figure 1:  ’t Hooft partition functions for massive vacua of $SU(6)$
$\mathcal{N}=1^{*}$ SYM theory, and their $SL(2,{\mathbb{Z}})$
transformations. There are $\sigma(6)=1+2+3+6=12$ supersymmetric gapped vacua
and these phases are in 1:1 correspondence with the symmetry breaking pattern
$(\mathbb{Z}_{N}^{[1]})_{4\mathrm{d}}\rightarrow(\mathbb{Z}_{n}^{[1]})_{4\mathrm{d}}$
enriched with the $\mathbb{Z}_{n}^{[1]}$ level-$k$ SPT state. Since $N=6$ is a
square-free integer, all massive phases are connected by $\mathsf{S}$ and
$\mathsf{T}$ transformations.

The analyses of the remaining two cases are similar, so let us focus on the
case of $\Phi^{\prime}_{i}=\bm{1}_{2}\otimes J^{(3)}_{i}$, where the gauge
group becomes

$SU(6)\xrightarrow{\mathrm{Higgs}}SU(2)\times\mathbb{Z}_{3}.$ (66)

The Higgs expectation value $\Phi^{\prime}_{i}=\bm{1}_{2}\otimes J^{(3)}_{i}$
is compatible with the magnetic fluxes
$\int_{\Sigma}\frac{2\pi}{6}B_{4\mathrm{d}}\in\pi\mathbb{Z}$, so the partition
function should contain the factor $\delta_{6}[2B_{4\mathrm{d}}]$. The
possible ’t Hooft partition functions for gapped phases are then given by

$\delta_{6}[2B_{m},3\widetilde{B}_{e}+kB_{m}],$ (67)

with $k\sim k+2$. We note that these two states are related by the
$\mathsf{T}$ operation since

$\mathsf{T}\colon\delta_{6}[2B_{m},3\widetilde{B}_{e}]\longleftrightarrow\delta_{6}[2B_{m},3\widetilde{B}_{e}+3B_{m}]=\delta_{6}[2B_{m},3\widetilde{B}_{e}+B_{m}].$
(68)

Let us discuss why this is the case. To proceed, we need to study the
confinement dynamics of the effective $\mathcal{N}=1$ $SU(2)$ SYM theory, and
we should note that the $\theta$ angle for the $SU(2)$ gauge group becomes
$\theta_{\mathrm{eff}}=3\theta$ in the Higgsing (66). The $\mathsf{T}$
operation, $\theta\to\theta+2\pi$, acts as
$\theta_{\mathrm{eff}}\to\theta_{\mathrm{eff}}+6\pi=\theta_{\mathrm{eff}}+2\pi\bmod
4\pi$ and the chiral broken vacua for the $SU(2)$ theory are actually related
by the $\mathsf{T}$ transformation.

The complete list of the ’t Hooft partition functions for gapped phases are
listed in Fig. 1, and we also show how the $SL(2,\mathbb{Z})$ operations
relate those phases.101010This figure has the identical structure with the
$S,T$-duality orbit for $\mathcal{N}=4$ SYM of the gauge Lie algebra
$\mathfrak{g}=\mathfrak{su}(6)$ in Ref. Aharony:2013hda , and some readers may
be confused about the difference between them. The key difference is that Ref.
Aharony:2013hda  discusses the global structure of the gauge group itself,
which is about the kinematics not the dynamics, while we here discuss the
vacuum structure of the $SU(6)$ gauge theory. In particular, in Ref.
Aharony:2013hda , one constructs genuine line operators in
$(SU(N)/\mathbb{Z}_{n})_{k}$ theory, while in our case, we are depicting the
dyonic charges that are screened in the $SU(N)$ theory.  All the gapped states
are exchanged by $\mathsf{S}$ and $\mathsf{T}$ operations. This is a general
fact when $N$ is a square-free integer, and we demonstrate it here for $N=6$
as an example. In this figure, we use the fact that the ’t Hooft partition
function for this case can be written using a single delta functional. For
example, (67) contains two different delta functions, but it can be
equivalently rewritten as

$\delta_{6}[2B_{m},3\widetilde{B}_{e}+kB_{m}]=\delta_{6}[3\widetilde{B}_{e}+kB_{m}]$
(69)

for $k=1,2$ (Since $k\sim k+2$, we can always choose this way). This is
related to the fact that all the order $N$ subgroups of
$\widetilde{\mathbb{Z}}_{N}\times\mathbb{Z}_{N}$ is isomorphic to
$\mathbb{Z}_{N}$ when $N$ is square free.111111Let us prove this fact using
the ’t Hooft partition function (in a physically intuitive way). We start from
the ’t Hooft partition function,
$\delta_{N}[nB_{m},\frac{N}{n}\widetilde{B}_{e}+kB_{m}]$, and we assume that
$\gcd(n,N/n)=1$, which is always the case if $N$ is square free. The
$\mathsf{T}$ transformation shifts $k\to k+(N/n)$, and this surveys all
possible $k\sim k+n$ due to $\gcd(n,N/n)=1$. In particular, we can reach the
vacuum,
$\delta_{N}[nB_{m},\frac{N}{n}\widetilde{B}_{e}+B_{m}]=\delta_{N}[\frac{N}{n}\widetilde{B}_{e}+B_{m}]$,
and its $\mathsf{S}$ transformation gives one of the totally confined phase,
$\delta_{N}[\widetilde{B}_{e}-\frac{N}{n}B_{m}]$.

#### Massive vacua of $\mathcal{N}=1^{*}$ $SU(4)$ SYM theory (the case with
$N$ not square-free)

When $N$ contains squares in its prime factorization, the $SL(2,\mathbb{Z})$
structure of massive vacua has disconnected components. Let us discuss $N=4$
as the simplest example. The massive vacua are listed in the following table:

(74)

The analysis of the Higgs and totally confining phases is completely the same
as we have done for $N=6$, so let us here focus on the case
$\Phi^{\prime}_{i}=\bm{1}_{2}\otimes J_{i}^{(2)}$.

Figure 2:  Phases and ’t Hooft partition functions for the ${\cal N}=1^{*}$
SYM theory with $SU(4)$ gauge group. The marked points on the lattice indicate
the electromagnetic charges of deconfined dyonic lines in each phase. As $N=4$
is not square free, the web of $\mathsf{S}$ and $\mathsf{T}$ transformations
is disconnected.

The Higgs expectation value $\Phi^{\prime}_{i}=\bm{1}_{2}\otimes J_{i}^{(2)}$
causes the Higgsing of the gauge group,

$SU(4)\xrightarrow{\mathrm{Higgs}}\frac{SU(2)\times\mathbb{Z}_{4}}{\mathbb{Z}_{2}},$
(75)

and it breaks 1-form symmetry as $(\mathbb{Z}_{4}^{[1]})_{\rm
4d}\rightarrow(\mathbb{Z}_{2}^{[1]})_{\rm 4d}$. The $SU(2)$ gauge group is
confined in the infrared and it exhibits two vacua due to the spontaneous
breaking of emergent chiral symmetry. These two phases are distinguished by
the SPT actions of the unbroken $(\mathbb{Z}_{2}^{[1]})_{4\mathrm{d}}$
symmetry, and thus their $4$d partition functions are given by

$\mathcal{Z}[B_{4\mathrm{d}}]=\delta_{4}[2B_{4\mathrm{d}}]\exp\biggl{(}\frac{2\pi\mathrm{i}k}{2}\int\frac{1}{2}P_{2}(B_{4\mathrm{d}}/2)\biggr{)},$
(76)

with $k\sim k+2$. The corresponding ’t Hooft partition functions are given by

$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\delta_{4}[2B_{m},2\widetilde{B}_{e}+kB_{m}].$
(77)

Importantly, these chiral broken vacua are not related by the $\mathsf{T}$
transformation, $\widetilde{B}_{e}\to\widetilde{B}_{e}+B_{m}$, and each
chiral-broken vacuum is invariant under $\mathsf{T}$. To understand their
$\mathsf{T}$ invariance from the microscopic viewpoint, we should notice that
the effective $\theta$ angle of the $SU(2)$ theory is given by
$\theta_{\mathrm{eff}}=2\theta$ using the $\theta$ angle of the $SU(4)$
theory. Under the $2\pi$ shift of $\theta$, the effective $\theta$ angle is
shifted as $\theta_{\mathrm{eff}}\to\theta_{\mathrm{eff}}+4\pi$, and each
chiral-broken vacua of $\mathcal{N}=1$ $SU(2)$ SYM theory is invariant under
this operation.

In Fig. 2, we give the complete list of the ’t Hooft partition functions for
gapped vacua and their relations under the $\mathsf{S}$ and $\mathsf{T}$
operations. We also show the set of deconfined lines of each gapped vacua.
Unlike the $SU(6)$ case, there are disconnected components in the duality web
of $\mathsf{S}$ and $\mathsf{T}$ operations for $SU(4)$ theory, and in
particular,
$\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]=\delta_{4}[2B_{m},2\widetilde{B}_{e}]$
is invariant under both $\mathsf{S}$ and $\mathsf{T}$ operations. The presence
of disconnected components is a general feature for cases with $N$ not square
free, and $SU(4)$ is an illustrative example.

##  5 Summary and discussion

In this paper, we introduce the temporal gauging for $4$d QFTs with
$\mathbb{Z}_{N}^{[1]}$ symmetry and define the ’t Hooft partition function.
This operation does not respect the $4$d Lorentz invariance, but it introduces
the spatial $\widetilde{\mathbb{Z}}_{N}^{[1]}\times\mathbb{Z}_{N}^{[1]}$
symmetry, and thus the spatial Wilson and ’t Hooft lines become genuine line
operators (while all the temporal line operators are no longer genuine). This
allows us to justify the classification of gapped phases using the Wilson–’t
Hooft criterion, and we establish its $1$-to-$1$ correspondence with the
spontaneous breakdown of $4$d $\mathbb{Z}_{N}^{[1]}$ symmetry enriched with
the SPT phase of the unbroken $1$-form symmetry.

In other words, our argument justifies the use of dyonic line operators to
classify the SPT phases for the vacua of $4$d gauge theories. This may be
reminiscent of the use of the string order parameter PhysRevB.40.4709  to
characterize the Haldane gap or AKLT state Haldane:1983ru ; Affleck:1987vf .
The Kennedy–Tasaki (KT) transformation kennedy1992hidden ; Kennedy:1992ifl ;
oshikawa1992hidden  maps this nonlocal string order parameter to a local
correlation function, and the SPT nature of the AKLT state can be understood
as the spontaneous breaking of hidden (or dual)
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry (see Ref. Li:2023mmw  for its
field-theoretic description from a modern viewpoint). In this context, we may
say that the temporal gauging of $4$d $\mathbb{Z}_{N}^{[1]}$ symmetry gives a
suitable KT transformation for $4$d gauge theories to classify their gapped
phases solely by spontaneous symmetry breaking. Honestly, it is quite
astonishing that ’t Hooft had already introduced all the essential ingredients
in Refs. tHooft:1977nqb ; tHooft:1979rtg ; tHooft:1981bkw  before any of these
developments.

In this paper, we compute the ’t Hooft partition function for the simplest
$4$d topological states and make the connection with the Wilson–’t Hooft
criterion. As the general $4$d topological states are known to be described by
the $2$-group gauge theories Lan:2018vjb ; Johnson-Freyd:2020usu ;
Thorngren:2020aph , it would be an interesting future study to compute the ’t
Hooft partition functions of these topological states, which would give us
better understanding of $4$d gapped phases.

###### Acknowledgements.

The work of Y. T. was supported by Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant numbers, 22H01218, and by Center for Gravitational
Physics and Quantum Information (CGPQI) at Yukawa Institute for Theoretical
Physics. The work of M. Ü. was supported by U.S. Department of Energy, Office
of Science, Office of Nuclear Physics under Award Number DE-FG02-03ER41260.

##  Appendix A Duality equation on $T^{4}$ and classification of $4$d gapped
phases

In the main text, we have taken the point of view that the mutual locality
condition (13) is nothing but the statement that the spontaneous breakdown of
1-form symmetry be consistent with 4d Lorentz invariance. Here, we review ’t
Hooft’s original derivation of this condition tHooft:1979rtg , which is quite
insightful and also interesting for its elementary character.

We take the Euclidean spacetime manifold to be a flat four-torus with
$L_{\mu=1,\ldots,4}$ the circumferences of the various circles in
$T^{4}=(S^{1})^{4}$. With this topology, the background gauge fields $A$,
$B_{m}$, $\widetilde{B}_{e}$ are associated with triplets of mod $N$ integers
$\vec{n}=(n_{1},n_{2},n_{3})\equiv(n_{14},n_{24},n_{34})$,
$\vec{m}=(m_{1},m_{2},m_{3})\equiv(n_{23},n_{31},n_{12})$,
$\vec{e}=(e_{1},e_{2},e_{3})\equiv(e_{23},e_{31},e_{12})$ via

$\displaystyle B_{4\mathrm{d}}$
$\displaystyle=B_{m}+A\wedge\frac{\mathrm{d}\tau}{L_{4}}=\sum_{i<j}n_{ij}\frac{\mathrm{d}x_{i}\wedge\mathrm{d}x_{j}}{L_{i}L_{j}}+\sum_{i}n_{i4}\frac{\mathrm{d}x_{i}\wedge\mathrm{d}\tau}{L_{i}L_{4}},$
(78) $\displaystyle\widetilde{B}_{e}$
$\displaystyle=\sum_{i<j}e_{ij}\frac{\mathrm{d}x_{i}\wedge\mathrm{d}x_{j}}{L_{i}L_{j}}.$
(79)

Accordingly, we shall write the ordinary and ’t Hooft partition functions as

$\displaystyle\mathcal{Z}[B_{4\mathrm{d}}]\equiv\mathcal{Z}[\vec{n},\vec{m}],\quad\mathcal{Z}_{\text{tH}}[\widetilde{B}_{e},B_{m}]\equiv\mathcal{Z}_{\text{tH}}[\vec{e},\vec{m}],$
(80)

and the relation between them as121212When we use the definition (6), the
canonical normalization is given by
$\frac{1}{|H^{0}(M_{3};\mathbb{Z}_{N})|}=\frac{1}{N}$ instead of
$\frac{1}{N^{3}}$, and the inverse operation
$\int{\mathcal{D}}\widetilde{B}_{e}$ has the normalization factor
$\frac{|H^{0}(M_{3};\mathbb{Z}_{N})|}{|H^{1}(M_{3};\mathbb{Z}_{N})|}=\frac{1}{N^{\beta_{1}(M_{3})-1}}=\frac{1}{N^{2}}$.
Here, we follow the original normalization by ’t Hooft that assigns $1/N^{3}$
for the ’t Hooft partition function. The following argument basically works in
both normalizations, but we would like here to note that the normalization of
(18) becomes $1$ instead of $N^{\beta_{1}(M_{3})-1}$ in this convention.

$\displaystyle\mathcal{Z}_{\text{tH}}[\vec{e},\vec{m}]=\frac{1}{N^{3}}\sum_{\vec{n}}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}\vec{e}\cdot\vec{n}\biggr{)}\,\mathcal{Z}[\vec{n},\vec{m}].$
(81)

4d Lorentz invariance provides a important constraint on the ’t Hooft
partition function. In particular, let us consider the Euclidean Lorentz
transformation

$\Lambda=\begin{pmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\\\
0&0&1&0\end{pmatrix},$ (82)

then the magnetic flux is transformed as
$(n^{\prime}_{ij})=\Lambda^{t}(n_{ij})\Lambda$. This has the effect of
interchanging the pairs $(n_{1},n_{2})$ and $(m_{1},m_{2})$ as we have

$\begin{pmatrix}-m^{\prime}_{2}&n^{\prime}_{1}\\\
m^{\prime}_{1}&n^{\prime}_{2}\end{pmatrix}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\begin{pmatrix}-m_{2}&n_{1}\\\
m_{1}&n_{2}\end{pmatrix}\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$ (83)

while keeping $n_{3},m_{3}$ fixed. Using the notation
$\tilde{q}\equiv(q_{1},q_{2})$ for a three-vector
$\vec{q}=(q_{1},q_{2},q_{3})$, the covariance of the ordinary partition
function under the above Lorentz transformation reads (we also have
$L_{1}\leftrightarrow L_{2},L_{3}\leftrightarrow L_{4}$, but this is
suppressed in our notation)

$\displaystyle\mathcal{Z}[\tilde{n},n_{3};\tilde{m},m_{3}]=\mathcal{Z}[\tilde{m},n_{3};\tilde{n},m_{3}].$
(84)

After Fourier transformation, we then obtain

$\displaystyle\mathcal{Z}_{\text{tH}}[\tilde{e},e_{3};\tilde{m},m_{3}]=\frac{1}{N^{2}}\sum_{\tilde{e}^{\prime},\tilde{m}^{\prime}}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}(\tilde{e}\cdot\tilde{m}^{\prime}-\tilde{m}\cdot\tilde{e}^{\prime})\biggr{)}\,\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}],$
(85)

which is what ’t Hooft calls the “duality relation.”

To obtain the constraints on the gapped vacua, ’t Hooft makes a technical
assumption: If the vacuum is gapped, then the ratio
$\mathcal{Z}_{\text{tH}}[\vec{e},\vec{m}]/\mathcal{Z}_{\text{tH}}[\vec{0},\vec{0}]$
should approach either $0$ or $1$ as $L_{\mu=1,\ldots,4}\to\infty$. Although
this seems to be a nontrivial assumption, we have checked in (18) that it is
actually valid for the $\mathbb{Z}_{N/n}$ topological states with a
$\mathbb{Z}_{n}$ SPT phase. It would be nice if we could prove/disprove it for
general $4$d topological states with $\mathbb{Z}_{N}^{[1]}$.

Under the above assumption, a gapped phase is then characterized by the set of
all fluxes that are ‘light,’ i.e., the set of fluxes $(\vec{e},\vec{m})$ with
$\mathcal{Z}_{\text{tH}}[\vec{e},\vec{m}]/\mathcal{Z}_{\text{tH}}[\vec{0},\vec{0}]\to
1$. The possible sets of light fluxes are strongly constrained by the duality
relation (85). As shown in Ref. tHooft:1979rtg , any two light fluxes
$(\vec{e},\vec{m}),(\vec{e}\,^{\prime},\vec{m}^{\prime})$ with
$e_{3}=e_{3}{}^{\prime},m_{3},=m_{3}{}^{\prime}$ must satisfy

$\displaystyle\tilde{e}\cdot\tilde{m}^{\prime}-\tilde{m}\cdot\tilde{e}^{\prime}=0\mod
N,$ (86)

and there are either exactly $N^{2}$ or $0$ light fluxes out of the possible
$N^{4}$ fluxes with given $e_{3},m_{3}$.

[Proof] Let us fix the normalization
$\mathcal{Z}_{\text{tH}}[\vec{0},\vec{0}]=1$. Suppose
$(\tilde{e},e_{3};\tilde{m},m_{3})$ is a light flux. Let us first establish
that $(\tilde{0},e_{3};\tilde{0},m_{3})$ is also light. To see this, we note
that

$\displaystyle 1$
$\displaystyle=\mathcal{Z}_{\text{tH}}[\tilde{e},e_{3};\tilde{m},m_{3}]=\frac{1}{N^{2}}\sum_{\tilde{e}^{\prime},\tilde{m}^{\prime}}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}(\tilde{e}\cdot\tilde{m}^{\prime}-\tilde{m}\cdot\tilde{e}^{\prime})\biggr{)}\,\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}]$
$\displaystyle\leq\frac{1}{N^{2}}\sum_{\tilde{e}^{\prime},\tilde{m}^{\prime}}\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}]=\mathcal{Z}_{\text{tH}}[\tilde{0},e_{3};\tilde{0},m_{3}]\leq
1,$ (87)

where the second step uses the duality relation, the third step uses the
positivity of $\mathcal{Z}_{\text{tH}}$, and the fourth step uses the duality
relation again. So indeed
$\mathcal{Z}_{\text{tH}}[\tilde{0},e_{3};\tilde{0},m_{3}]=1$.

Replacing the final inequality with an equality in (87), we find

$\displaystyle\frac{1}{N^{2}}\sum_{\tilde{e}^{\prime},\tilde{m}^{\prime}}\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}]=\mathcal{Z}_{\text{tH}}[\tilde{0},e_{3};\tilde{0},m_{3}]=1.$
(88)

Clearly, this can hold only if precisely $N^{2}$ of the $N^{4}$ terms on the
left-hand side are equal to $1$, with all the remaining terms equal to $0$. In
other words, we have established that precisely $N^{2}$ among the $N^{4}$
fluxes $(\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3})$ are light.

Let us now look at the duality relation again,

$\displaystyle
1=\mathcal{Z}_{\text{tH}}[\tilde{e},e_{3};\tilde{m},m_{3}]=\frac{1}{N^{2}}\sum_{\tilde{e}^{\prime},\tilde{m}^{\prime}}\exp\biggl{(}\frac{2\pi\mathrm{i}}{N}(\tilde{e}\cdot\tilde{m}^{\prime}-\tilde{m}\cdot\tilde{e}^{\prime})\biggr{)}\,\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}].$

As exactly $N^{2}$ of the
$\mathcal{Z}_{\text{tH}}[\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3}]$
equal 1 while all the others vanish, this equality can be true iff
$\exp\bigl{(}\frac{2\pi\mathrm{i}}{N}(\tilde{e}\cdot\tilde{m}^{\prime}-\tilde{m}\cdot\tilde{e}^{\prime})\bigr{)}=1$
for each light flux $(\tilde{e}^{\prime},e_{3};\tilde{m}^{\prime},m_{3})$.
$\square$

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