Non-invertible symmetries along 4d RG flows

Jeremias Aguilera Damiaa, Riccardo Argurioa, Francesco Benini${}^{b,\,c,\,d}$,
Sergio Benvenutid, Christian Copetti${}^{b,\,d}$ and Luigi Tizzanoa

a Physique Théorique et Mathématique and International Solvay Institutes
Université Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium
b SISSA, Via Bonomea 265, 34136 Trieste, Italy
c ICTP, Strada Costiera 11, 34151 Trieste, Italy
d INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

###### Abstract

We explore novel examples of RG flows preserving a non-invertible duality
symmetry. Our main focus is on $\mathcal{N}=1$ quadratic superpotential
deformations of 4d $\mathcal{N}=4$ super-Yang-Mills theory with gauge algebra
$\mathfrak{su}(N)$. A theory that can be obtained in this way is the so-called
$\mathcal{N}=1^{*}$ SYM where all adjoint chiral multiplets have a mass. Such
IR theory exhibits a rich structure of vacua which we thoroughly examine. Our
analysis elucidates the physics of spontaneous breaking of self-duality
symmetry occurring in the degenerate gapped vacua. The construction can be
generalized, taking as UV starting point a theory of class $\mathcal{S}$, to
demonstrate how non-invertible duality symmetries exist in a variety of
$\mathcal{N}=1$ SCFTs. We finally apply this understanding to prove that the
conifold theory has a non-invertible duality symmetry.

###### Contents

  1. 1 Introduction
  2. 2 Duality-preserving flows in 4d $\boldsymbol{\mathcal{N}=4}$ SYM
    1. 2.1 Global variants and line operators
    2. 2.2 Non-invertible self-duality defects
    3. 2.3 Duality-preserving RG flows
  3. 3 Gapped flows: $\boldsymbol{\mathcal{N}=1^{*}}$
    1. 3.1 $\mathfrak{su}(2)$ gauge algebra
    2. 3.2 $\mathfrak{su}(N)$ gauge algebra
    3. 3.3 Duality symmetries on generic global variants
    4. 3.4 Examples: $\mathfrak{su}(3)$ and $\mathfrak{su}(4)$
    5. 3.5 Comments on Coulomb vacua
    6. 3.6 Cubic anomalies and invariant vacua
  4. 4 Gapless flows: $\boldsymbol{\mathcal{N}=2}$ orbifolds and the conifold
    1. 4.1 Dualities from class $\mathcal{S}$
    2. 4.2 Non-invertible duality defects on the conifold
    3. 4.3 Holographic aspects
    4. 4.4 $\mathcal{N}=1$ non-invertible symmetry from Seiberg duality
  5. A TQFTs for $\boldsymbol{\mathcal{N}=1}$ SYM
  6. B Local order parameters for $\boldsymbol{\mathcal{N}=1^{*}}$
  7. C TQFTs and topological manipulations

##  1 Introduction

Global symmetries in quantum field theory (QFT) are an indispensable tool,
especially at strong coupling. As such, symmetry considerations oftentimes
have inspired far-reaching insights into non-perturbative physics.

The modern understanding of $p$-form global symmetries, pioneered in [1],
begins with the notion of topological operators $\mathfrak{D}_{a}(\gamma)$
supported on codimension $p+1$ submanifolds that act on $p$-dimensional
operators. From this topological perspective, it is natural to consider
symmetries that do not obey the standard multiplication rules of a group, but
that instead exhibit a more general categorical fusion algebra:

$\mathfrak{D}_{a}\times\mathfrak{D}_{b}=\sum\nolimits_{c}\mathcal{N}_{ab}^{c}\,\mathfrak{D}_{c}\;.$
(1.1)

The expression on the right-hand-side offers two important points of departure
from the standard discussion of symmetry groups. First, it allows for the
presence of multiple operators $\mathfrak{D}_{c}$. Second, the coefficients
$\mathcal{N}^{c}_{ab}$ are topological quantum field theories (TQFTs) instead
of ordinary $c$-numbers. This type of symmetries are commonly referred to as
_non-invertible_. Their existence has been appreciated for a long time in the
context of both $d=2$ conformal field theories (CFTs) [2, 3] and $d=3$ TQFTs
[4, 5], where many new insights have appeared recently [6, 7, 8, 9, 10, 11,
12, 13]. The generalization of these concepts to $d\geq 3$ QFTs has been the
subject of intense research over the past few years, see for instance [14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

The starting point of this work is the recent discovery that $\mathcal{N}=4$
super-Yang-Mills (SYM) theory exhibits a particular class of non-invertible
symmetries implemented by the so called _self-duality_ defects [14, 19, 15,
20]. The basic idea behind the construction of these defects is as follows. An
$\mathcal{N}=4$ SYM theory $\mathsf{T}_{\rho}(\tau_{\mathrm{YM}})$ with gauge
algebra $\mathfrak{g}$ is labelled by a choice $\rho$ of gauge group and by
its complexified gauge coupling $\tau_{\mathrm{YM}}$. The theory enjoys
Montonen-Olive $SL(2,\mathbb{Z})$ duality [35]. The $S$ generator identifies
theories with $\tau_{\mathrm{YM}}^{\prime}=-\frac{1}{\tau_{\mathrm{YM}}}$ and
Langlands dual gauge group $\rho^{\prime}=\rho^{L}$, while $T$ identifies
$\tau_{\mathrm{YM}}^{\prime}$ with $\tau_{\mathrm{YM}}+1$ and turns on a
dynamical discrete theta angle for $\rho$ [36]. The theory also admits
topological manipulations
$\varphi_{g}:\mathsf{T}_{\rho}(\tau_{\mathrm{YM}})\to\mathsf{T}_{\rho_{g}}(\tau_{\mathrm{YM}})$
which correspond to a generalized gauging of the 1-form symmetry. We will
discuss them in depth in Section 2. These act only on the global form $\rho$
and generically give rise to an interface between two inequivalent gauge
theories. For special values $\tau_{\mathrm{YM}}^{*}$ of $\tau_{\mathrm{YM}}$
it might happen that $\mathsf{T}_{\rho_{g}}(\tau_{\mathrm{YM}}^{*})$ is dual
to $\mathsf{T}_{\rho}(\tau_{\mathrm{YM}}^{*})$ by some element $g$ of
$SL(2,\mathbb{Z})$. Denoting the duality interface by
$I_{g}:\mathsf{T}_{\rho}(\tau_{\mathrm{YM}})\to\mathsf{T}_{\rho_{g}}(g\cdot\tau_{\mathrm{YM}})$
we see that $\tau_{\mathrm{YM}}^{*}$ needs to be invariant under a discrete
subgroup of $SL(2,\mathbb{Z})$. We can then define a defect
$\mathfrak{D}_{g}:\mathsf{T}_{\rho}(\tau_{\mathrm{YM}}^{*})\to\mathsf{T}_{\rho}(\tau_{\mathrm{YM}}^{*})$
by

$\mathfrak{D}_{g}=\varphi^{\dagger}\circ
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This defect is non-invertible since:

$\mathfrak{D}_{g}\times\overline{\mathfrak{D}}_{g}=\mathcal{C}\,,$ (1.3)

with $\mathcal{C}$ a 3d condensation defect of the 1-form symmetry [37]. At
$\tau_{\mathrm{YM}}^{*}=i$, $\mathfrak{D}$ implements a non-invertible
$S$-duality symmetry while at $\tau_{\mathrm{YM}}^{*}=e^{\frac{2\pi i}{3}}$ it
implements a non-invertible $ST$-triality symmetry.

The main goal in this paper is to explore how the non-invertible symmetry
implemented by $\mathfrak{D}$ behaves under RG flows triggered by mass
deformations. Our primary focus is on $\mathcal{N}=1$ preserving deformations:
decomposing the $\mathcal{N}=4$ vector multiplet into one vector and three
chiral $\mathcal{N}=1$ multiplets, we study what happens when some or all of
the chiral multiplets get a mass. Since the $SL(2,\mathbb{Z})$ modular group
has a non-trivial action on the supercharges of $\mathcal{N}=4$ SYM [38], the
only way to preserve the duality/triality symmetry is to combine
$\mathfrak{D}$ with a suitable $R$-symmetry rotation inside the maximal torus
of the $SU(4)_{R}$ $R$-symmetry group that is also explicitly broken by the
massive deformation. In practice this amounts to a discrete rotation inside
the superconformal $U(1)_{R}$. The fact that $\mathcal{N}=1$ mass deformations
can preserve a duality symmetry was already appreciated long ago [39]. A
similar idea was also exploited in the construction of 4d S-folds [40, 41,
42].

When all three chiral multiplets get a mass, one ends up with the so called
$\mathcal{N}=1^{*}$ SYM theory [43, 44, 45, 46]. The structure of vacua of
this theory is quite rich, and comprises both gapped and gapless vacua. In
this work we present a detailed description of how the non-invertible self-
duality defects are realized in these vacua. An important point is that all
gapped vacua are organized into orbits of the _spontaneously broken
duality/triality symmetry_. A spontaneously broken non-invertible symmetry can
relate ground states featuring different physical properties. More precisely,
we find that the domain walls associated to such a spontaneously broken
symmetry can interpolate between different realizations of the $1$-form
symmetry, e.g., between a Higgs and a confining phase. Similar examples of
this kind have already appeared in the literature [6, 7], remarkably in the
tricritical Ising model in two dimensions.111The tricritical Ising model
admits a relevant duality-preserving deformation by the $\epsilon^{\prime}$
operator. With a certain sign of the deformation the theory flows to a gapped
phase with three vacua. These should be interpreted as a direct sum
$\mathbb{Z}_{2}\oplus\text{trivial}$ representing the SSB of the duality
symmetry. We thank Yifan Wang for pointing this out to us. Our work provides
an explicit example in four dimensions.222See also [15] for a related analysis
in the context of pure $\mathcal{N}=1$ $SO(3)$ SYM.

We exhaustively classify the TQFTs describing the gapped vacua and establish a
bijective correspondence between gapped phases, global forms of the gauge
group, and TQFTs with $\mathbb{Z}_{N}$ 1-form symmetry. We provide detailed
examples of these features for theories based on the Lie algebra
$\mathfrak{su}(N)$ with $N=2$, 3, and 4. For gapless vacua in
$\mathfrak{su}(N)$ $\mathcal{N}=1^{*}$ SYM with $N\geq 3$, we describe how to
realize non-invertible symmetries in a Coulomb vacuum using a single $U(1)$
gauge field, while more general cases remain to be explored in future work. As
a final consistency check, we compute the anomalies associated to duality and
triality symmetries following [47, 48].333See also [49] for similar
computations. We conclude that, for all values of $N$ with a self-duality
symmetric ground state, the anomaly trivializes. This follows from a
nontrivial cancellation between the pure duality anomaly and the cubic anomaly
associated to the superconformal $U(1)_{R}$.

Our analysis can be readily adapted to other interesting classes of RG flows
that we explore. It is known that the self-duality defects are not limited to
$\mathcal{N}=4$ SYM and exist also in more general $\mathcal{N}=2$ SCFTs [50,
51, 52, 53]. As shown in [51, 52], for those theories it is more convenient to
consider the class $\mathcal{S}$ formulation. We generalize previous analysis
by considering theories whose UV curve is a torus with regular punctures
associated to fundamental matter. As discussed above, giving mass to the
adjoint scalars in the $\mathcal{N}=2$ vector multiplets allows one to define
$\mathcal{N}=1$ preserving RG flows which also preserve the non-invertible
self-duality symmetry defects. In this context, we discuss the RG flow whose
IR fixed point is the $\mathcal{N}=1$ conifold theory (also known as Klebanov-
Witten theory) [54]. Thanks to our construction, we can demonstrate that the
conifold theory inherits a self-duality symmetry defect. To strengthen this
perspective, we study the holographic realization of the model in type IIB
string theory and show how it matches the expectations from field theory. In
the special case of the conifold theory with
$\mathfrak{su}(2)\times\mathfrak{su}(2)$ gauge algebra, one can derive the
action of S-duality on the global forms from Seiberg duality of
$\mathfrak{so}(4)$ super-QCD. The resulting defect could be called a non-
invertible Seiberg-duality symmetry defect. Giving evidence of S-duality for
higher rank in purely $\mathcal{N}=1$ field theoretic terms is an interesting
problem. Our construction can be generalized to class $\mathcal{S}$ theories
without a Lagrangian description since many details of their symmetry
structure have already been studied [55, 56, 57]. This opens up the
possibility to discover non-invertible duality defects in a wide variety of
strongly coupled $\mathcal{N}=1$ SCFTs.

We point out that our strategy to construct novel examples of non-invertible
self-duality defects is in principle stable under the effect of mass
deformations that break supersymmetry completely. An interesting example is
the deformation by the bottom component of the $\mathcal{N}=4$ Konishi
supermultiplet. This point certainly deserves further study, in particular it
would be interesting to understand its relevance for the IR physics of
$N_{f}=4$ adjoint QCD.

The paper is organized as follows. In Section 2 we review the construction of
non-invertible self-duality defects in $\mathcal{N}=4$ SYM and their action on
global variants. We then introduce the duality action on massive deformations
and describe the construction of non-invertible symmetries preserved along the
RG flow. In Section 3 we discuss the realization of self-duality symmetries on
massive vacua of $\mathcal{N}=1^{*}$ SYM and formulate a simple criterion to
establish when these symmetries are spontaneously broken, including a
description of the physical consequences of such spontaneous symmetry breaking
(SSB) process. We then comment on some applications to gapless (Coulomb) vacua
and the cubic anomalies of self-duality symmetries. In Section 4 we study
generalizations of our construction in which the starting point is an
$\mathcal{N}=2$ SCFT. We give an explicit derivation of the self-duality
defects for the $\mathbb{Z}_{r}$ orbifolds of $\mathcal{N}=4$ SYM using its
class $\mathcal{S}$ description, and discuss their fate under massive RG
flows. Our main example is the flow to the $\mathcal{N}=1$ conifold theory. We
discuss the holographic description of such a duality and its relation to
$\mathcal{N}=1$ Seiberg duality. Technical details, mainly involving formal
manipulations of TQFTs with $\mathbb{Z}^{[1]}_{N}$ 1-form symmetry, are
relegated to the appendices.

##  2 Duality-preserving flows in 4d $\boldsymbol{\mathcal{N}=4}$ SYM

We begin with a review of several aspects of $\mathcal{N}=4$ theories which
are instrumental to our work. We start by reviewing the classifications of
global variants for a pure gauge theory based on the
$\mathfrak{g}=\mathfrak{su}(N)$ gauge algebra, following [36]. We then combine
this with Montonen-Olive duality to describe in detail the non-invertible
duality and triality defects of $\mathcal{N}=4$ SYM. Finally, we outline the
general strategy to introduce duality-preserving RG flows and give several key
examples.

###  2.1 Global variants and line operators

Consider a four-dimensional gauge theory based on the simple Lie algebra
$\mathfrak{g}$ and with connected gauge group $G$. We denote the universal
cover of $G$ by $\widetilde{G}$ and the center of the latter by $\Gamma$. Then
the gauge group is $G=\widetilde{G}/\Pi$ and its center is $\Gamma/\Pi$, where
$\Pi\subset\Gamma$ is a subgroup. Different choices of $\Pi$ define different
global variants of the gauge group based on the same Lie algebra. However, in
order to completely characterize the gauge theory, in general one should also
specify the set of allowed line operators [36]. Each line falls in a conjugacy
class specified by a pair
$(a_{\mathrm{e}},b_{\mathrm{m}})\in\widehat{\Gamma}\times\Gamma$, where
$\widehat{\Gamma}=\\{\text{homomorphisms}:\Gamma\to\mathbb{R}/\mathbb{Z}\\}$
is the Pontryagin dual to $\Gamma$ (and is isomorphic to it). The set
$\mathscr{L}$ of allowed conjugacy classes of lines is subject to a version of
Dirac quantization (ensuring that correlation functions of line operators are
local) which is expressed in terms of the natural pairing:

$0=\bigl{\langle}(a_{\mathrm{e}},b_{\mathrm{m}}),(c_{\mathrm{e}},d_{\mathrm{m}})\big{\rangle}=a_{\mathrm{e}}(d_{\mathrm{m}})-c_{\mathrm{e}}(b_{\mathrm{m}})\;.$
(2.1)

For example, if $\Gamma=\mathbb{Z}_{N}$ then
$(a_{\mathrm{e}},b_{\mathrm{m}})=(n,m)\mod N$ and the quantization condition
requires that for each pair of line operators $(n,m)$ and
$(n^{\prime},m^{\prime})$:

$nm^{\prime}-mn^{\prime}=0\mod N\;.$ (2.2)

The set of allowed classes should also be maximal. Satisfaction of the two
conditions corresponds to the choice of a Lagrangian subgroup $\mathscr{L}$ of
$\widehat{\Gamma}\times\Gamma$. The set of Wilson lines, namely of elements in
$\mathscr{L}$ of the form $(\hat{\gamma},0)$, is fixed by the global variant
$G$: $\hat{\gamma}\in\widehat{\Gamma}$ are such that
$\Pi\subset\ker(\hat{\gamma})$ (they form a subgroup isomorphic to the center
$\Gamma/\Pi$ of $G$). Even after imposing this condition, for a given global
variant there can be physically different choices of allowed line operators
(i.e., different choices of $\mathscr{L}$). From the Lagrangian point of view,
the different theories are distinguished by various types of theta terms [36].

In this section we will mostly focus on four-dimensional $\mathcal{N}=4$
supersymmetric444Supersymmetry will play a role only when discussing
dualities. Everything that pertains to the choice of global variant and line
operators applies to generic gauge theories. However one should remember that,
in the presence of matter fields, only those global variants such that the
matter representation is invariant under $\Pi$ are allowed. Here we will have
in mind the case that the matter representation is invariant under the full
$\Gamma$. Yang–Mills theory with Lie algebra $\mathfrak{su}(N)$. In this case
the maximal center is $\Gamma=\mathbb{Z}_{N}$ and the gauge group is
$SU(N)/\Pi$ with $\Pi=\mathbb{Z}_{k}$ and $k$ is a divisor of $N$. When $\Pi$
is trivial, the gauge group is $SU(N)$ and the set of allowed line operators
corresponds to the Lagrangian subgroup

$\mathscr{L}_{1,0}=\bigl{\\{}(a_{\mathrm{e}},b_{\mathrm{m}})=(n,0)\text{ mod
}N\bigm{|}n\in\mathbb{Z}\bigr{\\}}\;.$ (2.3)

These are the standard Wilson line operators in representations of $N$-ality
$n$, and in this paper we denote them by $D_{(n,0)}$. When
$\Pi=\mathbb{Z}_{N}$, the possible sets of allowed classes of line operators
(i.e., the possible Lagrangian subgroups) are given by

$\mathscr{L}_{N,\ell}=\bigl{\\{}(a_{\mathrm{e}},b_{\mathrm{m}})=(\ell
m,m)\text{ mod }N\bigm{|}m\in\mathbb{Z}\bigr{\\}}\;,$ (2.4)

where for each value of $\ell\in\mathbb{Z}_{N}$ we have a different possible
choice. Indeed there are distinct theories, that we dub
$\bigl{(}PSU(N)=SU(N)/\mathbb{Z}_{N}\bigr{)}{}_{\ell}$, whose line operators
have charges as in (2.4). We denote such line operators by $D_{(\ell m,m)}$.
Importantly, the $N$ distinct theories are related to each other by the Witten
effect [58]. A shift $\theta\to\theta+2\pi$ of the theta parameter induces a
shift
$(a_{\mathrm{e}},b_{\mathrm{m}})\to(a_{\mathrm{e}}+b_{\mathrm{m}},b_{\mathrm{m}})$
of the labels, implying the following identification between different
$PSU(N)_{\ell}$ theories:

$PSU(N)^{\theta+2\pi}_{\ell}=PSU(N)^{\theta}_{\ell+1}\;.$ (2.5)

In the general case, $\Pi=\mathbb{Z}_{k}$ with $k$ a divisor of $N$, and we
write $N=kk^{\prime}$. The possible Lagrangian subgroups are

$\mathscr{L}_{k,\ell}=\bigl{\\{}(a_{\mathrm{e}},b_{\mathrm{m}})=n\,(k,0)+m\,(\ell,k^{\prime})\text{
mod }N\bigm{|}n,m\in\mathbb{Z}\bigr{\\}}\;,$ (2.6)

with $\ell\in\mathbb{Z}_{k}$. They correspond to the possible classes of line
operators in the theories $\bigl{(}SU(N)/\mathbb{Z}_{k}\bigr{)}{}_{\ell}$. A
shift $\theta\to\theta+2\pi$ relates different theories according to:

$\bigl{(}SU(N)/\mathbb{Z}_{k}\bigr{)}^{\theta+2\pi}_{\ell}=\bigl{(}SU(N)/\mathbb{Z}_{k}\bigr{)}^{\theta}_{\ell+k^{\prime}}\;.$
(2.7)

Note that when $\gcd(k,k^{\prime})=p>1$, there exist $p$ sets of theories with
the same gauge group that are not related by shifts of $\theta$. For instance,
when $N=4$ the two theories $\bigl{(}SU(4)/\mathbb{Z}_{2}\bigr{)}{}_{0}$ and
$\bigl{(}SU(4)/\mathbb{Z}_{2}\bigr{)}{}_{1}$ are not related by a shift of
$\theta$. The total number of distinct theories (for fixed $\theta$) is given
by the sum of the divisors of $N$:

$\text{number of theories}=\sigma_{1}(N)=\sum_{d\,|N}\,d\;.$ (2.8)

This number will come up again in later discussions.

For 4d $\mathcal{N}=4$ SYM with gauge algebra $\mathfrak{su}(N)$, theories
with different global form are related by $SL(2,\mathbb{Z})$ S-duality. The
duality group is generated by the operations
$T=\big{(}\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\big{)}$ and
$S=\big{(}\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\big{)}$. They act by
fractional linear transformations on the complexified gauge coupling

$\tau_{\text{YM}}=\frac{\theta}{2\pi}+\frac{4\pi i}{g_{\mathrm{YM}}^{2}}$
(2.9)

as follows, $T:\tau_{\text{YM}}\mapsto\tau_{\text{YM}}+1$ and
$S:\tau_{\text{YM}}\mapsto-\frac{1}{\tau_{\text{YM}}}$. They also act on the
labels $\big{(}\begin{smallmatrix}a_{\mathrm{e}}\\\
b_{\mathrm{m}}\end{smallmatrix}\big{)}$ of lines as described by their matrix
form, and consequently on the Lagrangian subgroup $\mathscr{L}$ as [36,
59]555Note that $S:\mathscr{L}_{k,0}\mapsto\mathscr{L}_{k^{\prime},0}$ because
$p=k$ and then $0^{-1}=0$ in $\mathbb{Z}_{1}$.

$T:\mathscr{L}_{k,\ell}\,\mapsto\,\mathscr{L}_{k,\ell+k^{\prime}}\;,\qquad
S:\mathscr{L}_{k,\ell}\,\mapsto\,\mathscr{L}_{N/p,\;-k^{\prime}(\ell/p)^{-1}_{\text{mod
}k/p}}\quad\text{with }p=\gcd(k,\ell)\;.$ (2.10)

The group that acts faithfully on the set of global forms is
$PSL(2,\mathbb{Z}_{N})$.666The theories form orbits under
$PSL(2,\mathbb{Z}_{N})$, one orbit for each square divisor $d^{2}$ of $N$.
Each orbit contains a number of theories equal to
$\psi\bigl{(}N/d^{2}\bigr{)}$, where
$\psi(n)=n\prod_{p|n}\bigl{(}1+1/p\bigr{)}$ is the Dedekind psi function and
the product is over all primes $p$ that divide $n$. The orbit of
$\mathscr{L}_{k,\ell}$ is identified by $d=\gcd(k,k^{\prime},\ell)$. We will
come back to it in Section 2.2.

Another way to connect gauge theories with different global form, and that
applies to more general 4d gauge theories than $\mathcal{N}=4$ SYM because it
does not rely on duality, is the following [1]. We note that the gauge
theories are characterized by a 1-form symmetry. For simplicity we restrict to
gauge algebra $\mathfrak{su}(N)$ and to matter whose representation is
invariant under the maximal center $\Gamma$. Then the 1-form symmetry is
$\mathbb{Z}_{N}$ for the global variants $SU(N)$ and $PSU(N)$, while more
generally it is the Pontryagin dual to

$\mathscr{L}_{k,\ell}=\Bigl{(}\mathbb{Z}_{k^{\prime}}\times\mathbb{Z}_{N/\gcd(k^{\prime},\ell)}\Bigr{)}\text{\Large$/$}\mathbb{Z}_{k^{\prime}/\gcd(k^{\prime},\ell)}\,\cong\,\mathbb{Z}_{N/\gcd(k,k^{\prime},\ell)}\times\mathbb{Z}_{\gcd(k,k^{\prime},\ell)}$
(2.11)

and it has order $N$ in all cases. The first equality is apparent from (2.6)
where the quotient comes from the relation $k(\ell,k^{\prime})\sim(\ell k,0)$
mod $N$, while the second isomorphism is shown in [59].777For $\mathcal{N}=4$
SYM, as it should, the 1-form symmetry is invariant (up to isomorphisms) under
S-duality. We can then associate to each theory its partition function in the
presence of a background 2-form discrete gauge field $B$ for the 1-form
symmetry. Different global variants are related by partially or completely
gauging the 1-form symmetry. It also turns out that there is an anomaly in the
space of couplings [60] that involves the $\theta$ angle and the 1-form
symmetry. For instance, for gauge group $SU(N)$ we have that under
$\theta\to\theta+2\pi$ the partition function on $X$ changes as

$Z(B)\;\to\;\exp\biggl{(}\frac{2\pi
i(N-1)}{2N}\int_{X}\mathfrak{P}(B)\biggr{)}\;Z(B)\;,$ (2.12)

where $B\in H^{2}(X,\mathbb{Z}_{N})$,
$\mathfrak{P}\\!:\\!H^{2}(X,\mathbb{Z}_{N})\to
H^{4}\bigl{(}X,\mathbb{Z}_{N\gcd(N,2)}\bigr{)}$ is the Pontryagin square
operation, and $\mathfrak{P}(B)$ is an even integer class on spin manifolds,
which we will assume for the rest of the paper. Indeed, we can always add to
the theory the following local counterterm in the background field $B$ (that
we can think of as an invertible TQFT on $X$):

$\text{SPT}_{k}(B)=\exp\biggl{(}\frac{2\pi
ik}{2N}\int_{X}\mathfrak{P}(B)\biggr{)}\;.$ (2.13)

This produces a background-dependent phase.

We can thus refine the classification of global forms by keeping track of the
coupling to the background field, in terms of the partition functions $Z(B)$.
For simplicity888We either assume that $N$ does not have square divisors and
thus the 1-form symmetry is $\mathbb{Z}_{N}$ in all global forms of
$\mathfrak{su}(N)$, or we restrict to the orbit of $SU(N)$. When $N$ is not
prime, one can also partially gauge the 1-form symmetry, i.e., restrict the
sum in (2.14) to $b\in H^{2}(X,\mathbb{Z}_{k})$ for a divisor $k$ of $N$. We
will discuss some additional details of a non-prime $N$ in an example below.
we restrict to the case that the 1-form symmetry is $\mathbb{Z}_{N}$. We then
introduce two topological manipulations $\sigma$ and $\tau$ [1, 61, 62] that
transform the partition function as follows:

$\displaystyle\sigma$ $\displaystyle:\quad$ $\displaystyle[\sigma Z](B)$
$\displaystyle=\frac{1}{\sqrt{\bigl{\lvert}H^{2}(X,\mathbb{Z}_{N})\bigr{\rvert}}}\sum_{b\,\in
H^{2}(X,\mathbb{Z}_{N})}\exp\biggl{(}\frac{2\pi i}{N}\int_{X}b\cup
B\biggr{)}\;Z(b)\;,$ (2.14) $\displaystyle\tau$ $\displaystyle:\quad$
$\displaystyle[\tau Z](B)$ $\displaystyle=\exp\biggl{(}\frac{2\pi
i}{2N}\int_{X}\mathfrak{P}(B)\biggr{)}\;Z(B)\;.$

The operation $\sigma$ corresponds to gauging the 1-form symmetry, while
$\tau$ acts by adding the local counterterm $\text{SPT}_{1}(B)$. Together,
they generate the group $SL(2,\mathbb{Z}_{N})$.999Studying the operations
$\sigma,\tau$ on generic (not necessarily spin) oriented 4-manifolds, one
obtains a central extension of $SL(2,\mathbb{Z})$ where the extension is by
stacking with the invertible TQFTs $\bigl{[}\mathbb{Z}_{N}^{(2)}\bigr{]}_{k}$
[62, 52]. Here $\bigl{[}\mathbb{Z}_{N}^{(2)}\bigr{]}_{k}$ is the
$\mathbb{Z}_{N}$ 2-form Dijkgraaf-Witten theory with partition function
$Y_{k}=|H^{2}|^{-1/2}\sum_{b\in H^{2}}\exp\Bigl{(}\frac{2\pi
ik}{2N}\int_{X}\mathfrak{P}(b)\Bigr{)}$ for $k\in\mathbb{Z}_{N}^{*}$. For
instance, one finds that $(\sigma\tau)^{3}=Y_{1}$. On simply-connected spin
manifolds all $Y_{k}$ reduce to 1 [19], and the faithful action of
$\sigma,\tau$ reduces to $SL(2,\mathbb{Z}_{N})$. In any case, one can consider
the action of $\sigma,\tau$ on theories coupled to $B$ modulo stacking with
invertible TQFTs, which is $SL(2,\mathbb{Z}_{N})$.

The topological operations $\sigma$ and $\tau$ affect the spectrum of line
operators, and this fact can be used to understand how they act on global
variants. We use the notations $SU(N)_{p}$ and $PSU(N)_{k,p}$ where the extra
label $p$ indicates the counterterm (2.13). We also define the following
combined action:

$\Phi_{k}\equiv\sigma\,\tau^{k}\;.$ (2.15)

The $SU(N)_{0}$ theory has genuine Wilson line operators $D_{(n,0)}$ supported
on a curve $\gamma$ which we can always couple to the $\mathbb{Z}_{N}$
background field as follows:

$D_{(n,0)}(B)=D_{(n,0)}[\gamma]\;\exp\biggl{(}\frac{2\pi
in}{N}\int_{\Sigma}B\biggr{)}\;,\qquad\qquad\partial\Sigma=\gamma\;.$ (2.16)

This gives $D_{(n,0)}$ the correct charge under $\mathbb{Z}_{N}$ gauge
transformations. One can also consider non-genuine ’t Hooft operators
$D^{U}_{(0,m)}[\gamma]$ which need to live at the boundary of the surface
operators $U_{\mathrm{e}}[\Sigma]$ that generates the electric 1-form symmetry
[1], in other words, the lines $D^{U}_{(0,m)}$ are in twisted sectors. If we
act with $\Phi_{0}$ on $SU(N)_{0}$, the correlation functions of
$U_{\mathrm{e}}[\Sigma]$ become trivial and the roles of $D_{(n,0)}$ and
$D^{U}_{(0,m)}$ get exchanged, namely the ’t Hooft operators $D_{(0,m)}$
become genuine line operators while the Wilson lines $D^{U}_{(n,0)}$ become
non-genuine. To understand what is the effect of $\Phi_{k>0}$ on $SU(N)_{0}$
we first need to consider a $\mathbb{Z}_{N}$ background gauge transformation
$B\to B+\delta\lambda$. From (2.16) we have

$D_{(n,0)}(B)\,\to\,\exp\biggl{(}\frac{2\pi
in}{N}\int_{\gamma}\lambda\biggr{)}\,D_{(n,0)}(B)\;,$ (2.17)

expressing the $\mathbb{Z}_{N}$ charge. On the other hand, the insertion of an
’t Hooft line operator $D^{U}_{(0,m)}$ supported on $\gamma$ corresponds to a
variation of the background field by a $\Delta B$ such that $\delta(\Delta
B)=m\operatorname{PD}[\gamma]$, where $\operatorname{PD}[\gamma]$ is a
delta-3-form localized on $\gamma$ (its Poincaré dual). As a consequence, the
counterterm $\text{SPT}_{k}(B)$ ceases to be gauge invariant and instead:

$\text{SPT}_{k}(B)\,\to\,\exp\biggl{(}-\frac{2\pi
ikm}{N}\int_{X}\operatorname{PD}[\gamma]\cup\lambda\biggr{)}\,\text{SPT}_{k}(B)\;.$
(2.18)

This shows that acting with $\Phi_{k}$ on $SU(N)_{0}$ gives a global form in
which the line operators $D_{(km,m)}\equiv D_{(km,0)}D_{(0,m)}$ are gauge
invariant and thus genuine. This fact characterizes the global variant
$PSU(N)_{k}$ with Lagrangian subgroup $\mathscr{L}_{N,k}$.

As an example, the set of actions for $\mathfrak{su}(2)$ $\mathcal{N}=4$ SYM
is:

${SU(2)_{0}}$${PSU(2)_{0,0}}$${PSU(2)_{1,0}}$${SU(2)_{1}}$${PSU(2)_{0,1}}$${PSU(2)_{1,1}}$$\scriptstyle{\tau}$$\scriptstyle{\sigma}$$\scriptstyle{\tau}$$\scriptstyle{\tau}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma}$
(2.19)

###  2.2 Non-invertible self-duality defects

For $\mathcal{N}=4$ SYM, one can discuss the interplay between the topological
action described above and the action of $SL(2,\mathbb{Z})$ S-duality
discussed around (2.9). We will do that for Lie algebra $\mathfrak{su}(N)$. As
explained in [38] and reviewed above, the action of $SL(2,\mathbb{Z})$
generates duality orbits connecting theories with different global structure.
The action on lines is

$S:(n,m)\,\mapsto\,(-m,n)\;,\qquad\qquad T:(n,m)\,\mapsto\,(n+m,m)\;.$ (2.20)

For instance, the duality orbits for $N=2,3,4$ are the following.

$\mathfrak{su}(2)$:

${SU(2)}$${PSU(2)_{0}}$${PSU(2)_{1}}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{S}$
(2.21)

The theories $PSU(2)_{0}$ and $PSU(2)_{1}$ are usually denoted as $SO(3)^{+}$
and $SO(3)^{-}$.

$\mathfrak{su}(3)$:

${PSU(3)_{1}}$${SU(3)}$${PSU(3)_{0}}$${PSU(3)_{2}}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{T}$
(2.22)

$\mathfrak{su}(4)$:

$(SU(4)/\mathbb{Z}_{2})_{0}\quad\;\;$$\scriptstyle{S,T}$ (2.23)
${PSU(4)_{1}}$${SU(4)}$${PSU(4)_{0}}$${PSU(4)_{2}}$${{(SU(4)/\mathbb{Z}_{2})_{1}}}$${PSU(4)_{3}}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{T}$$\scriptstyle{S}$$\scriptstyle{T}$$\scriptstyle{T}$

As described in [20], one can refine the description by taking into account
the coupling to a $\mathbb{Z}_{N}$ background field $B$ and combine the
actions of $S,T$ with those of $\sigma,\tau$. For instance, in the case of
$\mathfrak{su}(2)$ $\mathcal{N}=4$ SYM one obtains:

${SU(2)_{0}}$${PSU(2)_{0,0}}$${PSU(2)_{1,0}}$${SU(2)_{1}}$${PSU(2)_{0,1}}$${PSU(2)_{1,1}}$$\scriptstyle{T,\tau}$$\scriptstyle{S,\sigma}$$\scriptstyle{\tau}$$\scriptstyle{T}$$\scriptstyle{S,\tau}$$\scriptstyle{\sigma}$$\scriptstyle{S}$$\scriptstyle{T,\sigma}$
(2.24)

Similar tables, of increasing complexity, can be drawn for groups of higher
rank (see [20]).

The topological and duality operations can be used to construct self-duality
defects. As an example, consider the theory $SU(2)_{0}$. From (2.24) we see
that

$[\sigma
S\,Z_{SU(2)_{0}}](\tau_{\mathrm{YM}},B)=Z_{SU(2)_{0}}\Bigl{(}-\tfrac{1}{\tau_{\mathrm{YM}}},B\Bigr{)}\;.$
(2.25)

In particular, for $\tau_{\mathrm{YM}}=i$ the operation $\sigma S$ maps the
theory to itself and therefore it becomes a symmetry, implemented by a
topological defect in the theory. The construction generalizes to any value of
$N$. Let us denote by $\mathsf{T}_{\rho}(\tau_{\text{YM}})$ the set of global
forms of $\mathcal{N}=4$ SYM with Lie algebra $\mathfrak{su}(N)$, where $\rho$
labels the global form. The family of theories is acted upon by duality
transformations implemented by topological interfaces
$I_{g}:\mathsf{T}_{\rho}(\tau_{\mathrm{YM}})\to\mathsf{T}_{\rho_{g}}(g\cdot\tau_{\mathrm{YM}})$,
where $g\in SL(2,\mathbb{Z})$ acts both on the global form and the coupling.
Meanwhile, the global form $\rho_{g}$ can also be reached by a sequence of
$\sigma$ and $\tau$ operations, implemented by another topological interface
$\varphi_{g}:\mathsf{T}_{\rho}(\tau_{\mathrm{YM}})\to\mathsf{T}_{\rho_{g}}(\tau_{\mathrm{YM}})$.
The action of the modular group on the conformal manifold of $\mathcal{N}=4$
SYM has fixed points at orbifold singularities on the fundamental domain
$\mathcal{F}=H^{+}/SL(2,\mathbb{Z})$, located at

$\tau^{*}_{\mathrm{YM}}=i,\;e^{2\pi i/3},\;i\infty\;.$ (2.26)

At these points the stabilizer group $H=\bigl{\\{}g\in
SL(2,\mathbb{Z})\bigm{|}g\cdot\tau^{*}_{\mathrm{YM}}=\tau^{*}_{\mathrm{YM}}\bigr{\\}}$
is101010We have also included the action of charge conjugation $C$ which acts
trivially on $\mathcal{F}$.

$H=\mathbb{Z}_{4},\;\mathbb{Z}_{6},\;\mathbb{Z}\;,$ (2.27)

and is generated by $S$, $ST$, $T$, respectively. At the special points $H$
becomes a symmetry and for each of its elements we can construct a defect
$\mathfrak{D}_{g}=\varphi^{\dagger}_{g}\circ
I_{g}:\mathsf{T}_{\rho}(\tau^{*}_{\mathrm{YM}})\mapsto\mathsf{T}_{\rho}(\tau^{*}_{\mathrm{YM}})$.
The defect $\mathfrak{D}_{g}$ satisfies the following relations [20]:

$\displaystyle\mathfrak{D}_{g}\times\mathfrak{D}_{h}$
$\displaystyle=\mathcal{N}_{g,h}\;\mathfrak{D}_{gh}$ (2.28)
$\displaystyle\mathfrak{D}_{g}\times\overline{\mathfrak{D}}_{g}$
$\displaystyle=\mathcal{C}\;,$

where $\mathcal{N}_{g,h}$ are some 3d TQFTs, and $\overline{\mathfrak{D}}_{g}$
is the orientation reversal of $\mathfrak{D}_{g}$. The topological operator
$\mathcal{C}$ is known as a 3d condensate, or higher gauging [37], of the
1-form symmetry on a 3d submanifold.

###  2.3 Duality-preserving RG flows

After reviewing the self-duality symmetries of $\mathcal{N}=4$ SYM, we address
the question of constructing relevant deformations that trigger an RG flow
while preserving those symmetries.

In the language of $\mathcal{N}=1$ supersymmetry, the field content of
$\mathcal{N}=4$ SYM is given by a vector multiplet
$V=(A_{\mu},\lambda_{\alpha})$ and three chiral multiplets
$\Phi_{i=1,2,3}=(\phi_{i},\psi_{i\alpha})$ transforming in the adjoint
representation of the gauge group. The theory has a superpotential

$W=\sqrt{2}\,\operatorname{Tr}\Phi_{1}[\Phi_{2},\Phi_{3}]\;.$ (2.29)

The R-symmetry group is $SU(4)_{R}$, although only $U(1)\times SU(3)$ is
manifest in the $\mathcal{N}=1$ description. The $SL(2,\mathbb{Z})$ duality
group also acts on the supercharges $Q^{A}_{\alpha}$, where $A$ is an index in
the fundamental representation of $SU(4)_{R}$ [38]. Given
$M=\big{(}\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\big{)}$ in
$SL(2,\mathbb{Z})$ that acts as
$\tau_{\mathrm{YM}}\mapsto\frac{a\tau_{\mathrm{YM}}+b}{c\tau_{\mathrm{YM}}+d}$
on the coupling, its action on the supercharges is

$M\cdot Q_{\alpha}^{A}=\biggl{(}\frac{\lvert
c\,\tau_{\mathrm{YM}}+d\rvert}{c\,\tau_{\mathrm{YM}}+d}\biggr{)}^{1/2}Q^{A}_{\alpha}\;.$
(2.30)

This action was chosen so as to commute with $SU(4)_{R}$. The square root
signifies that the group that acts on the supersymmetries is a double cover of
$SL(2,\mathbb{Z})$, centrally extended by $S^{4}=C^{2}=(-1)^{F}$ which changes
the sign of all fermions. The extension is called the metaplectic group
$Mp(2,\mathbb{Z})$. Notice that $(-1)^{F}$ is also a central element of
$SU(4)_{R}$. The one in (2.30) is a $U(1)$ action, and the one on
$\overline{Q}$ is its conjugate. Indeed, an operator $\mathcal{O}$ is said to
have charge $q$ under modular transformations if it transforms as

$M\cdot\mathcal{O}=\biggl{(}\frac{\lvert
c\,\tau_{\mathrm{YM}}+d\rvert}{c\,\tau_{\mathrm{YM}}+d}\biggr{)}^{q/2}\mathcal{O}\;,$
(2.31)

and thus $Q^{A}_{\alpha}$ has charge 1. For a $\mathbb{Z}_{k}$ subgroup of
$SL(2,\mathbb{Z})$ occurring at an invariant value of $\tau_{\mathrm{YM}}$,
its action on the supercharges is

$M\cdot Q_{\alpha}^{A}=e^{-\frac{i\pi}{k}}\,Q_{\alpha}^{A}\;.$ (2.32)

The action of the modular group on the field strength $F$ and its dual
$\widetilde{F}$ is the standard one described by the matrix representation:
$\big{(}\begin{smallmatrix}F\\\ \widetilde{F}\end{smallmatrix}\big{)}\mapsto
M\big{(}\begin{smallmatrix}F\\\ \widetilde{F}\end{smallmatrix}\big{)}$. This
implies that the operator
$\frac{1}{\sqrt{\operatorname{\mathbb{I}m}\tau}}(F-\tau\widetilde{F})$ has
charge 2.

For the choice we made of the modular action, the real scalars $\phi_{I}$
(with $I$ an index in the rank-2 antisymmetric representation of $SU(4)_{R}$)
are invariant, while the fermions have charge 1 as the $Q$’s.

A simple class of superpotential deformations that we will analyze in this
paper is:

$\delta W=\sum^{3}_{i=1}m_{i}\operatorname{Tr}\Phi^{2}_{i}\;.$ (2.33)

These superpotentials partially break supersymmetry. If all the masses
$m_{i}\neq 0$, the resulting theory only retains $\mathcal{N}=1$ supersymmetry
and it is typically referred to as the $\mathcal{N}=1^{*}$ theory. When
$m_{1}=m_{2}\neq 0$ and $m_{3}=0$, the theory has $\mathcal{N}=2$
supersymmetry. If instead $m_{1}=m_{2}=0$ but $m_{3}\neq 0$, the theory flows
to a $\mathcal{N}=1$ superconformal fixed point with a conformal manifold
[63].

We ask whether the superpotential deformations (2.33) are also invariant under
the action of the duality defects $\mathfrak{D}_{g}$ introduced above. We
notice that the superspace coordinates $\theta$ are also charged under
$S$-duality, in particular they transform in the opposite way with respect to
the supercharges (2.32), and therefore

$M\cdot d^{2}\theta=e^{-\frac{2\pi i}{k}}\,d^{2}\theta\;.$ (2.34)

It follows that the superpotential should transform as $M\cdot W=e^{\frac{2\pi
i}{k}}W$ in order to lead to an invariant Lagrangian, however the chiral
superfields $\Phi_{i}$ are neutral. To remedy this, we combine
$\mathfrak{D}_{g}$ with a suitable R-symmetry rotation $R_{\varphi}$ inside
the maximal torus of $SU(4)_{R}$ such that

$\mathfrak{D}_{\varphi}=R_{\varphi}\times\mathfrak{D}_{g}$ (2.35)

is a symmetry of the deformed theory. The complex scalar fields $\phi_{i}$,
the fermions $\psi_{i\alpha}$ and the gaugino $\lambda_{\alpha}$ have the
following charges under the Cartan subalgebra of $SU(4)_{R}$:

$\begin{array}[]{c|c|c|c|c|c|c|c}&\phi_{1}&\phi_{2}&\phi_{3}&\psi_{1}&\psi_{2}&\psi_{3}&\lambda\\\
\hline\cr U(1)_{1}&1&0&0&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\\\
U(1)_{2}&0&1&0&-\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\\\
U(1)_{3}&0&0&1&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
\end{array}$ (2.36)

Using these charge assignments, we can discuss what non-invertible duality
symmetries (2.35) are preserved by the superpotential deformations (2.33) and
along the resulting RG flows. For the class of examples we consider we can
always choose $R_{\varphi}$ to lie in the diagonal subgroup of the three
Cartans.

#### Example I

Let us first consider the case $m_{1}=m_{2}=0$ and $m_{3}\equiv m\neq 0$:

$\delta W=m\operatorname{Tr}\Phi^{2}_{3}\;.$ (2.37)

This example preserves $\mathcal{N}=1$ supersymmetry. Under an S-duality
transformation the integrated superpotential deformation
$\int\\!d^{2}\theta\,\delta W$ picks up a phase $e^{-\frac{2\pi i}{k}}$.
Looking at (2.36), we deduce its charges under the Cartan subgroup of
$SU(4)_{R}$ (with the above conventions subtended):

$\left[\int d^{2}\theta\delta W\right]_{R}=(-1,-1,1)\,,$ (2.38)

where $(p,q,r)$ are the charges under the Cartans $U(1)_{1},\,U(1)_{2}$ and
$U(1)_{3}$ respectively. The charge assignement follows from
$[d^{2}\theta]_{R}=(-1,-1,-1)$ and table (2.36).

We thus see that we can compensate the effect of the duality transformation by
a superconformal $U(1)_{R}$ rotation $R_{-\frac{2\pi}{k}}$, thus:

$\mathfrak{D}_{\frac{2\pi}{k}}=R_{-\frac{2\pi}{k}}\times\mathfrak{D}\;,$
(2.39)

is a symmetry of the deformed theory.

#### Example II

Let us now consider the case $m_{1}=m_{2}=m\neq 0$ and $m_{3}=0$

$\delta W=m(\operatorname{Tr}\Phi^{2}_{1}+\operatorname{Tr}\Phi^{2}_{2})\;,$
(2.40)

which is $\mathcal{N}=2$ preserving. Under the Cartan subgroup of $SU(4)_{R}$
we now have

$\left[\int d^{2}\theta\delta W\right]_{R}=(1,-1,-1)\oplus(-1,1,-1)\;.$ (2.41)

Since both terms transform homogeneously under the broken superconformal
$U(1)_{R}$ we can again find that

$\mathfrak{D}_{\frac{2\pi}{k}}=R_{-\frac{2\pi}{k}}\times\mathfrak{D}\;,$
(2.42)

is preserved under the RG flow.

#### Example III

In our last example we study the case $m_{1}=m_{2}=m_{3}=m\neq 0$

$\delta
W=m(\operatorname{Tr}\Phi^{2}_{1}+\operatorname{Tr}\Phi^{2}_{2}+\operatorname{Tr}\Phi^{2}_{3})\;.$
(2.43)

This is the most general RG flow of this class that we discuss here, leading
to the $\mathcal{N}=1^{*}$ theory. As before, $\delta W$ behave in the same
way under S-duality. Under the Cartan subgroup of $SU(4)_{R}$ we now have

$\left[\int d^{2}\theta\delta
W\right]_{R}=(1,-1,-1)\oplus(-1,1,-1)\oplus(-1,-1,1)\;.$ (2.44)

All terms transform homogeneously under the broken superconformal $U(1)_{R}$
and we find again that

$\mathfrak{D}_{\frac{2\pi}{k}}=R_{-\frac{2\pi}{k}}\times\mathfrak{D}\;,$
(2.45)

is a symmetry of the $\mathcal{N}=1^{*}$ theory.

#### Example IV.

Finally we can consider a non-supersymmetric deformation triggered by the
Konishi operator:111111The possible interest of such deformation was pointed
out, to the best of the authors’ knowledge, by J. Kaidi and K. Ohmori.

$\delta S=\int\mu\,\mathcal{O}_{\text{Konishi}}\,,$ (2.46)

which at weak coupling is just
$\mathcal{O}_{\text{Konishi}}=\sum_{i=1}^{3}\operatorname{Tr}(\phi_{i}\phi_{i}^{\dagger})$.
This deformation at $\tau_{\mathrm{YM}}=i$ preserves both the non-invertible
duality symmetry $\mathfrak{D}$, which satisfies
$\mathfrak{D}^{4}\sim(-)^{F}\,\mathcal{C}$, as well as the full $SU(4)_{R}$
R-symmetry. This flow is relevant for small enough values of $N\leq 4$ [64].
The flow is intrinsically strongly coupled and the theory in the IR cannot be
gapped due to the cubic ’t Hooft anomaly of $SU(4)_{R}$. The correct
description is either given by SSB of the chiral symmetry due to gaugino
condensation (possibly stacked with an SPT) or a strongly coupled CFT. If the
deformation happens to interpolate smoothly between weak and strong coupling
$\tau_{\mathrm{YM}}$ the CFT should be the same as the conjectured one for
pure $SU(N)$ gauge theory with $N_{f}=4$ adjoint Weyl fermions.

Let us conclude this section with a brief remark about the $k=4$ case, namely
the element $S\in SL(2,\mathbb{Z})$. Notice that, even if both $S$ and
$R_{-\frac{2\pi}{4}}$ are $\mathbb{Z}_{8}$ such that
$S^{4}=R_{\frac{2\pi}{4}}^{4}=(-)^{F}$, the preserved combined action
$\mathfrak{D}_{\frac{2\pi}{k}}$ defines only a $\mathbb{Z}_{4}$ symmetry. In
the IR, starting from $\tau_{YM}=i$, the preserved symmetry is a non-
invertible $\mathbb{Z}_{4}$ times fermion parity instead of a nontrivial
extension $\mathbb{Z}_{8}$. This will have consequences for the cancellation
of some cubic anomalies which we examine in Section 3.6.

##  3 Gapped flows: $\boldsymbol{\mathcal{N}=1^{*}}$

In this section we will focus on mass deformations of the form (2.43) leading
to the ${\cal N}=1^{*}$ SYM theory. Such deformations completely break the
$SU(4)_{R}$ symmetry of the UV theory but preserve both the non-invertible
duality and triality symmetries $\mathfrak{D}$ for $\tau_{\mathrm{YM}}=i$ or
$\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$, respectively. Recall that the
preserved defects are combined with a discrete $R_{-\frac{2\pi}{k}}$
R-symmetry rotation.

The $\mathcal{N}=1^{*}$ SYM theory has been intensively studied in the past
[43, 44, 45, 65, 46]. It has a discrete set of vacua, some of which are
completely gapped while others include Coulomb phases. Let us briefly review
them here. By a suitable rescaling of the chiral fields, the F-term equations
read:

$[\Phi_{i},\Phi_{j}]=\epsilon_{ijk}\;\Phi_{k}\,.$ (3.1)

Since the $\Phi_{i}$ are $N\times N$ traceless matrices, solutions to the
above equations are given in terms of $N$-dimensional reducible
representations of the Lie algebra $\mathfrak{su}(2)$. For any positive
integer $N$, the algebra $\mathfrak{su}(2)$ has a unique (up to isomorphisms)
irreducible representation of that dimension. Therefore each classical vacuum
corresponds to a partition of $N$ into positive integers. For a generic
partition $N=\sum_{d=1}^{N}n_{d}\,d$ with $k$ non-vanishing terms, the gauge
algebra is Higgsed to

$\bigoplus_{d}\mathfrak{su}(n_{d})\;\oplus(k-1)\ \mathfrak{u}(1)\,.$ (3.2)

The two extreme cases are $d=N$, $n_{d}=1$, where the gauge group is fully
Higgsed, and $d=1$, $n_{d}=N$ where the $\mathfrak{su}(N)$ gauge symmetry is
unbroken.

Quantum mechanically, the dynamics is that of a pure ${\cal N}=1$ gauge theory
with gauge group (3.2). The $\mathfrak{su}(n_{d})$ factor flows to a confining
phase with $n_{d}$ degenerate gapped vacua, while each Abelian factor hosts a
Coulomb phase. The latter are always present unless $k=1$, accordingly gapped
vacua are in correspondence with divisors $d$ of $N$. In this case, the gauge
group is Higgsed to $\mathfrak{su}(N/d)$ leading to $N/d$ confining vacua.
Then the total number of gapped vacua is

$\text{number of gapped vacua}=\sum_{d|N}\;N/d=\sum_{d|N}\;d\ .$ (3.3)

This is equal to the number $\sigma_{1}(N)$ of global variants of the gauge
group, that we defined in (2.8). We will see that there is indeed a natural
one-to-one correspondence between gapped vacua and global variants, as they
are both labeled by a choice of line operators.

According to ’t Hooft’s classification of gapped phases in gauge theories,
vacua are characterized by the condensation of dyons. For $N$ prime, there are
exactly $N+1$ gapped vacua because of (3.3). These consist of a perturbative
Higgs vacuum, where $D_{(1,0)}$ condenses, and $N$ confining vacua where
$D_{(s,1)}$ condense, with $s=0$ corresponding to confinement of purely
electric charges and $s=1,\ldots,N-1$ describing different patterns of oblique
confinement. We will denote these vacua by $H$ and $C^{(s)}$ respectively.
Using a modern terminology, whenever a genuine line operator condenses the
1-form symmetry is spontaneously broken and the corresponding phase hosts a
non-invertible TQFT. On the other hand, condensation of non-genuine line
operators leads to vacua described by an SPT.

When $N$ has non trivial divisors, the structure is more involved due to the
occurrence of combined Higgs/confining phases. More precisely, take a divisor
$d$ of $N$ and consider vacua in which the gauge symmetry is Higgsed to
$\mathfrak{su}(N/d)$. Then, quantum mechanically, there are $N/d$ confining
vacua characterized by patterns of condensation of $\mathfrak{su}(N/d)$ dyons.
These vacua will be denoted by $HC^{(s)}_{d,N/d}$, with $s=0,\ldots,N/d-1$.
They exhibit a partially broken one-form symmetry and host a non-trivial TQFT
on top of an SPT.

One of the main purposes of this work is to show that the vacua discussed
above, which are characterized by different physical properties, are
nevertheless related by non-invertible duality and triality symmetries. The
latter are hence generically spontaneously broken along the RG flow triggered
by the mass deformation. For the sake of clarity, we will first describe the
vacua of the $\mathcal{N}=1^{*}$ theories based on the Lie algebra
$\mathfrak{su}(2)$. Then we will present a general discussion, and carry out
an explicit analysis of the $\mathfrak{su}(3)$ and $\mathfrak{su}(4)$
$\mathcal{N}=1^{*}$ gauge theories.

###  3.1 $\mathfrak{su}(2)$ gauge algebra

The $\mathcal{N}=1^{*}$ SYM theory with gauge algebra $\mathfrak{su}(2)$
possesses three degenerate gapped vacua, $\\{H,\,C^{(0)},\,C^{(1)}\\}$ where
the corresponding line operators $\\{D_{(1,0)},\,D_{(0,1)},\,D_{(1,1)}\\}$
condense. These vacua are in one-to-one correspondence with the global
variants $SU(2)$, $PSU(2)_{0}$ and $PSU(2)_{1}$ respectively. In the $SU(2)$
global structure, the fundamental Wilson line $D_{(1,0)}$ is the unique
genuine line. In the Higgs vacuum $H$, the $\mathbb{Z}_{2}$ 1-form symmetry is
spontaneously broken and it is realized by a $\mathbb{Z}_{2}$ gauge theory. In
the remaining two vacua, the $\mathbb{Z}_{2}$ 1-form symmetry is unbroken and
the low energy dynamics is described by a $\mathbb{Z}_{2}$ SPT. This analysis
can be easily repeated for other choices of global structure.

We will now focus on two special values of the UV modular parameter
$\tau_{\mathrm{YM}}$, namely $\tau_{\mathrm{YM}}=i$ and
$\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$ where the duality element $S$ and
the triality element $ST$ become non invertible global symmetries,
respectively.

The action of $S$ and $ST$ on the spectrum of line operators is summarized by
the following diagrams:

${D_{(1,0)}}$${D_{(0,1)}}$${D_{(1,1)}}$$\scriptstyle S$$\scriptstyle S$
${D_{(1,0)}}$${D_{(0,1)}}$${D_{(1,1)}}$$\scriptstyle ST$$\scriptstyle
ST$$\scriptstyle ST$ (3.4)

We thus find that, at $\tau_{\mathrm{YM}}=i$ and under the action of the non-
invertible global symmetry defect associated to $S$-duality, the three
degenerate vacua form a doublet $\\{H,C^{(0)}\\}$ and a singlet
$\\{C^{(1)}\\}$. On the other hand, at $\tau_{\mathrm{YM}}=e^{\frac{2\pi
i}{3}}$, these vacua form a triplet, with its elements permuted under the
action of $ST$-triality.

In this example we found two or more degenerate vacua with different physical
properties that are permuted under the action of a non-invertible symmetry. We
interpret this phenomenon as a spontaneous breaking of non-invertible
symmetry. On the other hand, if a vacuum happens to be a singlet (such as
$\\{C^{(1)}\\}$ for $\tau_{YM}=i$), the corresponding non-invertible symmetry
is unbroken in such a vacuum.

Whenever a symmetry, even if non-invertible, is spontaneously broken, it is
important to identify a corresponding local order parameter. In fact, these
symmetries have an invertible action on local operators and we expect to
characterize a spontaneously-broken phase in a standard way.

Let us recall that, in order for the symmetry to be preserved by a massive
deformation, the topological defect is constructed by composing a modular
transformation (either $S$ or $ST$) with a discrete R-symmetry rotation acting
on the matter fields. From equation (2.32), we find this rotation to be by a
$k$-th root of unity, with $k=4$ for $S$-duality at $\tau_{\mathrm{YM}}=i$ and
$k=6$ for $ST$-triality at $\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$. The
simplest chiral operator which is non-trivially charged under this action is

$\mathcal{O}=\operatorname{Tr}\Phi_{i}^{2}\;,$ (3.5)

for some value of $i=1,2,3$. In our conventions, chiral superfields have
vanishing modular weights and charge 1 under the diagonal R-symmetry rotation
used to construct the topological defects, hence having

$\displaystyle\mathfrak{D}_{-\frac{2\pi}{4}}\;\mathcal{O}$
$\displaystyle=e^{i\pi}\;\mathcal{O}\;\mathfrak{D}_{-\frac{2\pi}{4}}\quad,$
$\displaystyle\quad(\tau_{\mathrm{YM}}=i)$ (3.6)
$\displaystyle\mathfrak{D}_{-\frac{2\pi}{6}}\;\mathcal{O}$
$\displaystyle=e^{\frac{2\pi
i}{3}}\;\mathcal{O}\;\mathfrak{D}_{-\frac{2\pi}{6}}\quad,$
$\displaystyle\quad(\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}})$

Expectation values of the order parameter (3.5) need to comply with the above
selection rules when evaluated on the different vacua. Furthermore,
$\langle\mathcal{O}\rangle=\partial_{m}\,W$, with $W$ the superpotential for
which an exact expression in any of the gapped vacua has been derived in [44,
46]:

$W^{p,q,s}(\tau_{\mathrm{YM}})\equiv\bigl{\langle}HC^{(s)}_{p,q}\big{|}W\big{|}HC^{(s)}_{p,q}\bigr{\rangle}=\frac{N^{3}}{24}m^{3}\operatorname{\mathbb{I}m}(\tau_{\mathrm{YM}})\biggl{[}E_{2}(\tau_{\mathrm{YM}})-\frac{p}{q}E_{2}\biggl{(}\frac{p\tau_{\mathrm{YM}}+s}{q}\biggr{)}\biggr{]}\;.$
(3.7)

Here the relevant values of $\\{p,q,s\\}$ are $H=\\{2,1,0\\}$,
$C^{(0)}=\\{1,2,0\\}$ and $C^{(1)}=\\{1,2,1\\}$ and $E_{2}(x)$ denotes the
quasi-modular second Eisenstein series. By exploiting the modular properties
of these functions, one can easily show that (for details see Appendix B)

$\displaystyle\mathcal{O}^{1,2,0}\left(\tau_{\mathrm{YM}}=i\right)=e^{i\pi}\mathcal{O}^{2,1,0}(\tau_{\mathrm{YM}}=i)\;,\quad\mathcal{O}^{1,2,1}(\tau_{\mathrm{YM}}=i)=0\;,$
(3.8) $\displaystyle\mathcal{O}^{1,2,1}\left(\tau_{\mathrm{YM}}=e^{\frac{2\pi
i}{3}}\right)=e^{\frac{2\pi
i}{3}}\mathcal{O}^{1,2,0}\left(\tau_{\mathrm{YM}}=e^{\frac{2\pi
i}{3}}\right)=e^{\frac{4\pi
i}{3}}\mathcal{O}^{2,1,0}\left(\tau_{\mathrm{YM}}=e^{\frac{2\pi
i}{3}}\right)\;,$

hence in perfect agreement with (3.6). The remarkable fact that these
functions satisfy the required properties only at $\tau_{\mathrm{YM}}=i$ and
$\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$ should be regarded as a direct
consequence of the presence of non-invertible symmetry defects at these
specific points in the UV conformal manifold.

#### Action on lines and TQFTs.

We now give a detailed description of gapped phases in $\mathfrak{su}(2)$
$\mathcal{N}=1^{*}$ SYM theory. Vacua with different physical properties are
mapped into each other by broken duality/triality symmetries. This is due to
the non-invertibility of such global symmetries which act non-trivially on the
line operators. In the following we will focus on the case of $S$-duality. The
analysis can be repeated easily for the $ST$-triality symmetry.

If we choose the global variant $SU(2)$, the symmetry defect acts implementing
both $S$ and $\sigma$. It follows that a genuine line is necessarily mapped to
a non-genuine line, carrying a topological surface in order to render it gauge
invariant as depicted in Figure (3.9).121212Indeed, $S$ takes the Wilson line
(the genuine line of $SU(2)$) to the ’t Hooft line (the genuine line of
$PSU(2)_{0}$), while $\sigma$ brings back to the global variant $SU(2)$, hence
making the ’t Hooft line no longer genuine.

$D_{(1,0)}$$\mathbb{Z}_{2}$SPT$\leadsto$$\mathfrak{D}$$\mathbb{Z}_{2}$SPT$D_{(0,1)}$$\mathfrak{D}$
(3.9)

This implies that vacua related by the action of the non-invertible duality
symmetry $\mathfrak{D}$ can be physically inequivalent. For instance, in the
Higgs vacuum $|H\rangle$ the (genuine) fundamental Wilson line condenses,
$\langle H|D_{(1,0)}|H\rangle\neq 0$, so as to play the role of an order
parameter for the spontaneously broken 1-form symmetry $\mathbb{Z}_{2}^{[1]}$.
This topological order is described by a $\mathbb{Z}_{2}$ gauge theory. The
duality defect maps a genuine line $D_{(1,0)}$ into a non-genuine line
$D_{(0,1)}$ corresponding to a vacuum $S\,|H\rangle=|C^{(0)}\rangle$. Of
course, the fact that $\langle H|D_{(1,0)}|H\rangle\neq 0$ implies that
$\langle C^{(0)}|D_{(0,1)}|C^{(0)}\rangle\neq 0$. However, since the ’t Hooft
line is not genuine in the $SU(2)$ global variant, $D_{(0,1)}$ cannot serve as
an order parameter for the 1-form symmetry. Hence $\mathbb{Z}_{2}^{[1]}$ is
preserved in $|C^{(0)}\rangle$ and confinement takes place. There is no
topological order and the effective topological theory is accounted by a
$\mathbb{Z}_{2}^{[1]}$ SPT phase. The fact that the duality symmetry defect
$\mathfrak{D}$ maps a genuine order parameter to a non-genuine (twisted) order
parameter gives strong evidence for the spontaneous breaking of a non-
invertible symmetry.

We now explicitly construct the topological theories describing each gapped
vacuum. As explained above, the Higgs vacuum hosts a $\mathbb{Z}_{2}$ gauge
theory, as it should be for a phase with spontaneously broken
$\mathbb{Z}_{2}^{[1]}$ 1-form symmetry. Including the conventional
normalization, the partition function for this theory is

$Z[\mathbb{Z}_{2}](B)=\frac{1}{\sqrt{\lvert
H^{2}(X,\mathbb{Z}_{2})\rvert}}\sum_{b\in
H^{2}(X,\mathbb{Z}_{2})}e^{\frac{2\pi i}{2}\int b\cup B}=\sqrt{\lvert
H^{2}(X,\mathbb{Z}_{2})\rvert}\;\;\delta_{H^{2}(X,\mathbb{Z}_{2})}(B)\;.$
(3.10)

Because of the delta function in cohomology, this theory does not suffer from
any ambiguity in the form of an SPT factor. On the other hand, the vacua
$C^{(0)}$ and $C^{(1)}$ are characterized by a $\mathbb{Z}_{2}$ SPT phase.
There are two possibilities for such a topological phase, namely

$\text{SPT}(B)_{-k}\;:\;\;-\frac{2\pi ik}{4}\int\mathfrak{P}(B)\;,\qquad
k=0,1\;.$ (3.11)

The question of which value of $k$ in (3.11) corresponds to $C^{(0)}$ depends
on a choice of local counterterm in the UV and, as such, is not universal. We
choose this counterterm in such a way that to $C^{(0)}$ is assigned
$\text{SPT}_{0}$, i.e., the trivially gapped vacuum.

In fact, since $S$-duality maps $|H\rangle$ to $|C^{(0)}\rangle$, and the
symmetry defect involves the action of $\sigma$ (given by gauging the
$\mathbb{Z}_{2}^{[1]}$ 1-form symmetry), then the choice is fixed by the
following computation:

$[\sigma Z](B)=\frac{1}{\sqrt{\lvert H^{2}\rvert}}\sum_{b\in
H^{2}(X,\mathbb{Z}_{2})}Z[\mathbb{Z}_{2}](b)\;e^{\frac{2\pi
i}{2}\\!\int\\!b\cup B}=1=\text{SPT}_{0}\;.$ (3.12)

We can similarly determine which topological phase corresponds to $C^{(1)}$.
This state is a singlet under $S$-duality and it turns out that this condition
is enough to fix its low energy description. The only $\mathbb{Z}_{2}^{[1]}$
SPT phase that is left invariant by $\sigma$ is precisely $\text{SPT}_{1}$:
$\bigl{[}\sigma Z[\text{SPT}_{1}]\bigr{]}(B)=Z[\text{SPT}_{1}](B)$.
Alternatively, we can use the property that $C^{(0)}$ and $C^{(1)}$ are mapped
to each other by the action of the modular transformation $T$. In the UV, this
is accounted for by a shift $\theta\to\theta+2\pi$, under which the partition
function gets a background dependent phase $\exp\left(\frac{2\pi
i}{4}\\!\int\\!\mathfrak{P}(B)\right)$. We then conclude that, having assigned
$\text{SPT}_{0}$ to $C^{(0)}$, there is no other choice but assigning
$\text{SPT}_{1}$ to $C^{(1)}$. We can finally summarize the analysis with the
following formal expression for the topological field theory assigned to each
vacuum:

$\text{TQFT}=\begin{cases}\mathbb{Z}_{2}\bigoplus\text{SPT}_{0}\qquad\qquad&\text{for
}\\{H,\,C^{(0)}\\}\;,\\\ \text{SPT}_{1}\qquad&\text{for
}C^{(1)}\;.\end{cases}$ (3.13)

The duality symmetry $S$ is spontanously broken in the vacua
$\\{H,C^{(0)}\\}$, while it is preserved in the vacuum $C^{(1)}$. Note that
the ones in (3.13) are the only topological field theories with
$\mathbb{Z}_{2}$ 1-form symmetry. Each gapped vacuum of $SU(2)$
$\mathcal{N}=1^{*}$ SYM theory realizes one of them. We will see that this is
a general feature also for theories with gauge group $SU(N)$.

The story is similar for the triality symmetry at
$\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$. It is clear that the TQFT
assignment to the three vacua must be the same as the one for
$\tau_{\mathrm{YM}}=i$. The triality symmetry in the $SU(2)$ theory is
implemented by composing the UV duality $ST$ with a topological manipulation
$\sigma\tau$. A quick computation shows that the three $\mathbb{Z}_{2}^{[1]}$
TQFTs form a unique orbit under the action of triality:

${\mathbb{Z}_{2}}$${\text{SPT}_{0}}$${\text{SPT}_{1}}$$\scriptstyle\sigma\tau$$\scriptstyle\sigma\tau$$\scriptstyle\sigma\tau$
(3.14)

signalling the spontaneous breaking of the non-invertible triality symmetry.

#### Comments on the other global variants.

Let us discuss the other global variants associated with the
$\mathfrak{su}(2)$ algebra, namely $PSU(2)_{0}$ and $PSU(2)_{1}$ (also
commonly referred to as $SO(3)^{+}$ and $SO(3)^{-}$, respectively). Notice
that the condensation of dyons is a dynamical process, not affected by the
choice of global variant. On the other hand, the realization of the 1-form
global symmetry in each phase, either being preserved or broken, does depend
on the global aspects of the theory. Consequently, the effective low-energy
description of each vacuum changes when discussing $PSU(2)_{0}$ or
$PSU(2)_{1}$. The descriptions will actually be reshuffled, as we still expect
a one-to-one correspondence between the three vacua and the three possible
topological field theories with $\mathbb{Z}^{[1]}_{2}$ 1-form symmetry. For
instance, in each global variant there will always be a vacuum where the
condensing dyon corresponds to the genuine line, leading to a $\mathbb{Z}_{2}$
topological order.

Global variants are simply related by the topological manipulations $\sigma$
and $\tau$: starting from $SU(2)$, $PSU(2)_{0}$ is obtained by acting with
$\sigma$, while $PSU(2)_{1}$ by acting with $\sigma\tau$. Note that here we
are not combining these operations with an $S$ duality, so that we do not
change the behavior of local operators in each vacuum, only the TQFT
describing them. In practice, we perform the corresponding topological
manipulations on the effective topological field theories.

Let us begin with $G=PSU(2)_{0}$. We act with $\sigma$ on the topological
theories describing the three vacua of $SU(2)$. We get $\text{SPT}_{0}$ for
$H$, the $\mathbb{Z}_{2}$ gauge theory for $C^{(0)}$, and $\text{SPT}_{1}$ for
$C^{(1)}$. The ’t Hooft line $D_{(0,1)}$ becomes a genuine operator, and in
the vacuum $C^{(0)}$ where it condenses, the $\mathbb{Z}_{2}$ 1-form symmetry
is spontaneously broken and described by a $\mathbb{Z}_{2}$ gauge theory.
Since $H=S\cdot C^{(0)}$, we conclude that this phase is now realized by a
trivial SPT phase. Finally, since $D_{(1,1)}$ is invariant under $S$-duality,
the vacuum $C^{(1)}$ is accounted for by $\text{SPT}_{1}$.

Consider finally $G=PSU(2)_{1}$. This variant is interesting because it is
invariant under $S$-duality, and thus the $S$-duality defect becomes
invertible. The effective description in each of the three vacua is obtained
by acting with $\sigma\tau$ on the phases of $G=SU(2)$, leading to
$\text{SPT}_{0}$ for $H$, $\text{SPT}_{1}$ for $C^{(0)}$, and the
$\mathbb{Z}_{2}$ gauge theory for $C^{(1)}$. Alternatively, one can assign the
$\mathbb{Z}_{2}$ gauge theory to $C^{(1)}$ by the argument based on the
condensation of the genuine line $D_{(1,1)}$, and the SPT phases to $H$ and
$C^{(0)}$ by the argument below.

In $PSU(2)_{1}$ there is a mixed anomaly between the invertible duality
symmetry and the $\mathbb{Z}_{2}$ 1-form symmetry. From the perspective of the
UV theory, this stems from the fact that $PSU(2)_{1}$ is not exactly invariant
in the presence of a 1-form symmetry background. Indeed the action on the
Lagrangian subgroup is $S\,\big{(}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}-1\\\
1\end{smallmatrix}\big{)}\cong\big{(}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{)}$. The equivalence $\big{(}\begin{smallmatrix}-1\\\
1\end{smallmatrix}\big{)}\cong\big{(}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{)}$ is implemented by a shift of the $\theta$ angle,
and this corresponds to an ’t Hooft anomaly

$I_{\text{anom}}=\frac{2\pi i}{4}\int A_{S}\cup\mathfrak{P}(B)\;,$ (3.15)

where $A_{S}\in H^{1}(X,\mathbb{Z}_{4})$ a the S-duality gauge field. Indeed,
this reproduces the phase factor arising when acting with $S$ on
$\text{SPT}_{0}$ (vacuum $H$) to get $\text{SPT}_{1}$ (vacuum
$C^{(0)}$).131313As reviewed in Appendix A, vacua related by a broken
invertible symmetry display different SPTs if the broken symmetry participates
in a mixed anomaly. On the other hand, this is also the anomaly of the
$\mathbb{Z}_{2}$ gauge theory in the vacuum $C^{(1)}$.

We can summarize the content of this Section in the tables
below:141414Condensed lines which are non-genuine are indicated by a
superscript $U$, reflecting the fact that they are attached to a $U$ surface
for the $\mathbb{Z}_{N}$ 1-form symmetry. Vacua which are connected by
spontaneous symmetry breaking (SSB) of the duality symmetry are indicated by
the same background color.

$\displaystyle\begin{array}[]{c|b|b|c}\lx<EMAIL_ADDRESS>\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&C^{(1)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{\phantom{U}}&\pagecolor{LightCyan}D_{(0,1)}^{U}&D_{(1,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{2}&\pagecolor{LightCyan}\text{SPT}_{0}&\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}\hskip
40.00006pt\begin{array}[]{c|b|b|c}\lx<EMAIL_ADDRESS>\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&C^{(1)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{\phantom{U}}&D_{(1,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{LightCyan}\mathbb{Z}_{2}&\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}$ (3.26)
$\displaystyle\begin{array}[]{c|b|b|c}\lx<EMAIL_ADDRESS>\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&C^{(1)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{U}&D_{(1,1)}^{\phantom{U}}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{LightCyan}\text{SPT}_{1}&\mathbb{Z}_{2}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}$ (3.32)

We also include a table for the triality symmetry at
$\tau_{\mathrm{YM}}=e^{\frac{2\pi i}{3}}$ in the $SU(2)$ variant. Note that
all three vacua form a single orbit under the symmetry action, though the
effective topological field theories remain the same as in the previous case:

$\begin{array}[]{c|b|b|b}\lx<EMAIL_ADDRESS>\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{LightCyan}C^{(1)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{\phantom{U}}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{LightCyan}D_{(1,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{2}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{LightCyan}\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\\
\hline\cr\hline\cr\end{array}$ (3.33)

###  3.2 $\boldsymbol{\mathfrak{su}(N)}$ gauge algebra

Here we generalize the previous analysis to higher-rank gauge algebras
$\mathfrak{su}(N)$. We exploit some interesting patterns that were already
encountered in the simple case of $\mathfrak{su}(2)$, in particular the
correspondence between gapped vacua and the classification of topological
field theories with a given 1-form symmetry, and, more importantly, the
realization of the duality action in terms of simple topological manipulations
performed over such topological field theories.

#### TQFTs with $\boldsymbol{\mathbb{Z}_{N}}$ 1-form symmetry.

For global variants with $\Gamma=\mathbb{Z}_{N}$ 1-form symmetry, such as
$SU(N)$, the TQFTs describing the gapped vacua are completely characterized by
the symmetry-breaking pattern $\mathbb{Z}_{N}\rightarrow H=\mathbb{Z}_{m}$.
Not all global variants fall in this category: the 1-form symmetry may take
the form $\Gamma=\mathbb{Z}_{k}\times\mathbb{Z}_{k^{\prime}}$ as it happens in
$SU(N)/\mathbb{Z}_{k}$ with $N=kk^{\prime}$. In the following we will focus on
the case $\Gamma=\mathbb{Z}_{N}$, and then make some comments about more
complicated examples at the end.

A TQFT with $\mathbb{Z}_{N}$ 1-form symmetry can be specified by the symmetry-
breaking pattern $\mathbb{Z}_{N}\rightarrow\mathbb{Z}_{m}$. The broken
symmetry is described by a $\mathbb{Z}_{N}/\mathbb{Z}_{m}$ gauge theory, while
an SPT phase $\alpha_{m}$ is assigned to the preserved subgroup
$\mathbb{Z}_{m}$.151515More precisely, one should also specify the coupling to
the $\mathbb{Z}_{N}$ background field. The choice of coupling can be absorbed
by a rescaling $B\to rB$, with $\gcd(r,N)=1$. The latter is determined by a
class

$\alpha_{m}\,\in\,H^{4}\bigl{(}B^{2}\mathbb{Z}_{m},\,U(1)\bigr{)}=\mathbb{Z}_{\gcd(2,m)\,m}\;.$
(3.34)

As generator we take the Pontryagin square operation
$\mathfrak{P}\\!:\\!H^{2}(X,\mathbb{Z}_{m})\to
H^{4}\bigl{(}X,\mathbb{Z}_{\gcd(2,m)\,m}\bigr{)}$:

$\alpha_{m}(b)=\frac{2\pi i}{2m}\,\mathfrak{P}(b)\qquad\text{with}\qquad b\in
H^{2}(X,\mathbb{Z}_{m})\;.$ (3.35)

On spin manifolds, $\mathfrak{P}(b)$ is an even class and thus there are only
$m$ SPTs given by $\ell\,\alpha_{m}$, regardless of whether $m$ is even or
odd.161616More precisely, for spin theories these classes should be read from
$\Omega_{4}^{\text{spin}}(B^{2}\mathbb{Z}_{N})=\mathbb{Z}\oplus\mathbb{Z}_{N}$,
where the first generator is the signature $\sigma(X)$. See, e.g., [18] for a
derivation. For $H=\mathbb{Z}_{N}$ we have the SPTs

$\text{SPT}_{\ell}(B)=\exp\left(\frac{2\pi
ip}{2N}\int\mathfrak{P}(B)\right)\qquad\text{with }\ell=0,\dots,N-1\;.$ (3.36)

On the other hand, for $H=\\{0\\}$ the TQFT is a $\mathbb{Z}_{N}$ gauge theory
with partition function171717In order to fix the normalization of the various
TQFTs, we work under the assumption that the spacetime manifold is closed and
simply connected.

$Z[\mathbb{Z}_{N}](B)=\frac{1}{\sqrt{\lvert
H^{2}(X,\mathbb{Z}_{N})\rvert}}\,\sum_{b\in
H^{2}(X,\mathbb{Z}_{N})}\\!\\!\exp\biggl{(}\frac{2\pi i}{N}\\!\int\\!b\cup
B\biggr{)}=\sqrt{\lvert
H^{2}(X,\mathbb{Z}_{N})\rvert}\;\;\delta_{H^{2}}(B)\;.$ (3.37)

For $N$ prime, the cases above exhaust all realizations of the 1-form
symmetry, and are in correspondence with the $N+1$ gapped vacua (3.3).

More generaly, for $N=kk^{\prime}$ but still considering a global variant with
$\Gamma=\mathbb{Z}_{N}$, partially broken phases with $H=\mathbb{Z}_{k}$ and
SPT given by $\ell\alpha_{k}$ are described by theories that we dub
$\mathbb{Z}_{N|k}^{\ell}$ with $\ell\in\mathbb{Z}_{k}$, whose partition
function is

$\displaystyle Z[\mathbb{Z}_{N|k}^{\ell}](B)$
$\displaystyle=\exp\biggl{(}\frac{2\pi
i\ell}{2Nk^{\prime}}\\!\int\\!\mathfrak{P}(B)\biggr{)}\,Z[\mathbb{Z}_{k^{\prime}}](B)$
(3.38) $\displaystyle=\sqrt{\lvert
H^{2}(X,\mathbb{Z}_{k^{\prime}})\rvert}\;\exp\biggl{(}\frac{2\pi
i\ell}{2Nk^{\prime}}\\!\int\\!\mathfrak{P}(B)\biggr{)}\;\delta_{H^{2}(X,\mathbb{Z}_{k^{\prime}})}(B)\;.$

The partition function of the $\mathbb{Z}_{k^{\prime}}$ gauge theory vanishes
unless $B=k^{\prime}\tilde{B}$ with $\tilde{B}\in H^{2}(X,\mathbb{Z}_{k})$,
and in that case the SPT phase takes the well-defined form
$\exp\bigl{(}\frac{2\pi i\ell}{2k}\\!\int\\!\mathfrak{P}(\tilde{B})\bigr{)}$.
Some details concerning these topological field theories are presented in
Appendix C.

There is a correspondence between the TQFTs we described and global variants
of the $\mathfrak{su}(N)$ gauge theory with 1-form symmetry $\mathbb{Z}_{N}$.
Both are classified by doublets $(H,\alpha)$ where $H\subset\mathbb{Z}_{N}$
and $\alpha\in H^{4}\bigl{(}B^{2}H,U(1)\bigr{)}$. Indeed the data $(H,\alpha)$
determine how to reach a given global variant starting from the electric
$SU(N)$ via discrete gauging, but also specify the possible patterns of
spontaneous symmetry breaking. As the set of global variants is also
isomorphic to the gapped vacua of the $\mathcal{N}=1^{*}$ theory, one might
expect that each $\mathbb{Z}_{N}$ gauge theory is realized exactly once in a
certain massive vacuum of $\mathcal{N}=1^{*}$ with gauge group $G=SU(N)$. This
expectation is indeed correct, as we will see.

As mentioned at the beginning, there are other global variants in which the
1-form symmetry takes the form
$\Gamma=\mathbb{Z}_{k}\times\mathbb{Z}_{k^{\prime}}$ with $N=kk^{\prime}$.
This describes some of the variants
$G=\bigl{(}SU(N)/\mathbb{Z}_{k}\bigr{)}_{s}$.181818The form of the symmetry
group $\Gamma$ depends on whether $k$ and $k^{\prime}=N/k$ are coprime. This
seems in contrast with the lore that gauging a $\mathbb{Z}_{k}$ 1-form
symmetry yields a dual $\mathbb{Z}_{k}$ 1-form symmetry in $d=4$. This
subtletly arises because the short exact sequence
$\mathbb{Z}_{k}\rightarrow\mathbb{Z}_{N}\rightarrow\mathbb{Z}_{k^{\prime}}$
does not split when $k$ and $k^{\prime}$ are not coprime. Consequently, the
the $\mathbb{Z}_{N}$ background field splits into $\mathbb{Z}_{k}$ and
$\mathbb{Z}_{k^{\prime}}$ fields which are non-trivially correlated through
the Bockstein homomorphism $\beta$:  $\delta
B_{\mathbb{Z}_{k}}=\beta\bigl{(}B_{\mathbb{Z}_{k^{\prime}}}\bigr{)}\qquad\text{where}\qquad\beta:H^{j}(X,\mathbb{Z}_{k^{\prime}})\to
H^{j+1}(X,\mathbb{Z}_{k})\;.$ (3.39) With this condition, the discrete torsion
fails to be gauge invariant:  $\frac{2\pi
is}{2k}\int\mathfrak{P}\bigl{(}B_{\mathbb{Z}_{k}}+\delta\lambda_{\mathbb{Z}_{k}}\bigr{)}=\frac{2\pi
is}{2k}\int\mathfrak{P}\bigl{(}B_{\mathbb{Z}_{k}}\bigr{)}-\frac{2\pi
is}{k}\int\beta\bigl{(}B_{\mathbb{Z}_{k^{\prime}}}\bigr{)}\cup\lambda_{\mathbb{Z}_{k}}\;,$
(3.40) signalling a mixed anomaly. Upon gauging $B_{\mathbb{Z}_{k}}$, this
anomaly forces the dual symmetry $\check{B}_{\mathbb{Z}_{k}}$ to sit in an
extension with a (subgroup of) the broken $\mathbb{Z}_{k^{\prime}}$ symmetry
[66]. Carefully characterizing the anomaly-free subgroup allows one to
reproduce the result in (2.11) [59]. In this case the most general unbroken
subgroup is of the form $H=\mathbb{Z}_{r}\times\mathbb{Z}_{s}$, for
which191919For a general Abelian group $A$, one has that
$H^{4}\bigl{(}B^{2}A,U(1)\bigr{)}=\widehat{\Gamma}(A)$, with $\Gamma(A)$ the
universal quadratic group of $A$ [67, 68], see also [69].

$H^{4}\bigl{(}B^{2}H,U(1)\bigr{)}=\mathbb{Z}_{\gcd(2,r)\,r}\times\mathbb{Z}_{\gcd(2,s)\,s}\times\mathbb{Z}_{\gcd(r,s)}$
(3.41)

generated by the Pontraygin squares of the two factors and by the cup product
$B_{\mathbb{Z}_{r}}\cup B_{\mathbb{Z}_{s}}$, respectively. The classification
in this case is more involved, partly due to the fact that we can also have
mixed anomalies. The correspondence between TQFTs and gapped vacua breaks
down, and it turns out to be simpler to fix this data through discrete gauging
starting from the $SU(N)$ variant.

#### Assigning TQFTs to vacua.

Let us here list the TQFTs that describe the massive vacua of the $SU(N)$
$\mathcal{N}=1^{*}$ theory. As already explained in Section 3.1, in the Higgs
vacuum $H$ the Wilson line $D_{(1,0)}$ — which is a genuine line operator —
condenses and the 1-form symmetry is completely broken. This is matched in the
IR by a $\mathbb{Z}_{N}$ gauge theory. We summarize the properties of this
vacuum by the triplet

$\bigl{(}H,\;\langle D_{(1,0)}\rangle\neq 0,\;\mathbb{Z}_{N}\bigr{)}\;.$
(3.42)

In each of the confining vacua $C^{(p)}$ ($p=0,\dots,N-1$) the non-genuine
dyon $D_{(p,1)}$ condenses. Since this is not a genuine line, the 1-form
symmetry is preserved. The IR physics is reproduced by a $\mathbb{Z}_{N}$ SPT
phase. The standard choice is to assign to $C^{(p)}$ the theory
$\text{SPT}_{-p}(B)$. This is in accordance with pure $\mathcal{N}=1$ SYM,
which can be approached from $\mathcal{N}=1^{*}$ by sending $m\to\infty$ and
$g_{\mathrm{YM}}\to 0$ while keeping the dynamical scale $\Lambda=m^{3}e^{2\pi
i\tau}$ fixed. We stress that this choice is affected by UV counterterms of
the form $\mathfrak{P}(B)$, which do not affect the physics in the Higgs
vacuum because they become trivial in the $\mathbb{Z}_{N}$ gauge theory. We
summarize these vacua by

$\bigl{(}C^{(p)},\;\langle D_{(p,1)}\rangle\neq
0,\;\text{SPT}_{-p}\bigr{)}\;.$ (3.43)

In the mixed vacua — that we indicate as $HC^{\ell}_{k^{\prime},k}$ — the
condensation pattern is more involved. The $\mathbb{Z}_{k}$ subgroup of the
1-form symmetry is confined, and only the genuine lines $D_{(rk,0)}$ which are
uncharged under it may get an expectation value. As reviewed in Section 2, the
generic lattice describing such a process is generated by

$D_{(\ell,k^{\prime})}\oplus D_{(k,0)}$ (3.44)

with $\ell\in\mathbb{Z}_{k}$. To this lattice we associate the partially
broken TQFT $\mathbb{Z}_{N|k}^{-\ell}$, and thus the vacua are summarized by

$\Bigl{(}HC_{k^{\prime},k}^{\ell},\;\langle
D_{(\ell,k^{\prime})}\rangle\oplus\langle D_{(k,0)}\rangle\neq
0,\;\mathbb{Z}_{N|k}^{-\ell}\Bigr{)}\;.$ (3.45)

For instance, for $\ell=0$ the lattice of condensed lines is generated by
$D_{(0,k^{\prime})}\oplus D_{(k,0)}$ and the TQFT in the IR is a pure
$\mathbb{Z}_{k^{\prime}}$ gauge theory.

Having understood the case $G=SU(N)$, we can study other global variants by
discrete gauging. We can reach the theory
$\bigl{(}SU(N)/\mathbb{Z}_{q}\bigr{)}_{s}$ from $SU(N)$ by gauging a
$\mathbb{Z}_{q}$ subgroup of the electric 1-form symmetry after stacking with
an $\text{SPT}_{s}^{\mathbb{Z}_{q}}=\exp\bigl{(}\frac{2\pi
is}{2q}\int\mathfrak{P}(B_{\mathbb{Z}_{q}})\bigr{)}$ phase:

$\displaystyle
Z_{\left(\rule{0.0pt}{4.20004pt}SU(N)/\mathbb{Z}_{q}\right){}_{s}}(B_{\mathbb{Z}_{p}},B_{\mathbb{Z}_{q}})$
$\displaystyle=\frac{1}{\sqrt{\lvert
H^{2}(X,\,\mathbb{Z}_{q})\rvert}}\;\sum_{b_{\mathbb{Z}_{q}}\in
H^{2}(X,\,\mathbb{Z}_{q})}\;Z_{SU(N)}\bigl{(}B_{\mathbb{Z}_{p}}+p\,b_{\mathbb{Z}_{q}}\bigr{)}$
(3.46) $\displaystyle\quad\times\exp\biggl{(}\frac{2\pi
is}{2q}\int\mathfrak{P}\bigl{(}b_{\mathbb{Z}_{q}}\bigr{)}+\frac{2\pi i}{q}\int
b_{\mathbb{Z}_{q}}\cup B_{\mathbb{Z}_{q}}\biggr{)}\;.$

#### SSB of duality symmetries.

As in Section 3.1, the gapped vacua form orbits under the action of the
duality/triality symmetries. Here we will mostly focus on the action of $S$
duality, while similar statements for $ST$ triality can be similarly obtained.

The condensation patterns of dyons — hence gapped vacua — are associated to
Lagragian sublattices $\mathscr{L}_{k,\ell}$ of line operators spanned by
$D_{(\ell,k^{\prime})}\oplus D_{(k,0)}$ ($k^{\prime}=N/k$). In addition, there
is a natural action (2.10) of $S$ duality on these lattices.

The relation (2.10) determines the duality orbits comprising several subsets
of vacua, together with the occurrence of singlets labelled by duality
invariant sublattices, which chracterize the spontaneous breaking of duality
symmetry at $\tau_{\mathrm{YM}}=i$ in the $SU(N)$ $\mathcal{N}=1^{*}$ theory.

The local order parameter $\mathcal{O}=\operatorname{Tr}\Phi_{i}^{2}$ was
described in (3.5), and is charged under the action of $\mathfrak{D}$. The
non-trivial relations between expectation values taken by this operator in
different vacua can be obtained using the modular properties of the IR
superpotential (3.7) listed in Appendix B.

We can also study the realization of the duality symmetry in the IR by
inspecting its action on the low energy TQFTs. For instance, in the $SU(N)$
Higgs vacuum the low energy TQFT is a $\mathbb{Z}_{N}$ gauge theory which
describes the SSB of the 1-form symmetry by the condensation of the Wilson
line $D_{(1,0)}$. This is mapped by the duality defect into a non-genuine ’t
Hooft line and the dual vacuum is in a $\mathbb{Z}_{N}$ SPT phase. Below we
characterize such action on vacua completely and find that it matches with the
UV duality action (2.10) on the lines.

#### Symmetry action on TQFTs.

The non-invertible duality symmetry acts in the IR on the $SU(N)$ variant by a
discrete gauging $\sigma$ of the $\mathbb{Z}_{N}$ 1-form symmetry. For
consistency, this should mimic the UV action of $S$-duality on the dyons:

${\mathscr{L}}$${\mathscr{L}^{\prime}}$${\text{TQFT}(B)}$${\text{TQFT}^{\prime}(B)}$$\scriptstyle{S}$IRIR$\scriptstyle{\sigma}$
(3.47)

It is simple to show the following actions of $\sigma$:

$\displaystyle\sigma\cdot\mathbb{Z}_{N}(B)$ $\displaystyle=\text{SPT}_{0}(B)$
(3.48) $\displaystyle\sigma\cdot\text{SPT}_{\ell}(B)$
$\displaystyle=\begin{cases}\mathbb{Z}_{N}(B)\,,\qquad&\text{if }\ell=0\text{
mod }N\\\
\text{SPT}_{-\ell^{\raisebox{0.43054pt}{$\scriptscriptstyle-1$}}}(B)\,,\qquad&\text{if
}\ell\neq 0\text{ mod }N\;\;\text{and}\;\;\gcd(\ell,N)=1\\\
\mathbb{Z}_{N|k}^{\ell}(B)\,,\qquad&\text{if
}\gcd(\ell,N)=k^{\prime}\;.\end{cases}$

When $N$ is a prime number, the action of $\sigma$ on the TQFTs is compatible
with the action of $S$ on the condensed dyons, since
$SD_{(\ell,1)}=D_{(-1,\ell)}$ and the latter generates the same lattice as
$D_{(-\ell^{\raisebox{0.43054pt}{$\scriptscriptstyle-1$}},1)}$ (where
$\ell^{-1}$ is taken in $\mathbb{Z}_{N}$).

The action (3.48) implies that, in a vacuum described by a $\mathbb{Z}_{N}$
SPT, the non-invertible duality symmetry is spontaneously broken unless the
equation $\ell^{2}=-1\text{ mod }N$ has a solution.202020One can compare with
the discussion in Section 3.1. This is related to the existence of a global
variant $G=PSU(N)_{\ell}$ in which the duality symmetry becomes invertible.
Indeed, symmetry defects of this kind have been dubbed “non-intrinsic” in [70]
(see also [71]).

A similar story applies to mixed vacua in which the $\mathbb{Z}_{N}$ 1-form
symmetry is spontaneously broken to $\mathbb{Z}_{k}$. As detailed in Appendix
C, one finds

$\sigma\cdot\mathbb{Z}_{N|k}^{\ell}(B)=\mathbb{Z}_{N|\,N/p}^{\tilde{\ell}}(B)\quad\text{with}\quad
k^{\prime}=N/k\;,\quad
p=\gcd(\ell,k)\;,\quad\tilde{\ell}=-k^{\prime}(\ell/p)^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}_{\text{mod
}k/p}$ (3.49)

with the proviso of footnote 5. This is consistent with the duality action
(2.10) on dyonic lattices. As a byproduct, for $N=k^{2}$ one finds

$\sigma\cdot\mathbb{Z}_{k^{2}|k}^{0}(B)=\mathbb{Z}_{k^{2}|k}^{0}(B)\;.$ (3.50)

This implies that the vacuum $HC_{k,k}^{0}$, described by a $\mathbb{Z}_{k}$
gauge theory, is invariant under the full modular group212121Indeed using
(2.14) and (3.38), in general $\tau$ maps $\mathbb{Z}_{N|k}^{\ell}$ to
$\mathbb{Z}_{N|k}^{\ell+k^{\prime}}$, and thus $\mathbb{Z}_{k^{2}|k}^{0}$ is
also $\tau$-invariant. and in particular it preserves the duality symmetry.

More generally, the presence of duality-invariant vacua is in correspondence
with the existence of TQFTs which are self-dual under discrete gauging of the
$\mathbb{Z}_{N}$ 1-form symmetry. A thorough study of this problem was
initiated in [14] and completed in [72], finding the following condition:

$\exists\;k^{\prime},s,\tilde{\ell}\qquad\text{such that}\qquad
N=(k^{\prime})^{2}s\qquad\text{and}\qquad\tilde{\ell}^{2}+1=0\text{ mod }s\;.$
(3.51)

Indeed, identifying $k=k^{\prime}s$ and $\ell=k^{\prime}\tilde{\ell}$, one can
prove that (3.51) is equivalent to

$\exists\;k,\ell\quad\text{such that}\quad
k=\frac{N}{p}\;,\quad\ell=-p\,(\ell/p)^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}_{\text{mod
}k/p}\mod k\;,\quad\text{with}\quad p=\gcd(\ell,k)\;,$ (3.52)

i.e., to the condition that there exists a phase $\mathbb{Z}_{N|k}^{\ell}$
invariant under $\sigma$ according to (3.49). Note that for $N$ prime one
necessarily has to set $k^{\prime}=1$ and $s=N$, thus reproducing the analysis
above. On the other hand, for $N=k^{2}$ there is the solution $s=1$ that leads
to (3.50). Another instance where these TQFTs appear is in $\mathfrak{su}(8)$,
for which $k^{\prime}=2$, $s=2$ and $\ell=\pm 2$ provide solutions. Indeed,
the global variant $G=\bigl{(}SU(8)/\mathbb{Z}_{4}\bigr{)}_{2}$ has a duality-
invariant lattice of genuine line operators generated by the dyons
$D_{(4,0)}\oplus D_{(2,2)}$.222222The same problem can be studied for a
generic one-form symmetry group $\Gamma^{[1]}$ and one is lead to similar
conclusions. These results will appear in [73].

###  3.3 Duality symmetries on generic global variants

In section 3.2, we have shown that the action of $S$-duality on dyons is
reproduced by a discrete gauging in the low-energy TQFTs assigned to the vacua
of $G=SU(N)$ $\mathcal{N}=1^{*}$ SYM. For different choices of global
variants, such as $G=PSU(N)_{k}$, the technique becomes slightly more
involved. In this section we refine our approach from section 2, to construct
non-invertible defects associated with generic modular transformations over
different global variants. This procedure allows to detect the occurrence of
global variants for which these defects become invertible and to diagnose the
presence of mixed anomalies of the latter with the one-form symmetry.
Unfortunately, a systematic approach comprising all possible ranks of the
gauge group is out of the scope of this paper, so we will restrict ourselves
to $N$ prime.

First recall that, in theories with a $\mathbb{Z}_{N}$ one-form symmetry,
there is a natural action of $SL(2,\,\mathbb{Z}_{\gcd(2,N)N})$ [14].232323The
$\gcd$ factor is only important on non-spin manifolds and corresponds to
$\tau^{2N}=1$, while if $\omega_{2}(TM)=0$ $\tau^{N}=1$. We will thus ignore
it in our analysis. For $N$ prime this group has three generators: $\sigma$,
$\tau$ (which we already have introduced in Section 2)

$\displaystyle\sigma$ $\displaystyle:\ $ $\displaystyle\left[\sigma
Z\right](B)=\frac{1}{\sqrt{H^{2}(X,\mathbb{Z}_{N})}}\sum_{b\;\in
H^{2}(X,\mathbb{Z}_{N})}\exp\left(\frac{2\pi i}{N}\int b\cup
B\right)\;Z(b)\,,$ (3.53) $\displaystyle\tau$ $\displaystyle:\ $
$\displaystyle\left[\tau Z\right](B)=\exp\left(\frac{2\pi
i}{2N}\int\mathfrak{P}(B)\right)\;Z(B)\;,$

and $\nu(u)$ defined by

$\nu(u):\ \left[\nu(u)Z\right](B)=Z(uB)\,\ \ u\in\mathbb{Z}_{N}^{\times}.$
(3.54)

Global variants $G$ of $\mathfrak{su}(N)$ are classified by the quotient
$SL(2,\,\mathbb{Z}_{N})/\Gamma_{\infty}$ [51, 52], where

$\Gamma_{\infty}=\left\\{M\in SL(2,\,\mathbb{Z}_{N}):\ M=\begin{pmatrix}u&k\\\
0&u^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\end{pmatrix}\right\\}$
(3.55)

is the parabolic subgroup of $SL(2,\mathbb{Z}_{N})$. In this matrix
representation, the first column encodes the charges of genuine line
operators, while the second describes symmetry generators.

The topological $SL(2,\,\mathbb{Z}_{N})$ acts by right multiplication
($\sigma$ and $\tau$ being the analog of $S$ and $T$ matrices), while the
duality group by left multiplication. The condition that the matrix for $G$ is
modular implies that the lattice of genuine lines is Lagrangian, while the
quotient takes care of the choice of lattice generator(s).

In the case of $\mathfrak{su}(N)$, for $N$ prime, the global variants can be
represented by

$SU(N)=\mathbbm{1}\,,\ \ \ PSU(N)_{r}=\begin{pmatrix}r&-1\\\
1&0\end{pmatrix}\,.$ (3.56)

To define a duality symmetry $\mathfrak{D}$ for a variant $G$ and a duality
$M\in SL(2,\mathbb{Z})$ we must solve the matrix equation

$M\cdot G\cdot\Sigma=G$ (3.57)

which fixes the topological manipulation $\Sigma$ that undoes the duality
action.

Sometimes the global variant $G$ might be $M-invariant$. This implies that
(3.57) is solved by $\Sigma\in\Gamma_{\infty}$. Hence the $M$-duality defect
is invertible on $G$, but the presence of a nontrivial parabolic element
$\Sigma=\tau^{x}\nu(u)$ implies that the partition function transforms as

$M\cdot Z(B)=\exp\left(\frac{2\pi ix}{2N}\int\mathfrak{P}(B)\right)\;Z(uB)\,.$
(3.58)

When $M=S$, $N=2$ and $u,x=1$ this just becomes the familiar anomaly between
one-form symmetry and zero-form symmetry

$I=\frac{2\pi i}{4}\int S\cup\mathfrak{P}(B)\,,$ (3.59)

as was first noticed in [20].242424In the $N>2$ case a similar interpretation
is possible, however now the zero-form symmetry comes with an action
$\rho\,\in\,\text{Aut}(\mathbb{Z}_{N}):B\to uB$ making the symmetry a split
2-group [69]. Likewise, the anomaly must now be classified by an appropriate
class $I\in H^{5}(B^{2}\mathbb{Z}_{N}\rtimes_{\rho}B\mathbb{Z}_{4},\,U(1))$.

Whenever $u\neq 1$, the rescaling implemented by $\nu(u)$ is important, as it
makes the stacking operation to have an order compatible with $M$. Consider
for instance the case of $S$-duality and gauge group $PSU(N)_{u}$ for $N$
prime. Then, in order for the duality defect to be invertible, we need for the
topological manipulation $\Sigma$ to square to charge conjugation, i.e.
$\Sigma^{2}=C$. Furthermore, it imposes a condition over the discrete torsion
$u$, namely $u^{2}+1=0$ mod $N$. Indeed in these cases the $PSU(N)_{u}$
variant has an invertible self-duality symmetry which acts via the
element252525Using the matrix formulation one finds
$\Sigma=\begin{pmatrix}u^{-1}&1\\\ 0&u\end{pmatrix}$, which can be decomposed
as $\begin{pmatrix}u^{-1}&0\\\ 0&u\end{pmatrix}\,\begin{pmatrix}1&u\\\
0&1\end{pmatrix}$. This corresponds to the element $\tau^{u}\,\nu(u)$ with the
right action. It is also possible to check the result directly by using the
explicit formula for the $PSU(N)_{u}$ partition function (3.46).

$\Sigma_{u}=\tau^{u}\;\nu(u)\in\Gamma_{\infty}\,.$ (3.60)

As emphasized at the begining of this subsection, this simple representation
does not apply when $N$ is not prime (notably, some Lagrangian lattices have
two generators). In which case the correct transformation $\Sigma$ must be
computed by brute force.

Finally, notice that, strictly speaking, the general construction depicted
above does apply for $\mathcal{N}=4$ SYM. However, one only needs to stack
with an appropriate $R$-symmetry rotation, as described in section 2.

###  3.4 Examples: $\mathfrak{su}(3)$ and $\mathfrak{su}(4)$

With these tools at hand we can provide some explicit examples for the SSB in
the $\mathcal{N}=1^{*}$ vacua. A detailed exposition of the $\mathfrak{su}(2)$
case has been provided in section 3.1. We therefore proceed to describe the
examples $\mathfrak{su}(3)$ and $\mathfrak{su}(4)$, which are slightly more
complicated and lead to some additional important insight. As before, we
mostly focus on the realization of the duality symmetry at
$\tau_{\mathrm{YM}}=i$ and make some comments regarding triality on the way.

$\underline{\mathfrak{su}(3)}$

In this case we have four global variants $SU(3),\,PSU(3)_{0}\,,PSU(3)_{1}$
and $PSU(3)_{2}$, which form two orbits under S-duality. The non-invertible
symmetry here will be always spontaneously broken. In fact, the map (2.10)
yields

${D_{(1,0)}}$${D_{(0,1)}}$${D_{(1,1)}}$${D_{(2,1)}}$$\scriptstyle{S}$$\scriptstyle{S}$

and we find

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{SU(3)}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|y}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{YellowT}C^{(2)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}^{U}&\pagecolor{YellowT}D_{(2,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{3}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{YellowT}\text{SPT}_{2}&\pagecolor{YellowT}\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\\
\hline\cr\hline\cr\end{array}\end{subarray}\ \ \ \ \
\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{PSU(3)}_{1}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|y}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{YellowT}C^{(2)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}&\pagecolor{YellowT}D_{(2,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}SPT_{0}&\pagecolor{LightCyan}\text{SPT}_{2}&\pagecolor{YellowT}\mathbb{Z}_{3}&\pagecolor{YellowT}\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

This case is similar to the $\mathfrak{su}(2)$, since $N=3$ is a prime number.
Notice that the duality defect on $PSU(3)_{1}$ is implemented by the
topological manipulation $\tau\;\sigma\;\tau^{2}$ instead of just $\sigma$.
This follows either from direct computation or from the matrix formalism we
have introduced in section 3.3. It is simple to check that this transformation
on the TQFTs reproduces the action of $S$ duality.

Notice that the $\mathfrak{su}(3)$ theories also have a Coulomb vacuum,
corresponding to the partition $3=1+2$. We will comment on the physics in such
vacua in section 3.5. The case of generic prime $N=p$ is also very similar. In
the $SU(N)$ variant we have a spontaneously broken duality symmetry unless we
are in a confining $C^{(r)}$ vacuum with $r^{2}+1=0\mod N$, in which case both
the duality symmetry and the one-form symmetry are preserved. Other non-
invariant variants work similarly, but with some of the TQFTs permuted, while
in the invariant variant(s) the invertible action in the IR is implemented by
(3.60).

For the $ST$ symmetry the orbits instead are

${D_{(1,\,0)}}$${D_{(0,1)}}$${D_{(2,1)}}$$\scriptstyle{ST}$$\scriptstyle{ST}$$\scriptstyle{ST}$
${D_{(1,1)}}$$\scriptstyle{ST}$ (3.61)

In the $SU(3)$ variant the symmetry is implemented by $\sigma\tau^{-1}$ and we
find

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{SU(3)}\ \ \
\mathfrak{D}_{-\frac{2\pi}{6}}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|c|b}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&C^{(1)}&\pagecolor{LightCyan}C^{(2)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}&\pagecolor{LightCyan}D_{(0,1)}^{U}&D_{(1,1)}^{U}&\pagecolor{LightCyan}D_{(2,1)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{3}&\pagecolor{LightCyan}\text{SPT}_{0}&\text{SPT}_{2}&\pagecolor{LightCyan}\text{SPT}_{1}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

Which is consistent with the action on dyons.

$\underline{\mathfrak{su}(4)}$

Finally we consider the case of $\mathfrak{su}(4)$. Since $N$ is not prime the
theory has two new gapped vacua: $HC_{2\,,\,2}^{(0)}$ and $HC_{2\,,\,2}^{(1)}$
which correspond to the condensation of the dyons $D_{(2,0)}\oplus D_{(0,2)}$
and $D_{(1,2)}$ respectively. According to (2.10), their orbits under $S$ are

${D_{(1,1)}}$${D_{(2,0)}\;,\;D_{(0,2)}}$${D_{(1,0)}}$${D_{(0,1)}}$${D_{(2,1)}}$${D_{(3,1)}}$${D_{(1,2)}}$$\scriptstyle{S}$$\scriptstyle{S}$$\scriptstyle{S}$$\scriptstyle{S}$
(3.62)

By assigning the corresponding TQFTs for these vacua, we find the following
table

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{SU(4)}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|g|y|g|c}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{GreenT}C^{(2)}&\pagecolor{YellowT}C^{(3)}&\pagecolor{GreenT}HC_{2\,,\,2}^{(1)}&HC_{2\,,\,2}^{(0)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}^{U}&\pagecolor{GreenT}D_{(2,1)}^{U}&\pagecolor{YellowT}D_{(3,1)}^{U}&\pagecolor{GreenT}D_{(1,2)}^{U}&D_{(2,0)}\oplus
D_{(0,2)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{4}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{YellowT}\text{SPT}_{3}&\pagecolor{GreenT}\text{SPT}_{2}&\pagecolor{YellowT}\text{SPT}_{1}&\pagecolor{GreenT}\mathbb{Z}_{4\;|\;2}^{(1)}&\mathbb{Z}_{2}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

Notice the presence of phases in which $\mathbb{Z}_{4}$ is broken down to
$\mathbb{Z}_{2}$ spontaneously. To show that the vacua transform in the
correct way under the gauging of the one-form symmetry, we use the results of
section 3.2. For the doublets $\\{H,C^{(0)}\\}$ and $\\{C^{(1)},C^{(3)}\\}$ it
follows directly from the first three lines in (3.48). Regarding the doublet
$\\{C^{(2)},HC_{2,2}^{(1)}\\}$ and the singlet $HC_{2,2}^{(0)}$, their
relation derives from the last line in (3.48) and (3.49). In particular,
notice that

$\sigma\cdot\text{SPT}_{2}(B)=\mathbb{Z}_{4\;|\;2}^{(1)}(B)\,\,\,{\rm
and}\,\,\,\sigma\cdot\mathbb{Z}_{2}(B)=\mathbb{Z}_{2}(B)$ (3.63)

There are three other variants to consider, of which two are obtained from the
table above by gauging $\mathbb{Z}_{4}$ with a discrete torsion and the last
one by gauging only $\mathbb{Z}_{2}$ instead. They give the following
configurations

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{PSU(4)}_{1}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|g|y|g|c}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{GreenT}C^{(2)}&\pagecolor{YellowT}C^{(3)}&\pagecolor{GreenT}HC_{2\,,\,2}^{(1)}&HC_{2\,,\,2}^{(0)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}&\pagecolor{GreenT}D_{(2,1)}^{U}&\pagecolor{YellowT}D_{(3,1)}^{U}&\pagecolor{GreenT}D_{(1,2)}^{U}&D_{(2,2)}\oplus
D_{(0,2)}^{U}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{LightCyan}\text{SPT}_{3}&\pagecolor{YellowT}\mathbb{Z}_{4}&\pagecolor{GreenT}\text{SPT}_{1}&\pagecolor{YellowT}\mathbb{Z}_{4\;|\;2}^{(1)}&\pagecolor{GreenT}\text{SPT}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

and

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle\textbf{PSU(4)}_{2}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|g|y|g|c}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{GreenT}C^{(2)}&\pagecolor{YellowT}C^{(3)}&\pagecolor{GreenT}HC_{2\,,\,2}^{(1)}&HC_{2\,,\,2}^{(0)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}^{U}&\pagecolor{GreenT}D_{(2,1)}&\pagecolor{YellowT}D_{(3,1)}^{U}&\pagecolor{GreenT}D_{(1,2)}^{U}&D_{(2,0)}^{U}\oplus
D_{(0,2)}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\text{SPT}_{0}&\pagecolor{LightCyan}\mathbb{Z}_{4\;|\;2}^{(1)}&\pagecolor{YellowT}\text{SPT}_{3}&\pagecolor{GreenT}\mathbb{Z}_{4}&\pagecolor{YellowT}\text{SPT}_{1}&\pagecolor{GreenT}\text{SPT}_{2}&\mathbb{Z}_{2}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

On the former the IR duality symmetry acts through the transformation
$\Sigma(PSU(4)_{1})=\sigma\tau^{2}\sigma\tau$ while on the latter it is
generated by
$\Sigma(PSU(4)_{2})=\sigma\tau^{2}\sigma_{\mathbb{Z}_{2}}\tau_{\mathbb{Z}_{2}}$.
With $\sigma_{\mathbb{Z}_{2}}$ being the gauging of the $\mathbb{Z}_{2}$
subgroup and $\tau_{\mathbb{Z}_{2}}$ the stacking of the nontrivial
$\mathbb{Z}_{2}$ SPT.

Not surprisingly, the most involved case is the one in which the duality
symmetry is invertible corresponding to the global variant
$[SU(4)/\mathbb{Z}_{2}]_{0}$. To find the table of vacua we must gauge the
$\mathbb{Z}_{2}$ subgroup of $\mathbb{Z}_{4}$. Since $\mathbb{Z}_{4}$ is a
non-trivial extension

$1\longrightarrow\mathbb{Z}_{2}\longrightarrow\mathbb{Z}_{4}\longrightarrow\mathbb{Z}_{2}\longrightarrow
1$ (3.64)

then the backgrounds $C_{e}$ for the $\mathbb{Z}_{2}$ subgroup and $B_{e}$ for
the $\mathbb{Z}_{2}$ quotient are related by (see footnote 18)

$\delta C_{e}=\beta(B_{e})\,,$ (3.65)

with $\beta$ the Bockstein map associated to the above sequence (explicitly
$\beta(B_{e})=\frac{1}{2}\delta B_{e}$). According to [66], gauging the
$\mathbb{Z}_{2}$ subgroup results in a theory with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ one-form symmetry and a mixed anomaly

$I=\pi i\int B_{m}\;\beta(B_{e})=\pi i\int B_{e}\;\beta(B_{m})$ (3.66)

$B_{m}\in H^{2}(X,\mathbb{Z}_{2})$ being the gauge field for the dual one-form
symmetry. The mixed anomaly (3.66) cannot be matched by a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT, so we expect the
$[SU(4)/\mathbb{Z}_{2}]_{0}$ vacua in $\mathcal{N}=1^{*}$ to always display
(partial) SSB of the one-form symmetry. The theory also has a $\mathbb{Z}_{2}$
zero-form symmetry, $\mathfrak{s}$, which acts as a $\mathbb{Z}_{2}$
automorphism exchanging $B_{e}$ with $B_{m}$.

Let us first give the table of the results and then describe in detail some
subtleties of the gauging procedure. We find

$\begin{subarray}{c}\scalebox{1.0}{\mbox{$\displaystyle[\textbf{SU(4)}/\mathbb{Z}_{2}]_{0}$}}\\\\[2.5pt]
\begin{array}[]{c|b|b|y|g|y|g|c}\text{Vacuum}&\pagecolor{LightCyan}H&\pagecolor{LightCyan}C^{(0)}&\pagecolor{YellowT}C^{(1)}&\pagecolor{GreenT}C^{(2)}&\pagecolor{YellowT}C^{(3)}&\pagecolor{GreenT}HC_{2\,,\,2}^{(1)}&HC_{2\,,\,2}^{(0)}\\\
\hline\cr\hline\cr\text{Cond.
line}&\pagecolor{LightCyan}D_{(1,0)}^{U}&\pagecolor{LightCyan}D_{(0,1)}^{U}&\pagecolor{YellowT}D_{(1,1)}^{U}&\pagecolor{GreenT}D_{(2,1)}^{U}&\pagecolor{YellowT}D_{(3,1)}^{U}&\pagecolor{GreenT}D_{(1,2)}^{U}&D_{(2,0)}\oplus
D_{(0,2)}\\\
\hline\cr\text{TQFT}&\pagecolor{LightCyan}\mathbb{Z}_{2}&\pagecolor{LightCyan}\mathbb{Z}_{2}&\pagecolor{YellowT}\mathbb{Z}_{2}&\pagecolor{GreenT}\mathbb{Z}_{2}&\pagecolor{YellowT}\mathbb{Z}_{2}&\pagecolor{GreenT}\mathbb{Z}_{2}&\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\\
\hline\cr\text{SSB}\;\mathfrak{D}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{LightCyan}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{YellowT}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\pagecolor{GreenT}\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\checkmark\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}&\text{\texttimes}\\\
\hline\cr\hline\cr\end{array}\end{subarray}$

The lattices generated by the condensed dyons in each vacuum always contains
at least one local line (not counting the identity). This is consistent with
the fact that at least a $\mathbb{Z}_{2}$ must be spontaneously broken. In the
last vacuum, in which we have a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ theory,
the duality symmetry is unbroken, but acts nontrivially as an outer
automorphisms exchanging the two $\mathbb{Z}_{2}$ factors. To understand the
matching of the mixed anomaly we need to carefully follow the discrete gauging
procedure and understand the coupling of the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry to the $\mathbb{Z}_{2}$ gauge
theory. We describe $C_{e}$ in (3.65) as a torsor over
$H^{2}(X,\,\mathbb{Z}_{2})$, by choosing a splitting
$C_{e}=C_{e}^{\beta}+C_{e}^{\prime}$, with $C_{e}^{\prime}\in
H^{2}(X,\mathbb{Z}_{2})$ and $\delta C_{e}^{\beta}=\beta(B_{e})$. We can gauge
the one-form symmetry as usual by summing over $C_{e}^{\prime}$. This means
that

$\left[\sigma_{\mathbb{Z}_{2}}\cdot
Z\right](B_{e},\,B_{m})=\frac{1}{\sqrt{H^{2}(X,\,\mathbb{Z}_{2})}}\sum_{c^{\prime}\,\in\,H^{2}(X,\,\mathbb{Z}_{2})}e^{\pi
i\int c\,\cup B_{m}}\;Z(B_{e}+2c)\,\ \ \ c=C^{\beta}_{e}+c^{\prime}$ (3.67)

The fact that the above has a mixed $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
anomaly follows from considering the gauge transformation $B_{m}\to
B_{m}+\delta\lambda$. This kind of mixed anomaly allows for a Bardeen-like
counterterm:

$2\pi i\,K(B_{e},\,B_{m})=\frac{2\pi i}{4}\int B_{e}\,\cup\,B_{m}\,.$ (3.68)

With this notation we find:

$S\cdot
Z_{[SU(4)/\mathbb{Z}_{2}]_{0}}(B_{e},B_{m})=Z_{[SU(4)/\mathbb{Z}_{2}]_{0}}(B_{m},B_{e})\;e^{2\pi
i\;K(B_{e},B_{m})}\equiv\mathfrak{s}\cdot
Z_{[SU(4)/\mathbb{Z}_{2}]_{0}}(B_{e},B_{m})$ (3.69)

Where we have chosen to include the counterterm in the definition of
$\mathfrak{s}$. Thus the invertible $S$-duality is implemented by the discrete
$\mathbb{Z}_{2}$ symmetry. This is the action that we must match in the IR
vacua

To consistently assign vacua we must compute the action of the
$\mathbb{Z}_{2}$ gauging on the $SU(4)$ TQFTs. We find

$\left[\sigma_{\mathbb{Z}_{2}}\cdot\text{SPT}_{k}\right](B_{e},\,B_{m})=\sqrt{H^{2}(X,\,\mathbb{Z}_{2})}\;\delta^{\mathbb{Z}_{2}}(B_{m}+kB_{e})\;e^{\frac{\pi
ik}{4}\int\mathfrak{P}(B_{e})+\pi i\int C^{\beta}_{e}\cup(B_{m}+kB_{e})}\,.$
(3.70)

This is a $\mathbb{Z}_{2}$ gauge theory coupled to $B_{e}+kB_{m}$. The
Pontryagin square term is in principle ill defined, however this is resolved
by the term $kC_{e}^{\beta}\cup B_{e}$. The mixed anomaly is instead encoded
in the coupling $C_{e}^{\beta}\cup B_{m}$. On the other hand

$\left[\sigma_{\mathbb{Z}_{2}}\cdot\mathbb{Z}_{4}\right](B_{e},\,B_{m})=\sqrt{H^{2}(X,\,\mathbb{Z}_{2})}\;\delta^{\mathbb{Z}_{2}}(B_{e})\;e^{\pi
i\int C^{\beta}_{e}\cup B_{m}+2\pi i\,K(B_{e},\,B_{m})}\;,$ (3.71)

is the $\mathfrak{s}$ image of $\sigma_{\mathbb{Z}_{2}}\text{SPT}_{0}$. To see
it explicitly we use the Bardeen-like counterterm to write $\pi i\int
C^{\beta}_{e}\cup B_{m}+2\pi i\,K(B_{e},\,B_{m})=\pi i\int C_{m}^{\beta}\cup
B_{e}$, with $\delta C_{m}^{\beta}=\beta(B_{m})$. We also have that

$\left[\sigma_{\mathbb{Z}_{2}}\cdot\mathbb{Z}_{4\,|\,2}^{(1)}\right](B_{e},\,B_{m})=\sqrt{H^{2}(X,\,\mathbb{Z}_{2})}\;\delta^{\mathbb{Z}_{2}}(B_{e})e^{\frac{\pi
i}{2}\int\mathfrak{P}(B_{m})+\pi i\int C_{m}^{\beta}\cup B_{e}}\;.$ (3.72)

describing the $\mathfrak{s}$ action on
$\sigma_{\mathbb{Z}_{2}}\text{SPT}_{2}$. Finally:

$\left[\sigma_{\mathbb{Z}_{2}}\cdot\mathbb{Z}_{2}\right](B_{e},\,B_{m})=|H^{2}(X,\mathbb{Z}_{2})|\delta^{\mathbb{Z}_{2}}(B_{e})\;\delta^{\mathbb{Z}_{2}}(B_{m})\;e^{\pi
i\int C_{e}^{\beta}\cup B_{m}}\;,$ (3.73)

is the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory which is invariant
under $\mathfrak{s}$.

###  3.5 Comments on Coulomb vacua

All $\mathfrak{su}(N)$ theories for $N\geq 3$ also have gapless Coulomb vacua.
Such vacua often appear dressed by SPTs and/or discrete gauge theories that
follow from the precise Higgs/confinement pattern. Let us focus on the pure
Coulomb vacuum, which describes the partition $N=1+(N-1)$ in which the $SU(N)$
symmetry is Higgsed to $U(1)$. Since at $\tau_{\mathrm{YM}}=i$ there is a non-
invertible duality symmetry in the UV, this must also be matched in the
Coulomb vacua.

It is known that pure Maxwell theory hosts a wealth of non-invertible defects
coming from composing a modular $SL(2,\mathbb{Z})$ transformation with the
gauging of a discrete subgroup $\Gamma\subset U(1)^{[1]}_{e}\times
U(1)^{[1]}_{m}$ of the one-form symmetry [14, 74]. Once the subgroup $\Gamma$
and the modular transformation $M$ are identified, this fixes the Coulomb
coupling of the Maxwell theory $\tau_{C}$. In a pure Coulomb vacuum the one-
form symmetry is spontaneously broken, while the duality symmetry is
preserved. Here $M=S$, while the choice of $\Gamma$ is dictated by the UV
global variant $G$, with the convention that the one-form symmetry of the
$SU(N)$ variant is associated to the subgroup $\mathbb{Z}_{N,e}^{[1]}\subset
U(1)^{[1]}_{e}$ generated by

$U_{l}=e^{\frac{il}{N}\int\tilde{F}}\,,\ \ \ \tilde{F}=\frac{1}{e^{2}}\star
F+i\frac{\theta}{2\pi}F\,.$ (3.74)

Gauging the electric one-form symmetry projects onto Wilson lines $W_{n}$ with
$n=Nm$ but also allows for fractionalized ’t Hooft lines $T_{m^{\prime}/N}$.
We can restore the standard quantization by defining $A=\frac{1}{N}A^{\prime}$
and $\tilde{A}=N\tilde{A}^{\prime}$ for the dual gauge field. The action in
the $A^{\prime}$ variables has a coupling $\tau_{C}^{\prime}=\tau_{C}/N^{2}$.
By composing with S duality one finds that a duality symmetry defect exists if

$\tau_{C}=-\frac{N^{2}}{\tau_{C}}\,,\ \ \ \tau_{C}^{(e)}=iN\,.$ (3.75)

The magnetic gauging can be treated similarly, finding $\tau_{C}^{(m)}=i/N$.
The above argument fixes the Maxwell coupling $\tau_{C}=iN$ in the pure
Coulomb vacuum of $SU(N)$ $\mathcal{N}=1^{*}$. If we want to treat other
global variants we can do so by the appropriate discrete gauging. We can do it
directly at the level of the action by writing

$S=\int\frac{1}{2e^{2}}(dA-b)\wedge\star(dA-b)+\frac{i}{2\pi}G\wedge(Nb-
dl)+\frac{iNp}{4\pi}\,b\wedge b+\frac{iN}{2\pi}b\wedge C\,,$ (3.76)

where $b$ is the dynamical $\mathbb{Z}^{[1]}_{N,e}$ gauge field and $C$ is the
background field for the dual one-form symmetry. $G$ is also a dynamical field
which acts as a Lagrange multiplier restricting $b$ to be in $\mathbb{Z}_{N}$
by its equation of motion.

We set the theta angle in the electric frame to zero for simplicity.262626This
is sufficient to deal with duality transformations, since the self-dual
coupling is purely imaginary. We define shifted variables $b^{\prime}=b-dA$
and $G^{\prime}=G+pdA$, so that the action reads:

$\displaystyle S=$ $\displaystyle\int\frac{1}{2e^{2}}b^{\prime}\wedge\star
b^{\prime}+\frac{iNp}{4\pi}b^{\prime}\wedge
b^{\prime}+i\frac{N}{2\pi}b^{\prime}\wedge(C+G^{\prime})+\frac{iN}{2\pi}dA\wedge
G^{\prime}-\frac{i}{2\pi}G^{\prime}\wedge dl\,,$ (3.77)

where we have simplified the terms that are $2\pi\mathbb{Z}$-valued.
Integrating out $l$ sets $G^{\prime}=dV$ and the $\frac{1}{2\pi}dA\wedge dV$
term can also be dropped. After this we can integrate out $b^{\prime}$ by its
equations of motion:

$\frac{1}{e^{2}}\star b^{\prime}+\frac{Ni}{2\pi}(dV+pb^{\prime}+C)=0\,,$
(3.78)

which sets
$b^{\prime}=-i(\frac{1}{e^{2}}\star+\frac{iNp}{2\pi})^{-1}\frac{N}{2\pi}(dV+C)$.
The action becomes:

$S=\int\frac{1}{2e^{2}}\frac{1}{\frac{(2\pi)^{2}}{e^{4}N^{2}}+p^{2}}(dV+C)\wedge\star(dV+C)-\frac{iNp}{4\pi}\frac{1}{\frac{(2\pi)^{2}}{e^{4}N^{2}}+p^{2}}(dV+C)\wedge(dV+C)\,.$
(3.79)

If we start from the self-dual coupling $\frac{1}{e^{2}}=\frac{N}{2\pi}$ then
the complexified coupling $\tau_{C}=\frac{2\pi i}{e^{2}}+\frac{\theta}{2\pi}$
is given by:

$\tau_{C}^{(e)}=\frac{N}{(1+p^{2})}(i-p)\,.$ (3.80)

This is in an electric duality frame (since $dV$ couples electrically to $C$).
In the magnetic frame the answer instead reads

$\tau_{C}^{(m)}=\frac{1}{N}(i+p)\,,$ (3.81)

and reproduces the naive answer for $PSU(N)_{0}$ (magnetic gauging)
$\tau_{C}=i/N$. This reasoning allows to fix the coupling $\tau_{C}$ in the
pure Coulomb vacua with different global forms. Let us conclude with some
comments on vacua which are not purely Coulomb. One example is the partition
$4=1+1+2$ for $SU(4)$. In this vacuum the $SU(4)$ is Higgsed down to
$SU(2)\times U(1)$, with the $SU(2)$ ultimately confining. This is an
interesting question which we do not address here.

###  3.6 Cubic anomalies and invariant vacua

Lastly we perform some further consistency checks based on ’t Hooft anomalies.
It is known [47, 48] (see also [75, 49] for a recent discussion) that the
$SL(2,\mathbb{Z})$ group — or more appropriately its spin cover
$\text{Mp}(2,\mathbb{Z})$ — can have nontrivial ’t Hooft anomalies. The ’t
Hooft anomaly for a single Maxwell field, $I_{\text{Maxwell}}$, for a given
cyclic subgroup $\mathbb{Z}_{k}$ of the metaplectic group is related to the
chiral anomaly of a Weyl fermion $I_{\text{Weyl}}$ by:

$I_{\text{Maxwell}}=56\,I_{\text{Weyl}}\;.$ (3.82)

The latter must be carefully defined. Since $S^{4}=(-1)^{F}$, the correct
structure to consider is

$\text{spin}-Mp(2,\mathbb{Z})=\frac{\text{Spin}(d)\times
Mp(2,\mathbb{Z})}{\mathbb{Z}_{2}}\;,$ (3.83)

and the full anomaly is captured by a bordism group

$\Omega^{\text{spin}-Mp(2,\mathbb{Z})}_{5}(\text{pt})=\mathbb{Z}_{32}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{9}\;.$
(3.84)

The first two summands stem from the $S$ cubic anomaly, while the third from
the $ST$ one. On the other hand, if the symmetry group was only
$SL(2,\mathbb{Z})\times\mathbb{Z}_{2}^{F}$, anomalies would be classified by
[49]

$\Omega^{\text{spin}}_{5}(BSL(2,\mathbb{Z}))=\mathbb{Z}_{4}\oplus\mathbb{Z}_{9}\,,$
(3.85)

the first factor stemming from the subgroup generated by $S$ while the second
from $ST$. The anomaly for the $\mathcal{N}=4$ theory can be computed on the
Coulomb branch, where we have $\text{rank}(G)=N-1$ $\mathcal{N}=4$ vector
multiplets. Thus

$I_{\mathcal{N}=4}=60\,(N-1)\,I_{\text{Weyl}}\;.$ (3.86)

On the other hand, the chiral anomaly for the $U(1)_{R}$ is given by272727Here
we use a normalization in which the charges of all fermions are integers and
the periodicity of the $U(1)_{R}$ is $2\pi$. This amounts to doubling all the
entries of table (2.36) and gives charges $(3,-1,-1,-1)$ to fermions in the
vector multiplet and in the chiral multiplets respectively.

$I_{\text{chiral}}=-\sum_{i=1}^{4}q_{i}^{3}(N^{2}-1)I_{\text{Weyl}}=-24\,(N^{2}-1)\,I_{\text{Weyl}}\;,$
(3.87)

the minus sign stemming from the opposite action of $SL(2,\mathbb{Z})$ and
$U(1)_{R}$ on the fermions. Given a cyclic subgroup $\mathbb{Z}_{k}$ of
$Mp(2,\mathbb{Z})$, the presence of such cubic anomalies would prevent the
existence of an invariant gapped vacuum for the symmetry.282828One might be
concerned that such cubic anomalies could have a different description when
the symmetry is non-invertible. The best way to think about it is to work on a
manifold with trivial $H^{2}(X)$, so that we are blind to the one-form
symmetry and the duality defect looks invertible. In such case we can turn on