Disconnected gauge groups in the infrared

Guillermo Arias-Tamargoa<EMAIL_ADDRESS>and Mario De Marcob
<EMAIL_ADDRESS>
aTheoretical Physics Group, The Blackett Laboratory, Imperial College London,

Prince Consort Road London, SW7 2AZ, UK

bMathematics Institute, Uppsala University,
Box 480, SE-75106 Uppsala, Sweden

Abstract

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the
global form of the gauge group but not its perturbative dynamics. In this
work, we study the Seiberg-Witten solution of theories resulting from the
gauging of charge conjugation in 4d $\mathcal{N}=2$ theories with $SU(N)$
gauge group and fundamental hypermultiplets. The basic idea of our procedure
is to identify the $\mathbb{Z}_{2}$ action at the level of the SW curve and
perform the quotient, and it should also be applicable to non-lagrangian
theories. We study dynamical aspects of these theories such as their moduli
space singularities and the corresponding physics; in particular, we explore
the complex structure singularity arising from the quotient procedure. We also
discuss some implications of our work in regards to three problems: the
geometric classification of 4d SCFTs, the study of non-invertible symmetries
from the SW geometry, and the String Theory engineering of theories with
disconnected gauge groups.

###### Contents

  1. 1 Introduction and summary
  2. 2 Discrete gauging: UV perspective
    1. 2.1 Gauging the outer automorphism of $SU(N)$
    2. 2.2 Matter fields, lagrangians and global symmetry
  3. 3 Discrete gauging: IR perspective
    1. 3.1 General strategy
    2. 3.2 $\widetilde{SU}(3)$ Super Yang-Mills
    3. 3.3 $\widetilde{SU}(3)+6\,(F\oplus\overline{F})$
    4. 3.4 $\widetilde{SU}(4)+8\,(F\oplus\overline{F})$
    5. 3.5 $\widetilde{SU}(5)+10\,(F\oplus\overline{F})$
    6. 3.6 Higher ranks
  4. 4 BPS Spectra
    1. 4.1 BPS charges from the Seiberg-Witten solution
    2. 4.2 BPS spectra and charge conjugation: general results
    3. 4.3 Explicit example: BPS spectrum of $\widetilde{SU}(3)$
    4. 4.4 BPS spectra for $\mathbb{Z}_{k}$ groups
  5. 5 Discussion and outlook
    1. 5.1 On the classification of rank-2 geometries
    2. 5.2 Comments on SymTFT for non-invertible defects
    3. 5.3 String and M-theory uplift
    4. 5.4 Open problems
  6. A  The case of $SU(2)+4\,F$
  7. B Explicit check of the monodromy algebra

##  1 Introduction and summary

In recent years, it has been appreciated that when studying the behavior of a
Quantum Field Theory (QFT), the global form of the gauge group, and not only
its Lie algebra, can play a very important role [1]. A classic example is the
case of $SU(2)$ and $SO(3)$, which have a different spectrum of line
operators, periodicity of the $\theta$ angle, etc.; which in turn translate
into stark differences in their IR dynamics. In the modern language of
generalised global symmetries [2] it is their differences in higher form
symmetries and anomalies that gives rise to the change in behaviour. In this
particular example, the different “global form” of the gauge group refers to
the fundamental group $\pi_{1}(G)$ and the two QFTs are related by gauging a
(discrete) 1-form symmetry.

In this work, we shall be focusing on a slightly different situation, namely,
we will concern ourselves with the gauging of a discrete 0-form symmetry,
leading to a different $\pi_{0}(G)$ for the gauge group, that displays now
disconnected components. Examples of this type of operation feature frequently
in theoretical physics, from the GSO projection in String Theory to various
dualities in lattice systems (see e.g. [3, 4, 5]). Quantum Gravity
considerations also lead in a natural way to disconnected gauge groups: we
know that our world is described at low energies by some gauge theory, and
very often these have discrete global symmetries. But we also know that global
symmetries in quantum gravity, including discrete ones, should be either
broken or gauged when going to high energies [6, 7, 8, 9, 10], which in the
latter case will lead us to said type of group. In particular, we will study
the case where the discrete symmetry we gauge corresponds to charge
conjugation.

Charge conjugation has the particularity that its action doesn’t commute with
a gauge transformation. As a global symmetry, this has the implication that
the total symmetry of the theory under consideration will form a non-trivial
2-group structure [11, 12]. When gauging it, it means that the total gauge
group will be a semidirect product. This has several interesting consequences.
In [13, 14, 9] (and similarly in spirit [15, 16]) it was shown that some of
the topological defects of the theory are acted upon by charge conjugation,
and thus the gauge invariant combination one can build from them is non-
invertible in any dimension (see [17, 18, 19, 20, 21] for recent reviews of
this fascinating and fast-developing subject). The spectrum of gauge invariant
local operators is also modified; as an example, the baryon and anti-baryon of
$SU(N)+N_{f}$ fundamentals become the same local operator after gauging charge
conjugation. When restricting to the case of supersymmetric gauge theories
with 8 supercharges, tools such as indices or Hilbert Series allow for an in-
depth study of the spectrum of local operators at the quantum level [22, 23,
24, 25]. Indeed, these works showed that the disconnected gauge groups provide
the first known examples of 4d $\mathcal{N}=2$ SCFTs whose Coulomb branch is
not freely generated.

The discussion above concerns only kinematical aspects of QFTs with
disconnected gauge groups. Our aim in this work is to begin the investigation
of their dynamics. A zero-th order observation in this regard is that since
the discrete gauging is changing the global form of the gauge group but not
its Lie algebra, all perturbative quantum corrections (Feynman diagrams, etc.)
should be identical. Instead, something more interesting might come by looking
at non-perturbative phenomena. Once again, we will invoke supersymmetry as a
simplifying assumption that will give us the tools necessary to address this
question in an exact way.

In the seminal work by Seiberg and Witten [26, 27] it was found that 4d
$\mathcal{N}=2$ SUSY imposes sufficient constraints on the theory so that the
low energy EFT in the Coulomb branch can be computed exactly. The idea is that
the metric entering the kinetic term of the vector multiplet scalars at low
energy is encoded in an auxiliary object, the Seiberg-Witten (SW) curve and
differential (see e.g. [28, 29, 30, 31] for pedagogical reviews of the topic).
In the example of pure $SU(2)$ SYM, the curve and differential would be

$\displaystyle
y^{2}=(x^{2}+u_{2})^{2}-4\Lambda^{4},\qquad\lambda=x\,d\left[\log\left(\frac{x^{2}+u_{2}-y}{x^{2}+u_{2}+y}\right)\right]$
(1.1)

where $u_{2}$ is the Coulomb branch parameter, $(x,y)$ are the complex
variables parametrizing the curve and differential, and $\Lambda$ is a scale
involving the UV cutoff and gauge coupling. From these, one computes the
periods $a$ and $a_{D}$ by integrating the SW differential over the $A$ and
$B$ cycles of the curve, and from those one finds the metric of the low energy
NLSM (which in the $SU(2)$ would have only one component),

$\displaystyle\tau(a)=\frac{\partial a_{D}}{\partial a}\,.$ (1.2)

This metric has singularities whenever the SW curve on top of the Coulomb
branch has a degeneration. The interpretation is that the low energy EFT fails
because there is an extra particle becoming massless. Famously, in the $SU(2)$
case these singularities occur at $u=\pm 2\Lambda^{2}$ and correspond to a
monopole and a dyon becoming massless.

The question we ask is: how are the SW curve and differential modified after
the discrete gauging of a 0-form symmetry? This problem was addressed in [32]
in the rank 1 case. The basic idea is that the extension of the gauge group
results in a quotient of the moduli space (morally speaking, “doubling the
size” of the gauge orbits results in “half the amount” of physically distinct
configurations). From the lagrangian, we know how the symmetry acts on the
Coulomb branch VEVs and how to take the quotient. But the full SW solution is
given by a fibration over the Coulomb branch, and therefore to be able to
consistently take the quotient we need to find the correct gluing isomorphism
that identifies the fibers on top of the corresponding points in the parent
theory Coulomb branch $\mathcal{M}_{\text{CB}}$. In [32] it was observed that
for the rank 1 case the only discrete 0-form symmetries one can gauge are
combinations of $U(1)_{R}$ R-symmetry, $SL(2,\mathbb{Z})$ (for fixed values of
the coupling) and the of outer automorphism of the flavor group
$\text{Out}(F)$. Our starting point will be instead $SU(N)$ SQCD theories for
higher ranks, which has an extra symmetry not present at rank 1, the outer
automorphism of the gauge group $\text{Out}(G)$, whose gauging is compatible
with supersymmetry [22, 23]. This is a $\mathbb{Z}_{2}$ that acts as charge
conjugation, and that leads us to the disconnected group
$SU(N)\rtimes\mathbb{Z}_{2}$ and which henceforth we will be denoted as
$\widetilde{SU}(N)$.

Let us anticipate what are the main results of our analysis. We are indeed
able to compute the SW solution for $\widetilde{SU}(N)$ theory, focusing for
simplicity in either the pure SYM case or the conformal case (although from
the latter we can straightforwardly integrate out part of the flavours
introducing the corresponding mass parameter in the curve and taking the
appropriate decoupling limit). We are not able to give a closed expression for
generic $N$, because there appear to be several sources of non-generic-ness,
as we will discuss in the main text. The most important one is that, from the
$SU(4)$ case on, the Seiberg-Witten geometry is not a complete intersection
anymore. Nevertheless, our procedure is straightforward to generalise to
arbitrary rank. As checks of our proposals, we are able to match the global
symmetry seen by the curve with the expectation from the lagrangian, as well
as the monodromy at infinity due to the 1-loop $\beta$-function.

We also identify the BPS particles that become massless at different points of
the Coulomb branch by studying the vanishing cycles of our SW curves. Once
again, we cannot perform the analysis for generic $N$, but we do discuss the
universal features that we expect to hold in all cases. Most surprisingly, we
find the existence of a new singularity in the origin of the moduli space,
which does not descend from a massless particle of $SU(N)$ before the discrete
gauging, and which plays a very important role in making the solution
consistent. Moreover, using the fact that our method for analyzing the
spectrum is purely geometrical, we will be able to (conjecturally) extend this
analysis beyond the $\mathbb{Z}_{2}$ gauging in the $SU(N)+F$ case. Indeed,
under some technical assumptions, we will describe the effect of the discrete-
gauging of a $\mathbb{Z}_{k}$ zero-form symmetry on the BPS spectrum of a
generic, not necessarily lagrangian, 4$d$ $\mathcal{N}=2$ theory.111We should
remark that for these cases the consistency of the discrete gauging with
supersymmetry is an assumption rather than known from a lagrangian
construction.

There are three particular problems where our results can help to shed some
light. The first concerns the bottom-up classification of 4d $\mathcal{N}=2$
SCFTs. In the rank 1 case [33, 34, 35, 36] it turned out that several of the
allowed SW geometries correspond to discrete gaugings [32]. At rank 2 [37, 38,
39, 40, 41] (see also [42, 43, 44, 45, 46]) a puzzle occurs: the scaling
dimensions of the Coulomb branch parameters of $\widetilde{SU}(3)$ with 6
fundamentals are $(2,6)$ [22, 23, 24] and the flavor group is of rank three,
which at a first glance is not allowed [39]. Interestingly, it’s the
aforementioned new singularity at the origin of the Coulomb branch that
explains how the $\widetilde{SU}(3)$ theory relates to the classification of
Argyres and Martone.

The second problem concerns the study of generalized global symmetries with
the formalism of the Symmetry Topological Field Theory (SymTFT) [47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. This is an auxiliary topological
field theory in one dimension higher generalising the concept of anomaly
inflow, that encodes all the symmetries of the theory of interest. In [48] it
was shown how the SymTFT for the 1-form symmetry can be obtained from the
Seiberg-Witten solution, which has the implication that the deep IR knows
about the fundamental group of the gauge group (see also [61]). Given that the
theories we study have a non-invertible 1-form symmetry, it’s interesting to
check whether the method of [48] (which amounts to finding the BPS particles
that can screen the relevant line operators) can detect it: we find that the
answer is no, and that the non-invertibility should be otherwise encoded in
the geometry.

The third one concerns the String Theory engineering of theories with
disconnected gauge groups. As showed by Witten [62], there is an intimate
relation between the Seiberg-Witten solution and the geometry of the type IIA
or M-theory setup that engineers the theory. Starting from a setup of M5
branes wrapping a curve $\mathcal{C}$ (by specifying the curve one is
specifying the theory as in the class $\mathcal{S}$ construction [63]), in the
IR the system will flow to a single M5 wrapping a certain cover of
$\mathcal{C}$ which is precisely the Seiberg-Witten curve. In principle, it is
easy to reverse engineer this process in order to find what is the UV curve
wrapped by the M5 branes from the SW solution; however, as we shall see, there
is an additional complication for the disconnected gauge groups.

The rest of this paper is organized as follows. In section 2 we review the
lagrangian construction of $\mathcal{N}=2$ SYM theories based on disconnected
gauge groups. In section 3 we obtain their Seiberg-Witten geometry, including
some checks for the validity of our proposal. In section 4 we compute the
spectrum of particles that become massless at the various singularities in the
Coulomb branch, and identify their electric and magnetic charges from the
monodromies in the SW curve. We finish in section 5 with a discussion of our
results in regards to the three aforementioned problems: the classification of
rank 2 geometries, the derivation of the SymTFT from the IR, and the uplift of
the SW geometry to M-theory; as well as some additional open questions. We
relegate to the appendices some auxiliary computations. In appendix A we
reproduce the already known results in the rank 1 case of $SU(2)+4F$ using our
(slightly different) methods. In appendix B we give the details of the proof
of consistency of the monodromies in our curves.

##  2 Discrete gauging: UV perspective

The main goal of this work is to explore the consequences of the discrete
gauging of the outer automorphism of the gauge group in the Seiberg-Witten
geometry of the 4d $\mathcal{N}=2$ SYM theory with $SU(N)$ gauge group and
fundamental flavours. With this in mind, the first step is to understand the
UV lagrangian. Our point of view will be to consider the disconnected Lie
group built as an extension of $SU(N)$ by its outer automorphism, which we
denote $\widetilde{SU}(N)$, and then study the supersymmetric gauge theories
we can build with it. This is equivalent to starting with a $SU(N)$
(supersymmetric) gauge theory and gauging the 0-form global symmetry
corresponding to its outer automorphism.

###  2.1 Gauging the outer automorphism of $SU(N)$

There are in principle infinitely many ways to build a group extension of
$SU(N)$ by any other group $G$, specified by building a short exact sequence
of the form

$\displaystyle 1\to SU(N)\to\widetilde{G}\to G\to 1\,.$ (2.1)

We will focus on the very simple case where the group we use to build the
extension is $\mathbb{Z}_{2}$ and the sequence splits,

$\displaystyle 1\to
SU(N)\rightleftarrows\widetilde{SU}(N)\rightleftarrows\mathbb{Z}_{2}\to 1\,.$
(2.2)

In order words, $\widetilde{SU}(N)$ is a semidirect product,

$\displaystyle\widetilde{SU}(N)=SU(N)\rtimes_{\Theta}\mathbb{Z}_{2}\,.$ (2.3)

Moreover, we can identify the $\mathbb{Z}_{2}$ with the group of outer
automorphisms of $SU(N)$. This is the symmetry of the Dynkin diagram of
$SU(N)$ that acts by exchanging the roots
$\alpha_{j}\leftrightarrow\alpha_{N-1-j}$ (see Figure 1). This action on the
root system can be lifted to the group and used to specify the homomorphism
$\Theta$ involved in the definition of the semidirect product (2.3),222A
detailed analysis of all the possible ways to extend $SU(N)$ by its group of
outer automorphisms was carried out in [24]. It was found that if $N$ is odd,
$\Theta$ is uniquely defined; while if $N$ is even, there are two
possibilities, one of which coincides with the $N$ odd case. The two
extensions were dubbed $\widetilde{SU}(N)_{I,II}$. When analysing the moduli
space of 4d $\mathcal{N}=2$ theories with these disconnected gauge groups, it
was found that while the Higgs branch does depend on the choice of extension,
the Coulomb branch does not (see also [25, 22]). Therefore we will restrict to
the type $I$ extension common to both $N$ even and odd, which is defined by
the homomorphism (2.4).

$\displaystyle\Theta_{1}(g)=g\,,\quad\Theta_{-1}(g)=(g^{-1})^{T}=g^{*}\,,$
(2.4)

where $g\in SU(N)$. Elements in $\widetilde{SU}(N)$ can be described as pairs
$(g,\epsilon)$ with $g\in SU(N)$ and $\epsilon\in\mathbb{Z}_{2}$, with the
product given by

$\displaystyle(g_{1},\epsilon_{1})\cdot(g_{2},\epsilon_{2})=(g_{1}\Theta_{\epsilon_{1}}(g_{2}),\epsilon_{1}\epsilon_{2})\,.$
(2.5)

$\cdots$$\bf\mathbb{Z}_{2}$ Figure 1: The outer automorphism of the gauge
group $SU(N)$ is a (0-form) global symmetry of the gauge theory when the rank
is higher than 1. Intuitively it corresponds to charge conjugation, as matter
in the fundamental representation is mapped to the antifundamental (and
viceversa).

Sometimes it is useful to use directly the fundamental representation, which
has dimension $2N$ and where the two connected components of the group can be
written as

$\displaystyle\widetilde{SU}(N)=\left\\{\left(\begin{array}[]{cc}g&0\\\
0&g^{*}\end{array}\right),\,g\in
SU(N)\right\\}\cup\left\\{\left(\begin{array}[]{cc}0&g\\\
g^{*}&0\end{array}\right),\,g\in SU(N)\right\\}\,,$ (2.10)

and where the element that generates the $\mathbb{Z}_{2}$ and which takes us
from one connected component of the group to the other is

$\displaystyle(\text{Id},-1)\to\left(\begin{array}[]{cc}0&\mathds{1}_{N}\\\
\mathds{1}_{N}&0\end{array}\right)\,.$ (2.13)

We will also need to use the adjoint representation. In this case, as opposed
to the fundamental, we don’t need to double the size of the matrices when
going to the extended group. This can be seen to be the case because the
Dynkin labels of the adjoint of $SU(N)$ are $[1,0,\dots,0,1]$ which is
symmetric under the $\mathbb{Z}_{2}$ outer automorphism (see [64, section
VI]). Instead, the adjoint of the connected group survives directly in
$\widetilde{SU}(N)$. The consequence of the extension is that we have another
$N^{2}-1$ dimensional representation, built by tensoring the adjoint with the
fundamental of the $\mathbb{Z}_{2}$, called the pseudo-adjoint (by analogy
with the pseudo-vector representation of $O(N)$ groups); however, we will not
need it in the rest of the paper.

Before moving on to building theories based on these gauge groups, there are
two important remarks that we’d like to make. The first is that, as is
manifest by the matrix representation (2.10), one can exchange the roles of
$g$ and $g^{*}$ by multiplication with elements of $\widetilde{SU}(N)$
(concretely, by conjugation with the element (2.13)). In this sense we can
understand that when building the group extension we are gauging charge
conjugation. The second is that, while the fundamental representation of
$SU(N)$ is complex, after the discrete gauging, the new fundamental
representation (2.10) is real; a fact that will become important to build
lagrangians in the following section. An intuitive way to understand this is
to think of this representation as the one corresponding to the Dynkin labels
$[1,0,\dots,0]\oplus[0,\dots,0,1]$, except it’s made irreducible by the
$\mathbb{Z}_{2}$ that mixes the two summands. One can also rigorously check
the reality of this representation by e.g. computing the Frobenius-Schur
indicator [24].

###  2.2 Matter fields, lagrangians and global symmetry

The usual way to construct an $\mathcal{N}=2$ supersymmetric gauge theory is
to start with $\mathcal{N}=1$ superfields and lagrangian, and then impose the
necessary restrictions so that one has extended supersymmetry (usually by
means of looking at the $R$-symmetry). Let’s focus first on the
hypermultiplets. The story is completely analogous to the textbook case of
$SU(2)$: we start with a collection of two different $\mathcal{N}=1$ chiral
superfields

$\displaystyle Q_{I}^{a}\,,\quad\widetilde{Q}^{I}_{a}\,,$ (2.14)

where $a=1,\dots,N$ is a gauge index and $I$ a flavor index. But, since the
fundamental representation of $\widetilde{SU}(N)$, as presented above, is
real, we can just reorganize them as

$\displaystyle Q_{I}^{a}\,,\quad a=1,\dots,2N\,,\text{ and }I=1,\dots
N_{f}\,,$ (2.15)

with $N_{f}$ even. Again due to the reality of the representation, we have an
invariant tensor $\delta_{ab}$ which is symmetric. We can use it to build the
quadratic term in the superpotential.

$\displaystyle W\,\supset\,M^{IJ}\,Q^{a}_{I}\,Q^{b}_{J}\,\delta_{ab}\,,$
(2.16)

where the mass matrix satisfies $M^{IJ}=M^{JI}$. From this, we already see
that SYM with $N_{f}$ flavours in the fundamental of $\widetilde{SU}(N)$ has a
$USp(N_{f})$ global symmetry.

In order to build the cubic term of the superpotential, we need also a vector
multiplet. It is straightforward to construct the corresponding superfields
transforming in the adjoint representation, since as we saw in the previous
section it trivially survives the discrete gauging. In order to be able to
contract the indices with the presentation (2.15) we construct the following
block-diagonal matrix,

$\displaystyle(\Phi)_{ab}=\left(\begin{array}[]{cc}\varphi&0\\\
0&\varphi^{*}\end{array}\right)\,,$ (2.19)

where $\varphi$ is the $\mathcal{N}=1$ chiral in the $SU(N)$ vector multiplet
and now $\Phi$ is a $2N\times 2N$ matrix. Now we can build the cubic term in
the superpotential,

$\displaystyle W\,\supset\,g^{IJ}\,Q^{a}_{I}\,\Phi_{ab}\,Q^{b}_{J}\,.$ (2.20)

We have introduced a matrix of couplings $g^{IJ}$ whose values are fixed by
$\mathcal{N}=2$ supersymmetry. The matrix couplings satisfy $g^{IJ}=-g^{JI}$
as a consequence of $\Phi_{ab}$ being antisymmetric, again indicating a
$USp(N_{f})$ global symmetry.

Now that we have the superpotential $W$, the only thing left to write down the
complete lagrangian are the kinetic terms. The gauge kinetic term is
completely identical to the $SU(N)$ case, and the same happens for the Kahler
potential of the chiral part of the vector multiplet. Once again, this is
because the adjoint representation of $SU(N)$ and $\widetilde{SU}(N)$ are the
same; and indeed it was to be expected that anything involving the gauge
field, which is a connection taking values on the Lie _algebra_ of the group,
is not sensitive to the discrete gauging. Lastly, we also write the Kahler
potential for the hypermultiplets in the standard fashion, after taking two
copies of the $\mathcal{N}=1$ vector superfield analogously to (2.19),

$\displaystyle K=Q^{a}_{I}\,e^{V_{ab}}\,Q^{b}_{J}\,\delta^{IJ}\,.$ (2.21)

To summarize, the only difference between $SU(N)$ and $\widetilde{SU}(N)$
$\mathcal{N}=2$ SYM theories lies in the reality properties of the fundamental
representation. From the $F$ and the $\overline{F}$ of $SU(N)$ we build the
fundamental of $\widetilde{SU}(N)$, which we’ll denote as
$(F\oplus\overline{F})$ (keeping in mind that it’s irreducible), which
requires reorganizing the fields as in (2.15) and (2.19). It is important to
note that the number of degrees of freedom doesn’t change with this
reorganization, as it should be with a discrete gauging.

There are two main aspects of the discussion of the lagrangian that will
become very important when finding the SW solution. The first is the global
symmetry. In $SU(N)+N_{f}\,F$ SYM, the global symmetry is $U(N_{f})$. As we
saw, after the discrete gauging, the meson matrix appearing in the quadratic
term of the superpotential becomes symmetric, and the global symmetry of
$\widetilde{SU}(N)+N_{f}\,(F\oplus\overline{F})$ is reduced to $USp(N_{f})$.
Since the SW solution is sensitive to the rank of the global symmetry, we will
use this as one of the checks of our geometries. The second concerns the
perturbative renormalization of the gauge coupling. As we just mentioned, the
gauge field sees only the Lie algebra, and as a result all Feynman rules and
loop diagrams are the same after the discrete gauging. This means that the
1-loop correction to the gauge coupling is identical in $SU$ and
$\widetilde{SU}$, which will fix for us the monodromy at infinity in the SW
curve. It also has the consequence that the condition for the vanishing of the
$\beta$-function is not modified, hence
$\widetilde{SU}(N)+\,2N\,(F\oplus\overline{F})$ is superconformal.

##  3 Discrete gauging: IR perspective

Now that we know the lagrangian, we are in a position to study the Seiberg-
Witten geometries for the theories with $\widetilde{SU}(N)$ gauge groups. We
will begin by giving, in Section 3.1, the general prescription, that is based
on considerations coming just from the rigidity of the IR effective theory (in
fact, they could be applied also to the non-Lagrangian case). We will then use
this method, in the following subsections, to obtain the curve and
differential for many discrete gaugings
$SU(N)+F\to\widetilde{SU}(N)+(F\oplus\overline{F})$, concentrating on the
cases $N=3,4,5$. Finally, in Section 3.6, we will give our general predictions
on the features of Seiberg-Witten curves for the generic
$SU(N)+F\to\widetilde{SU}(N)+(F\oplus\overline{F})$ discrete-gauging.

###  3.1 General strategy

There are two ingredients that we need in order to compute the SW solution for
$\widetilde{SU}(N)$ theories: the solution for the theory before the discrete
gauging (following the terminology of [32], we will sometimes refer to it as
the “parent” theory), and the action of the $\mathbb{Z}_{2}$ on the Coulomb
branch.
In what follows, we will denote with $r$ the rank of the considered theory,
with $\mathcal{M}_{\text{CB}}\cong\mathbb{C}^{r}$ the Coulomb branch of the
parent theory, and with $\widetilde{\mathcal{M}_{\text{CB}}}$ the Coulomb
branch of the daughter theory.

  * •

The SW solution of the $SU(N)$ theory with $N_{f}=2N$ fundamental flavors is
well known [65, 66],

$y^{2}=P_{N}^{2}(x)-f\prod_{j=1}^{2N}(x+g\mu+\hat{m_{i}}),\qquad\lambda=\Big{(}x+(g-1)\mu\Big{)}d\left[\log\left(\frac{P_{N}(x)-y}{P_{N}(x)+y}\right)\right]\,.$
(3.1)

Here, $(x,y)$ are the complex variables that parametrize the curve,
$\hat{m_{i}}$ and $\mu\equiv\frac{1}{2N}\sum_{i=1}^{2N}\hat{m_{i}}$ are the
masses of the fundamental flavours, $f=1-g^{2}$, $g=g(\tau)$ is a function of
the UV coupling $\tau$ (whose explicit form in terms of Jacobi
$\theta$-functions we won’t need, but we reproduce here for convenience),

$\displaystyle
g(\tau)=\frac{\theta_{2}(0,\tau)^{4}+\theta_{1}(0,\tau)^{4}}{\theta_{2}(0,\tau)^{4}-\theta_{1}(0,\tau)^{4}}\,,$
(3.2)

and $P_{N}(x)$ is the characteristic polynomial of the VEV of the adjoint
scalar in the vector multiplet that parametrizes the Coulomb branch,

$\displaystyle
P_{N}(x)=\text{det}\left(\mathds{1}x-\langle\Phi\rangle\right)\,.$ (3.3)

For convenience, we will often repackage the masses as $m_{i}\equiv
g\mu+\hat{m_{i}}$, with now $\sum_{i}m_{i}\neq 0$.
We will also need the SW data for the $SU(N)$ pure SYM theories [65, 66],

$y^{2}=P_{N}^{2}(x)-4\Lambda^{2N},\qquad\lambda=x\,d\left[\log\left(\frac{P_{N}(x)-y}{P_{N}(x)+y}\right)\right],$
(3.4)

with $\Lambda$ the dynamical scale related to the UV cutoff $\Lambda_{UV}$ and
the UV coupling $\tau_{UV}$ as

$\displaystyle\Lambda^{2N}=\Lambda_{UV}^{2N}e^{2\pi i\,\tau_{UV}}\,.$ (3.5)

  * •

The action $\Gamma:\mathcal{M}_{\text{CB}}\to\mathcal{M}_{\text{CB}}$ of the
outer automorphism on the Coulomb branch parameters follows in a
straightforward way from the analysis of the lagrangian in the previous
section. We define

$u_{n}\equiv\text{Tr}\left(\langle\Phi^{n}\rangle\right),\qquad
n=2,\dots,N\,,$ (3.6)

with $\Phi$ the adjoint scalar in the $\mathcal{N}=2$ vector multiplet. More
explicitly, this field transforms in the adjoint representation of the Lie
algebra of $SU(N)$. Therefore, the action of the $\mathbb{Z}_{2}$ is the
tangent map to the action on the Lie group (2.4),

$\displaystyle\Phi\to-\Phi^{T}\,.$ (3.7)

As a consequence, on the Coulomb branch, we have that

$u_{n}\to(-1)^{n}u_{n}\,.$ (3.8)

In what follows, we expect to be able to lift the action of $\Gamma$ on the
$\mathcal{M}_{\text{CB}}$ to an action $\widetilde{\Gamma}$ on the total space
$\mathcal{F}$ of the SW geometry:

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}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope
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}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.31387pt}{25.11642pt}\pgfsys@lineto{17.21114pt}{25.11642pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.2472pt}{6.34558pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\widetilde{\pi}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.98888pt}{-7.42528pt}\pgfsys@lineto{17.21114pt}{-7.42528pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.41112pt}{-7.42528pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.87361pt}{-3.32251pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{/\Gamma}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (3.9)

where $\widetilde{\mathcal{F}}$ and $\widetilde{\mathcal{M}_{\text{CB}}}$
denote, respectively, the total space of the SW geometry and the Coulomb
branch of the theory after the discrete gauging; the horizontal arrows are the
orbifold maps on $\mathcal{F}$ and $\mathcal{M}_{\text{CB}}$; and the vertical
arrows $\pi,\,\widetilde{\pi}$ are the projection of the SW families on the
Coulomb branches.

For $\Gamma$ to be a (spontaneously broken) symmetry on the CB, it must
preserve the special Kähler metric. Let’s see the consequences of this
statement. Let

$a_{i}(u)\equiv\int_{A_{i}(u)}\lambda(u),\qquad
a_{D,i}(u)\equiv\int_{B_{i}(u)}\lambda(u),\qquad i=1,...,N-1$ (3.10)

be the “special coordinates” on the CB, with $u$ denoting a point in the CB
and with $\left\\{A_{i}(u),B_{i}(u)\right\\}_{i=1}^{N-1}$ being a basis of
$H_{1}(\pi^{-1}(u),\mathbb{Z})$ with intersection product

$\displaystyle\langle A_{i},A_{j}\rangle=0,\quad\langle
B_{i},B_{j}\rangle=0,\quad\langle A_{i},B_{j}\rangle=\delta_{ij}\,.$ (3.11)

We can locally invert the expressions $a_{i}(u)$ to obtain $u(a_{i})$ and use
$a_{i}$ as (local) coordinates on the CB. In these coordinates, the special
Kähler metric reads $ds^{2}=\tau_{ij}\,da^{i}\,da^{j}_{D}$, with

$\tau_{ij}=\frac{\partial a_{D,j}\rvert_{u=u(a)}}{\partial a_{i}}\,,\quad
i,j=1,...,N-1$ (3.12)

being related to the effective couplings of the $U(1)^{r}$ theory on the CB.

Requiring that $\Gamma$ is an isometry of the CB therefore requires that the
periods (3.10) are preserved by its action. By Torelli’s theorem, this is
shown to imply that the fibers above $u$ and $\Gamma(u)$ must be isomorphic,

$\displaystyle\pi^{-1}(u)\cong\pi^{-1}\left(\Gamma(u)\right)\,,$ (3.13)

and that $\Gamma$ lifts to an automorphism $\widetilde{\Gamma}$ of
$\mathcal{F}$ (as in (3.9)).

In fact, the action of $\widetilde{\Gamma}$ on the fibral coordinates $x,y$
must be non-trivial. This can be understood as follows: if we are in a mass-
deformed phase of the SCFT point, $\Gamma$ will, in general, also exchange
points where different BPS states have vanishing mass. If the two states are
supported, respectively, on $\gamma\in H^{1}(\pi^{-1}(u),\mathbb{Z})$ and
$\gamma^{\prime}\in H^{1}(\pi^{-1}(\Gamma(u)),\mathbb{Z})$ (where
$\gamma\neq\gamma^{\prime}$) then the fibral action of $\widetilde{\Gamma}$
identifies $\gamma$ with $\gamma^{\prime}$, namely

$\widetilde{\Gamma}_{\ast}\gamma=\gamma^{\prime},$ (3.14)

with $\widetilde{\Gamma}_{\ast}:H_{1}(\pi^{-1}(u),\mathbb{Z})\to
H_{1}(\pi^{-1}\left(\Gamma(u)\right),\mathbb{Z})$ being the action induced by
$\widetilde{\Gamma}$ on the degree-one homologies.

As a final comment, we notice that, if the parent theory is conformal, in
order to preserve $\mathcal{N}=2$ supersymmetry, the $\widetilde{\Gamma}$
action has to commute with the $U(1)_{R}$ $R$-symmetry. In the context of SW
geometry, the $U(1)_{R}$ $R$-symmetry gets promoted to a quasi-homogeneous
$\mathbb{C}^{*}$ action, that acts with weights $[u_{n}]$ on the CB
coordinates and with weights $[x],[y]$ on the fibral coordinates. Asking
$\widetilde{\Gamma}$ to commute with $\mathbb{C}^{*}$ hence, means that
$[\Gamma(\cdot)]=[\cdot]$. This strongly restricts the possible actions in the
fibral coordinates. In general, a biholomorphic change of coordinates on
$\mathbb{C}^{3}\ni(x,y,u)$ looks like

$\widetilde{\Gamma}(x)=cx+f_{x}(y,u),$ (3.15)

with $c$ a complex constant, and similarly for $y$. One can easily see that
there are no polynomial combinations of $y,u$ that have the same $U(1)_{R}$
charge of $x$ and that can then appear at the r.h.s. of (3.15) if we ask
$\widetilde{\Gamma}(x)$ to have a well-defined $U(1)_{R}$ charge (and the same
argument holds for $\widetilde{\Gamma}(y)$). Finally, recall that the outer
automorphism is a $\mathbb{Z}_{2}$ symmetry, meaning that the action of
$\widetilde{\Gamma}$ must square to 1. This further restricts the constant $c$
to be $\pm 1$ (independently for $x$ and for $y$). In the end, we conclude
that

$\displaystyle\widetilde{\Gamma}(x,y,u_{n})=(\pm x,\pm y,(-1)^{n}u_{n})\,.$
(3.16)

Note that whilst our argument requires the theory to be conformal (so that the
$R$-symmetry is not anomalous), the form of the action (3.16) should also hold
for the non-conformal case. This is because one can introduce masses for the
hypermultiplets and integrate them out without disturbing the discrete action.

Let’s now proceed to compute the SW solution for $\widetilde{SU}(N)$ in
explicit examples. The task amounts to, firstly, extract the
$\widetilde{\Gamma}$ action in different cases; and secondly, compute the SW
data of the “daughter” theory by taking the quotient, namely

  * •

The new SW geometry,
$\widetilde{\mathcal{F}}\equiv\mathcal{F}/\widetilde{\Gamma}$, with canonical
projection $\widetilde{\pi}$ on the CB induced by the one of $\mathcal{F}$
(3.9).

  * •

The new SW differential $\widetilde{\lambda}$ obtained expressing $\lambda$ in
terms of the $\widetilde{\Gamma}$-invariant coordinates.

###  3.2 $\widetilde{SU}(3)$ Super Yang-Mills

We begin with the example of pure $\widetilde{SU}(3)$ SYM. The starting SW
data of the parent theory are the particularization of (3.4),

$y^{2}=P_{3}^{2}(x)-4\Lambda^{6},\qquad\lambda=x\,d\left[\log\left(\frac{P_{3}(x)-y}{P_{3}(x)+y}\right)\right]\,,$
(3.17)

with

$\displaystyle P_{3}(x)=x^{3}+u_{2}x+u_{3}$ (3.18)

and $\Lambda$ the dynamical scale. The action of $\Gamma$ on the CB parameters
is

$\displaystyle\Gamma(u_{2},u_{3})=(u_{2},-u_{3})\,.$ (3.19)

There is only one choice that preserves the SW data (3.17),

$\displaystyle\widetilde{\Gamma}(x,y,u_{2},u_{3})=(-x,y,u_{2},-u_{3})\,,$
(3.20)

such that $P_{3}$ picks up a sign that ends up cancelling in the SW curve and
differential.

To compute the quotient, we introduce the invariant coordinates

$a\equiv x^{2},\quad b\equiv xu_{3},\quad u_{6}\equiv u_{3}^{2}\,.$ (3.21)

and their relation

$au_{6}-b^{2}=0\,.$ (3.22)

We now have to express the SW data (4.16) in terms of the invariant
coordinates. The SW curve, expressed in terms of the invariant coordinates,
reads

$y^{2}=\left(a+u_{2}\right)\left(a\left(a+u_{2}\right)+2b\right)-4\Lambda^{6}+u_{6}\,.$
(3.23)

Equations (3.23) and (3.22) together define the full SW geometry for
$\widetilde{SU}(3)$ as a complete intersection inside
$\mathbb{C}^{5}\ni(y,a,b,u_{2},u_{6})$, with the canonical projection on the
CB being realized as the projection on $(u_{2},u_{6})$. Note that it is very
important to remember the second equation, as (3.23) alone looks like a genus
one curve at first glance.

In order to compute the SW differential, we can use that, for the $SU(3)$ SYM
case [67],

$\lambda=\frac{x}{y}dP_{3}=\frac{3x^{2}+u_{2}}{y}xdx=\frac{(3a+u_{2})}{2y}da\,,$
(3.24)

and hence can be promoted to a Seiberg-Witten differential for the discrete-
gauged curve:

$\widetilde{\lambda}=\frac{(3a+u_{2})}{2y}da\,.$ (3.25)

We can massage the result, solving (3.22) for $a$ and plugging the solution
into (3.23), obtaining

$y^{2}=\frac{\left(b^{3}+bu_{2}u_{6}+u_{6}^{2}\right)^{2}}{u_{6}^{3}}-4\Lambda^{6},$
(3.26)

where we can now easily see that for $u_{6}\neq 0$ we have a genus two curve
as expected. We note that (3.26) is _not_ in the canonical form
$y^{2}=\mathcal{P}(x,u_{j})$, with $\mathcal{P}(x,u_{j})$ a polynomial in
$(x,u_{j})$, but displays some poles at $u_{6}=0$. For the pure $SU(3)$ theory
(and for its daughter theory $\widetilde{SU}(3)$) this might not seem a big
issue: we indeed also have a point at infinity in the CB where other CB
coefficients diverge. However, as we will see in Section 3.3, this will also
happen in the conformal case, where there is no singularity at infinity.
Indeed, the pole signals the fact that the true SW geometry is a complete
intersection given by two equations in $\mathbb{C}^{5}$, not by one equation
in $\mathbb{C}^{4}$. We can only impose $a=\frac{b^{2}}{u_{6}}$ in the patch
$u_{6}\neq 0$ from (3.22).

###  3.3 $\widetilde{SU}(3)+6\,(F\oplus\overline{F})$

Let’s consider now the superconformal case. The SW data, setting to zero the
mass parameters $\mu$ and $m_{i}$ in (3.1) (for $N=3$), leads to333 The SW
curve for $SU(3)$ with six fundamental flavors is often written [68] in the
form,
$y^{2}+x^{6}+(v+ux)^{2}+x^{3}(v+ux)\tau,\qquad\lambda=v\frac{dx}{y}+u\frac{xdx}{y}.$
One can match the notation of [68] with ours by the following change of
coordinates:  $v_{\text{there}}=iu_{3},\quad
u_{\text{there}}=\frac{iu_{2}}{(f-1)^{1/6}},\quad\tau_{\text{there}}=\frac{2i}{\sqrt{f-1}},\quad
y_{\text{there}}=y,\quad x_{\text{there}}=(f-1)^{1/6}x$

$y^{2}=P_{3}^{2}(x)-fx^{6},\quad\lambda=xd\Bigg{[}\log\left(\frac{P_{3}(x)-y}{P_{3}(x)+y}\right)\Bigg{]},$
(3.27)

with again $P_{3}(x)=x^{3}+u_{2}x+u_{3}$. Once again, there is only one choice
of $\widetilde{\Gamma}$ that can preserve the SW data (3.27),

$\widetilde{\Gamma}(x,y,u_{2},u_{3})=(-x,y,u_{2},-u_{3})\,.$ (3.28)

We note an interesting difference between the rank 1 case [32] (which we recap
in appendix A) and our case. Here, (3.28) is an automorphism of the SW
geometry (3.27) even without imposing any tuning on the UV coupling $\tau$.
Indeed, the outer automorphism of the gauge group is a symmetry of the theory
for all the values of the UV coupling, rather than a duality that gets
enhanced to a symmetry for specific values of $\tau$.

Now we proceed as in the pure SYM case. The invariant coordinates are

$a\equiv x^{2},\quad b\equiv xu_{3},\quad u_{6}\equiv u_{3}^{2}\,,$ (3.29)

and satisfy

$au_{6}-b^{2}=0.$ (3.30)

We then express the SW geometry in terms of (3.29), obtaining

$y^{2}=-4a^{3}f+a^{3}+2u_{2}\left(a^{2}+b\right)+2ab+au_{2}^{2}+u_{6}.$ (3.31)

Equations (3.31) and (3.30) define the SW geometry of the daughter theory,
with the canonical projection on the Coulomb Branch base being the projection
over $(u_{2},u_{6})$.

We can again massage the expression for the SW geometry. We proceed first by
solving (3.30) to get $a=\frac{b^{2}}{u_{6}}$, and then we plug it in (3.31),

$y^{2}=\frac{b^{6}(1-4f)+2b^{4}u_{2}u_{6}+b^{2}u_{6}^{2}\left(2b+u_{2}^{2}\right)+2bu_{2}u_{6}^{3}+u_{6}^{4}}{u_{6}^{3}}.$
(3.32)

Let us now concentrate on the SW differential. In the parent theory, we can
rewrite it as

$\lambda=\frac{2u_{2}x+3u_{3}}{y}dx\,.$ (3.33)

This is already, up to numerical constants, in the canonical form of [68]. We
now want to obtain $\widetilde{\lambda}$, by expressing $\lambda$ in terms of
invariant coordinates. One can check that

$\widetilde{\lambda}=\lambda\left(x=x(a,b,u_{6}),u_{3}=u_{3}(a,b,u_{6})\right)=\frac{u_{2}+\frac{3b}{2a}}{y}da\,.$
(3.34)

We can massage (3.34) as well, imposing $a=\frac{b^{2}}{u_{6}}$, obtaining

$\widetilde{\lambda}=\frac{2bu_{2}}{u_{6}y}db+\frac{3}{y}db\,.$ (3.35)

We want now to express the result in coordinates where the SW differential
[68] is $\lambda=u_{2}Xdb/Y+u_{6}db/Y$, with the curve in the form
$u_{6}^{k}Y^{2}+P(b,u)=0$. This can be achieved uniquely by taking
$y=\frac{Y}{u_{6}}$. In these coordinates, the curve takes the nicer, yet not
“polynomial” form

$Y^{2}=\frac{b^{6}(1-4f)+2b^{4}u_{2}u_{6}+b^{2}u_{6}^{2}\left(2b+u_{2}^{2}\right)+2bu_{2}u_{6}^{3}+u_{6}^{4}}{u_{6}}\,.$
(3.36)

where we highlighted in red the non-canonical coefficient $u_{6}$ on the
r.h.s. of the equation. This is responsible for the new singularity in the CB
after the discrete gauging that we advertised in the introduction, and which
we’ll discuss at lenght in Section 4.

It turns out that the presence of this coefficient is good news. In the
classification of Argyres and Martone [39] there is no theory with rank 3
global symmetry (as we expect from the lagrangian analysis from Section 2; and
will check momentarily in the curve) and CB parameters of scaling dimension
$\left\\{2,6\right\\}$. But that reference restricts to geometries such that
the coefficients $f_{i}(u)$ in the r.h.s. of the SW curve in canonical form
are polynomials of the CB parameters. This coefficient, leading to a pole in
the r.h.s., is what allows us to escape the the classification of [39]. We
hence see that the discrete gauging of charge conjugation in $SU(3)+6$ flavors
is a new rank-two SCFT beyond the classification of [39], and more generally,
that there can be rank 2 SCFTs with non-polynomial coefficients.

Let’s now compute the number of flavor parameters of the discrete-gauged
theory from the IR. The mass-deformed curve before the discrete gauging is
(3.1), which we recall here for convenience:

$y^{2}=P_{3}^{2}(x)-f\prod_{i=1}^{6}(x-m_{i})\,.$

We then see that requiring that (3.28) is a symmetry selects just the even
terms in the product $\prod_{i=1}^{6}(x-m_{i})$, and we are left with three
mass deformations. In particular, the degree-one mass Casimir invariant
$\mu=\frac{1}{6}\sum_{i=1}^{6}m_{i}$ has to be set to zero, which is the IR
avatar of the fact that the $\mathbb{Z}_{2}$ outer automorphism acts non-
trivially on the $\mathfrak{u}(1)$ center factor of the flavor group. It is
important to remark that $\mu\to 0$ is also important to get an invariant SW
differential in the mass-deformed theory. We can understand better the final
flavor group by noticing that setting to zero $\sum_{i=1}^{6}m_{i}$ reduces
first the flavor group from $\mathfrak{u}(6)\to\mathfrak{su}(6)$. Then,
setting to zero the odd-degree Casimirs of $\mathfrak{su}(6)\cong A_{5}$ leads
us to a Lie algebra compatible with $C_{3}$ (or $D_{3}$). This is precisely
the flavour symmetry we expected from the lagrangian, which constitutes one of
the checks of our computation.

###  3.4 $\widetilde{SU}(4)+8\,(F\oplus\overline{F})$

Let’s now increase the rank and compute the SW curve for the SCFT with gauge
group $\widetilde{SU}(4)$. The SW curve and differential of the parent theory
are

$\displaystyle
y^{2}=P_{4}^{2}(x)-fx^{8},\quad\lambda=x\,d\Bigg{[}\log\left(\frac{P_{4}(x)-y}{P_{4}(x)+y}\right)\Bigg{]}\,,$
(3.37)

where

$\displaystyle P_{4}(x)=x^{4}+u_{2}x^{2}+u_{3}x+u_{4}\,.$ (3.38)

The action of the $\mathbb{Z}_{2}$ on the base is

$\displaystyle\Gamma(u_{2},u_{3},u_{4})=(u_{2},-u_{3},u_{4})\,.$ (3.39)

We notice that the action of the fibral coordinates cannot be the same as in
$\widetilde{SU}(3)$. Flipping the sign of the coordinate $x$ now leads to
$P_{4}$ being invariant and the SW differential getting an overall sign. In
order to compensate it, also $y$ needs to pick a sign so that (3.37) is
preserved,

$\displaystyle\widetilde{\Gamma}(x,y,u_{2},u_{3},u_{4})=(-x,-y,u_{2},-u_{3},u_{4})\,.$
(3.40)

From this point we proceed analogously to the rank 2 case. The ring of
invariant coordinates has generators,

$\displaystyle a_{1}$ $\displaystyle\equiv$ $\displaystyle x^{2},\quad
a_{2}\equiv y^{2},\quad a_{3}\equiv xy,$ $\displaystyle u_{6}$
$\displaystyle\equiv$ $\displaystyle u_{3}^{2},\quad b_{1}\equiv xu_{3},\quad
b_{2}\equiv yu_{3}\,.$ (3.41)

To find all the relations between them, we first build the matrix

$\mathcal{M}=\left(\begin{array}[]{ccc}a_{1}&a_{3}&b_{1}\\\
a_{3}&a_{2}&b_{2}\\\ b_{1}&b_{2}&u_{6}\\\ \end{array}\right)\,,$ (3.42)

and then set all its $2\times 2$ minors equal to zero,

$\text{Min}_{2\times 2}(\mathcal{M})=0\,.$ (3.43)

We also have the equation (3.37), which in terms of (3.4) becomes

$\displaystyle
a_{2}+a_{1}^{4}(f-1)=2a_{1}^{3}u_{2}+2a_{1}u_{2}(b_{1}+u_{4})+(b_{1}+u_{4})^{2}+a_{1}^{2}(2b_{1}+u_{2}^{2}+2u_{4})\,.$
(3.44)

Together with (3.43), (3.44) defines the SW geometry of $\widetilde{SU}(4)$.
We note that the SW geometry is of complex dimension four, and is not a
complete intersection inside the ambient space
$\mathbb{C}^{8}\ni(a_{1},a_{2},a_{3},b_{1},b_{2},u_{6},u_{2},u_{4})$ since the
number of equations that identify it is bigger than its codimension in
$\mathbb{C}^{8}$.

Lastly, the SW differential before the discrete gauging can be put into the
following form,

$\displaystyle\lambda=-\left(8u_{4}+6u_{3}x+4x^{2}u_{2}\right)\frac{dx}{y}\,.$
(3.45)

Note that, as expected, all the terms are invariant under $\widetilde{\Gamma}$
(3.40). Therefore,

$\displaystyle\widetilde{\lambda}=-\left(4u_{4}+3b_{1}+2u_{2}a_{1}\right)\frac{da_{1}}{a_{3}}\,,$
(3.46)

where we have used that, from the relations (3.43), $dx/y=da_{1}/2a_{3}$.

We finish with three remarks. First, note that while in this case we have not
given the expression for the curve after trying to plug the solutions of
(3.43) into (3.44), it is clear that doing so would lead to pole
singularities, similarly to the $\widetilde{SU}(3)$ case. Second, notice that
the lift of the action of the $\mathbb{Z}_{2}$ to the full SW fibration in
this case includes flipping also the sign of the coordinate $y$, which results
in the quotient geometry involving several more generators and relations. As
mentioned in the introduction, this is one of the reasons that make difficult
obtaining a closed form expression for the SW solution of $\widetilde{SU}(N)$
for generic $N$. It is easy to convince oneself that the different action on
the coordinate $y$ will depend on whether $N$ is even or odd. And last, we see
that, as in the $\widetilde{SU}(3)$ case, the flip of the sign of the
coordinate $x$ prevents us from having mass deformations corresponding to odd-
degree Casimirs of the flavour algebra. This is compatible with either the
$C_{4}$ or $D_{4}$ subalgebras of the original $A_{7}$, which is consistent
with the flavour symmetry of the lagrangian $USp(8)$.

###  3.5 $\widetilde{SU}(5)+10\,(F\oplus\overline{F})$

Next, we will repeat the analysis we just outlined for $SU(4)$ and $SU(3)$
groups for the $SU(5)+10F$ theory. This case is particularly interesting since
the CB of the discretely gauged theory is not freely generated and displays
complex structure singularities. Indeed, the CB parameters of the parent
theory are the Casimir invariants $u_{2},u_{3},u_{4},u_{5}$ of the UV theory
adjoint complex scalar. The discrete gauging leaves even-degrees Casimir
invariant, while it acts on $u_{3}$ and $u_{5}$ as

$u_{3}\to-u_{3},\qquad u_{5}\to-u_{5}.$ (3.47)

The new CB, after the discrete gauging, is hence described by five invariant
coordinates $u_{2},u_{4},u_{6}\equiv u_{3}^{2},u_{10}\equiv
u_{5}^{2},u_{8}\equiv u_{3}u_{5}$ satisfying

$u_{10}u_{6}-u_{8}^{2}=0\,.$ (3.48)

We see that (3.48) is the equation of an $A_{1}$ Du Val singularity, and the
CB of the daughter theory is, as complex manifold, $\mathbb{C}^{2}\times
A_{1}$, where $u_{2},u_{4}$ span the $\mathbb{C}^{2}$ factor.

We now want to obtain the SW solution for the discrete gauging of the charge
conjugation of $SU(5)+10F$. We will proceed as in the previous cases, first
lifting (uniquely) the charge conjugation action to the SW geometry of the
parent theory $\mathcal{F}$, and then performing the quotient to obtain
$\widetilde{\mathcal{F}}$. The SW solution for $SU(5)+10F$ is

$y^{2}=P_{5}^{2}(x)-fx^{10},\quad\lambda=xd\left[\log\left(\frac{P_{5}(x)-y}{P_{5}(x)+y}\right)\right]\,,$
(3.49)

with

$\displaystyle P_{5}(x)=x^{5}+u_{2}x^{3}+u_{3}x^{2}+u_{4}x+u_{5}.$ (3.50)

The only possible lift of the action $\Gamma$ to $\mathcal{F}$ is

$\widetilde{\Gamma}(x,y,u_{2},u_{3},u_{4},u_{5})=(-x,y,u_{2},-u_{3},u_{4},-u_{5})\,.$
(3.51)

The ring of invariant coordinates is generated by

$\displaystyle a_{1}$ $\displaystyle\equiv$ $\displaystyle x^{2},\quad
b_{1}\equiv xu_{3},\quad b_{2}\equiv xu_{5},$ $\displaystyle u_{6}$
$\displaystyle\equiv$ $\displaystyle u_{3}^{2},\quad u_{10}\equiv
u_{5}^{2},\quad u_{8}\equiv u_{3}u_{5}.$ (3.52)

The relations satisfied by (3.5) are found by first constructing the following
matrix,

$\mathcal{M}=\left(\begin{array}[]{ccc}a_{1}&b_{1}&b_{2}\\\
b_{1}&u_{6}&u_{8}\\\ b_{2}&u_{8}&u_{10}\\\ \end{array}\right)\,,$ (3.53)

and then setting to zero its two by two minors,

$\text{Min}_{2\times 2}(\mathcal{M})=0.$ (3.54)

We note that (3.48) corresponds to the relation associated to the minor
obtained by removing the first row and the first column of (3.53).

As in the previous cases, the next step is to express the SW geometry
$\mathcal{F}$ in terms of the invariants coordinates (3.5):

$\displaystyle y^{2}=$ $\displaystyle
a_{1}^{3}\left(2b_{1}+u_{2}^{2}+2u_{4}\right)+a_{1}^{2}\left(2u_{2}\left(b_{1}+u_{4}\right)+2b_{2}+u_{6}\right)$
$\displaystyle+a_{1}\left(2b_{1}u_{4}+2b_{2}u_{2}+u_{4}^{2}+2u_{8}\right)+a_{1}^{5}(-(f-1))+2a_{1}^{4}u_{2}+2b_{2}u_{4}+u_{10}$
(3.55)

The equation (3.5), together with (3.54), describes the SW geometry
$\widetilde{\mathcal{F}}$ of the daughter theory
$\widetilde{SU}(5)+10(F\oplus\overline{F})$. The projection on the Coulomb
branch basis
$\widetilde{\pi}:\widetilde{\mathcal{F}}\twoheadrightarrow\widetilde{\mathcal{M}_{\text{CB}}}$
is realized as the projection on the $u_{n}$ coordinates, that satisfy the
non-trivial relation (3.48). If we restrict ourselves to a Zariski-open subset
of $\widetilde{\mathcal{M}_{\text{CB}}}$, we can obtain an expression for the
SW geometry that explicitly shows that the genus of the SW curve is four. In
the $u_{6}\neq 0$ open set of $\widetilde{\mathcal{M}_{\text{CB}}}$, we find

$\displaystyle y^{2}$ $\displaystyle=$
$\displaystyle\frac{b_{1}^{10}(1-f)+2b_{1}^{8}u_{2}u_{6}+2b_{1}^{7}u_{6}^{2}+b_{1}^{6}\left(u_{2}^{2}+2u_{4}\right)u_{6}^{2}+2b_{1}^{5}u_{2}u_{6}^{3}}{u_{6}^{5}}+$
$\displaystyle\frac{b_{1}^{4}u_{6}^{3}\left(2b_{2}+2u_{2}u_{4}+u_{6}\right)+2b_{1}^{3}u_{4}u_{6}^{4}+b_{1}^{2}u_{6}^{4}\left(2b_{2}u_{2}+u_{4}^{2}+2u_{8}\right)}{u_{6}^{5}}+2b_{2}u_{4}+u_{10}$

with the Coulomb branch parameters $u_{n}$ satisfying (3.48).

Let’s now concentrate on the SW differential: one can rewrite it, using that
$y^{2}=P_{5}^{2}-fx^{10}$ and hence $dy=d(P_{5}^{2}+fx^{10})/2y$, as

$\lambda=2(-2x^{5}+x^{2}u_{3}+2xu_{4}+3u_{5})\frac{dx}{y}\,.$ (3.57)

Once again, all the terms appearing in the rightmost side of (3.57) are
explicitly invariant under $\widetilde{\Gamma}$. Consequently, we can easily
re-express $\lambda$ in terms of the invariant coordinates (3.5) obtaining in
this way the SW differential $\widetilde{\lambda}$ of the daughter theory:

$\widetilde{\lambda}=\left(2a_{1}u_{2}+3b_{1}+4u_{4}+5b_{2}/a_{1}\right)da_{1}\,.$
(3.58)

All together, (3.58), (3.53) and (3.5) define the SW solution of the discrete
gauging of the outer automorphism of $SU(5)+10$ flavors. This is the first
explicit example of a SW solution for a 4d $\mathcal{N}=2$ theory with non-
freely generated Coulomb branch!

We can conclude this section by performing, as in the previous cases, the
counting of the mass parameters of the daughter theory. The mass-deformed
curve of $SU(5)+10F$ is

$y^{2}=P_{5}^{2}(x)-f\prod_{i=1}^{10}(x-m_{i})=P_{5}^{2}(x)+Q(x,m).$ (3.59)

We see from (3.59) that $\widetilde{\Gamma}$ extends to an automorphism on the
of the extended Coulomb branch when $Q(-x,m)=Q(x,m)$. Consequently, we have to
set to zero all the odd-degree mass Casimir invariants of the starting flavor
symmetry, selecting in this way a $C_{5}$ subalgebra of the original
$\mathfrak{su}(10)\cong A_{9}$ flavor algebra. This is, as in the previous
cases, in agreement with the computation we performed in Section 2.

###  3.6 Higher ranks

Even if we are not able to provide a closed form expression for
$\widetilde{SU}(N)$, we can comment on the general structure of the discrete
gauging procedure for generic rank. As we saw, the action on the coordinates
$(x,y,u_{n})$ can be extracted just with geometric methods, but we can, a
posteriori, find a motivation for it. Recall the action of the outer
automorphism on the UV adjoint scalar $\Phi\to-\Phi^{T}$ and the generic form
of the SW solution for $SU(N)+N_{f}F$ [65],

$y^{2}=P_{N}^{2}(x)-4\Lambda^{2N-N_{f}}\prod_{i=1}^{N_{f}}(x-m_{i}),\qquad\lambda=x\,d\left[\log\left(\frac{P_{N}(x)-y}{P_{N}(x)+y}\right)\right]\,,$
(3.60)

with $P_{N}(x)=\text{det}(\mathds{1}x-\Phi)$ and (3.1) for the conformal case.
Consequently, since [69] $x$ has to transform as444It would be interesting to
apply our geometric techniques to the 5d discrete gauging setup of [70, 71,
72, 69]. In that case, we are interested in the Higgs branch of the theory
obtained by putting M-theory on a Calabi-Yau space. Similarly to our setup,
the physics on the 5d moduli space of vacua can be understood in terms of the
complex deformation of a geometric space (in that case the Calabi-Yau
variety). the eigenvalues of $\Phi$, and hence as $\Phi$, we have
$[x]=[\Phi]$, and hence we must always have $x\to-x$. Furthermore, we have
$u_{n}\propto\langle\Phi^{n}\rangle$ and hence the discrete-gauging acts as
$u_{j}\to(-1)^{n}u_{n}$. This leads to $P_{N}\to(-1)^{N}P_{N}$. By looking to
(3.60), we see that the curve is already preserved by this action, but, to
make the SW differential invariant w.r.t. to $\widetilde{\Gamma}$, we need to
impose $y\to(-1)^{N+1}y$.

For all $N$, we then reflect at least two coordinates among $x,y,u_{n}$, and
hence in general the final SW geometry is not a hypersurface equation, with
the Coulomb branch not freely generated and displaying complex structure
singularities for $N\geq 5$. Curiously, we notice that for $N\geq 7$, the
Coulomb branch of the daughter theory can be non-Gorestein (e.g., isomorphic
to $\mathbb{C}^{3}\times\frac{\mathbb{C}^{3}}{\mathbb{Z}_{2}}$, with the
$\mathbb{Z}_{2}$ acting as $-\mathds{1}$ on the coordinates) in the particular
case of $SU(7)$.

We conclude with a remark on the flavor symmetry: in all the cases, $x\to-x$
forces the odd-degree symmetric polynomials of the masses appearing in the
expansion of $\prod_{i=1}^{N_{f}}(x-m_{i})$ to be zero. If $N$ is odd, this
automatically leads to a $C_{N_{f}/2}$ flavour symmetry. If $N$ is even, this
criteria cannot be used to distinguish between the subalgebras $C_{N_{f}/2}$
or $D_{N_{f}/2}$. In principle, one could get more information by studying the
explicit action of $\mathbb{Z}_{2}$ on the Weyl group of the flavor symmetry,
but we postpone this analysis to future work.

These differences between $N$ even and odd are reminiscent of the two
possibilities for the discrete gauging of the outer automorphism for even $N$
studied in [24]. The two disconnected gauge theories $\widetilde{SU}(2N)_{I}$
and $\widetilde{SU}(2N)_{II}$ have precisely a $USp(N_{f})$ or $SO(N_{f})$
global symmetry respectively. It would be interesting to understand if this
apparent correspondence is coincidental or there is a deeper reason for it.

##  4 BPS Spectra

In this section, we will comment on the BPS spectra of the analyzed theories,
furnishing a complete computation in the $\widetilde{SU}(3)$ SYM case. We will
start, in Section 4.1 briefly reviewing how to compute the generators of the
BPS spectrum of a 4d $\mathcal{N}=2$ theory from its SW solution. Then, in
Section 4.2, we will list our general results for the BPS spectra of
$\widetilde{SU}(3)+N_{f}(F\oplus\overline{F})$, discovering the existence of a
new singularity $u_{fixed}\in\widetilde{\mathcal{M}_{\text{CB}}}$ at the
points that are images of fixed points of $\Gamma$. We will then explicitly
check our results, in Section 4.3 computing the spectrum of
$\widetilde{SU}(3)$ SYM. Finally, in Section 4.4, with a general argument
based upon the monodromies of the IR gauge coupling, we will predict the
existence of singularities analogous to $u_{fixed}$ in the, up to mild
assumptions, general situation of a discrete gauging of a $\mathbb{Z}_{k}$
symmetry of a generic, non necessarily lagrangian, 4d $\mathcal{N}=2$ theory.

###  4.1 BPS charges from the Seiberg-Witten solution

In Seiberg-Witten theory, a BPS particle is identified with particular
elements $\nu$ of $H_{1}(\pi^{-1}(u),\mathbb{Z})$. Let’s fix a symplectic
system of generators $\left\\{A_{i},B_{i}\right\\}_{i=1}^{r}$ of
$H_{1}(\pi^{-1}(\mathbf{u}),\mathbb{Z})$, satisfying (3.11); then, the central
charge of the BPS particle associated with the cycle
$\nu=\sum_{i=1}^{r}g_{i}B_{i}+q_{i}A_{i}$ is

$\mathcal{Z}(\nu)=\int_{\nu}\lambda=\sum_{i=1}^{r}g_{i}a_{D,i}+q_{i}a_{i}\,,$
(4.1)

with $g_{i},q_{i}$ valued in $\mathbb{Z}$ being, respectively, the magnetic
and electric charges of the particle. For a BPS particle, we have the
condition

$m(\nu)=\absolutevalue{Z(\nu)}\,,$ (4.2)

with $m(\nu)$ the mass of the particle. Consequently, the vanishing of
$Z(\nu)$ selects a complex codimension-one locus on the Coulomb branch where
the considered particle becomes massless. The union of all such loci coincides
with the singular locus $\mathcal{S}$ of the Coulomb branch special metric,
and $(a_{D,i},a_{i})$ become bad coordinates when we approach $\mathcal{S}$.
Physically, this is due to the fact that, along the naive RG flow, we have
integrated out at least one particle whose mass, in an analytic neighborhood
of the singularity, is comparable to the one of the Coulomb branch modes.

More precisely, let $\mathcal{S}_{j}$ be one of the irreducible components of
$\mathcal{S}\subset\mathbb{C}^{r}$, where the particle associated to a cycle
$\nu_{j}\in H_{1}(\pi^{-1}(u),\mathbb{Z})$ becomes massless. Let
$\gamma_{j}\in\pi_{1}(\mathbb{C}^{r}\setminus\mathcal{S})$ be a path that goes
around $\mathcal{S}_{j}$ once, without linking with any other component
$S_{k}\neq S_{j}$. Then, $(a_{D,i},a_{i})$ develop the following monodromy
along $\gamma_{j}$,

$(a_{D,i},a_{i})^{T}\longrightarrow M(\gamma_{j})(a_{D,i},a_{i})^{T}\,.$ (4.3)

Here, we are picking a choice of convention, which matches the one of [73],
such that the monodromy matrix acts on the charges from the right,

$\displaystyle(g_{i},q_{i})\to(g_{i},q_{i})M(\gamma_{j})\,,$ (4.4)

where $M(\gamma_{j})\in Sp(2r,\mathbb{Z})$555Physically, the condition that
$M(\gamma_{j})$ lies in the symplectic group is due to the fact that (4.6)
realizes an electro-magnetic duality in the IR theory and the Dirac pairing
$\langle\nu_{1},\nu_{2}\rangle\equiv(\mathbf{g}_{1},\mathbf{q}_{1})^{T}\left(\begin{array}[]{cc}0&\mathds{1}\\\
-\mathds{1}&0\\\ \end{array}\right)(\mathbf{g}_{2},\mathbf{q}_{2})$ (4.5) must
be preserved by such transformations.. The monodromy matrix associated to a
particle of magnetic and electric charges
$(\mathbf{g},\mathbf{q})=(g_{i},q_{i})$, with $i=1,...,r$, is [73]

$M(\nu)=\left(\begin{array}[]{cc}\mathds{1}_{r}-\mathbf{q}\otimes\mathbf{g}&-\mathbf{q}\otimes\mathbf{q}\\\
\mathbf{g}\otimes\mathbf{g}&\mathds{1}_{r}+\mathbf{g}\otimes\mathbf{q}\\\
\end{array}\right)\,.$ (4.6)

From the perspective of the SW geometry, (4.6) is re-obtained by noticing that
$(a_{D,i},a_{i})$ are the periods of the Seiberg-Witten differential
$\lambda$, that is a holomorphic one-form along $\pi^{-1}(u)$, varying
holomorphically (hence without monodromies) on the Coulomb branch.
Consequently, the monodromies of the $\lambda$ periods, as functions of the
Coulomb branch coordinates $u$, must come from the monodromies of our choice
of basis $\\{B_{i}(u),A_{i}(u)\\}$ on the Coulomb branch. The theory
regulating the monodromies of a choice of basis $\\{B_{i}(u),A_{i}(u)\\}$ of
the mid-dimensional homology in a family $\mathcal{F}$ of deformed varieties
is well known [74], and is regulated by the so-called Picard-Lefschetz
formula.

Let us recall the Picard-Lefschetz formula for the case in which the fiber of
the SW geometry is a complex curve. Let $\nu_{j}\in
H_{1}(\pi^{-1}(u),\mathbb{Z})$ be a one-cycle vanishing along a component
$\mathcal{S}_{j}\subset\mathcal{S}\subset\mathbb{C}^{r}$. Then, for each
$\beta\in H_{1}(\pi^{-1}(u),\mathbb{Z})$, we have the following monodromy
transformation going once around $\gamma_{j}$,

$\beta\to\beta-\langle\beta,\nu\rangle\nu\,.$ (4.7)

Equation (4.7) then induces a linear action on $H_{1}(\pi^{-1}(u),\mathbb{Z})$
that exactly coincides with the one of (4.6).

It might also be the case that, along $\mathcal{S}_{j}$, many BPS particles
become massless at the same time. If this happens, the monodromy is
$M(\mathcal{S}_{j})=\prod_{k=1}^{n}M(U_{k})$, with $k=1,...,n$ indexing the
BPS particles simultaneously massless on $\mathcal{S}_{j}$. In this case, we
can not use the Picard-Lefschetz formula (4.7), but we are still able, by
carefully tracking the motion of the one-cycles of the curve (e.g. using the
code of [75]), to obtain the action $M(\mathcal{S}_{j})$ on
$H_{1}(\pi^{-1}(u),\mathbb{Z})$. More explicitly, in our case all the SW
curves are written in hyperelliptic form $y^{2}=R_{2N}(x,u_{j},m_{i})$, with
$R_{2N}$ a polynomial in $x$ with coefficients depending on masses and Coulomb
branch parameters. Consequently, the projection on the $x$ coordinates induces
a double cover of the projective line $\mathbb{P}^{1}\supset\mathbb{C}\ni x$
and the roots $x_{k}(u_{j},m_{i})$ of $R_{2N}$ are the branching points of the
double cover. We can then construct the one-cycles of the curve by fibrering
the solutions $y_{\pm}=\pm\sqrt{R_{2N}}$ over a real path joining two of the
$x_{k}$. Therefore, we identify a vanishing cycle from requiring that (at
least) a pair of roots $x_{k}$ coincide.

In practice, one picks a generic complex line $\mathcal{L}$ through a
basepoint on the Coulomb branch (which is to say fixing a value for $u_{j}$)
and then computes the monodromies around the points in
$\mathcal{L}\cap\mathcal{S}$. Using the result of Van Kampen and Zariski (see
e.g. [76]), if the slice is generic, we can then compute a system of
generators for the BPS particles with the following procedure:

  1. 1.

Identify the vanishing cycles on top of the considered point
$u_{\text{sing}}\in\mathcal{L}\cap\mathcal{S}$ by looking to which branching
points of the hyperelliptic curve double cover collide for
$u=u_{\text{sing}}$.

  2. 2.

Compute the monodromy for the other cycles using the Picard-Lefschetz formula
(4.7), or by brute force, e.g. using the code of [75] to visualize the
monodromies of the hyperelliptic double cover branching points.

  3. 3.

Compute the charges of the BPS particle becoming massless at the points in
which $\mathcal{L}$ intersects $\mathcal{S}$ by comparing with (4.6).

  4. 4.

Select a system of generators (over $\mathbb{Z}$) out of the aforementioned
charges.

The monodromies that found on $\mathcal{L}$ by this procedure must satisfy the
following topological condition:

$M_{N}...M_{1}=M_{loop},$ (4.8)

with $M_{loop}$ being the monodromy computed on a large anticlockwise loop
inside $\mathcal{L}$ starting from some basepoint $u_{b}$. The matrices in the
l.h.s. of (4.8) are ordered according to the sequence in which they get
touched by the anticlockwise rotation of a semi-infinite ray starting at the
basepoint. This condition, which we have presented in the mathematical
language (as it applies to any hyperelliptic curve, regardless of the
context), has the simple physical interpretation that the monodromy picked up
when going around all the singularities on the Coulomb branch has to match the
monodromy at infinity computed from the 1-loop $\beta$-function.

###  4.2 BPS spectra and charge conjugation: general results

In this section, we gather the general results for the BPS spectra of the
$\mathbb{Z}_{2}$ gauging of $SU(N)+F$ flavors, which we will later verify
explicitly in the example of $SU(3)$ in Section 4.3.

We remark that all the results in this section are based purely on the
geometry of the SW solution. Hence, working with the assumption that the
$\mathbb{Z}_{2}$ action on the Coulomb branch is accompanied by a
$\mathbb{Z}_{2}$ action on the fiber, we believe that our results can be
applied also to a more generic gauging of an abelian zero-form symmetry. We’ll
use this insight to extend our results to the case of a $\mathbb{Z}_{k}$
gauging of a generic 4d $\mathcal{N}=2$ theory in Section 4.4.

Let $\mathcal{L}$ be a slice on the Coulomb branch $\mathcal{M}_{\text{CB}}$
of the parent theory. The summary of our results is as follows:

  * •

For $SU(N)+F$ theories [73, 67], if $(g_{i},p_{i})$ and
$(g_{i}^{\prime},p_{i}^{\prime})$ are the charges of two particles becoming
massless at two points $p,p^{\prime}=\Gamma(p)$ of the parent theory Coulomb
branch, then

$M_{p}=V_{N}M_{p^{\prime}}V_{N},\qquad V_{N}^{2}=\mathbbm{1}_{2r}$ (4.9)

with $V_{N}$ the electromagnetic duality realizing the $\mathbb{Z}_{2}$ action
on the parent theory at the level of the SW fiber.

  * •

We expect that, if a BPS particle becomes massless at a point
$\widetilde{p}\in\widetilde{\mathcal{M}_{\text{CB}}}$ whose preimages are
$p,p^{\prime}=\Gamma(p)$, then either

$M_{p}=M_{\widetilde{p}}\quad\text{or}\quad
M_{\widetilde{p}}=M_{p^{\prime}}=V_{N}M_{\widetilde{p}}V_{N},$ (4.10)

where, for the rightmost equality, we used (4.9). This is because going once
around $\widetilde{p}$ corresponds, as observed in [32], to go once around,
say $p$. Consequently, the monodromies are identified up to the action of the
electromagnetic duality $V_{N}$ (that signals the fact that we could have
picked $p^{\prime}$ instead of $p$ as the preimage of $\widetilde{p}$).

  * •

If the considered slice $\mathcal{L}$ is an invariant subset of the parent
theory Coulomb branch under the $\mathbb{Z}_{2}$ action, from the 1-loop
computation we must have

$M_{loop,daughter}^{2}=M_{loop,parent}.$ (4.11)

as a consequence of the squaring of the Coulomb branch coordinates.

  * •

If $u_{0}\in\mathcal{M}_{\text{CB}}$ is a fixed point of the $\mathbb{Z}_{2}$
action, it gains a monodromy after the discrete gauging even if there is no
BPS particle becoming massless at $u_{0}$ in the parent. The monodromy is
given by $M_{\text{fixed}}=V_{N}$.666At this point we are asuming that the
overall choice of duality frame between the parent and daughter theory is the
same. If that is not the case and they are related by some $B\in
Sp(2r,\mathbb{Z})$ then we would have $M_{\text{fixed}}=BV_{N}B^{-1}$ We prove
this as follows. If the line $\mathcal{L}_{*}$ of the parent theory inherits
the $\mathbb{Z}_{2}$ action (and hence has a fixed point), we can split its
intersections with $\mathcal{S}$ into two sets of points:
$u_{*,1},...,u_{*,n}$ and $-u_{*,1},...,-u_{*,n}$. The corresponding
monodromies are, by (4.9), $M_{1},\dots,M_{n}$ and
$V_{N}M_{1}V_{N},\dots,V_{N}M_{n}V_{N}$. Consequently, using (4.8), we have

$\displaystyle V_{N}M_{1}\cancel{V_{N}V_{N}}M_{2}V_{N}\cdots
V_{N}M_{n}V_{N}M_{1}M_{2}\cdots M_{n}$
$\displaystyle=\left(V_{N}M_{1}M_{2}\cdots
M_{n}\right)\left(V_{N}M_{1}M_{2}\cdots M_{n}V_{N}\right)$
$\displaystyle=M^{2}=M_{loop,parent}\,,$ (4.12)

with $M\equiv V_{N}M_{1}M_{2}...M_{n}$, with $M_{loop,parent}$ being the
monodromy around a big anticlockwise loop inside $\mathcal{L}_{*}$ and we
remark that we used that $V_{N}^{2}=\mathbbm{1}$. By combining (4.11) and (• ‣
4.2), we must have

$M_{loop,daughter}=M=V_{N}M_{1}M_{2}\cdots M_{n}.$ (4.13)

We can now consider the topological condition (4.8) for the monodromies in the
daughter theory, obtaining

$M_{\text{fixed}}M_{1}\cdots M_{n}=M_{loop,daughter}$ (4.14)

where, up to a global choice of electromagnetic duality frame, exactly the
same monodromies $M_{j}$, $j=1,\dots,N$ of the parent theory appear due to
(4.10). By inverting (4.14), and using (4.13), we find

$M_{\text{fixed}}=V.$ (4.15)

Indeed, the preimage $u_{0}$ of $u_{\text{fixed}}$ under the quotient map is a
smooth point of $\mathcal{M}_{\text{CB}}$, hence, in the parent theory,
$M_{u_{0}}=\mathds{1}$. Consequently, using the argument of [32], we must have
$M_{\text{fixed}}^{2}=\mathds{1}_{r}$. The interesting fact is that we do not
need, and indeed we must not pick the trivial choice
$M_{\text{fixed}}=\mathds{1}_{r}$. This predicts the existence of a
singularity analogous to $(u_{\text{fixed}},M_{\text{fixed}})$ also for the
$\mathbb{Z}_{k}$ gauging of a zero-form symmetry of a general (even non-
lagrangian) 4d $\mathcal{N}=2$ theory in Section 4.4.

  * •

The last situation that may occur is when a fixed point of the parent theory
is inside $\mathcal{S}$. We postpone the general analysis of the monodromies
of these points to future work.

In particular, (4.10) permits to say that the generators of the BPS quiver
computed from a generic slice of the Coulomb branch are preserved by the
$\mathbb{Z}_{2}$ quotient.

###  4.3 Explicit example: BPS spectrum of $\widetilde{SU}(3)$

In this section, we will compute explicitly the BPS spectrum of the theory
$\widetilde{SU}(3)$. We recall, for convenience, the SW solution that we
obtained in Section 3.2:

$y^{2}=-\frac{\left(b^{3}+bu_{2}u_{6}+u_{6}^{2}\right)^{2}}{u_{6}^{3}}+4\Lambda^{6}.$
(4.16)

The singular locus $\mathcal{S}_{SU(3)}$ of the parent $SU(3)$ SYM theory is

$\displaystyle\Delta_{SU(3)}$ $\displaystyle\propto$ $\displaystyle
216u_{2}^{3}\left(4\Lambda^{6}+u_{3}^{2}\right)+729\left(u_{3}^{2}-4\Lambda^{6}\right)^{2}+16u_{2}^{6}$
(4.17) $\displaystyle=$
$\displaystyle\left(108\Lambda^{6}-108\Lambda^{3}u_{3}+4u_{2}^{3}+27u_{3}^{2}\right)\left(108\Lambda^{6}+108\Lambda^{3}u_{3}+4u_{2}^{3}+27u_{3}^{2}\right).$

One can explicitly check, taking the discriminant of the r.h.s. of (4.16)
w.r.t $b$, that the singular locus of the daughter theory is double-covered by
the singular locus of the parent one being

$\Delta_{\widetilde{SU}(3)}\propto
216u_{2}^{3}\left(4\Lambda^{6}+u_{6}\right)+729\left(u_{6}-4\Lambda^{6}\right)^{2}+16u_{2}^{6}=\Delta_{SU(3)}(u_{3}\to\sqrt{u_{6}})\,.$
(4.18)

We can observe the following facts by matching the two discriminant loci:

  * •

The double-cover map, as expected, is the one that sends $u_{2}\to u_{2}$ and
$u_{3}\to u_{6}=u_{3}^{2}$, and is induced by the $\mathbb{Z}_{2}$ action on
the parent theory Coulomb branch;

  * •

$\Delta_{SU(3)}$ displays two irreducible components,

$f_{1}=108\Lambda^{6}-108\Lambda^{3}u_{3}+4u_{2}^{3}+27u_{3}^{2},\qquad
f_{2}=f_{1}\rvert_{u_{3}\to-u_{3}}.$ (4.19)

These two components are identified by the $\mathbb{Z}_{2}$ discrete gauging
action into the single irreducible component of the r.h.s. of (4.18)

Finally, we fix the same symplectic basis of one-cycles as in [73], see Figure
2.

$x_{1}^{+}$$x_{2}^{-}$$x_{3}^{+}$$x_{2}^{+}$$x_{3}^{-}$$x_{1}^{-}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}A_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}A_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}B_{1}}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}B_{2}}$
Figure 2: Choice of symplectic basis of 1-cycles, drawn in one of the two
sheets of the SW curve. Note that the $B$ cycles (in red) only intersect the
$A$ cycles (in blue) once, as they cross the branch cut and come back through
the second sheet. Their orientations are such that $\langle
A_{i},B_{i}\rangle=+1$.

We are now ready to check the expectations of Section 4.2 with an explicit
computation. It is important to notice that the slices $u_{3}=\text{const.}$
and $u_{6}=\text{const.}$ are generic in their respective Coulomb branches and
intersect the respective bifurcation loci in six points. On the opposite, the
slice $u_{2}=\text{const.}$ are non-generic in both the $SU(3)$ and
$\widetilde{SU}(3)$ Coulomb branches, as they intersect the bifurcation locus,
respectively, just in four and two points.

Let us start recalling the results of [73]. We concentrate on the
$u_{2}=\text{const}$ slice in the $SU(3)$ Coulomb branch, where the coordinate
$u_{3}$ is free to vary and describes a complex plane that intersects
$\mathcal{S}$ in six points, that we call $u_{3,1},...,u_{3,6}$. The charges
of the BPS particle becoming massless at a certain $u_{3,k}$ can be computed
using the procedure outlined before (4.8).

For the considered line, we obtain

$\displaystyle q_{3,w1}=(1,0,1,0),\quad q_{3,2}=(1,1,0,1),\quad
q_{2,3}=(0,-1,1,-1),$ $\displaystyle q_{3,4}=(1,0,-1,1),\quad
q_{3,5}=(0,-1,0,1),\quad q_{3,6}=(1,1,-1,0).$ (4.20)

The points particles $u_{3,1},u_{3,2},u_{3,3}$ are related by the
$\mathbb{Z}_{3}$ action $u_{2}\to e^{\frac{2\pi i}{3}}u_{3}$, and consequently
[73, 67], their charges are related as

$\displaystyle q_{3,1}$ $\displaystyle=$ $\displaystyle q_{3,2}U,\quad
q_{3,2}=q_{3,3}U,\quad q_{3,3}=q_{3,1}U,$ $\displaystyle q_{3,4}$
$\displaystyle=$ $\displaystyle q_{3,6}U,\quad q_{3,5}=q_{3,4}U,\quad
q_{3,6}=q_{3,5}U,$ (4.21)

with $U$ being the following electro-magnetic duality transformation777Our
conventions match those of [73], that differ from the ones of [67] where this
effect is explicitly mentioned. The convention of [67] can be matched with our
convention using $U=\eta^{-1}U_{\text{there}}\eta$, with
$\eta=\left(\begin{array}[]{cccc}0&1&0&-1\\\ 1&0&1&0\\\ 0&0&0&-1\\\
0&0&-1&0\\\ \end{array}\right)$

$U=\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&-1&0&0\\\ 0&0&-1&1\\\ 0&0&-1&0\\\
\end{array}\right)$ (4.22)

One can explicitly check that $q_{3,2}$ and $q_{3,4}$ can be written as
combinations of the other charge vectors. Computing the intersection pairing
between $q_{3,1},q_{3,3},q_{3,5},q_{3,6}$ one finds

$\left(\begin{array}[]{cccc}0&-1&0&-2\\\ 1&0&-2&0\\\ 0&2&0&1\\\ 2&0&-1&0\\\
\end{array}\right)\,.$ (4.23)

Let us then pass to the $u_{3}=\text{const.}$ slice in the SU(3) Coulomb
branch. On this slice, the coordinate $u_{2}$ is free to vary and describes a
complex line that intersects $\mathcal{S}$ in four points that we depict in
figure Fig. 3,

$\mathrm{Re}(u_{3})$$\mathrm{Im}(u_{3})$$u_{2,1}$$u_{2,3}$$u_{2,2}$$u_{2,4}$

$\mathrm{Re}(u_{6})$$\mathrm{Im}(u_{6})$$u_{+}$$u_{-}$

Figure 3: Singular points on the $u_{2}=$ const. slice of the $SU(3)$ and
$\widetilde{SU}(3)$ Coulomb branch, respectively. We color code the
$\mathbb{Z}_{2}$ orbits in the parent theory and to which point they descend
after the discrete gauging.

that we call $u_{2,1},u_{2,2},u_{2,3},u_{2,4}$. One can check that, for our
choice of the basis of one-cycles, the charges $q_{2,k}$ of the BPS particles
becoming massless at $u_{2,k}$ are

$q_{2,1}=(0,-1,0,1),\quad q_{2,2}=(-1,-1,1,0),\quad q_{2,3}=(-1,0,-1,0),\quad
q_{2,4}=(0,-1,1,-1),$ (4.24)

where we ordered the points as in Fig. 3. We remark that the number of
intersection points $u_{2,k}$ is less than the one of the $u_{3}=\text{const}$
slice, hence the $u_{2}=\text{const}$ slice is non-generic. However, one can
check that (4.3) are linearly independent over $\mathbb{Z}$ and the Smith
normal form of their Dirac pairing is the same of (4.23). As pointed out in
[67], the four singularities can be divided into two pairs, each pair being a
$\mathbb{Z}_{2}$ orbit. Then, as predicted by (4.9), the electromagnetic
charges of the BPS becoming massless are related by the right action
$q_{2,1}=q_{2,3}V$ and $q_{2,2}=q_{2,4}V$, where

$V\equiv\left(\begin{array}[]{cccc}0&0&0&-1\\\ 0&0&1&0\\\ 0&1&0&0\\\
-1&0&0&0\\\ \end{array}\right)\,.$ (4.25)

Let us now consider the daughter $\widetilde{SU}(3)$ theory, we can start from
the slice at $u_{6}=\text{const}$. By explicit computation, one can check that
there are six points $\widetilde{u}_{2,k}$ where the slice intersects
$\mathcal{S}$. The preimages of $\widetilde{u}_{2,k}$ inside
$\mathcal{M}_{\text{CB}}$ are not fixed points of the $\mathbb{Z}_{2}$
quotient action888We are considering the slice $u_{6}=c_{0}$, this corresponds
to two different slices $u_{3}=\pm\sqrt{c_{0}}$ with an empty intersection
that get identified by the $\mathbb{Z}_{2}$ action. and that, as predicted in
Section 4.2, the charges of the massless BPS particles at
$\widetilde{u}_{2,k}$ are exactly (4.3). We can again isolate the generators,
and compute the BPS quiver intersection matrix by computing their Dirac
paring, obtaining (4.23). Consequently, we just showed that the BPS quiver is
preserved by the discrete-gauging procedure.

To conclude, we consider the more interesting slice at $u_{2}=\text{const}$,
(with $u_{6}$ free to vary). There are just two points, $u_{+}$ and $u_{-}$,
depicted in Fig. 3 where we have a massless BPS particle, and the two charges
are

$q_{-}=(-1,-1,1,0),\qquad q_{+}=(0,-1,0,1).$ (4.26)

Apart from these “classic” BPS particle degeneration, the
$u_{2}=\text{const.}$ slice displays a peculiar feature: it intersects the
locus $u_{6}=0$ where the SW geometry (4.16) develops a pole in the Coulomb
branch coordinates. This point is peculiar also because its preimage inside
$\mathcal{M}$ is a fixed point of the $\mathbb{Z}_{2}$-quotient map. This is
due to the fact that, contrary to the $u_{3}=\text{const}$ slice, the
$u_{2}=\text{const}$ slice is an $\mathbb{Z}_{2}$ invariant subset of the
parent theory Coulomb branch, and hence, after the quotient, possesses a fixed
point at $u_{6}=0,u_{2}=\text{const}$. We can again compute its monodromy by
looking graphically at how the branching points of the hyperelliptic double
cover get permuted going anticlockwise around $u_{6}=0$ obtaining

$M_{\text{fixed}}=V.$ (4.27)

Stated differently, the result is the following: a non-singular fixed point of
the $\mathbb{Z}_{2}$ action of the parent theory gets mapped to a singular
point $p_{\text{fixed}}$ of the daughter theory Coulomb branch. The monodromy
$M_{\text{fixed}}$ around $p_{\text{fixed}}$ is exactly the electromagnetic
duality $V$ realizing the $\mathbb{Z}_{2}$ action in the parent theory.
On top of the general proof in Section 4.2, we can understand heuristically
the situation as follows. We should expect something peculiar to happen on the
$\widetilde{\mathcal{L}_{u_{6}}}\equiv\left\\{u_{2}=\text{const}\right\\}$
slice: we can consider the restriction of Riemann-Hilbert problem associated
with the SW solution to $\widetilde{\mathcal{L}_{u_{6}}}$. The slice is the
$\mathbb{Z}_{2}$ quotient of the $\mathcal{L}_{u3}\equiv u_{2}=\text{const}$
slice of the parent theory. By passing from $\mathcal{L}_{u_{3}}$ to
$\widetilde{\mathcal{L}_{u_{6}}}=\mathcal{L}_{u_{3}}/\mathbb{Z}_{2}$ we halved
the number of points in which we have a BPS massless particle (as we
identified them pairwise during the quotient). Hence, the Riemann-Hilbert
problem restricted to the considered slice is ill-defined, unless we introduce
something else to balance the fact that we halved the number of points
supporting a monodromy and that we changed the monodromy at infinity as in
(4.11). Indeed, heuristically this boils down to the fact that, having two
choices in (4.10), we have multiple equivalent ways, depending on the choices
branch cuts of $x_{k}(u,m)$, to relate the monodromies of the daughter theory
to those of the parent one. In some sense, $V_{\text{fixed}}$ permits us to
move between these equivalent choices.

###  4.4 BPS spectra for $\mathbb{Z}_{k}$ groups

In this section, we generalize the presence of an object analogous to the pair
$(u_{fixed},M_{fixed})$ that we introduced around (4.15) for the
$\mathbb{Z}_{k}$ discrete-gauging, with $k>2$, of a generic
$\mathcal{N}=2,\mathcal{D}=4$ theory. The argument goes in the same way as in
the $\mathbb{Z}_{2}$ lagrangian case: in the parent theory, we have an
electromagnetic duality $\mathcal{E}$ generalizing the action of $V$, with

$\mathcal{E}^{k}=\mathds{1}_{2r}$ (4.28)

with $r$ the rank of the theory. Notice that (4.28) is a simplyfing
assumption, as in general we could have e.g. $\mathcal{E}^{k}=-\mathds{1}$.

Figure 4: Clustering of singular points in a Coulomb branch slice
$\mathcal{L}$ of a theory enjoying a $\mathbb{Z}_{k}$ symmetry (here $k=4$).
The singular points are indicated in red, and the first "cluster" is formed by
the three rightmost red points. The light-blue point corresponds to a base-
point for an anticlockwise loop obtained composing the tiny blue loops. Hence,
the monodromy on the big loop is the product of the monodromies associated
with the red points. The monodromies are ordered from the right to the left
according to the sequence in which the red points get hit when we rotate
anticlockwise the green ray emanating from the light-blue point.

Let’s now consider a complex line $\mathcal{L}$ inside
$\mathcal{M}_{\text{CB}}$, such that
$\Gamma\rvert_{\mathcal{L}}\in\text{Aut}(\mathcal{L})$. Then, calling $U$ the
coordinate on $\mathcal{L}$, being the $\Gamma$ action linear, we have just
one fixed point at $U=0$. The points $U_{i}\in\mathcal{L}$ where we have
massless BPS particles organize, as in Figure 4, into $k$ clusters (related by
the action of $\Gamma$), that become more and more separated as we take large
values for the Coulomb branch parameters that we fixed999E.g., in the $SU(3)$
example of Section 4.3, we are taking $\absolutevalue{u_{2}}>>1$ for the
$u_{2}=\text{const.}$ slice of the parent theory Coulomb branch. in order to
obtain $\mathcal{L}\subset\mathcal{M}_{\text{CB}}$. Let’s call $U_{t}$, with
$t=1,...n_{cluster}$ the points belonging to the first cluster, and consider a
point $\Gamma^{j}(U_{t})$ belonging to the $j$-th cluster. $\Gamma^{j}(U_{t})$
is obtained from $U_{t}$ applying $j$ times the map $\Gamma$, then

$q_{\Gamma^{j}(U_{t})}=q_{U_{t}}\mathcal{E}^{j},\quad
M_{\Gamma(U_{t})}=\mathcal{E}^{-j}M_{U_{t}}\mathcal{E}^{j}.$ (4.29)

The monodromy on a big anticlockwise loop on $\mathcal{L}$ is

$M_{loop,parent}=(\mathcal{E}^{k-1})^{-1}M_{cluster}\mathcal{E}^{k-1}...M_{cluster},$
(4.30)

with $M_{cluster}=\prod_{t=1}^{n_{cluster}}M_{U_{t}}$. By using
$\mathcal{E}^{k}=\mathds{1}$, (4.30) can be rewritten as

$M_{loop,parent}=(\mathcal{E}M_{cluster})^{k}.$ (4.31)

The monodromy of the daughter theory on a big anticlockwise loop is related to
the one of the parent theory by

$M_{loop,daughter}^{k}=M_{loop,parent}$ (4.32)

The only way in which (4.31) and (4.32) are consistent is that

$M_{loop,daughter}=\mathcal{E}M_{cluster},$ (4.33)

where we used that the “roots of units” ambiguity given by taking the $k-$th.
root is not present because we are looking for integer-valued
solutions101010One might be worried when some of the roots of unit are real,
in that case they must be $-1$, and this simply encodes the fact that, on a
given $U_{t}$, both $\nu$ and $-\nu$ vanishes at the same time. In other
words, in that case $\mathcal{E}$ can be equally re-defined as
$-\mathcal{E}$.. Now, by using that again, for $U\neq 0$ the $\mathbb{Z}_{k}$
action $\widetilde{\Gamma}$ identifies different curves (rather than taking a
quotient of the same fiber of the parent theory SW solution) of the parent
theory SW family, we have $\widetilde{M_{t}}=M_{t}$ at points
$\widetilde{U}_{t}\in\widetilde{\mathcal{L}}$. Consequently, we get

$M_{loop,daughter}=M_{fixed}\prod_{t=1}^{n_{cluster}}M_{t}$ (4.34)

and hence, taking (4.33) and (4.34) together, we have

$M_{fixed}=\mathcal{E}.$ (4.35)

We can reformulate the result as follows:
Let $\mathcal{L}$ be a slice of the parent theory Coulomb branch that inherits
the $\mathbb{Z}_{k}$ discrete-gaugin action. Let $U$ being its coordinate, and
$\widetilde{U}$ the coordinate on
$\widetilde{\mathcal{L}}=\mathcal{L}/\mathbb{Z}_{k}$. Finally, let
$\mathcal{E}$ be the electromagnetic duality realizing the $\mathbb{Z}_{k}$
action at the level of the fibers of the SW geometry of the parent theory.
Then, if $\mathcal{E}^{k}=\mathds{1}$, we get a monodromy equals to
$\mathcal{E}$ at $\widetilde{U}=0$.

To conclude, we note that this result is again compatible with the argument of
[32]: the preimage under the $\mathbb{Z}_{k}$ quotient of
$u_{fixed}\in\widetilde{\mathcal{M}_{\text{CB}}}$ is, in our framework, a
smooth point $u_{0}\in\mathcal{M}_{\text{CB}}$ fixed by the $\mathbb{Z}_{k}$
action. Consequently, going once around $u_{\text{fixed}}$ corresponds to go
$k$ times around $u_{0}$, hence

$M_{\text{fixed}}^{k}=M_{u_{0}}=\mathds{1}_{r},$ (4.36)

where in the last passage we used that $u_{0}$ is a smooth point of
$\mathcal{M}_{\text{CB}}$. Our argument shows that, if
$\mathcal{E}^{k}=\mathds{1}_{r}$, the monodromy of $u_{fixed}$ is non-trivial
and corresponds to $\mathcal{E}$.

##  5 Discussion and outlook

Let us summarize our results. We have found the SW solutions for theories with
disconnected gauge groups $\widetilde{SU}(N)$ starting from those of the
theories with $SU(N)$ gauge group. Our method consists of finding the lift of
the (known) action $\Gamma$ on the Coulomb branch to the full SW geometry, by
requiring that $\widetilde{\Gamma}$ 1) commutes, in the conformal case, with
the $U(1)_{R}$ action, 2) is an automorphism of the SW geometry, and 3)
preserves the SW differential. Then, one can express the SW solution before
the discrete gauging in terms of the $\mathbb{Z}_{2}$ invariant coordinates
and impose the new relations that come from the quotient.

From these solutions, we also studied the singularities on the Coulomb branch
where extra particles become massless. Most of these are inherited in a
straightforward way from the theory before the discrete gauging; however, our
SW solutions show the existence of “new” points where the naive EFT fails that
descend from smooth $\Gamma$-invariant points of Coulomb branch of the theory
with gauge group $SU(N)$. We also computed the monodromy $M_{\text{fixed}}$
around these “new” points to be exactly the electromagnetic duality $V_{N}$
realizing $\widetilde{\Gamma}$ in the theory before the discrete gauging. We
then rephrased the appearance of these new singularities in terms of the
interplay between the Riemann-Hilbert problem and the discrete-gauging
procedure, suggesting them to be present also for the discrete-gauging of non-
lagrangian theories.

Let’s now proceed to discuss, as mentioned in the introduction, some
implications of our results.

###  5.1 On the classification of rank-2 geometries

As we discussed in detail in the main text, one of the consequences of the
quotient at the level of the SW geometry is that it can lead to poles on the
Seiberg-Witten curve. For convenience, we reproduce here the curve for
$\widetilde{SU}(3)+6(F\oplus\overline{F})$,

$\displaystyle y^{2}$
$\displaystyle=-4fa^{3}+a^{3}+2u_{2}\left(a^{2}+b\right)+2ab+au_{2}^{2}+u_{6}\,,$
$\displaystyle b^{2}$ $\displaystyle=au_{6}\,.$ (5.1)

The presence of $1/u_{6}$ terms when substituting $a$ in terms of $b$ into the
first equation is what explains why this theory, with Coulomb branch
parameters of dimensions $(2,6)$ and rank 3 global symmetry, did not appear in
the classification of Argyres and Martone [39].

It is interesting to remark that the new singularity at $u_{6}=0$ with
$u_{2}=const$, produced by the discrete gauging, turns out to not be very
exotic. Indeed, we can plug $u_{6}=0$ into (5.1), which results in $b=0$ and
$a$ being a free coordinate. Then, the first equation becomes

$\displaystyle y^{2}=(1-4f)a^{3}+2u_{2}a^{2}+au_{2}^{2}\,,$ (5.2)

which is a very simple elliptic curve. Since the genus has dropped to 1, and
the only CB parameter has scaling dimension 2, we can identify this as the
$SU(2)$ theory.

More generally, our computation shows that it is quite easy to find theories
where the SW geometry shows coefficients which are non-polynomial in the
Coulomb branch parameters, and as a consequence, extending the classification
of [39] to geometries where poles are allowed is an interesting open problem.

###  5.2 Comments on SymTFT for non-invertible defects

One of the interesting features of theories with disconnected gauge groups is
that, if the matter fields live only in the adjoint representation, they have
a non-invertible 1-form symmetry [9, 13, 12]. It is therefore interesting to
explore whether or not it can be seen from the SW solution. In [50], a
procedure was found to extract the SymTFT for the (grouplike) 1-form symmetry
from the SW geometry. One starts from a BF theory in 5d that schematically
looks like

$\displaystyle S_{\text{SymTFT}}=2\pi i\,N\int_{M^{5}}B\wedge dC\,,$ (5.3)

where $B$ and $C$ are 2-form fields that serve as a background connection for
the electric and magnetic 1-form symmetry. Then, the idea is to find which are
the local particles that can screen the relevant line operators. In the end,
this results in a very simple recipie: the coefficients that we need to fix
the SymTFT can be extracted from the BPS quiver by computing the Smith Normal
Form of the intersection pairing between the vanishing cycles. In the example
of $SU(3)$, the intersection matrix is (4.23), which has Smith Normal Form

$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&3&0\\\
0&0&0&3\\\ \end{array}\right)\,.$ (5.8)

This is telling us that the relative theory has a
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ 1-form symmetry and upon a choice of
polarization there remains a $\mathbb{Z}_{3}$ either electric or magnetic
1-form symmetry; this is encoded on the SymTFT

$\displaystyle S_{\text{SymTFT}}=6\pi i\int_{M^{5}}B\wedge dC\,.$ (5.9)

Now, as we have discussed in Section 4, the intersection matrix between the
vanishing cycles is preserved when performing the quotient. This seems to lead
us to the conclusion that the SymTFT should be the same; however, we know that
this cannot be all as the 1-form symmetry of $\widetilde{SU}(3)$ is non-
invertible. Indeed, one should note that the analysis of the screening of line
operators is the same for the connected and disconnected gauge groups, and
what gives rise to the non-invertibility is the way one builds gauge invariant
topological defects after the discrete gauging. Therefore, it is not a
contradiction that the procedure of [50] doesn’t detect this. Investigating
what is the way to generalize that procedure in such a way as to be able to
detect the non-invertibility from the SW curve is a very interesting open
problem. It is tempting to guess that the new singularity introduced by the
quotient, which precisely accounts for the ambiguity of choice of pre-image of
a point in the parent theory Coulomb branch, should play an important role in
this.

###  5.3 String and M-theory uplift

It is well known that there is an intimate relation between the Seiberg-Witten
solution of a theory and it’s engineering in String Theory [62]. Therefore,
one can try to use this relation to engineer the theories with disconnected
gauge groups.111111In this direction, it should be noted that the case of $O$
groups (which are built from gauging the outer automorphism of a $D$-type
algebra) is known and $\widetilde{SU}(N)$ seems to be a more complicated
problem [77, 78].

The pure $SU(N)$ gauge theory can be realized in type IIA as the world-volume
theory of $N$ D4 suspended between two vertical NS5 branes. Both the D4s and
the NS5s get uplifted in M-theory to M5 branes, that recombine into a single
M5 brane with a non-trivial worldvolume [62]. In particular, the recombined M5
brane expands to infinity at the points corresponding to the NS5 locations. In
this setup, the SW curve (3.60) describes the worldvolume of the recombined M5
along the $\mathbb{C}^{2}\ni(x,y)$ (one, in general, has to add points at
infinity, obtaining the class-S construction [63]).

In this setup, the reflection of the coordinates $(x,y)$ has a physical
meaning. We can start with the $N$-odd case, in which just the coordinate $x$
gets reflected. We set, for simplicity, the masses to zero, but the arguments
proceed in the same way with the caveat of turning on only the mass-
deformations that respect the $\mathbb{Z}_{2}$-symmetry. The world-volume that
corresponds to the Coulomb branch of the daughter theory is then (3.60), with
the points $(x,y)$ and $(-x,y)$ identified. In this case, the reflection of a
single coordinates transforms $\mathbb{C}^{2}\ni(x,y)$ into
$\mathbb{C}^{2}\ni(x^{2},y)$, that is still a Calabi-Yau surface, and then it
makes sense for the setup to be still $\mathcal{N}=2$. The situation is
analogous for the $N$-even case, with the caveat that, since now the
$\mathbb{Z}_{2}$ action reflects both $x$ and $y$, the M5 brane is contained
in $\mathbb{C}^{2}/\mathbb{Z}_{2}\cong A_{1}$, the rank-one Du Val
singularity121212In our opinion, this setup is still different from the one
presented in [62], where an $A_{1}$ singularity corresponded to the presence
of D6 branes in the IIA limit. Indeed, in [62] the metric on the $A_{1}$
singularity is the Taub-NUT one, also called ALF metric. In our case, instead,
the most natural metric is the ALE one, that asymptotically reproduces the
flat orbifold $\mathbb{C}_{2}/\mathbb{Z}_{2}$..

The problem however, is that the reflection of $x,y$ for us is accompanied
with a reflection on the $u_{j}$. In our construction, that is at the
geometric level of the SW solution, we are treating the variables $u_{j}$ and
$(x,y)$ on an equal footing (e.g., by constructing invariants that mixes
$u_{j}$ and $(x,y)$). In the M-theory realization though, the $u_{j}$ and the
$(x,y)$ are completely different in nature, being the first parameters of the
physical system, while the second are space-time coordinates. One is reminded
of a similarity with S-folds in type IIB, where a spacetime action is combined
with a non-trivial action on the axio-dilaton $\tau$. However, in that
context, F-theory precisely geometrizes $\tau$ in such a way as to be able to
treat it as well as the coordinates on equal footing. At this stage, we leave
a deeper investigation of the M-theory realization of disconnected gauge
theories for future work.

###  5.4 Open problems

There are more open questions that our investigation opens the door to, beyond
the ones discussed above. Let us comment on a few of them:

  * •

One interesting point would be to delve into the dynamics of the theory from
of our SW solution, namely, use them to try to compute the prepotential. This
is a natural next step, as so far our investigation of the dynamics of the
disconnected gauge theory was just scratching the surface by looking at the
particles that become massless. Since, in the end, the SW technology is a
smart way to compute the instanton contributions to the partition function, it
would be interesting to try to use this to understand in greater generality
the instantons of disconnected gauge groups.

  * •

Related to the previous point, we have seen that the appearance of new
singularities descending from smooth points of the theory before the discrete
gauging theory fixed by the quotient action should be a very general feature
of any discrete gauging. Understanding the physics of these Coulomb branch
points, which at the moment we understand just as a natural requirement coming
from the Riemann-Hilbert picture, would be a tantalizing follow-up of this
work. Moreover, as we have discussed, we expect this effect to take place also
for discrete gaugings of non-lagrangian theories. Another direction worthy of
exploration would be to study the singularities that arise in this way in more
exotic examples. As we discussed above, for $\widetilde{SU}(N)$ the theory
living on that singularity was simply $SU(2)$, but more interesting theories
may be found in general.

  * •

We believe that the phenomena of complex structure singularities in the CB is
deserving of further study beyond out simple example. Typically in
mathematical literature [79], a possible way to resolve the complex structure
singularities of a given moduli space $\mathcal{M}$ parametrizing a family of
objects $\mathcal{F}$ is to find a smart, and in some sense “minimal”,
rephrasing of the definition of the objects in $\mathcal{F}$. For example, we
can construct the resolution of the $A_{1}\cong\mathbb{C}^{2}/\mathbb{Z}_{2}$
singularity as follows: consider the symmetric product of $\mathbb{C}^{2}$
with itself; the $\mathbb{Z}_{2}$ action on $\mathbb{C}^{2}$ induces a natural
$\mathbb{Z}_{2}$ action on the symmetric product. One of the invariant
components inside the symmetric product w.r.t the induced $\mathbb{Z}_{2}$
action is the $A_{1}$ singularity. Now, to produce the resolution we perform
the same procedure, but using the "Hilbert scheme of two points in
$\mathbb{C}^{2}$", that contains the symmetric product of $\mathbb{C}^{2}$
with itself as a dense open subset. After considering again the
$\mathbb{Z}_{2}$ invariant part of the Hilbert scheme, we get now the resolved
$A_{1}$ singularity. Stated differently, resolving the $A_{1}$ singularity is
equivalent to modify the notion of objects parametrized by it, passing from
varieties (in this case, couples of distinct antipodal points) to schemes. It
would be interesting to understand whether this logic applies to our
situation, and what would be the relevant “object” such that the complex
structure singularity of our examples becomes resolved.

  * •

Lastly, throughout the paper we have selected one of the two possible ways to
gauge charge conjugation in a $SU(2N)$ theory [24]. Understanding the SW
solution for the other possible discrete gauging, as well as the relation, if
there is one, with the apparent difference between the action on the
coordinates $x,y$, is another open problem.

### Acknowledgments

It’s a pleasure to thank Michele Del Zotto, Pietro Longhi, Kunal Gupta, Robert
Moscrop, Guglielmo Lockhart, Iñaki García-Etxebarria, Sergej Monavari,
Federico Carta and Amihay Hanany for discussions. We particularly thank
Roberto Valandro for comments on the draft. GAT is supported by the STFC
Consolidated Grants ST/T000791/1 and ST/X000575/1. The work of MDM is
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement No. 851931). MDM
thanks the “Fondazione Angelo Della Riccia” for financial support during the
initial stage of this work.

##  Appendix A  The case of $SU(2)+4\,F$

In this appendix, we will apply the IR geometric procedure of the main text to
the case of $SU(2)+4F$, reproducing the result of [32]. This will serve as a
non-trivial test for our method, and will show it at work in a different
setup, where the discrete symmetry arises promoting an S-duality of the UV
theory to a symmetry by finely tuning the UV coupling.

The SW data before the discrete gauging are [27]:

$y^{2}=(x-e_{1}u)(x-e_{2}u)(x+(e_{1}+e_{2})u),\quad\lambda=u\frac{dx}{y},$
(A.1)

with $e_{i}$ being defined by

$e_{1}-e_{2}=\left(\theta_{3}(0,\tau)\right)^{4},\quad
e_{3}-e_{2}=\left(\theta_{1}(0,\tau)\right)^{4},\quad
e_{1}-e_{3}=\left(\theta_{2}(0,\tau)\right)^{4},$ (A.2)

where $\theta_{j}$ are the Jacobi $\theta$-functions on the elliptic curve of
modular parameter $\tau$,

$\displaystyle\theta_{1}(0,\tau)=\sum_{n\in\mathbb{Z}}q^{1/2(n+1/2)^{2}},\quad\theta_{2}(0,\tau)=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{1/2n^{2}},\quad\theta_{3}(0,\tau)=\sum_{n\in\mathbb{Z}}q^{1/2n^{2}},$
(A.3)

with $q=e^{2\pi i\tau}$. The action of the $\mathbb{Z}_{2}$ symmetry sends
$u\to-u$, hence we do need a non-trivial fibral transformation
$(x,y)\to(x^{\prime},y^{\prime})$ in such a way to make $\lambda$ invariant
w.r.t. the $\Gamma$-quotient. For a $\mathbb{Z}_{2}$ quotient that respects
the $U(1)_{R}$ charges of $x,y$ (hence, that acts diagonally on $x,y$), we
have then two-possibilities:

  1. 1.

$x\to-x$ and $y\to y$, but this does not induces an automorphism of the SW
geometry (A.1) for any choice of the UV coupling $\tau$;

  2. 2.

$x\to x$ and $y\to-y$.

We see that also the second option is not an automorphism of the SW geometry
(A.1) for a generic choice of $\tau$. However, choosing $\tau$ such that,
e.g., $e_{2}=0$ (this corresponds to $\tau=i$), we get that

$\widetilde{\Gamma}(x,y,u)=(x,-y,-u)\,,$ (A.4)

preserves both the SW data in (A.1). We can then use (A.4) to quotient the
total SW geometry, and obtain the SW geometry and SW differential for the
discrete gauging of the $SU(2)+4F\oplus\overline{F}$ theory.

Now that we found the candidate $\widetilde{\Gamma}$ action, to perform the
quotient we first introduce the invariant coordinates

$a=y^{2},\quad b\equiv yu,\quad U\equiv u^{2}$ (A.5)

satisfying the relation

$aU-b^{2}=0.$ (A.6)

Firstly, we note that the new Coulomb branch coordinate is $U$, hence the
projection on $U$ describes the new SW geometry as a fibration of the new
Coulomb branch $\mathbb{C}\ni U$ and is identified with $\widetilde{\pi}$ in
(3.9). Secondly, we rewrite the initial (A.1) SW curve in terms of the
invariant coordinates (A.5), obtaining (after a numerical re-scaling of the
coordinates),

$a+x^{3}-Ux=0.$ (A.7)

We can now solve (A.7) for $a$ in terms of $(x,U)$, and plug the solution into
(A.6), obtaining

$-b^{2}+Ux(-U+x^{2})=0.$ (A.8)

We observe that the curve is not in Weierstrass form, but the SW differential
is already in the desired [68] canonical form $\lambda=udx/y=Udx/b$. To
produce the Weierstrass form, we have to rescale $x\to U^{f}X,b\to U^{g}B$. We
notice that, since the SW differential is already in the desired canonical
form $\lambda=udx/y=Udx/b$ it must be $f=g$. We then see that the choice
$f=-1$ put (A.8) in the Weierstrass form:

$B^{2}+X^{3}-U^{3}X=0,\qquad\lambda=U\frac{dX}{B}.$ (A.9)

We can observe that (A.9) is a type $III^{*}$ Kodaira singularity, as
predicted in [32] with slightly different methods.

To conclude and check our result we can count the mass deformations of the
theory after the discrete gauging. As in [32], these are the masses of the
parent theory that are invariant w.r.t. the outer-automorphism factor of the
discrete gauging. From a SW perspective, we ask ourselves what is the maximum
number of mass deformations for which $\widetilde{\Gamma}$ is again an
automorphism, or, equivalently, which subvariety of the SW family over the
extended Coulomb branch still enjoys $\widetilde{\Gamma}$ as automorphism. For
this, we already impose $e_{2}=0$ in the mass-deformed curve of [27]. By
explicit computation, one can check that the only term of the mass-deformed
SU(2) + 4 F curve that is not $\widetilde{\Gamma}$-invariant is

$\delta\propto(T_{1}-T_{3})u,$ (A.10)

with

$\displaystyle
T_{1}=\frac{1}{12}\sum_{i>j}m_{i}^{2}m_{j}^{2}-\frac{1}{24}\sum_{i}m_{i}^{4},\qquad
T_{3}=\frac{1}{2}\left(\prod_{i}m_{i}-T_{1}\right),$ (A.11)

with $i,j=1,...,4$. We then see that imposing (A.4) to be a symmetry of the
mass-deformed geometry forces $T_{1}=T_{3}$ and drops the rank of the flavor
group by one. This is compatible with the fact that the new expected flavor
algebra is $B_{3}$ [32]. To finely check the group is indeed $B_{3}$, we
follow [27] and define $T_{2}=\frac{1}{2}\left(-\prod_{i}m_{i}-T_{1}\right)$.
$T_{1},T_{2},T_{3}$ are permuted by the triality of the flavor group SO(8),
hence one of the three $\mathbb{Z}_{2}$ subgroups that folds $D_{4}$ to
$B_{3}$ exactly exchanges $T_{1}\leftrightarrow T_{3}$. In other words, we see
that imposing $T_{1}=T_{3}$ is equivalent to require the masses to lie into a
$B_{3}$ subalgebra of $D_{4}$, reproducing the expected result [32].

##  Appendix B Explicit check of the monodromy algebra

In this appendix, we will verify that the monodromies that we found in Section
4.3 satisfy (4.8). Starting from a base point $u_{6,b}$ with
$\text{Re}(u_{6,b})\to+\infty$, $\text{Im}(u_{6},b)=0$, and going
anticlockwise we have

$M_{fixed}M_{p_{+}}M_{p_{-}}=M_{loop,u_{6}},\qquad
M_{loop,u_{6}}=\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&-1&0&0\\\ 1&-1&-1&1\\\
2&1&-1&0\\\ \end{array}\right)$ (B.1)

One can check that $M_{loop,u_{6}}^{2}=M_{loop,u3}$, with

$M_{loop,u3}=\left(\begin{array}[]{cccc}-1&-1&0&0\\\ 1&0&0&0\\\ 2&4&0&-1\\\
-2&2&1&-1\\\ \end{array}\right),\qquad M_{loop,u3}=M_{3}M_{4}M_{1}M_{2}$ (B.2)

with $M_{j}$ being the monodromy around $q_{2,j}$ in (4.3), computed on an
anti-clockwise loop starting from a real-valued base-point with large negative
real part. We can understand the result by noticing that

$M_{loop,u3}=VM_{1}VVM_{2}VM_{1}M_{2}=(VM_{1}M_{2})^{2}=M_{loop,u6}^{2}$ (B.3)

Now, by using $M_{p_{+}}=M_{1},M_{p_{-}}=M_{2}$ (that comes from the fact that
the point $p_{+}$ is the $\mathbb{Z}_{2}$-quotient image of $u_{2,1}$ etc.),
we see that

$VM_{1}M_{2}=M_{loop,u6}=M_{fixed}M_{1}M_{2},$ (B.4)

and hence $M_{fixed}=V$.

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