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# Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry
gaugings

Elias Furrer♯, Horia Magureanu♭
♯ School of Mathematics, University of Birmingham,
Watson Building, Edgbaston, Birmingham, B15 2TT, United Kingdom
♭ Mathematical Institute, University of Oxford,
Andrew-Wiles Building, Woodstock Road, Oxford, OX2 6GG, United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Symmetries of Seiberg–Witten (SW) geometries capture intricate physical
aspects of the underlying 4d $\mathcal{N}=2$ field theories. For rank-one
theories, these geometries are rational elliptic surfaces whose automorphism
group is a semi-direct product between the Coulomb branch (CB) symmetries and
the Mordell-Weil group. We study quotients of the SW geometry by subgroups of
its automorphism group, which most naturally become gaugings of discrete 0-
and 1-form symmetries. Yet, new interpretations of these _surgeries_ become
evident when considering 5d $\mathcal{N}=1$ superconformal field theories.
There, certain CB symmetries are related to symmetries of the corresponding
$(p,q)$-brane web and, as a result, CB surgeries can give rise to (fractional)
S-folds. Another novel interpretation of these quotients is the folding across
dimensions: circle compactifications of the 5d $E_{2N_{f}+1}$ Seiberg theories
lead in the infrared to two copies of locally indistinguishable 4d SU(2) SQCD
theories with $N_{f}$ fundamental flavours. This extends earlier results on
holonomy saddles, while also reproducing detailed computations of 5d BPS
spectra and predicting new 5d and 6d BPS quivers. Finally, we argue that the
semi-direct product structure of the automorphism group of the SW geometry
includes mixed ‘t Hooft anomalies between the 0- and 1-form symmetries, and we
also present some new results on non-cyclic CB symmetries.

##  1 Introduction

Supersymmetric quantum field theories have received great attention over the
years due to their remarkable exact solutions and tractable non-perturbative
effects. This is particularly evident in the case of four-dimensional
$\mathcal{N}=2$ supersymmetric gauge theories, where the Seiberg–Witten (SW)
geometry Seiberg:1994rs ; Seiberg:1994aj fully solves the intricate dynamics
of the strong-coupling regimes. Besides offering a detailed description of the
vacuum moduli space Argyres:1995jj ; Argyres:1995xn ; Aspman:2020lmf ;
Aspman:2021vhs , SW geometry has been shown to capture various other physical
aspects such as flavour symmetries Minahan:1996cj ; Minahan:1996fg ;
Argyres:2016xua ; Caorsi:2018ahl , spectra of BPS states Bilal:1996sk ;
Ferrari:1996sv ; Bilal:1997st ; Magureanu:2022qym , renormalisation group
flows Eguchi:2002fc ; Argyres:2007tq ; Eguchi:2002nx ; Caorsi:2019vex ;
Argyres:2015ffa ; Argyres:2015gha ; Argyres:2016xmc , and even global
properties of these models Closset:2021lhd ; Argyres:2022kon ; Closset:2023pmc
.

###  1.1 Automorphisms of the SW geometry

In this paper, we extend the applications of SW geometry through a careful
analysis of its symmetries. We direct our attention to 4d rank-one
$\mathcal{N}=2$ theories, which also include Kaluza–Klein (KK) theories.
There, the total space of the SW curve fibred over the complex one-dimensional
Coulomb branch, or $U$-plane, can be identified as a _rational elliptic
surface_ (RES) Caorsi:2018ahl ; Closset:2021lhd . In many instances, quotients
of a RES by subgroups of the full automorphism group ${\rm Aut}(\mathcal{S})$
lead to other well-defined RES. As a result, a link between seemingly distinct
SW geometries is realised. Rather remarkably, these _surgeries_ lead to deep
physical insights, going beyond discrete symmetry gaugings, contrary to naive
expectations.

The full automorphism group $\text{Aut}(\mathcal{S})$ for such surfaces
$\mathcal{S}$ takes the form of a semidirect product KARAYAYLA20121 ;
Karayayla+2014+1772+1795 ; Bouchard:2007mf ,

${\rm Aut}(\mathcal{S})={\rm MW}(\mathcal{S})\rtimes{\rm
Aut}_{\sigma}(\mathcal{S})\leavevmode\nobreak\ .$ (1.1)

The first component, $\text{MW}(\mathcal{S})$, is the _Mordell–Weil group_ of
$\mathcal{S}$. Notably, this group has been argued to encode the 1-form
symmetry of the theory whose SW geometry is described by the RES $\mathcal{S}$
Closset:2021lhd (see also Caorsi:2019vex ; Mayrhofer:2014opa ;
Aspinwall:1998xj ; Apruzzi:2020zot ). The conjecture was further refined in
Closset:2023pmc , where a procedure for gauging 1-form symmetries directly
from the SW geometry was proposed and checked in all rank-one examples. This
procedure involves compositions of so-called _isogenies_ , which are generated
as quotients of the underlying rational elliptic surface by subgroups of the
Mordell-Weil group. Isogenies find another application in generating Galois
covers Cecotti:2015qha , which are constructions that relate the spectra of
BPS states between two theories. This topic will be the focus the our upcoming
work ACFMM:part1 .

The second factor appearing in (1.1) is the subgroup of automorphisms
preserving the zero section $\sigma$ of the SW elliptic fibration,
$\text{Aut}_{\sigma}(\mathcal{S})$. It consists of symmetries acting non-
trivially on the Coulomb branch, which, in the simplest case, are the cyclic
symmetries $U\mapsto e^{\frac{2\pi i}{n}}U$. These discrete 0-form symmetries
can have various physical origins, such as residual $R$-symmetries, charge
conjugation, or remnants of higher-form symmetries if the theory is obtained
by toroidal compactification from a higher-dimensional theory. Gaugings of
these discrete 0-form symmetries are naturally associated with a _folding_ of
the $U$-plane, which arises as a quotient of $\mathcal{S}$ by subgroups of
${\rm Aut}_{\sigma}(\mathcal{S})$, similar to how isogenies are defined.

Coulomb branch foldings corresponding to cyclic subgroups of ${\rm
Aut}_{\sigma}(\mathcal{S})$ have been extensively studied within the
classification programme of 4d $\mathcal{N}=2$ rank-one SCFTs Argyres:2015gha
; Argyres:2016yzz ; Caorsi:2019vex . Yet, new interpretations of such foldings
are revealed when considering the SW geometries of KK theories resulting from
the toroidal compactification of higher dimensional SCFTs. In this regard, we
discover intricate relationships between four and five-dimensional theories,
some of which have previously been suggested by detailed computations of BPS
spectra Closset:2019juk ; Jia:2021ikh ; Jia:2022dra ; DelMonte:2021ytz .
Additionally, for the five-dimensional theories admitting a $(p,q)$-brane web
description we show how CB surgery relates to brane-web folding Kim:2021fxx ;
Apruzzi:2022nax . Notably, $U$-planes contain more information than the
underlying brane web, allowing thus for symmetries not visible at the level of
the brane web.

Another new result of this work involves the semi-direct product structure of
(1.1), which captures a very subtle feature of the underlying theory. This
structure defines a non-trivial action $\text{Aut}_{\sigma}(\mathcal{S})$ on
$\text{MW}(\mathcal{S})$. In the simplest interpretation of these groups as 0-
and 1-form symmetries, the most natural interpretation of said action is that
of a mixed anomaly between the two symmetries. We show that this is indeed the
case and formulate the precise condition giving an anomaly. Lastly, we should
also mention that the group ${\rm Aut}_{\sigma}(\mathcal{S})$ is in many cases
non-cyclic. We study the action of these groups on the Coulomb branch geometry
and discuss its physical interpretation. In the rest of the introduction, we
expand on the results presented above.

###  1.2 Coulomb branch folding

As previously alluded to, it frequently occurs that quotients of the SW
geometry by subgroups of ${\rm Aut}(\mathcal{S})$ lead to new rational
elliptic surfaces, which are thus compatible with new 4d $\mathcal{N}=2$
models. The simplest such constructions are the discrete gaugings of 0- and
1-form symmetries. Nevertheless, a single quotient can often times be given
multiple physical interpretations: the same RES can describe several theories,
typically involving the fine-tuning of mass deformation parameters. We will
refer to the interpretations different from discrete gaugings as _geometric
transitions_. Table 1 summarises the various interpretations of these
surgeries.

|  Isogenies – $\bm{{\rm MW}(\mathcal{S})}$ |  Foldings – $\bm{{\rm Aut}_{\sigma}(\mathcal{S})}$
---|---|---
Discrete gaugings | 1-form sym. gauging | 0-form sym. gauging
Geometric | Galois covers | (Fractional) S-folds
transitions | Holonomy saddles
Table 1: Summary of the physical action of quotients by the subgroups of ${\rm
Aut}(\mathcal{S})$.

The simplest class of 4d $\mathcal{N}=2$ theories for which we can illustrate
the various surgery constructions are $\mathcal{N}=2$ supersymmetric QCD with
gauge group ${\rm SU}(2)$ and $N_{f}\leq 4$ massive fundamental
hypermultiplets. Their $U$-planes have been analysed in great detail in
Seiberg:1994rs ; Seiberg:1994aj ; Argyres:1995xn ; Nahm:1996di ; Kanno:1998qj
; Malmendier:2008db ; Aspman:2021vhs ; Aspman:2021evt ; Closset:2021lhd . Our
reasoning naturally extends to any 4d rank-one $\mathcal{N}=2$ SCFT, as well
as to the family of Kaluza–Klein theories obtained by circle compactifications
of the 5d $\mathcal{N}=1$ Seiberg $E_{n}$ theories, which we denote by
$D_{S^{1}}E_{n}$ Seiberg:1996bd ; Morrison:1996xf .

##### Cyclic symmetries of the $U$-plane.

Coulomb branch 0-form symmetries are generally manifestations of discrete
subgroups of the ${\rm U}(1)_{\mathtt{r}}$ $R$-symmetry. Gaugings of these
discrete symmetries have been extensively studied in the classification
programme of rank-one SCFTs Argyres:2015gha ; Argyres:2016yzz ; Caorsi:2019vex
. Nevertheless, the symmetries of the KK theories have not been explored yet
to the same extent. Although five-dimensional $\mathcal{N}=1$ SCFTs only
possess a ${\rm SU}(2)_{R}$ R-symmetry, their effective 4d $\mathcal{N}=2$ KK
theories can include discrete subgroups of the additional ${\rm
U}(1)_{\mathtt{r}}$ symmetry, which manifest themselves on the Coulomb branch
as accidental 0-form symmetries. We analyse in detail the possible
$\mathbb{Z}_{k}$ symmetries of the $D_{S^{1}}E_{n}$ theories and, using
geometric constraints from their underlying surfaces $\mathcal{S}$, classify
all allowed $\mathbb{Z}_{k}$ symmetries. These are listed in Table 2. Note
that the maximal cyclic symmetries of $D_{S^{1}}E_{n}$ for $n\geq 3$ are
precisely the centres of the simply connected flavour symmetry groups $E_{n}$,
i.e. $\mathbb{Z}_{9-n}\cong Z(E_{n})$.

$\bm{D_{S^{1}}E_{n}}$ | $\bm{Z(E_{n})}$ | $\bm{F_{\infty}}$ | $\bm{\\#}\textbf{ sing}$ | $\bm{\mathbb{Z}_{2}}$ | $\bm{\mathbb{Z}_{3}}$ | $\bm{\mathbb{Z}_{4}}$ | $\bm{\mathbb{Z}_{5}}$ | $\bm{\mathbb{Z}_{6}}$
---|---|---|---|---|---|---|---|---
$\bm{E_{0}}$ | $-$ | $I_{9}$ | $3$ |  | $\checkmark$ |  |  |
$\bm{E_{1}}$ | $\mathbb{Z}_{2}$ | $I_{8}$ | $4$ | $\checkmark$ |  | $\checkmark$ |  |
$\bm{\widetilde{E}_{1}}$ | ${\rm U}(1)$ | $I_{8}$ | $4$ |  |  |  |  |
$\bm{E_{2}}$ | ${\rm U}(1)$ | $I_{7}$ | $5$ |  |  |  |  |
$\bm{E_{3}}$ | $\mathbb{Z}_{6}$ | $I_{6}$ | $6$ | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$
$\bm{E_{4}}$ | $\mathbb{Z}_{5}$ | $I_{5}$ | $7$ |  |  |  | $\checkmark$ |
$\bm{E_{5}}$ | $\mathbb{Z}_{4}$ | $I_{4}$ | $8$ | $\checkmark$ |  | $\checkmark$ |  |
$\bm{E_{6}}$ | $\mathbb{Z}_{3}$ | $I_{3}$ | $9$ |  | $\checkmark$ |  |  |
$\bm{E_{7}}$ | $\mathbb{Z}_{2}$ | $I_{2}$ | $10$ | $\checkmark$ |  |  |  |
Table 2: Allowed $\mathbb{Z}_{k}$ automorphisms of the $U$-planes of the
$E_{n}$ KK theories which lead to well-defined quotients of the underlying SW
geometries. The red check is a symmetry that does not appear at the level of
the $(p,q)$-brane web of the 5d $E_{4}$ theory.

From the perspective of the underlying RES, all $U$-plane foldings described
in Table 2 are allowed, in the sense that the quotients lead to new well-
defined rational surfaces. Remarkably, the allowed – and therefore foldable –
symmetries of the $U$-plane almost coincide with the symmetries of the
underlying $(p,q)$ brane webs of the respective 5d SCFTs, where the folding
procedure corresponds to five-dimensional S-folding Kim:2021fxx ;
Apruzzi:2022nax ; Acharya:2021jsp ; Tian:2021cif . In fact, the interpretation
of CB foldings as S-folds was already apparent from the F-theoretic
construction of these models Aharony:2016kai ; Garcia-Etxebarria:2017ffg ;
Apruzzi:2020pmv ; Heckman:2020svr ; Giacomelli:2020gee ; Amariti:2023qdq ,
having been already discussed in Caorsi:2019vex for purely four-dimensional
SCFTs. Nevertheless, the link between the CB geometry and the brane-webs of
the five-dimensional models sheds new light on these S-folds.

The astute reader might have already noticed that Table 2 includes a
$\mathbb{Z}_{5}$ symmetry which exists on the Coulomb branch of the 4d
$D_{S^{1}}E_{4}$ theory, which does not manifest as a symmetry of the 5d brane
web. This symmetry is peculiar for the following reason. The fixed point of
the $\mathbb{Z}_{5}$ symmetry is a type $II$ singular fibre in Kodaira’s
classification (see Appendix A), and thus a CB folding would correspond to a
$\mathbb{Z}_{5}$ discrete gauging of the $H_{0}$ Argyres-Douglas theory
Argyres:1995jj ; Argyres:1995xn . However, the existence of this discrete
gauging was argued in Argyres:2016yzz to be rather speculative.

Five-dimensional $\mathbb{Z}_{k}$ S-folds involve cutting the 5-brane web into
$k$ slices, keeping only one of these slices, and inserting a specific 7-brane
at the fixed point of the $\mathbb{Z}_{k}$ action Kim:2021fxx . As recently
argued in Apruzzi:2022nax , such quotients can be generalised by keeping $k-1$
slices and only removing one, while inserting appropriate 7-brane
configurations at the fixed point. In the 4d KK theory, we understand this
procedure of _fractional folding_ as a composition of a folding and an
_unfolding_ , the latter being a formal inverse to a folding. This reproduces
the list of rank-one examples found in Apruzzi:2022nax .

##### Folding across dimensions.

The mathematical formalism of rational elliptic surfaces identifies a theory
by fixing one singular fibre of the SW geometry: the fibre at ‘infinity’,
$F_{\infty}$ Caorsi:2018ahl ; Closset:2021lhd . This singular fibre specifies
the UV physics of a field theory. An operation that can change the fibre at
infinity of a given RES $\mathcal{S}$ is the _quadratic twisting_ , which is a
particular reparametrisation of the Weierstraß invariants of the surface. Such
operations can thus give relations between theories of seemingly different
origins.

Of particular interest to us will be combinations of CB foldings and quadratic
twistings. These constructions are perfectly exemplified by the foldings of
the KK theories, which are described by $F_{\infty}=I_{n}$, where $I_{n}$ are
the multiplicative type Kodaira singularities (see Table 4 in Appendix A).
Coulomb branch $\mathbb{Z}_{k}$-foldings of these models would naturally
suggest the mapping $I_{n}\rightarrow I_{n/k}$, but our interest lies in the
slightly modified version $I_{n}\rightarrow I_{n/k}^{*}$. In this latter
example, the new fibre at infinity describes a 4d SQCD model, leading thus to
a link between 4d and 5d theories. Specifically, we show that this scenario
can only occur for $\mathbb{Z}_{2}$-foldings, being closely related to the
concept of _holonomy saddles_ Hwang:2017nop ; Hwang:2018riu . In particular,
such foldings imply that the circle compactification of particular 5d SCFTs
leads to two copies of locally indistinguishable 4d SQCD theories in the
infrared. This phenomenon was studied in Jia:2021ikh ; Jia:2022dra for the 5d
pure ${\rm SU}(2)$ gauge theory, and we extend this analysis to theories with
$2N_{f}$ flavours,

${\rm 5d}\leavevmode\nobreak\ {\rm SU}(2)+2N_{f}\text{ fund
}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ }2\text{ copies of 4d
}{\rm SU}(2)+N_{f}\text{ fund}\leavevmode\nobreak\ .$ (1.2)

The fact that there are always two copies is related to the rank of the gauge
group ${\rm SU}(2)$ Jia:2022dra , and it can be derived from the rationality
condition of the underlying surfaces. We prove the existence of these
$\mathbb{Z}_{2}$-foldings (1.2) by finding an explicit dictionary between the
4d and 5d mass parameters, and obtain the universal relation

$U_{2N_{f}+1}=\sqrt{\mathcal{K}_{N_{f}}u_{N_{f}}+\mathcal{P}_{N_{f}}}\leavevmode\nobreak\
,$ (1.3)

with $U_{2N_{f}+1}$ the CB parameter of the $D_{S^{1}}E_{2N_{f}+1}$ theory,
while $u_{N_{f}}$ is the SQCD Coulomb branch parameter, and
$\mathcal{K}_{N_{f}},\mathcal{P}_{N_{f}}$ are normalisation factors.

This analysis also leads to a conjecture on the BPS spectrum of these 5d
theories. For massless SQCD it generalises the results of Closset:2019juk ;
Jia:2022dra ; DelMonte:2021ytz , while for generic masses it gives a multitude
of new correspondences between 4d and 5d BPS quivers. Of particular interest
is the $N_{f}=3$ case, which would allow the computation of the BPS spectrum
for the non-toric BPS quivers of the $E_{7}$ theory. Furthermore, we expect
additional relations between the 5d $\mathcal{N}=1^{*}$ gauge theory and 4d
$\mathcal{N}=2^{*}$ SYM, as well as between the 6d E-string theory and
superconformal 4d $N_{f}=4$ SCQD. These results would not only predict 6d BPS
quivers Closset:2023pmc , but also lead to insights into their spectra of BPS
states.

###  1.3 Outline

The remainder of the paper is organised as follows. In Section 2, we present
the physical and mathematical ingredients that we aim to combine: we give a
short introduction to the $U$-plane of 4d rank-one $\mathcal{N}=2$ theories
and the formalism of rational elliptic surfaces $\mathcal{S}$. We further
introduce the two components of the automorphism group of $\mathcal{S}$.
Section 3 discusses the constraints and the explicit construction of discrete
gaugings in 4d and 5d theories, the relation to symmetries of $(p,q)$ brane
webs, as well as the construction of fractional foldings. We further comment
on non-cyclic symmetries of the Coulomb branch. Section 4 describes the
folding across dimensions and the relation to holonomy saddles. In Section 5,
we conjecture how mixed ’t Hooft anomalies are encoded in
$\text{Aut}(\mathcal{S})$. We conclude and give some directions for future
work in Section 6.

Several details and definitions of elliptic surfaces as well as modular forms
are given in Appendix A. In Appendix B, we provide a more general description
of how $U$-plane foldings can be obtained from the perspective of fundamental
domains. Appendix C discusses further aspects of non-cyclic Coulomb branch
symmetries by analysing several examples. Appendix D finally lists some
explicit expressions for the Seiberg–Witten curves. We also include a
Mathematica notebook with explicit curves for all $D_{S^{1}}E_{n}$ theories
encurves .

##  2 Symmetries of the Coulomb branch

In this section, we introduce the Coulomb branch of 4d rank-one
$\mathcal{N}=2$ theories and initiate the study of CB symmetries. Section 2.1
includes a gentle introduction to the formulation of the SW geometry as a
rational elliptic surface. Meanwhile, in Section 2.2 we discuss the
automorphisms of elliptic surfaces and the relations to isogenies and
foldings.

###  2.1 Introducing the $U$-plane

The low-energy dynamics on the Coulomb branch of four-dimensional
$\mathcal{N}=2$ supersymmetric quantum field theories is famously solved by
the Seiberg–Witten geometry Seiberg:1994rs ; Seiberg:1994aj . SW geometry
provides a beautiful synthesis of the notions of duality, elliptic curves and
modular forms. See Bilal:1995hc ; Tachikawa:2013kta ; DHoker:1999yni for some
reviews of the topic.

##### The Seiberg–Witten solution.

For any rank-one 4d $\mathcal{N}=2$ theory, the low-energy effective U(1)
field theory on the Coulomb branch $\mathcal{B}$ can be expressed in terms of
a scalar field $a$. The CB itself is parametrised by a gauge invariant
operator $U\in\mathcal{B}$, being thus one-complex dimensional for this class
of models.111For strictly 4d theories, the coordinate $U$ is more commonly
denoted by $u$. We also consider circle compactifications of 5d theories, and
use $U$ for both 5d and 4d Coulomb branch parameters. The celebrated
Seiberg–Witten solution expresses the scalar $a$ and its magnetic dual $a_{D}$
as periods of a differential form $\lambda$ over one-cycles of an elliptic
curve,

$a=\oint_{\gamma_{A}}\lambda\leavevmode\nobreak\ ,\qquad\qquad
a_{D}=\oint_{\gamma_{B}}\lambda\leavevmode\nobreak\ .$ (2.1)

These periods determine the low-energy effective gauge coupling $\tau$ as well
as the exact prepotential $\mathcal{F}$ as

$a_{D}={\partial\mathcal{F}\over\partial a}\leavevmode\nobreak\
,\qquad\quad\tau={\partial a_{D}\over\partial a}\leavevmode\nobreak\ .$ (2.2)

The SW geometry is then defined as the elliptic fibration of the SW curve over
the Coulomb branch, $E\to\mathcal{S}\to\mathcal{B}$. The total space of this
fibration, $\mathcal{S}$, forms a _rational elliptic surface_ (RES). We leave
to Appendix A more details about the structure of such surfaces (see also the
nice review schuttshioda , as well as Malmendier:2008yj ; Sen:1996vd ;
Banks:1996nj ; Minahan:1998vr ; He:2012kw ).

The elliptic fibres of the SW geometry can be brought into Weierstraß normal
form,

$y^{2}=4x^{3}-g_{2}(U,{\bm{m}})x-g_{3}(U,{\bm{m}})\leavevmode\nobreak\ ,$
(2.3)

where $g_{2}$ and $g_{3}$ are functions of the CB parameter $U$, as well as of
various mass parameters ${\bm{m}}$, and possibly dynamical scales and gauge
couplings that can appear in the theory. These fibres can become singular
above special loci on the CB, where the low-energy effective U(1) description
breaks down. The types of singularities that can appear on the $U$-plane are
covered by Kodaira’s classification of singular fibres, as we discuss in more
detail in Appendix A. In short, the various types of singular fibres can be
identified based on the orders of vanishing of the Weierstraß invariants
$g_{2},g_{3}$ and the discriminant $\Delta$. We should mention that a
configuration of singular fibres for a given theory will typically change as
the mass parameters vary.

The low-energy effective coupling $\tau$ (2.2) is realised as the complex
structure of the elliptic curve. This identifies the $\mathcal{J}$-invariant
$\mathcal{J}(U)$ of the elliptic curve (2.3) with the modular $J$-invariant
$J(\tau)$:222See Appendix A for the precise definitions of both
$\mathcal{J}(U)$ and $J(\tau)$.

$\mathcal{J}(U(\tau))=J(\tau)\leavevmode\nobreak\ ,$ (2.4)

For fixed parameters ${\bm{m}}$, the family of elliptic curves (2.3) is
parametrised by the coordinate $U$, and so the $\mathcal{J}$-invariant of the
corresponding RES becomes a rational function $\mathcal{J}(U)$ on the base
$\mathcal{B}$. This promotes $U$ to a function of $\tau$ through (2.4).
Solving this equation has been crucial for the study of topological
correlation functions Malmendier:2008db ; Griffin:2012kw ; Malmendier:2012zz ;
Malmendier:2010ss ; Korpas:2017qdo ; Moore:2017cmm ; Korpas:2019ava ;
Korpas:2019cwg ; Manschot:2021qqe ; Aspman:2021kfp ; Aspman:2022sfj ;
Aspman:2023ate , and for obtaining BPS quivers directly from the SW geometry
Closset:2021lhd ; Magureanu:2022qym ; Cecotti:2015qha ; he2015dessins ;
He:2013lha

While in the generic case of arbitrary ${\bm{m}}$ there are no analytical
solutions to (2.4), in many cases, the duality and modular properties of the
Coulomb branch of any 4d $\mathcal{N}=2$ theory $\mathcal{T}$ can be found by
rewriting (2.4) as $P_{\mathcal{T}}(X)=0$, where

$P_{\mathcal{T}}(X)=\left(g_{2}(X,{\bm{m}})^{3}-27g_{3}(X,{\bm{m}})^{2}\right)J(\tau)-g_{2}(X,{\bm{m}})^{3}\leavevmode\nobreak\
,$ (2.5)

is a polynomial specified by the RES of the theory $\mathcal{T}$
Aspman:2021vhs ; Aspman:2021evt ; Magureanu:2022qym .

##### Classes of 4d $\mathcal{N}=2$ theories.

The quantum field theories of interest throughout this paper are the 4d
$\mathcal{N}=2$ field theories with a one complex dimensional Coulomb branch.
We focus on two particular classes of such theories. The possibly simplest
class is 4d $\mathcal{N}=2$ supersymmetric QCD with gauge group SU(2) and
$N_{f}\leq 4$ fundamental hypermultiplets. The SW curves of these theories
have been studied in detail in Seiberg:1994rs ; Seiberg:1994aj ;
Argyres:1995xn ; Nahm:1996di ; Kanno:1998qj ; Malmendier:2008db ;
Aspman:2021vhs ; Aspman:2021evt ; Closset:2021lhd , and we present them
explicitly in Appendix D. The second family of interest is represented by the
Kaluza–Klein (KK) theories obtained by a circle compactification of the UV
completion of the 5d $\mathcal{N}=1$ SU(2) theories, with $N_{f}=n-1$
fundamental flavours Seiberg:1996bd ; Morrison:1996xf . These theories are the
simplest examples of 5d SCFTs, having $E_{n}$ flavour symmetry, where the low
flavour-rank groups are given by:

$\displaystyle E_{1}$ $\displaystyle=\text{SU}(2)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\widetilde{E}_{1}=$
$\displaystyle\text{U}(1)\leavevmode\nobreak\ ,$ (2.6) $\displaystyle E_{2}$
$\displaystyle=\text{SU}(2)\times\text{U}(1)\leavevmode\nobreak\ ,$
$\displaystyle\qquad E_{4}=$ $\displaystyle\text{SU}(5)\leavevmode\nobreak\ ,$
$\displaystyle E_{3}$
$\displaystyle=\text{SU}(3)\times\text{SU}(2)\leavevmode\nobreak\ ,$
$\displaystyle\qquad E_{5}=$ $\displaystyle\text{Spin}(10)\leavevmode\nobreak\
,$

while for $n=6$, $7$ and $8$, $E_{n}$ are the exceptional Lie groups. We will
denote their circle compactifications by $D_{S^{1}}E_{n}$.

These 5d rank-1 SCFTs are geometrically engineered in $M$-theory
compactifications on non-compact Calabi-Yau threefolds, which are the
canonical line bundles over del Pezzo surfaces $dP_{n}$ or the Hirzebruch
surface $\mathbb{F}_{0}$. More precisely, the $E_{n}$ theories with
$n=2,\dots,8$ are engineered on local $dP_{n}$, while the $E_{1}$ theory
corresponds to local $\mathbb{F}_{0}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.
The $E_{0}$ theory is engineered on local $\mathbb{P}^{2}$ and does not have a
gauge theory interpretation, but can be obtained from the $\widetilde{E}_{1}$
theory, which itself corresponds to local $\mathbb{F}_{1}\cong dP_{1}$.

From the $D_{S^{1}}E_{n}$ theories, by deformations, geometric-engineering
limits and RG flows one can obtain 4d SQCD, as well as the Argyres-Douglas
(AD) Argyres:1995jj ; Argyres:1995xn and Minahan-Nemeschansky (MN)
Minahan:1996fg ; Minahan:1996cj superconformal field theories in 4d. Their SW
geometries have been determined and analysed in great detail Eguchi:2002fc ;
Eguchi:2002nx ; Kim:2014nqa ; Huang:2013yta ; Ganor:1996pc ; Lawrence:1997jr ;
Aspinwall:1993xz ; Klemm:1996bj ; Chiang:1999tz ; Nekrasov:1996cz ;
Lerche:1996ni ; Katz:1997eq ; Hori:2000kt ; Hori:2000ck . We list the explicit
curves of the toric $D_{S^{1}}E_{n}$ theories in Appendix D. See also the
Mathematica notebook encurves for the toric as well as the non-toric curves.

The SW geometry of any rank-one 4d $\mathcal{N}=2$ SQFT is a rational elliptic
surface, being partially characterised by its configuration of singular
fibres. One particular dedicated singular fibre is the fibre at infinity
$F_{\infty}$, which corresponds to the point $U=\infty$ on the CB
Caorsi:2018ahl ; Closset:2021lhd . This singularity fixes the ‘UV physics’,
and will be singled out from the rest of the singular fibres. As such, we use
the notation $(F_{\infty};F_{v_{1}},\dots)$, for describing a RES with fibre
at infinity $F_{\infty}$, and bulk singular fibres $(F_{v_{1}},\ldots)$. For
the asymptotically free gauge theories with $N_{f}$ fundamentals for instance,
the point at infinity is characterised by the one-loop beta function
coefficient, leading to the following identifications Closset:2021lhd :

$\displaystyle F_{\infty}=I_{4-N_{f}}^{*}\,:$ $\displaystyle\qquad\text{4d
{\rm SU}(2) SQCD with $N_{f}$ fundamentals}\leavevmode\nobreak\ ,$ (2.7)
$\displaystyle F_{\infty}=I_{9-n}\,:$ $\displaystyle\qquad
D_{S^{1}}E_{n}\leavevmode\nobreak\ .$

The remaining possibilities describe 4d $\mathcal{N}=2$ SCFTs, such as the AD
and MN theories Caorsi:2019vex ; Closset:2021lhd ; Magureanu:2022qym ;
Argyres:1995jj ; Argyres:1995xn ; Minahan:1996fg ; Minahan:1996cj .

##### Higher-form symmetries.

Finally, let us comment on the higher-form symmetries of the theories
described above. The candidate theories that have 1-form symmetries are those
which allow a gauge theory description where the matter fields do not
transform under the centre of the gauge group Gaiotto:2014kfa . The first such
example is the pure 4d $\text{SU}(2)$ theory, which has a
$\Gamma^{(1)}=\mathbb{Z}_{2}$ 1-form symmetry. For future reference, let us
mention that this theory also has a mixed anomaly between the 1-form symmetry
and the $\mathbb{Z}_{8}\subset{\rm U}(1)_{\mathtt{r}}$ 0-form symmetry,
originating from the classical $R$-symmetry and broken by an ABJ anomaly
Cordova:2018acb . Another example of a purely 4d theory having a 1-form
symmetry is the $\mathcal{N}=2^{*}$ theory, which is the $\mathcal{N}=2$
preserving mass deformation of $\mathcal{N}=4$ SYM.

It is by now well-known that the 5d $E_{1}$ SCFT also exhibits a
$\Gamma^{(1)}=\mathbb{Z}_{2}$ 1-form symmetry. This might not come as a
surprise since the theory admits a deformation to the 5d $\mathcal{N}=1$
supersymmetric gauge theory with gauge algebra $\mathfrak{su}(2)$.
Consequently, the circle compactification to the $D_{S^{1}}E_{1}$ theory
yields both a $\mathbb{Z}_{2}$ 0-form and a 1-form symmetry.

Finally, the 5d $E_{0}$ theory has a $\Gamma^{(1)}=\mathbb{Z}_{3}$ 1-form
symmetry Albertini:2020mdx ; Morrison:2020ool , leading to both a
$\mathbb{Z}_{3}^{(1)}$ and $\mathbb{Z}_{3}^{(0)}$ 1-form and 0-form symmetries
in the 4d KK theory. In this case, however, there is a cubic anomaly in 5d,
leading to a mixed anomaly in the 4d theory Closset:2023pmc . We will discuss
the relation between the 0-form and 1-form symmetries in more detail in
Section 5.

###  2.2 Automorphisms of elliptic surfaces

The Seiberg–Witten geometry of rank-one 4d $\mathcal{N}=2$ theories is a
rational elliptic surface (RES), whose base corresponds to the $U$-plane,
together with the point at infinity. As previously explained, by a RES we mean
an elliptic fibration over the complex projective plane with a section
(referred to as the _zero-section_) and at least one singular fibre
schuttshioda . The main focus of this work will be on the group of
automorphisms of these surfaces, and on quotients of the SW geometry by such
automorphisms Donagi:1995cf ; Malmendier:2009au ; Argyres:2010py ;
Argyres:2022lah ; Argyres:2022fwy ; Xie:2022lcm ; Sakai:2023fnp ;
Argyres:2023tfx ; Arias-Tamargo:2023duo ; Argyres:2023eij .

Perhaps a natural interpretation of these quotients is as discrete gaugings of
the underlying field theories, but this dictionary can be further expanded.
The group of automorphisms of a RES $\mathcal{S}$ is isomorphic to the semi-
direct product KARAYAYLA20121 :

$\boxed{{\rm Aut}(\mathcal{S})={\rm MW}(\mathcal{S})\rtimes{\rm
Aut}_{\sigma}(\mathcal{S})\leavevmode\nobreak\ .}$ (2.8)

Here ${\rm MW}(\mathcal{S})$ is the Mordell–Weil group of $\mathcal{S}$, while
${\rm Aut}_{\sigma}(\mathcal{S})$ is the subgroup of automorphisms preserving
the zero section $\sigma$. We will analyse the two subgroups separately in the
rest of this section, and collect the precise definitions and relations in
Appendix A.2.

An important distinction between the two groups on the RHS of (2.8) is the
following: the group ${\rm Aut}_{\sigma}(\mathcal{S})$ is not fixed by the
singular fibres of $\mathcal{S}$, as it is the case for ${\rm
MW}(\mathcal{S})$.333This is not entirely accurate, as there are some
exceptions of configurations having the same singular fibres but different MW
groups, as we discuss later. The automorphisms ${\rm
Aut}_{\sigma}(\mathcal{S})$ induce automorphisms on the base
$\mathbb{P}^{1}\cong\\{U\\}$ of the SW geometry. Thus, these automorphisms
appear whenever the $U$-plane exhibits certain symmetries, which occur by
tuning the mass parameters of a theory. As such, perturbations of the mass
parameters can explicitly break the CB symmetry, while preserving the
configuration of singular fibres, leading thus to a different ${\rm
Aut}_{\sigma}(\mathcal{S})$ group. However, if the singular fibres are not
changed, the MW group is not modified.

####  2.2.1 Isogenies

Consider a section $P\in{\rm MW}(\mathcal{S})$ of the elliptic fibration
$\pi:\mathcal{S}\rightarrow\mathbb{P}^{1}$. For any such section we can define
an automorphism $t_{P}$ of $\mathcal{S}$ as the translation by $P$ on the
elliptic fibres Schuett2009EllipticS . On the smooth fibres $E$, this
automorphism acts as a simple translation by $P$, implemented through the
usual addition on elliptic curves. Note that this translation does not
preserve the zero-section of the elliptic surface, but can be thought of as an
automorphism of the underlying surface by forgetting the elliptic structure.

Our primary focus will be on the torsion sections of the MW group. For any
torsion section $P$ of finite order $k\in\mathbb{N}$, the automorphism $t_{P}$
will also be of finite order and will operate without any fixed points. For
the generic smooth fibre $E$ of the elliptic surface $\mathcal{S}$, the
quotient by the torsion subgroup $\langle t_{P}\rangle$ generated by $t_{P}$
leads to a $k-$_isogenous_ elliptic curve $\widetilde{E}$. Here, we define a
$k$-isogeny $\varphi_{k}:E\rightarrow\widetilde{E}$ as a $k$-to-1
homomorphism, mapping the marked point of $E$ to that of $\widetilde{E}$. An
isogeny is accompanied by a unique dual isogeny
$\hat{\varphi}_{k}:\widetilde{E}\to E$ such that their composition corresponds
to multiplication by $k$:

$\displaystyle\varphi_{k}\circ\hat{\varphi}_{k}=k\,\widetilde{E}\leavevmode\nobreak\
,\qquad\qquad\hat{\varphi}_{k}\circ\varphi_{k}=k\,E\leavevmode\nobreak\ ,$
(2.9)

where by $k\,E$ we mean the multiplication-by-$k$ map on the corresponding
elliptic curve.

Such quotients will also affect the singular fibres of the RES. Recall that in
the Weierstraß model, the singular fibres will be either nodal or cuspidal
curves and upon resolving these singularities the exceptional divisors will
intersect according to an extended ADE Dynkin diagram, as indicated in Table
4. To each fibre, we also associate an Euler number $e_{v}$, which is equal to
the order of vanishing of the discriminant $\Delta$. Then, an important
constraint for a rational elliptic surface is that:

$e(\mathcal{S})=\sum_{v}e(F_{v})=12\leavevmode\nobreak\ .$ (2.10)

Consider now a singular fibre of type $I_{n}$. Then, under a $k$-isogeny
generated by the $k$-torsion section $P$, for prime $k$, the action on the
singular fibre will depend on how it is intersected by $P$, as follows
Schuett2009EllipticS :

$\displaystyle\bullet\quad\text{For trivial intersections:}\,$ $\displaystyle
I_{n}\longrightarrow I_{nk}\leavevmode\nobreak\ ,\hskip 56.9055pt$ (2.11)
$\displaystyle\bullet\quad\text{For non-trivial intersections:}\,$
$\displaystyle I_{n}\longrightarrow I_{n/k}\leavevmode\nobreak\ .\hskip
56.9055pt$

Here, an intersection is said to be _non-trivial_ 444It can be shown that
whenever a $k$-torsion section, for prime $k$, intersects non-trivially an
$I_{n}$ fibre, then $k|n$ – see Corollary 7.5 of Schuett2009EllipticS .
whenever the section $P$ intersects the singular point of the fibre in the
Weierstraß model. We should also stress that the resulting isogenous elliptic
surface must have a torsion section of the same order for the dual isogeny to
exist. Importantly, while the Euler numbers of the singular fibres may change,
the isogeny does not affect the rationality condition Schuett2009EllipticS .
The explicit form of the isogeny can be determined using Vélu’s formula (see
e.g. Appendix A.2 of Closset:2023pmc for a review).

##### Semi-direct product structure.

Let us also briefly mention the semi-direct product structure of (2.8). Recall
that from any section $P\in{\rm MW}(\mathcal{S})$ we can define an
automorphism $t_{P}$ of $\mathcal{S}$. Then, given an element $\alpha\in{\rm
Aut}_{\sigma}(\mathcal{S})$, there is an action of ${\rm
Aut}_{\sigma}(\mathcal{S})$ on ${\rm MW}(\mathcal{S})$, defined by:

$\alpha\cdot t_{P}=t_{\alpha(P)}\leavevmode\nobreak\ .$ (2.12)

This subtle difference between a direct product and a semi-direct product will
be important in the context of mixed anomalies, which will be discussed in
Section 5.

####  2.2.2 Foldings

Let us now consider the automorphisms preserving the zero-section. As
mentioned above, any such automorphism $\tau:\mathcal{S}\to\mathcal{S}$
induces an automorphism on the base $\mathbb{P}^{1}\cong\\{U\\}$. We denote by
${\rm Aut}_{\mathcal{S}}(\mathbb{P}^{1})$ the collection of all induced
automorphisms on the CB. The group in $\text{Aut}_{\sigma}(\mathcal{S})$ (2.8)
is then a $\mathbb{Z}_{2}$ extension of ${\rm
Aut}_{\mathcal{S}}(\mathbb{P}^{1})$, where the $\mathbb{Z}_{2}$ acts as the
inversion on the elliptic fibre, i.e. it maps $(x,y)\mapsto(x,-y)$ in the
Weierstraß form (2.3) KARAYAYLA20121 :

$1\rightarrow\mathbb{Z}_{2}\rightarrow{\rm
Aut}_{\sigma}(\mathcal{S})\rightarrow\text{Aut}_{\mathcal{S}}(\mathbb{P}^{1})\rightarrow
1\leavevmode\nobreak\ .$ (2.13)

##### Induced automorphisms.

For any rational elliptic surface $\mathcal{S}$, the group ${\rm
Aut}_{\mathcal{S}}(\mathbb{P}^{1})$ is isomorphic to only one of the following
KARAYAYLA20121 :

  * •

The cyclic group $\mathbb{Z}_{k}$ with $k\leq 12$ and $k\neq 11$.

  * •

The Klein four-group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.

  * •

The dihedral group $D_{k}$ (with $2k$ elements) for $k=3,4$ or 6.

  * •

The alternating group $A_{4}$.

For the rational elliptic surfaces having constant $\mathcal{J}$-map, namely
the configurations having only elliptic elements and/or $I_{0}^{*}$ singular
fibres, there are a few more possibilities Karayayla+2014+1772+1795 , which,
however, will not be relevant for our purposes.

Note that the group of induced automorphisms is always a subgroup of the
symmetric group ${\rm Aut}_{\mathcal{S}}(\mathbb{P}^{1})\subset S_{k}$, with
$k$ being the number of singular fibres of $\mathcal{S}$. This is, of course,
just a statement about the maximal symmetry exchanging the $U$-plane
singularities.

As already alluded to, the configuration of singular fibres does not
automatically determine the group of automorphisms ${\rm
Aut}_{\sigma}(\mathcal{S})$. As an example, consider the $D_{S^{1}}E_{1}$
theory, for which there exists a point in the moduli space where the $U$-plane
singularities arrange themselves as the roots of $U^{4}=c$, for some
$c\in\mathbb{C}$ Closset:2021lhd . In this case, the group of induced
automorphisms is enhanced to ${\rm
Aut}_{\mathcal{S}}(\mathbb{P}^{1})=\mathbb{Z}_{4}$, while, otherwise, this is
only ${\rm Aut}_{\mathcal{S}}(\mathbb{P}^{1})=\mathbb{Z}_{2}$, for the same
configuration of singular fibres: $(I_{8};4I_{1})$. We will return to this
example in the following sections.

##### Base change.

Let us now consider quotients of rational elliptic surfaces by subgroups of
${\rm Aut}_{\sigma}(\mathcal{S})$. Generally, such quotients are well-defined
only if the generating automorphism $\alpha\in{\rm Aut}_{\sigma}(\mathcal{S})$
has the same order $k$ as the one induced on the base curve. The induced map
on the base is then a rotation $U\mapsto\omega_{k}\,U$ by the angle
$\frac{2\pi}{k}$, that is, $(\omega_{k})^{k}=1$. We should point out that
there exist CB symmetries which are not induced by automorphisms of the
underlying RES. Quotients by such automorphisms will not generate new RES, and
thus should also not be allowed physically.

Given the quotient projection
$\epsilon:\mathcal{S}\rightarrow\tilde{\mathcal{S}}=\mathcal{S}/\langle\alpha\rangle$,
for $\alpha\in{\rm Aut}_{\sigma}(\mathcal{S})$, there is an induced projection
map on the CB KARAYAYLA20121 :

$g_{k}:\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}\leavevmode\nobreak\
,\qquad\quad U\mapsto z=U^{k}\leavevmode\nobreak\ .$ (2.14)

The complete list of such quotients can be found in Table 6 of KARAYAYLA20121
for non-constant $\mathcal{J}$, and Table 6 of Karayayla+2014+1772+1795 for
constant $\mathcal{J}$-maps. In the mathematical literature, these
constructions are usually referred to as _base changes_ , and are important
for determining the MW groups of elliptic surfaces schuttshioda . Similar
quotient projections for non-cyclic symmetries have not been considered to the
same extent in the mathematical literature, and we discuss them further in
Section 3.5 and Appendix C.

Base changes are reflected as $k$-_foldings_ of the $U$-plane, according to
(2.14). Let us also mention that in this case, the degrees of the
$\mathcal{J}$-maps of the two rational elliptic surfaces are related by

${\rm deg}(\mathcal{J}_{\mathcal{S}})=k\cdot{\rm
deg}(\mathcal{J}_{\widetilde{\mathcal{S}}})\leavevmode\nobreak\ ,$ (2.15)

for $k>1$.555We define the degree of the rational function $\mathcal{J}(U)$ as
the maximum of the degrees of the numerator and denominator polynomials of
$U$. See Appendix A. As a result, these quotients are quite different from
isogenies, as the latter preserve the degree of the $\mathcal{J}$-map.
Foldings can be also understood from the perspective of fundamental domains,
where the identity (2.15) is particularly relevant – we leave this discussion
to Appendix B.

##### Quadratic twists.

Finally, there is an interesting interplay between base changes and quadratic
twists which allows for a richer structure when applied to Seiberg–Witten
geometry. A quadratic twist of an elliptic curve is another elliptic curve
obtained by rescaling the relative invariants $g_{2}$ and $g_{3}$ in the
Weierstraß model (see Appendix A.1). This transfers ‘stars’ across singular
fibres as follows:

$\displaystyle I_{n}\longleftrightarrow I_{n}^{*}\leavevmode\nobreak\ ,\qquad
II\longleftrightarrow IV^{*}\leavevmode\nobreak\ ,\qquad
III\longleftrightarrow III^{*}\leavevmode\nobreak\ ,\qquad
IV\longleftrightarrow II^{*}\leavevmode\nobreak\ .$ (2.16)

It is known that two elliptic curves with the same $\mathcal{J}$-invariant are
either isomorphic or quadratic twists of each other. Likewise, two elliptic
surfaces with the same $\mathcal{J}$-map have the same singular fibres, up to
a quadratic twist (or _transfer of star_). In particular, the smooth fibres
are isomorphic.

As we explain in more detail below, CB foldings of SQCD or KK theories are
associated with changes in the type of fibre at infinity $F_{\infty}$. The
only way to preserve the type of fibre at infinity is to combine the folding
with a simultaneous quadratic twist. This therefore gives two possibilities
for the behaviour of the type of fibre at infinity $F_{\infty}$:

$\displaystyle 1.\text{ Folding:}$ $\displaystyle\,I_{n}\mapsto
I_{n/k}^{*}\leavevmode\nobreak\ ,\text{ or }I_{n}^{*}\mapsto
I_{n/k}\leavevmode\nobreak\ ,$ (2.17) $\displaystyle 2.\text{ Folding with
quadratic twist:}$ $\displaystyle\,I_{n}\mapsto I_{n/k}\leavevmode\nobreak\
,\text{ and }I_{n}^{*}\mapsto I_{n/k}^{*}\leavevmode\nobreak\ .$

We will study these foldings in detail in Section 4.

Combining this base-change with a quadratic twist rescales the Weierstraß
invariants,

$g_{2}(U)\rightarrow f^{2}(z)\,g_{2}\left(z^{1\over
k}\right)\leavevmode\nobreak\ ,\qquad g_{2}(U)\rightarrow
f^{3}(z)\,g_{3}\left(z^{1\over k}\right)\leavevmode\nobreak\ .$ (2.18)

Then in order for the fibre at infinity $F_{\infty}$ to be preserved under a
base change, we need to apply a quadratic twist666Up to proportionality
constants. with $f(z)=z^{2-\frac{2}{k}}=U^{2k-2}$. Indeed, in this case
$g_{2}(U)$ becomes $g_{2}(z)\sim z^{4}$ and $g_{3}(U)$ becomes $g_{3}(z)\sim
z^{6}$ for all $k$. This generalises the discussion of the $\mathbb{Z}_{3}$
and $\mathbb{Z}_{2}$-foldings with quadratic twists of the $D_{S^{1}}E_{0}$
and $D_{S^{1}}E_{1}$ theories discussed in (Closset:2023pmc, , (4.23) &
(5.21)).

To summarise, any $k$-folding has the effect of mapping a rational elliptic
surface $\mathcal{S}$ to another surface $\widetilde{\mathcal{S}}$ such that
their $\mathcal{J}$-maps are related as (2.15). Without a quadratic twist,
this changes the fibre at infinity. Including the specific quadratic twist
$f(z)=z^{2-\frac{2}{k}}$ however, we can preserve the type of fibre at
infinity.

##  3 Folding the $U$-plane

As outlined in Section 2, the Coulomb branch of a theory $\mathcal{T}$ can
have certain discrete $\mathbb{Z}_{N}$ symmetries. This prompts a natural
question about the interpretation of the quotients of the SW geometry by such
symmetries. Note that the $U$-plane symmetries are generally subgroups of
$H\subset S_{n}$, where $n$ is the number of singularities, but quotients
$\mathcal{S}/H$ of the associated rational elliptic surface $\mathcal{S}$ by
such subgroups do not typically lead to other rational elliptic surfaces
KARAYAYLA20121 . We will thus consider mainly $\mathbb{Z}_{N}$ symmetries, and
comment on non-cyclic symmetries in Appendix C.

In Appendix B, we discuss how $U$-plane foldings can be obtained from both
modular and non-modular configurations. For the modular cases, we relate the
discrete $\mathbb{Z}_{N}$ symmetries to a pair of subgroups of
$\text{SL}(2,\mathbb{Z})$. Generic configurations are however non-modular, in
which case the $\mathbb{Z}_{N}$ symmetries can be read off from a consistent
choice of the fundamental domain. Our discussion in the appendix can be
applied to any rational elliptic surface, and can likely be generalised to
other (non-rational) elliptic surfaces, such as K3 surfaces, giving rise to
fundamental domains of index larger than 12 He:2012kw ; he2015dessins ;
He:2012jn ; He:2013lha .

The simplest interpretation of $U$-plane foldings is in terms of discrete
gaugings. Such a Coulomb branch analysis for the five-dimensional theories has
not been discussed in the literature, to the best of our knowledge.
Nevertheless, the gaugeable CB symmetries turn out to be related to the
symmetries of the corresponding $(p,q)$-brane webs, as we will show below.

###  3.1 Discrete gaugings of 4d theories

The continuous internal symmetries of 4d $\mathcal{N}=2$ superconformal field
theories are subgroups of the direct product ${\rm
U}(1)_{\mathtt{r}}\times{\rm SU}(2)_{R}\times F$ of the R-symmetry and the
flavour symmetry algebra, $F$. It was argued in Argyres:2015ffa ;
Argyres:2016yzz that discrete subgroups of $F$ cannot be gauged without adding
new degrees of freedom, while gaugings of subgroups of ${\rm SU}(2)_{R}$ break
(some of the) supersymmetry. As a result, the most general discrete gaugings
that one can thus consider are those of symmetries generated by
transformations:

$C=(\rho,\sigma,\varphi)\in{\rm U}(1)_{\mathtt{r}}\times{\rm
SL}(2,\mathbb{Z})\times{\rm Out}(F)\leavevmode\nobreak\ ,$ (3.1)

with ${\rm Out}(F)$ the outer automorphism group of the flavour algebra.777For
instance, the 4d SU(2) theory with $N_{f}=4$ fundamental hypermultiplets has
$F=\text{Spin}(8)$ flavour symmetry, with outer automorphism group
$\text{Out}(F)=S_{3}$ the triality group. See Tai:2010im ; Aspman:2021evt for
detailed discussions of this symmetry. The three generators must be chosen in
such a way that the $\mathcal{N}=2$ supersymmetry is preserved. Let us note
that the $\sigma$ and $\varphi$ factors do not act on the CB parameter, but
instead on the gauge coupling $\tau$ and the mass deformations.

##### Discrete gauging.

Consider first the generator $\sigma$ of some subgroup of the modular group,
$\mathbb{Z}_{k}\subset{\rm SL}(2,\mathbb{Z})$. It is straightforward to see
that the only such subgroups are
$\mathbb{Z}_{2},\mathbb{Z}_{3},\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$. Such
symmetries can only exist for fixed values of $\tau$, the only exception being
that of $\mathbb{Z}_{2}$, where the action of $\sigma$ on $\tau$ is trivial.

This transformation acts on the supercharges as a simple phase factor; as
such, in order to preserve the $\mathcal{N}=2$ supersymmetry, we need to
accompany it by a $\mathbb{Z}_{k}\subset{\rm U}(1)_{\mathtt{r}}$ action
generated by some element $\rho\in{\rm U}(1)_{\mathtt{r}}$. This action
manifests directly on the CB parameter $u$. Namely, the gauged theory will be
described by $\widetilde{u}=u^{r}$, for $r\in\mathbb{N}$ the smallest positive
integer necessary to build a ${\rm U}(1)_{\mathtt{r}}$ invariant operator
Argyres:2016yzz . That is:

$r=\frac{k}{\Delta_{u}}\leavevmode\nobreak\ ,$ (3.2)

with $\Delta_{u}$ being the scaling dimension of the CB parameter. This leads
to a new SCFT whose CB parameter has scaling dimension:

$\Delta_{\widetilde{u}}=r\Delta_{u}=k\leavevmode\nobreak\ .$ (3.3)

Finally, the discrete gauging must involve some ${\rm Out}(F)$ action relating
the mass parameters of the two theories. Note that here $\mathbb{Z}_{r}$ is
not the symmetry being gauged, but only a subgroup of it,
$\mathbb{Z}_{r}\subset\mathbb{Z}_{k}=\mathbb{Z}_{r\Delta_{u}}$. Nevertheless,
we will sometimes refer to such CB foldings as discrete gaugings.

In the language of rational elliptic surfaces, the CB scaling dimension is set
by the fibre at infinity as follows:

$\displaystyle\begin{tabular}[]{|c||c|c|c|c|c|c|c|}\hline\cr$\bm{F_{\infty}}$&$II$&$III$&$IV$&$I_{0}^{*}$&$II^{*}$&$III^{*}$&$IV^{*}$\\\
\hline\cr$\bm{\Delta_{u}}$&6&4&3&2&${6\over 5}$&${4\over 3}$&${3\over 2}$\\\
\hline\cr\end{tabular}$ (3.4)

Then, the discrete gauging relating the rational elliptic surfaces
$\mathcal{S}$ and $\widetilde{\mathcal{S}}=\mathcal{S}/\mathbb{Z}_{r}$, for
$\mathbb{Z}_{r}\subset{\rm Aut}_{\sigma}(\mathcal{S})$, will be one of the
following possibilities Caorsi:2019vex :

$\displaystyle\begin{tabular}[]{|c||c|c|c|c|c|c|}\hline\cr$\bm{r}$&5&4&3&2&2&2\\\
\hline\cr$\bm{F_{\infty}(\widetilde{\mathcal{S}})}$&$II^{*}$&$IV^{*}$&$I_{0}^{*}$&$IV$&$I_{0}^{*}$&$IV^{*}$\\\
\hline\cr$\bm{F_{\infty}(\mathcal{S})}$&$II$&$II$&$II$&$II$&$III$&$IV$\\\
\hline\cr\end{tabular}$ (3.5)

These follow from the physical constraint on the scaling dimensions (3.3). Out
of these, the $r=5$ case is not allowed, as there are no such quotients from
the perspective of rational elliptic surfaces KARAYAYLA20121 . The lists of
KARAYAYLA20121 can be used to find all 4d $\mathcal{N}=2$ SCFTs in the
classification of Argyres:2015ffa ; Argyres:2015gha ; Argyres:2016xmc ;
Argyres:2016xua , which was already detailed in Caorsi:2019vex .

Given a Coulomb branch with a $\mathbb{Z}_{r}$ symmetry, we distinguish two
distinct folding scenarios depending on the fixed point of the
$\mathbb{Z}_{r}$ symmetry. That is, we can either have a smooth fibre $I_{0}$,
or a singular fibre as the fixed point of the symmetry. We shall discuss the
two cases below and give a list of how these fixed points change under
discrete gaugings from the perspective of 4d $\mathcal{N}=2$ theories. In the
context of 5d SCFTs, these results can be interpreted as insertions of certain
7-branes in the geometry.

##### Foldings with smooth fibres at the fixed point.

Consider first the case where the fixed point of the $\mathbb{Z}_{r}$ symmetry
is not a CB singularity. This will correspond to gaugings of free ${\rm U}(1)$
theories. In particular, $\mathbb{Z}_{r}$ gaugings of a free vector multiplet
lead to so-called _frozen_ singularities:

$\displaystyle I_{0}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{r}\leavevmode\nobreak\ \leavevmode\nobreak\
}\begin{cases}I_{0}^{*}\,,&\text{ for }r=2\leavevmode\nobreak\ ,\\\
IV^{*}\,,&\text{ for }r=3\leavevmode\nobreak\ ,\\\ III^{*}\,,&\text{ for
}r=4\leavevmode\nobreak\ ,\\\ II^{*}\,,&\text{ for }r=6\leavevmode\nobreak\
.\end{cases}$ (3.6)

For a simple illustration, consider the $r=4$ case, for which there is a
single cover, namely:

$(IV^{*};4I_{1})\longrightarrow(II;III^{*},I_{1})\leavevmode\nobreak\ .$ (3.7)

This corresponds to the discrete gauging of the $A_{2}$ Argyres-Douglas theory
– sometimes also denoted $H_{2}$ or $(A_{1},D_{4})$ – by a
$\mathbb{Z}_{6}\subset{\rm U}(1)_{\mathtt{r}}$ symmetry, acting on the CB as a
$\mathbb{Z}_{4}$ symmetry Argyres:2016yzz . In Section 3.3, we will give an
alternative interpretation to these gaugings for 5d theories in terms of
insertions of 7-branes.

##### Foldings with singular fibres at the fixed point.

The second scenario is when the fixed point of the CB symmetry (which is
usually the origin of the $U$-plane) has a singularity. In the context of 4d
theories, we ought to mention the case of $\mathcal{N}=2$ ${\rm U}(1)$ gauge
theories with massless matter, whose CB is described by a $I_{n}$ cusp. The
$\mathbb{Z}_{2}$ gauging of such theories can only be implemented for even
$n=2n^{\prime}$ (being anomalous for odd $n$), in which case the CB transforms
as Argyres:2016yzz :

$I_{2n^{\prime}}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}I_{n^{\prime}}^{*}\leavevmode\nobreak\ .$ (3.8)

The daughter theories are ${\rm SU}(2)$ gauge theories with
$\Delta(\widetilde{u})=2$. Importantly, note that IR free theories cannot have
other $\mathbb{Z}_{k}$ global symmetries in ${\rm SL}(2,\mathbb{Z})$ with
$k>2$, as $\tau$ is allowed to vary on the Coulomb branch.

One can also have $II,III,$ or $IV$ singularities at the fixed point of the
discrete symmetry. In four dimensions, these singularities correspond to the
$H_{0},H_{1}$ and $H_{2}$ Argyres-Douglas theories, which admit the following
discrete gaugings Argyres:2016yzz :

$\displaystyle\begin{tabular}[]{|c|c||c|c|c|c|}\hline\cr$\bm{\mathcal{T}}$&{Flavour}&$\,\bm{\mathbb{Z}_{r}}\,$&$\bm{\mathcal{T}/\mathbb{Z}_{r}}$&{Deformation}&{Flavour}\\\
\hline\cr\hline\cr$II$&$-$&$\mathbb{Z}_{5}$&$II^{*}$&$(II^{*})$&$-$\\\
\hline\cr$III$&$A_{1}$&$\mathbb{Z}_{3}$&$III^{*}$&$(IV^{*},I_{1})$&$A_{1}$\\\
\hline\cr$IV$&$A_{2}$&$\mathbb{Z}_{2}$&$IV^{*}$&$(I_{0}^{*},2I_{1})$&$A_{2}$\\\
\hline\cr$IV$&$A_{2}$&$\mathbb{Z}_{4}$&$II^{*}$&$(III^{*},I_{1})$&$A_{1}$\\\
\hline\cr\end{tabular}$ (3.9)

Consider first the $H_{1}$ theory, with the massless singularity $III$.
Whenever the chiral deformation of the SW curve is frozen, the CB
singularities organise in a $\mathbb{Z}_{3}$ symmetric way. As a result, the
gauged theory simply freezes this deformation, without acting on the $A_{1}$
flavour symmetry. For the discrete gaugings of the $H_{2}$ AD theory, the
flavour symmetry changes only in the $\mathbb{Z}_{4}$ case Argyres:2016yzz .

A rather interesting example is the $\mathbb{Z}_{5}$ gauging of the $H_{0}$
theory proposed in Argyres:2016yzz . This symmetry is part of the ${\rm
U}(1)_{\mathtt{r}}$ R-symmetry and freezes the chiral deformation:

$II\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{5}\leavevmode\nobreak\ \leavevmode\nobreak\
}II^{*}\leavevmode\nobreak\ .$ (3.10)

This new theory, however, has been deemed as rather speculative, as it is not
connected by RG flows to any other 4d $\mathcal{N}=2$ SCFTs. We will come back
to this example when discussing gaugings of 5d SCFTs. Let us also mention that
the singular fibres $I_{0}^{*}$ or $IV^{*}$ can also appear as the fixed
points of a $\mathbb{Z}_{r}$ symmetry. Then, the (bulk) $U$-plane
singularities behave as follows Argyres:2016yzz :

$\displaystyle I_{0}^{*}\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{r}\leavevmode\nobreak\ \leavevmode\nobreak\
}\begin{cases}III^{*}\,,&\text{ for }r=2\leavevmode\nobreak\ ,\\\
II^{*}\,,&\text{ for }r=3\leavevmode\nobreak\ ,\\\ \end{cases}\qquad\quad
IV^{*}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}II^{*}\leavevmode\nobreak\ .$ (3.11)

As a final check, we consider the fibres of type $I_{n}^{*}$, for $n>0$. From
the classification of rational elliptic surfaces Persson:1990 ; Miranda:1990 ,
these are limited to $n\leq 4$ and, in four dimensions, they will either
describe ${\rm SU}(2)$ gauge theories with $N_{f}=4-n$ fundamentals, or,
alternatively $\mathbb{Z}_{2}\times{\rm U}(1)={\rm O}(2)$ gauge theories with
matter (i.e. the previously mentioned ${\rm U}(1)$ gauge theories from which a
$\mathbb{Z}_{2}$ symmetry was gauged; this is a mix of the $\mathbb{Z}_{2}$
charge conjugation and $\mathbb{Z}_{2}\subset{\rm U}(1)_{\mathtt{r}}$). In the
former case, the classical ${\rm U}(1)_{\mathtt{r}}$ R-symmetry is anomalous
being completely broken for $N_{f}>0$ with generic masses of the flavour
multiplets. However, in the massless case, a discrete
$\mathbb{Z}_{4(4-N_{f})}$ symmetry is anomaly-free. Since a $\mathbb{Z}_{2}$
subgroup of this acts the same as the centre of the ${\rm SO}(2N_{f})$ flavour
symmetry, the $u$-plane will only have a $\mathbb{Z}_{4-N_{f}}$ symmetry (or
$\mathbb{Z}_{2}$ for $N_{f}=0$). Let us note, however, that in these theories
there are mixed ‘t Hooft anomalies between the residual discrete R-symmetry
and the ${\rm SU}(2)_{R}$ symmetry, or with gravity, as well as a cubic
anomaly (see e.g. Cordova:2018acb ). Hence these residual R-symmetries cannot
be gauged.

###  3.2 Discrete gaugings of 5d theories

The next step in our analysis is to generalise the above picture for KK
theories. Five-dimensional $\mathcal{N}=1$ SCFTs only possess an ${\rm
SU}(2)_{R}$ R-symmetry, but the SW geometries we are considering correspond to
effective 4d $\mathcal{N}=2$ theories, which can include discrete subgroups of
the additional ${\rm U}(1)_{\mathtt{r}}$ symmetry. Such 0-form symmetries on
the CB arise as accidental R-symmetries of the KK theory. Note also that, as
opposed to the purely four-dimensional theories, the CB parameter $U$ is now
dimensionless, and thus there is no analogue for (3.3).

####  3.2.1 Discrete symmetries and the prepotential

Before we discuss explicit examples of foldings, we first constrain the cyclic
symmetries which lead to well-defined quotients of the SW geometry. These
quotients lead to a simple effect on the periods and the prepotential of the
respective theory.

##### Gaugeable $\bm{\mathbb{Z}}_{k}$ symmetries.

For the $D_{S^{1}}E_{n}$ theories, the fibre at infinity is of $I_{9-n}$ type,
and, thus, a $\mathbb{Z}_{k}$ discrete gauging must transform the fibre at
infinity as:

$\text{$\mathbb{Z}_{k}$ gaugings for }D_{S^{1}}E_{n}\,:\qquad
I_{9-n}^{\infty}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{k}\leavevmode\nobreak\ \leavevmode\nobreak\
}I_{(9-n)/k}^{\infty}\leavevmode\nobreak\ .$ (3.12)

Thus, it must be the case that $k|(9-n)$. Combining this with the fact that
the $U$-plane has $n+3$ simple singularities (i.e. $I_{1}$) for generic mass
parameters, the allowed $\mathbb{Z}_{k}$ symmetries are shown in Table 2. Note
that these conditions do not explicitly exclude the symmetries of
$D_{S^{1}}\widetilde{E}_{1}$, for which a more careful argument is needed. We
will come back to this matter in Section 3.2.3.

The symmetries listed in Table 2 are in precise agreement with the possible
induced cyclic automorphisms ${\rm
Aut}_{\mathcal{S}}(\mathbb{P}^{1})\cong\mathbb{Z}_{k}$ on the $\mathbb{P}^{1}$
base of the SW geometry KARAYAYLA20121 . For $n\geq 3$, the gaugeable
$\mathbb{Z}_{k}$ symmetries of the $D_{S^{1}}E_{n}$ theories are subgroups of
the centre of the simply connected flavour symmetry groups,888Note also that
the $D_{S^{1}}E_{n}$ theories are geometrically engineered on local $dP_{n}$
surfaces, where the del Pezzo surface $dP_{n}$ has degree (i.e. self-
intersection number of the canonical class) $9-n$.

$Z(E_{n})\cong\mathbb{Z}_{9-n}\leavevmode\nobreak\ .$ (3.13)

This relation (3.13) of course only holds for the theories whose maximal
flavour algebra does not involve a ${\rm U}(1)$ factor. Note also that the
folding by the centre of the simply connected flavour symmetry group gives a
map from the $D_{S^{1}}E_{n}$ theory to a $D_{S^{1}}E_{8}$ theory.

##### Periods and prepotential.

Such discrete quotients of the SW geometry do lead to tractable changes in the
prepotential and the periods. As shown in e.g. Closset:2021lhd , the Picard-
Fuchs differential equations for rank-one theories reduce to a single second
order differential equation, with the solutions being referred to as the
‘geometric periods’, i.e. the periods defined as:

$\displaystyle\omega_{D}=\frac{da_{D}}{dU}\leavevmode\nobreak\
,\qquad\quad\omega_{a}=\frac{da}{dU}\leavevmode\nobreak\ ,$ (3.14)

with $(a_{D},a)$ as introduced in (2.1). For the local $dP_{n}$ geometries, in
a conveniently chosen basis, these are the only two periods that receive
quantum corrections and coincide with the classical D-brane periods in the
large volume limit (see e.g. Aspinwall:2004jr ). Namely, we have
$\Pi_{D4}=a_{D}$, and $\Pi_{D2}=2a$, where $\Pi_{D4}$ and $\Pi_{D2}$ are the
periods associated with D4 and D2-branes wrapping the 4-cycle and a 2-cycle of
the local threefold, respectively. In this limit the so-called ‘mirror map’
Hori:2000ck ; Hori:2000kt becomes:

$a\approx\frac{1}{2\pi i}\log U\leavevmode\nobreak\
,\qquad\quad\omega_{a}=\frac{da}{dU}\approx\frac{1}{2\pi
i}\frac{1}{U}\leavevmode\nobreak\ ,$ (3.15)

in the region $U\rightarrow\infty$. Moreover, to leading order, we have
Closset:2021lhd :

$\Pi_{D4}\approx\frac{9-n}{2}\,a^{2}+\mathcal{O}(a)\leavevmode\nobreak\ ,$
(3.16)

where we neglect the contribution due to the mass parameters. This leads to
the following asymptotics for the dual geometric period:

$\displaystyle\omega_{D}\approx-\frac{9-n}{4\pi^{2}}\,\log
U+\ldots\leavevmode\nobreak\ .$ (3.17)

As the Coulomb branch of $\mathcal{T}=D_{S^{1}}E_{n}$ contains $n+3$
singularities for generic values of the mass parameters, the analytic
continuation of $(\omega_{D},\omega_{a})$ throughout the entire $U$-plane is
highly non-trivial. In the cases of interest, the $U$-plane shows a
$\mathbb{Z}_{k}$ symmetry, in which case it becomes simpler to solve for the
periods on the $z=U^{k}$ plane. Evidently, these would be the periods of the
folded theory, and we will refer to them as
$(\widetilde{\omega}_{D},\widetilde{\omega}_{a})$ and
$(\widetilde{a}_{D},\widetilde{a})$ Closset:2021lhd .

Consider a theory $\mathcal{T}=D_{S^{1}}E_{n}$ whose $U$-plane possesses a
$\mathbb{Z}_{k}$-symmetry, and let $\mathcal{T}/\mathbb{Z}_{k}$ denote the new
theory arising from the $\mathbb{Z}_{k}$-folding of the $U$-plane of
$\mathcal{T}$. We will also use $z=\widetilde{U}=U^{k}$. From the above
considerations, by matching asymptotics it is not difficult to see that there
is a region of the $U$-plane of $\mathcal{T}$ where the periods
$(\omega_{D},\omega_{a})$ are related to
$(\widetilde{\omega}_{D},\widetilde{\omega}_{a})$ by:

$\omega_{D}\approx k\,\widetilde{\omega}_{D}\leavevmode\nobreak\
,\qquad\quad\omega_{a}=\widetilde{\omega}_{a}\leavevmode\nobreak\ ,$ (3.18)

to leading order, where the additional factor in $\omega_{D}$ appears due to
the logarithmic term $\log(U^{k})$ inside the geometric period
$\widetilde{\omega}_{D}$. Thus, the leading order terms in the prepotential
will be related as follows:999The astute reader might have noticed that since
$\tau={\omega_{D}\over\omega}$, the identity (3.18) appears to lead to a
$k$-isogeny. This is, however, not the case, since our analysis only concerns
leading order terms.

$\mathcal{F}_{\mathcal{T}}\approx
k\mathcal{F}_{\mathcal{T}/\mathbb{Z}_{k}}\leavevmode\nobreak\ .$ (3.19)

This expression is identical to the one derived in Kim:2021fxx , in the
context of _five-dimensional S-folds_. There, the $\mathbb{Z}_{k}$-folding
procedure involved symmetries of the $(p,q)$-brane webs. We postpone the
discussion about the relation between the two constructions to Section 3.3.

####  3.2.2 Symmetry enhancements

In the remainder of this section, we will show how the folding procedure is
explicitly realised through some examples.

##### The $\bm{\mathbb{Z}_{3}}$-folding of $\bm{E_{0}}$.

The $U$-plane of the $D_{S^{1}}E_{0}$ theory is $\mathbb{Z}_{3}$ symmetric
and, thus, we can apply the folding procedure discussed above. The
$\mathbb{Z}_{3}$ symmetry identifies the three bulk $I_{1}$ singularities, and
thus the folded Coulomb branch $w=U^{3}$, corresponds to the transition:

$(I_{9}^{\infty};3I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{3}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{3}^{\infty};I_{1},IV^{*})\leavevmode\nobreak\ .$ (3.20)

This discrete gauging was also discussed in Closset:2023pmc . While the three
$I_{1}$ cusps are identified, the origin of the $w$-plane becomes a singular
point of type $IV^{*}$, as expected from (3.6). We also note that the new
configuration of singular fibres is identical to that of the massless
$D_{S^{1}}E_{6}$ configuration, which is modular with monodromy group
$\Gamma^{0}(3)$. The orbifold point $w=0$ corresponds to an elliptic point of
$\Gamma^{0}(3)$, and is smoothed out on the $\mathbb{Z}_{3}$ cover. This
folding can be also argued from the perspective of modular elliptic surfaces,
as detailed in Appendix B. We say, in particular, that
$\Gamma^{0}(9)\sqsubset\Gamma^{0}(3)$, i.e. $\Gamma^{0}(9)$ is a subgroup of
$\Gamma^{0}(3)$ which can be folded to the latter.

##### The $\bm{\mathbb{Z}_{2}}$-folding of $\bm{E_{1}}$.

The $U$-plane of the $D_{S^{1}}E_{1}$ theory has a $\mathbb{Z}_{2}$ symmetry
for generic values of the exponentiated inverse gauge coupling parameter
$\lambda$. One can interpret this from a gauge theory point of view as
follows: the CB parameter $U$ is the VEV of a fundamental Wilson line, which,
under multiplication by elements of the centre of the gauge group does not
change the physics as long as all fields are in the adjoint of the gauge group
Nekrasov:1996cz . Equivalently, the $E_{1}$ five-dimensional SCFT is known to
possess a $\mathbb{Z}_{2}$ 1-form symmetry, which, upon reduction on $S^{1}$
of the theory, becomes both a one and a 0-form symmetry Morrison:2020ool ;
Albertini:2020mdx .

Let us focus for now on the massless configuration, which is modular, with
monodromy group $\Gamma^{0}(8)$. The folding procedure corresponds to the
transition:

$(I_{8}^{\infty};I_{2},2I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{4}^{\infty};I_{1}^{*},I_{1})\leavevmode\nobreak\ ,$ (3.21)

where the new configuration can be interpreted as the massless
$D_{S^{1}}E_{5}$ configuration. Note also that this is again a modular
configuration, with monodromy group $\Gamma^{0}(4)$. Let us point out that
this $\mathbb{Z}_{2}$-folding is different from a 2-isogeny, which would
instead be a first step in gauging the 1-form symmetry Closset:2023pmc . The
difference can be understood, for instance, at the level of the monodromy
groups. Whilst isogenies preserve the index of the monodromy group inside
${\rm PSL}(2,\mathbb{Z})$, for $\mathbb{Z}_{k}$-foldings the index of the
monodromy group in the $\text{PSL}(2,\mathbb{Z})$ group enlarges by a factor
of $k$ – recall (2.15), and see also Appendix B.

##### Higher-form symmetries.

The previous two examples offer valuable insight into the origin of the
discrete symmetries on the Coulomb branch. The 5d $E_{0}$ and $E_{1}$ SCFTs
have $\mathbb{Z}_{3}$ and $\mathbb{Z}_{2}$ 1-form symmetries Morrison:2020ool
; Albertini:2020mdx , respectively, which upon compactification to four
dimensions turn into both 1-form and 0-form symmetries, depending on how the
line operators wrap the circle direction. For these examples, isogenies can be
thought of as discrete gaugings of 1-form symmetries, since the latter are
encoded in the MW group of the Seiberg–Witten geometry, as elaborated in
Closset:2021lhd ; Closset:2023pmc .

These isogenies should not be mistaken for _Galois covers_. The fundamental
distinction is that the SW geometry obtained after an isogeny corresponding to
a gauging of a 1-form symmetry will generally contain _undeformable_
singularities, in the sense of Argyres:2015ffa .101010Recall that an
undeformable or frozen $I_{n}$ singularity corresponds to a massless
hypermultiplet of charge $\sqrt{n}$. In contrast, when isogenies are
interpreted as Galois covers the resulting singularities will still allow a
deformation pattern. We will further analyse these in ACFMM:part1 .

Consider for now the $U$-plane of the $D_{S^{1}}E_{1}$ theory. The CB
singularities lie at the points $\pm 2\pm 2\sqrt{\lambda}$, for generic values
of the exponentiated inverse gauge coupling $\lambda$. Meanwhile, at
$\lambda=-1$, the $\mathbb{Z}_{2}$ 0-form symmetry enhances to
$\mathbb{Z}_{4}$. The existence of the short exact sequence

$0\rightarrow\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{4}\rightarrow\mathbb{Z}_{2}\rightarrow
0\leavevmode\nobreak\ ,$ (3.22)

would seem to suggest that the additional $\mathbb{Z}_{2}$ symmetry needed in
this enhancement is a manifestation of the centre of the simply-connected
group of the flavour symmetry algebra $\mathbb{Z}_{2}\subset{\rm SU}(2)$.
Note, in particular, that while the flavour symmetry group of the $E_{1}$ SCFT
is the centreless version ${\rm SO}(3)\cong{\rm SU}(2)/\mathbb{Z}_{2}$, the
Coulomb branch does have the full ${\rm SU}(2)$ symmetry – that is, there are
massive BPS states in the fundamental representation of ${\rm SU}(2)$
Closset:2021lhd . We give some further evidence for this claim below.

##### Characters and centre symmetries.

As discussed above, an interesting observation is that the maximal gaugeable
$\mathbb{Z}_{k}$ symmetries for the Coulomb branches of the $D_{S^{1}}E_{n}$
theories are precisely the centres (3.13) of the corresponding simply
connected flavour symmetry groups. Meanwhile, for $E_{2}$, the
$\mathbb{Z}_{2}$ centre symmetry of the ${\rm SU}(2)$ factor in the flavour
symmetry is ‘masked’ by the ${\rm U}(1)$ factor of the flavour symmetry. These
maximal $\mathbb{Z}_{k}$ symmetries shown in Table 2 can be found by fine-
tuning the mass parameters of the theories. Remarkably, an example of such a
fine-tuning is achieved for vanishing values of _all_ the characters of the
$E_{n}$ groups, as we describe below.111111This is not in disagreement with
the $D_{S^{1}}E_{2}$ case, as the vanishing of the characters in this case is
equivalent to the decoupling of the flavour hypermultiplet generating the
abelian ${\rm U}(1)$ symmetry.

Recall first that the flavour symmetry characters are defined as

$\chi_{\mathcal{R}}^{E_{n}}=\sum_{\rho\in\mathcal{R}}e^{2\pi
i\rho(\nu)}\leavevmode\nobreak\ ,$ (3.23)

for $\rho=(\rho_{i})$ the weights of some fundamental representation
$\mathcal{R}$ and $\nu=(\nu_{i})$ the $E_{n}$ flavour parameters. These Lie
algebra characters are invariant under Weyl group transformations, being
functions on the weight spaces of the Lie algebra to complex numbers (see
Chapter 13.5 of fuchs2003symmetries ). The weight zero characters $\nu=\bm{0}$
will thus lead to $\chi_{\mathcal{R}}=\dim(\mathcal{R})$, and correspond to an
enhancement of the flavour symmetry algebra on the Coulomb branch.

To better understand the physical interpretation of these characters, recall
the gauge theory definition of the CB parameter $U$ as the expectation value
of a half-BPS line Closset:2021lhd ; Closset:2023pmc . As such, the origin of
the extended CB corresponds to the vanishing of this Wilson line VEV, which is
responsible for the symmetry enhancement at the fixed point of the RG flow. In
the same way, the $\chi^{E_{n}}$ characters can also be interpreted as
_flavour Wilson lines_. Accordingly, field theory intuition would suggest that
the vanishing of all the $\chi^{E_{n}}$ characters should also be associated
with a symmetry enhancement. However, the precise reason still remains an open
question.

This symmetry enhancement can be made more precise by expressing the equations
$\chi=0$ in gauge theory language. In terms of the gauge theory parameters
$\lambda$ and $M_{i}$, the vanishing of the characters can be achieved as
follows:

$\displaystyle E_{1}\;:$ $\displaystyle(\mathbb{Z}_{4})\qquad$
$\displaystyle(\lambda)=(-1)\leavevmode\nobreak\ ,$ (3.24) $\displaystyle
E_{3}\;:$ $\displaystyle(\mathbb{Z}_{6})\qquad$
$\displaystyle(\lambda,M_{1},M_{2})=\left(e^{\frac{4\pi i}{6}},e^{\frac{5\pi
i}{6}},e^{\frac{11\pi i}{6}}\right)\leavevmode\nobreak\ ,$ $\displaystyle
E_{4}\;:$ $\displaystyle(\mathbb{Z}_{5})\qquad$
$\displaystyle(\lambda,M_{1},M_{2},M_{3})=\left(e^{\frac{2\pi
i}{5}},e^{\frac{0\pi i}{5}},e^{\frac{2\pi i}{5}},e^{\frac{4\pi
i}{5}}\right)\leavevmode\nobreak\ ,$ $\displaystyle E_{5}\;:$
$\displaystyle(\mathbb{Z}_{4})\qquad$
$\displaystyle(\lambda,M_{1},M_{2},M_{3},M_{4})=\left(e^{\frac{4\pi
i}{4}},e^{\frac{\pi i}{12}},e^{\frac{7\pi i}{12}},e^{\frac{13\pi
i}{12}},e^{\frac{19\pi i}{12}}\right)\leavevmode\nobreak\ .$

In all these examples, the coupling $\lambda$ satisfies $\lambda^{9-n}=1$,
while the masses $M_{i}$ lie on a unit circle and rotate into each other by a
constant phase. Note that we have no a priori reason to believe that
$\chi\equiv 0$ necessarily implies the striking result $|\lambda|=|M_{i}|=1$.
In fact, we will see that this result is not arbitrary, since such
configurations correspond to massless theories, where non-trivial holonomies
along the circle direction are turned on. These maximal (gaugeable) CB
symmetries for the $D_{S^{1}}E_{n}$ theories with $n=4,\dots,7$ are sketched
in Figure 1, and the CB foldings can be carried out explicitly as before.

Figure 1: Structure of the $U$-plane for $\chi_{i}=0$ for the $D_{S^{1}}E_{n}$
theories with $n=3,\dots,6$. The equations below the diagrams give the
discriminant locus.

Finally, it is also important to note that not all the $\mathbb{Z}_{k}$
symmetries of the $U$-plane discussed here are induced by automorphisms of the
elliptic surface. In Section C.1, we will discuss an example where these are
distinct symmetries.

##### 6d theories.

Let us briefly comment on the symmetries of $U$-planes originating from
toroidal compactifications of six-dimensional theories. For this class of
theories, we have $F_{\infty}=I_{0}$, and thus a folding will preserve the
width of the fibre at infinity.121212The width of a fibre of type $I_{n}$ or
$I_{n}^{\ast}$ is $n$. Thus, there are no a priori constraints on the
‘foldable’ symmetries of 6d theories. From the brane-web perspective we expect
that the 6d $E$-string CB will at least allow
$\mathbb{Z}_{2},\mathbb{Z}_{3},\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$ foldable
symmetries Kim:2021fxx .

Meanwhile, the M-string curve will always have a $\mathbb{Z}_{2}$ symmetry,
inherited from the 1-form symmetry of the theory Closset:2023pmc . We will
discuss the corresponding $\mathbb{Z}_{2}$ folding in Section 4. Let us also
note that the so-called relative curve of the M-string theory, containing six
$I_{2}$ fibres in the bulk, will have a $\mathbb{Z}_{6}$ symmetry. While the
$\mathbb{Z}_{3}$ folding of this configuration would consist of two copies of
the $\mathbb{Z}_{3}$ folded 4d $\mathcal{N}=2^{\ast}$ theory Argyres:2016yzz ,
the $\mathbb{Z}_{6}$ folding does not appear to lead to a RES.

####  3.2.3 Peculiar foldings

Let us now discuss the peculiar cases of the $\widetilde{E}_{1}$ and $E_{4}$
theories, as mentioned in Table 2.

##### Foldings of $\bm{\widetilde{E}_{1}}$.

The $\widetilde{E}_{1}$ theory is the UV completion of the 5d $\mathcal{N}=1$
$\text{SU}(2)_{\pi}$ gauge theory, thus sharing some similarities with the
$E_{1}$ theory. While both of their SW geometries have $F_{\infty}=I_{8}$, the
distinction is in the MW groups: only $D_{S^{1}}E_{1}$ has a $\mathbb{Z}_{2}$
1-form symmetry, and thus $\mathbb{Z}_{2}\subset{\rm MW}$ for any
configuration of $D_{S^{1}}E_{1}$. This is no longer the case for
$D_{S^{1}}\widetilde{E}_{1}$.

The previously presented folding argument (3.12) involved the condition
$k|(9-n)$, or $k|8$ for gaugeable $\mathbb{Z}_{k}$ symmetries of
$D_{S^{1}}\widetilde{E}_{1}$. From the SW curve, by explicit computation, one
can show that a $\mathbb{Z}_{4}$ symmetry cannot be realised for any value of
the gauge theory parameter $\lambda$. However, there exist configurations that
show a reflection symmetry along a line passing through the origin of the
$U$-plane. Such an example is the configuration $(I_{8};II,2I_{1})$, which is
modular Magureanu:2022qym and has a $\mathbb{Z}_{2}$ symmetry
$U\mapsto-\bar{U}$, where $\bar{U}$ is the complex conjugate of $U$. In this
case, the AD point is situated along the symmetry line, with the two $I_{1}$
singularities being identified under the reflection symmetry. Folding the
$U$-plane along this line, one expects a configuration having
$F_{\infty}=I_{4}$, as well as a bulk $I_{1}$ fibre and (at least) an elliptic
point. The only such rational elliptic surfaces in Persson’s classification
Persson:1990 ; Miranda:1990 are $(I_{4};I_{1},IV,III)$ and
$(I_{4};I_{1},III,2II)$, which are not promising candidates.

Note, moreover, that quotients of the underlying rational elliptic surface by
this automorphism do not generate a new rational elliptic surface (as per
Table 6 of KARAYAYLA20121 ). In fact, this reflection symmetry is the first
example of a symmetry that is _not induced_ from an automorphisms of the
elliptic surface preserving the zero section. This is because the
transformation $U\mapsto-\bar{U}$ is not a Möbius transformation. Thus, this
configuration does not exhibit a foldable $\mathbb{Z}_{2}$ symmetry. We can
also rule out such a $\mathbb{Z}_{2}$-folding from monodromy considerations;
that is, the monodromy group of the $(I_{8};II,2I_{1})$ configuration is not
an index 2 subgroup of any subgroup of $\rm{PSL}(2,\mathbb{Z})$
Magureanu:2022qym .

##### The $\bm{\mathbb{Z}_{5}}$-folding of $\bm{E_{4}}$.

As pointed out in Table 2, a rather peculiar symmetry is the $\mathbb{Z}_{5}$
symmetry of the $U$-plane of $D_{S^{1}}E_{4}$. This $\mathbb{Z}_{5}$ symmetry
manifests for vanishing characters, leading to the SW curve with Weierstraß
invariants

$g_{2}(U)=\frac{1}{12}U^{4}\leavevmode\nobreak\ ,\qquad\quad
g_{3}(U)=U\left(4-\frac{1}{216}U^{5}\right)\leavevmode\nobreak\ ,$ (3.25)

while the discriminant and $\mathcal{J}$-invariant read:

$\Delta(U)=U^{2}(U^{5}-432)\leavevmode\nobreak\
,\qquad\quad\mathcal{J}(U)=\frac{U^{10}}{12^{3}(U^{5}-432)}\leavevmode\nobreak\
.$ (3.26)

Thus, the $\mathbb{Z}_{5}$-folding of the $U$-plane is equivalent to the
transition:

$(I_{5};5I_{1},II)\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{5}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{1};I_{1},II^{*})\leavevmode\nobreak\ .$ (3.27)

In particular, at the fixed point of the $\mathbb{Z}_{5}$ symmetry, there is a
type $II$ singular fibre (see Figure 2). Consequently, this folding is
essentially describing a $\mathbb{Z}_{5}$ discrete gauging of the $H_{0}$
Argyres-Douglas theory. As already mentioned in Section 3.1, such a gauging is
however speculative, as there are no other RG flows known that could validate
this gauged theory Argyres:2016yzz .

(a)

(b)

(c)

(d)

Figure 2: The $\mathbb{Z}_{5}$-folding of the $D_{S^{1}}E_{4}$ theory. Top:
The $\mathbb{Z}_{5}$ symmetric configuration of $E_{4}$ with branch point
(purple) at $U=0$, which corresponds to the $II$ singular fibre of the $E_{4}$
configuration. From the fundamental domain, it is also clear that the branch
point is the fixed point of the folding, as it is identified under
$T:\tau\mapsto\tau+1$. By quotienting the $\mathbb{Z}_{5}$ symmetry, we find
the configuration $(I_{1};I_{1},II^{*})$ of the $D_{S^{1}}E_{8}$ theory. On
the $U$-plane, it is created by the image of a wedge (green) under that map.
Bottom: The resulting $U$-plane and fundamental domain contain the same branch
point, corresponding now to a $II^{*}$ singularity.

Curiously, there is another reason why such a discrete gauging might be
puzzling. Most gaugings outlined in Table 2 can be realised as five-
dimensional S-folds, which involve certain operations on the underlying
$(p,q)$-brane web. From a IIB perspective, the appearance of new singular
fibres at the origin of the fixed point of the symmetry corresponds to
insertions of 7-branes. However, since there is no 7-brane configuration that
would produce a deficit angle that corresponds to a $\mathbb{Z}_{5}$ symmetry,
such a discrete gauging has not been discussed in Kim:2021fxx . We will
revisit this construction in the following subsection.

###  3.3 Symmetries of $(p,q)$-brane webs

The foldable symmetries of the $U$-plane, as given in Table 2, almost coincide
with the symmetries of the underlying $(p,q)$-brane webs, which were recently
analysed in Kim:2021fxx ; Apruzzi:2022nax (see also Acharya:2021jsp ;
Tian:2021cif ). These symmetries were discussed for the case of vanishing mass
parameters of the 5d SCFTs, and the corresponding brane-web folding procedure
was referred to as _S-folding_ , being a generalisation of the four-
dimensional S-folds introduced in Garcia-Etxebarria:2017ffg ; Apruzzi:2020pmv
; Heckman:2020svr ; Giacomelli:2020gee . In this section, we review five-
dimensional S-folding and compare it to CB folding.

##### S-folds.

Consider a five-dimensional superconformal field theory $\mathcal{T}$, whose
brane web possesses a $\mathbb{Z}_{k}$ symmetry. That is, for a particular
choice of the deformation parameters131313We follow the setup of Kim:2021fxx ,
where the mass parameters are turned off. and of the axio-dilaton, the brane
web remains invariant under a $\mathbb{Z}_{k}\subset{\rm SL}(2,\mathbb{Z})$
action of the IIB string theory, combined with a $\mathbb{Z}_{k}$ spatial
symmetry action. In this case, the web can be split into $k$ distinct parts,
which are exchanged by the $\mathbb{Z}_{k}$ action. The S-fold
$\mathcal{T}/\mathbb{Z}_{k}$ of the theory is then a quotient by this
$\mathbb{Z}_{k}$ action, which produces a deficit angle at the fixed point of
the $\mathbb{Z}_{k}$ symmetry that corresponds to a 7-brane configuration. The
only possibilities for such 7-branes are listed below:

7-brane | $D_{4}$ | $E_{6}$ | $E_{7}$ | $E_{8}$
---|---|---|---|---
$\bm{\mathbb{Z}_{k}}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}_{4}$ | $\mathbb{Z}_{6}$
$\bm{\mathbb{M}_{F}}$ | $I_{0}^{*}$ | $IV^{*}$ | $III^{*}$ | $II^{*}$
(3.28)

In the last row of the above table, we also give the CB singularities that
reproduce the correct monodromy matrix $\mathbb{M}_{F}$ of the 7-brane. In
particular, the 7-brane insertions reproduce the singularities obtained from
the discrete gauging of a free vector multiplet (3.6), as previously
anticipated. Thus, these S-folds can only reproduce the CB foldings for which
the fixed point of the $\mathbb{Z}_{k}$ symmetry is a smooth $I_{0}$ fibre.
Importantly, however, S-folding and discrete 0-form symmetry gaugings are
distinct operations. In the former, the 7-brane configuration introduces new
degrees of freedom in the folded theory. As a result, the Kodaira singularity
located at the fixed point of the $\mathbb{Z}_{k}$ symmetry can be further
deformed, as opposed to the frozen singularity appearing in the discretely
gauged model.

What remains unclear at this stage is the precise relation between the CB
symmetries and the symmetries of the $(p,q)$-web. The link between the mirror
curve and the brane web can be made explicit through the amoeba projection of
the mirror curve Feng:2005gw . Recall from Hori:2000ck ; Hori:2000kt that the
CY3 mirror can be expressed as a double fibration over a complex plane $W$,
with the elliptic fibre given by the SW curve, which, in its most natural
form, is written in terms of two $\mathbb{C}^{*}$ variables, $t$ and $w$. The
amoeba projection is obtained as

$(t,w)\mapsto(r_{t},r_{w})=(\log|t|,\log|w|)\leavevmode\nobreak\ ,$ (3.29)

where $(r_{t},r_{w})$ are the amoeba coordinates, with $r_{t,w}\in\mathbb{R}$.
Then, the brane web appears as the spine of the amoeba projection - i.e. the
legs stretching out to infinity. To obtain the amoebæ, we need to fix both the
mass parameters as well as the Coulomb branch parameter $U$. We will implement
the amoeba projection algorithmically following bogdanov2016algorithmic .

##### The local $\mathbb{P}^{2}$ geometry.

For the toric phases of the $E_{n}$ theories with $n\leq 5$, the spine of the
amoeba projection is always dual to the corresponding $dP_{n}$ or
$\mathbb{F}_{0}$ geometry. Consider, for instance, the $\mathbb{P}^{2}$
geometry, whose mirror curve reads

${1\over w}+{1\over t}+wt-3U=0\leavevmode\nobreak\ .$ (3.30)

In the Weierstraß form of the curve, the $U$-plane singularities reside at
$U^{3}=1$. To find the loci of the singular points for the toric curve, one
computes the minors of the Newton polygon $F(t,w)=t+w+w^{2}t^{2}-3Utw$, for
fixed values of $U$. These lead to the following solutions:

$(1,1,1)\leavevmode\nobreak\ ,\qquad\quad\left(e^{2\pi i\over 3},e^{2\pi
i\over 3},e^{4\pi i\over 3}\right)\leavevmode\nobreak\
,\qquad\quad\left(e^{4\pi i\over 3},e^{4\pi i\over 3},e^{2\pi i\over
3}\right)\leavevmode\nobreak\ ,$ (3.31)

for the tuples $(t,w,U)$. These points turn out to be the fixed loci of the
$\mathbb{Z}_{3}^{\rm web}$ symmetry exchanging the $\mathbb{C}^{*}$
coordinates as follows:

$(t,w)\mapsto\left(w,{1\over wt}\right)\leavevmode\nobreak\ .$ (3.32)

This symmetry leaves the mirror curve invariant, and, from (3.29), it is clear
that it is preserved under the amoeba projection. Put differently, the
$\mathbb{Z}_{3}^{\rm web}$ symmetry of the brane web is nothing but the
symmetry inherited from (3.32).

On the other hand, the $\mathbb{Z}_{3}^{U}$ symmetry of the $U$-plane implies,
in particular, that under changes $U\rightarrow\rho U$, for $\rho^{3}=1$, the
mirror curve remains unchanged. To see this explicitly, we accompany this
action on $U$ by the coordinate rescaling $(t,w)\rightarrow\rho^{-1}(t,w)$
such that the curve becomes:

${1\over w}+{1\over t}+{wt\over\rho^{3}}-3U=0\leavevmode\nobreak\ ,$ (3.33)

and since $\rho^{3}=1$, this is identical to (3.30). Then, the
$\mathbb{Z}_{3}^{U}$ symmetry exchanges the points (3.31), as it ought to.
Note, however, that under the amoebæ projection, this symmetry is lost. The
amoebæ projections for various values of $U$ are depicted in Figure 3. Due to
the $\mathbb{Z}_{3}^{U}$ symmetry of the $U$-plane, for any $U\in\mathbb{C}$
the amoebæ projections for $U$, $e^{2\pi i\over 3}U$ and $e^{-{2\pi i\over
3}}U$ are identical.

Figure 3: Amoebæ for $D_{S^{1}}E_{0}$, for various points on the $U$-plane.
$U$-plane singularities lie at $U^{3}=1$, represented by red dots, while the
dotted lines are axes of symmetry. The $\mathbb{Z}_{3}^{U}$ symmetry is
preserved by the amoebæ projection projection: The amoebæ projections for $U$
and $U^{\prime}$ are identical if $U$ and $U^{\prime}$ are related by a
$\mathbb{Z}_{3}^{U}$ rotation.

We have established that the Coulomb branch $\mathbb{Z}_{3}^{U}$ symmetry is
inherently different from the brane-web symmetry $\mathbb{Z}_{3}^{\rm web}$.
This is perhaps not surprising, as the Coulomb branch symmetry is an artifact
of the KK theory, instead of the purely five-dimensional model. That is, on
the Coulomb branch, this symmetry appears as a subgroup of the ${\rm
U}(1)_{\mathtt{r}}$ symmetry of the effective 4d $\mathcal{N}=2$ KK theory.
Meanwhile, the brane-web symmetry $\mathbb{Z}_{3}^{\rm web}$ is inherited from
a symmetry of the mirror threefold.

Nevertheless, brane-web symmetries still appear in general from a fine-tuning
of the deformation parameters. Thus, a rather appealing question is whether a
symmetry of the brane-web would necessarily imply the existence of a Coulomb
branch symmetry. The reverse statement is clearly false, due to the above
considerations, as can also be seen from the lack of a $\mathbb{Z}_{5}$
symmetry for the brane-web of the $E_{4}$ SCFT. As the $E_{0}$ theory is too
simple from this perspective due to the lack of mass parameters, we will
present some evidence that the direct implication is true by considering the
$\mathbb{F}_{0}$ geometry.

##### The local $\bm{\mathbb{F}_{0}}$ geometry.

In the conventions of Closset:2021lhd , the mirror curve of the $E_{1}$ theory
can be written as

$w+{1\over w}+\sqrt{\lambda}\left(t+{1\over t}\right)-U=0\leavevmode\nobreak\
.$ (3.34)

The singular points of the curve are given by the following tuples for the
$(t,w,U)$ variables: $(-1,1,2-2\sqrt{\lambda})$, $(1,-1,-2+2\sqrt{\lambda})$,
$(-1,-1,-2-2\sqrt{\lambda})$ and $(1,1,2+2\sqrt{\lambda})$. There are multiple
symmetries for this curve. First, the symmetries

$\mathbb{Z}_{2}^{w}:\leavevmode\nobreak\ w\mapsto{1\over
w}\leavevmode\nobreak\ ,\qquad\quad\mathbb{Z}_{2}^{t}:\leavevmode\nobreak\
t\mapsto{1\over t}\leavevmode\nobreak\ ,$ (3.35)

leave the above singular points invariant, and are preserved by the amoebæ
projection. Then, the CB symmetry $\mathbb{Z}_{2}^{U}$ is generated by
$(t,w,U)\mapsto-(t,w,U)$, exchanging the curve singularities pairwise. Note
that this symmetry is not preserved by the amoebæ projection. Finally, the CB
shows a $\mathbb{Z}_{4}^{U}$ symmetry whenever $\lambda=-1$ Closset:2021lhd .
This symmetry acts as $(t,w,U)\to(w,-t,i\,U)$, but only a $\mathbb{Z}_{2}$
subgroup of this symmetry is preserved on the amoebæ.

Let us also note that the $\mathbb{Z}_{4}^{\rm web}$ symmetry of the brane-web
appears at $\lambda=1$, being generated by $(t,w,U)\to\left(w,{1\over
t},U\right)$. Importantly, this $\mathbb{Z}_{4}^{\rm web}$ symmetry does not
act on the CB parameter of the KK theory. Let us remark, however, that the
mass parameters of the SW curve are exponentiated 5d mass parameters, which
could also include non-trivial holonomies. Thus, from a five-dimensional point
of view, the values $|\lambda|=1$ would correspond to the same point on the
extended Coulomb branch, namely the SCFT point.

This example appears to suggest that the brane-web symmetries do indeed imply
the existence of similar symmetries on the Coulomb branch. This implication
could be proved through a chain of string theory dualities, as exemplified in
Acharya:2021jsp . In fact, equation (3.24) already shows that most, if not
all, of the CB symmetry enhancements can occur by only turning on some non-
trivial holonomies along the $S^{1}$ direction. Furthermore, such holonomies
can lead to new symmetry enhancements, which are not available in the brane-
web picture.

###  3.4 Fractional foldings

Five-dimensional $\mathbb{Z}_{k}$ S-folds involve cutting the 5-brane web into
$k$ slices (which are exchanged under the $\mathbb{Z}_{k}$ symmetry) and
keeping only one of these slices while inserting a specific 7-brane at the
fixed point of the $\mathbb{Z}_{k}$ action Kim:2021fxx . For the usual 5d
$S$-folds, the only possible 7-branes are $D_{4},E_{6},E_{7}$ and $E_{8}$,
which correspond to $\mathbb{Z}_{2},\mathbb{Z}_{3},\mathbb{Z}_{4}$ and
$\mathbb{Z}_{6}$ quotients, respectively, as listed in (3.28). In
Apruzzi:2022nax , a new type of S-folds was introduced, which makes use of
other types of 7-brane configurations:

7-brane | $H_{2}$ | $H_{1}$ | $H_{0}$
---|---|---|---
$\bm{\mathbb{Z}_{k}}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}_{4}$ | $\mathbb{Z}_{6}$
$\bm{\mathbb{M}_{F}}$ | $IV$ | $III$ | $II$
(3.36)

As before, we indicate the brane-web symmetries relevant for each insertion,
as well as the monodromy associated with the 7-branes Apruzzi:2020pmv . This
new construction was termed _fractional folding_ and can be realised as
follows. As before, consider a brane web with a $\mathbb{Z}_{k}$ symmetry (for
$k=3,4$ or 6) and split it into $k$ slices exchanged by this symmetry. Then,
the partial quotient procedure involves removing only one of these slices and
introducing an appropriate 7-brane configuration at the fixed point of the
$\mathbb{Z}_{k}$ action. The final configuration should be consistent with the
deficit angle of the 7-brane.

##### Fractional foldings on the $U$-plane.

To understand how this procedure could be implemented on the Coulomb branch,
recall that for a theory $\mathcal{T}$ with fibre at infinity
$F_{\infty}=I_{n}$, a $\mathbb{Z}_{k}$ symmetry is allowed only if $k$ divides
$n$. As a result, there must exist a positive integer $\widetilde{n}$, such
that

$n=\widetilde{n}\,k\leavevmode\nobreak\ .$ (3.37)

Then, we naively associate a partial contribution
$I_{\widetilde{n}}^{\infty}=I_{n/k}^{\infty}$ to every slice exchanged by the
$\mathbb{Z}_{k}$ symmetry of the original theory. Thus, under the partial
folding keeping $(k-1)$ slices, the new fibre at infinity becomes

$F_{\infty}=I_{n}\,\xrightarrow[\leavevmode\nobreak\
\text{partial}\leavevmode\nobreak\ ]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{k}\leavevmode\nobreak\ \leavevmode\nobreak\
}\,F_{\infty}=I_{(k-1)\widetilde{n}}=I_{n-{n\over k}}\leavevmode\nobreak\ .$
(3.38)

This change in the fibre at infinity is, of course, accompanied by the
previously mentioned change in the bulk fibres: the bulk singularities
appearing in one of the slices are exchanged with a $IV,III$ or $II$ fibre at
the fixed point of the $\mathbb{Z}_{k}$ symmetry (i.e. the origin of the
$U$-plane), for $k=3,4$ or 6, respectively. The remaining slices are glued
together, leading to a $\mathbb{Z}_{k-1}$ symmetry in the new theory.

The construction from the point of view of the rational elliptic surface looks
as follows. Let us denote by $\mathcal{S}_{n}$ a RES characterised by a fibre
at infinity $F_{\infty}=I_{n}^{\infty}$. In the case where
$\text{Aut}_{\sigma}(\mathcal{S}_{n})$ contains a $\mathbb{Z}_{k}$ group, the
quotient map
$f_{k}:\mathcal{S}_{n}\to\mathcal{S}_{\frac{n}{k}}=\mathcal{S}_{n}/\mathbb{Z}_{k}$
is well-defined, and results in a new RES with $F_{\infty}=I_{\frac{n}{k}}$.
Then, we interpret the construction of fractional foldings as a composition of
a folding and an _unfolding_ (i.e. an inverse of a folding): the surface
$\mathcal{S}_{n}$ has a fractional $k$-folding, if there exists a surface
$\mathcal{S}_{\frac{n}{k}(k-1)}$ with
$\text{Aut}_{\sigma}(\mathcal{S}_{\frac{n}{k}(k-1)})\supset\mathbb{Z}_{k-1}$,
such that $f_{k-1}(\mathcal{S}_{\frac{n}{k}(k-1)})=\mathcal{S}_{\frac{n}{k}}$.
Then the fractional folding is the composition

$\text{\textflorin}_{\frac{k}{k-1}}\coloneqq(f_{k-1})^{-1}\circ
f_{k}:\,\mathcal{S}_{n}\longrightarrow\mathcal{S}_{\frac{n}{k}(k-1)}\leavevmode\nobreak\
.$ (3.39)

This reproduces precisely the list of rank 1 examples discussed in
Apruzzi:2022nax , namely:

$\displaystyle\text{\textflorin}_{\frac{2}{3}}:$ $\displaystyle\quad
E_{6}:\,(I_{3}^{\infty};3I_{3})$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\,\,\dashrightarrow\,(I_{1}^{\infty};IV^{*},I_{3})}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\dashrightarrow\,\,}E_{7}:\,(I_{2}^{\infty};IV,2I_{3})\leavevmode\nobreak\
,$ (3.40) $\displaystyle\text{\textflorin}_{\frac{3}{4}}:$ $\displaystyle\quad
E_{1}:\,(I_{8}^{\infty};4I_{1})$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\,\,\dashrightarrow\,(I_{2}^{\infty};III^{*},I_{1})}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\dashrightarrow\,\,}E_{3}:\,(I_{6}^{\infty};III,3I_{1})\leavevmode\nobreak\
,$ $\displaystyle\text{\textflorin}_{\frac{5}{6}}:$ $\displaystyle\quad
E_{3}:\,(I_{6}^{\infty};6I_{1})$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\,\,\dashrightarrow\,(I_{1}^{\infty};II^{*},I_{1})}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\dashrightarrow\,\,}E_{4}:\,(I_{5}^{\infty};II,5I_{1})\leavevmode\nobreak\
,$

for the specific configurations with $\mathbb{Z}_{3}$, $\mathbb{Z}_{4}$ and
$\mathbb{Z}_{6}$ symmetries. For completeness, we also introduce the
intermediate step (3.39) of the above procedure. Note that the
$\text{\textflorin}_{\frac{5}{6}}$ fractional folding contains an unfolding of
the peculiar $\mathbb{Z}_{5}$ symmetry of the $D_{S^{1}}E_{4}$ theory, as
discussed in Section 3.2.3.

Since a $k$-unfolding maps the fibres at infinity as $I^{\infty}_{9-n}\mapsto
I^{\infty}_{k(9-n)}$, this maps a $D_{S^{1}}E_{n}$ theory to a
$D_{S^{1}}E_{9-k(9-n)}$ theory. We obtain the following Table 3 of theories
related by foldings and fractional foldings.

| $\displaystyle E_{0}$ | $\displaystyle E_{1}$ | $\displaystyle E_{2}$ | $\displaystyle E_{3}$ | $\displaystyle E_{4}$ | $\displaystyle E_{5}$ | $\displaystyle E_{6}$ | $\displaystyle E_{7}$ | $\displaystyle E_{8}$
---|---|---|---|---|---|---|---|---|---
$\displaystyle E_{0}$ |  |  |  |  |  |  | $\displaystyle\mathbb{Z}_{3}$ |  |
$\displaystyle E_{1}$ |  |  |  | $\displaystyle\text{\textflorin}_{\frac{3}{4}}$ |  | $\displaystyle\mathbb{Z}_{2}$ |  | $\displaystyle\mathbb{Z}_{4}$ |
$\displaystyle E_{2}$ |  |  |  |  |  |  |  |  |
$\displaystyle E_{3}$ |  |  |  |  | $\displaystyle\text{\textflorin}_{\frac{5}{6}}$ |  | $\displaystyle\mathbb{Z}_{2}$ | $\displaystyle\mathbb{Z}_{3}$ | $\displaystyle\mathbb{Z}_{6}$
$\displaystyle E_{4}$ |  |  |  |  |  |  |  |  | $\displaystyle\mathbb{Z}_{5}$
$\displaystyle E_{5}$ |  |  |  |  |  |  |  | $\displaystyle\mathbb{Z}_{2}$ | $\displaystyle\mathbb{Z}_{4}$
$\displaystyle E_{6}$ |  |  |  |  |  |  |  | $\displaystyle\text{\textflorin}_{\frac{2}{3}}$ | $\displaystyle\mathbb{Z}_{3}$
$\displaystyle E_{7}$ |  |  |  |  |  |  |  |  | $\displaystyle\mathbb{Z}_{2}$
$\displaystyle E_{8}$ |  |  |  |  |  |  |  |  |
Table 3: Matrix of $E_{n}$ theories related by $\mathbb{Z}_{k}$ foldings or
$\text{\textflorin}_{\frac{k}{k-1}}$ fractional foldings. The action is always
taken on the theories represented at the beginning of each row. We omit the
inverse directions.

It is again relatively straightforward to compute the prepotential of the new
theory, given the change in the fibre at infinity in (3.38). Indeed, from
(3.17), we have for the rank-one theories:

$\mathcal{F}_{\mathcal{T}^{k-1}/\mathbb{Z}_{k}}\approx{k-1\over
k}\mathcal{F}_{\mathcal{T}}\leavevmode\nobreak\ ,$ (3.41)

in agreement with Apruzzi:2022nax , where by
$\mathcal{T}^{k-1}/\mathbb{Z}_{k}$ we denote the $(k-1)/k$ partial quotient of
$\mathcal{T}$.

As it was the case for the 5d S-folds, the CB singularity appearing at the
fixed point of the $\mathbb{Z}_{k}$ symmetry can be further deformed, as it
corresponds to a 7-brane configuration. Thus, while the above prescription can
be used to reproduce the results of Apruzzi:2022nax , it would be interesting
to understand if a field-theoretic parallel to these partial foldings exists.
Importantly, in this construction, the new CB singularities should be
undeformable, similar to the comparison between S-folds and CB discrete
gaugings.

We point out that $\mathbb{Z}_{k-1}$ _unfoldings_ of (effective) 4d Coulomb
branches can be viewed as gaugings of discrete 2-form symmetries.
Nevertheless, this interpretation cannot describe a field theory analogue of a
partial $\mathbb{Z}_{k}$-folding, as $\mathbb{Z}_{k-1}$ is not a subgroup of
$\mathbb{Z}_{k}$. In general, these symmetries of course occur for a fine-
tuning of the masses. We leave this problem for future work.

###  3.5 Non-cyclic symmetries

The previously studied Coulomb branch singularities and their quotients all
correspond to cyclic subgroups of the induced automorphism group
$\text{Aut}_{\sigma}(\mathcal{S})$. As is clear from the full classification
KARAYAYLA20121 , this group $\text{Aut}_{\sigma}(\mathcal{S})$ also admits
_non-cyclic_ subgroups, such as the dihedral groups $D_{n}$ (of order $2n$)
with $n=2,3,4,6$, and the tetrahedral group $A_{4}$. The family of rational
elliptic surfaces $\mathcal{S}$ supporting non-cyclic symmetries is classified
by their singular fibres, as given in (KARAYAYLA20121, , Proposition 4.2.3).
In this section, we briefly discuss these non-cyclic symmetries, as well as
their physical interpretation, and present some examples for the
$D_{S^{1}}E_{n}$ theories. A more detailed study, including an organisation of
various symmetry groups preserving parts of the surface, can be found in
Appendix C.

Recall from Section 2.2.2 that $\text{Aut}_{\mathcal{S}}(\mathbb{P}^{1})$ is
defined as the automorphisms on the base $\mathbb{P}^{1}$ induced by all
elements of $\text{Aut}(\mathcal{S})$, being a subgroup of
$\text{Aut}_{\sigma}(\mathcal{S})$. These induced automorphisms act as Möbius
transformations

$\text{Aut}(\mathbb{P}^{1})\ni\varphi:U\mapsto\frac{aU+b}{cU+d}\leavevmode\nobreak\
,\qquad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{PGL}(2,\mathbb{C})\leavevmode\nobreak\ ,$ (3.42)

and, in particular, they preserve the $\mathcal{J}$-invariant of the surface
$\mathcal{S}$. The examples studied so far are the cases where
$\varphi(U)=\omega_{n}U$, with $\omega_{n}$ an $n$-th root of unity. Here, we
will study cases where the induced Möbius transformations are not cyclic.

The simplest example of a non-cyclic symmetry is the Klein four-group
$D_{2}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ KARAYAYLA20121 . This group acts by
Möbius transformations on $\mathbb{P}^{1}$ and is generated by $U\mapsto 1/U$
and $U\mapsto-U$, which correspond to the $\text{PGL}(2,\mathbb{C})$
transformations $s=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ and
$r=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$. Clearly, these are inversion and
reflection symmetries, with the inversion symmetry being non-cyclic.

This simplest non-cyclic abelian group is realised as the induced automorphism
group $\text{Aut}_{\mathcal{S}}(\mathbb{P}^{1})$ for many different surfaces
$\mathcal{S}$. Crucially, the inversion $U\mapsto 1/U$ necessarily
interchanges the fibre at infinity, defined by $U=\infty$, with a bulk
singularity $U=0$. In order for this inversion to be an induced symmetry, the
fibres corresponding to the singularities $U=0$ and $U=\infty$ must be of the
same Kodaira type. However, since we identify the fibre at infinity
$F_{\infty}$ as a characterisation of the ‘UV definition’ of a given theory,
any proper symmetry of a theory must preserve this fibre $F_{\infty}$.
Clearly, from (3.42) any Möbius transformation fixing $U=\infty$ is cyclic,
and consequently any non-cyclic automorphism can not be a proper physical
symmetry. Nonetheless, the existence of non-cyclic Coulomb branch symmetries
is rather curious and they are arguably worthwhile to explore. In the
following, we study two explicit examples of curves with both the ‘smallest’
and the ‘largest’ possible non-cyclic symmetry.

Non-cyclic abelian symmetry. Let us consider a surface whose $U$-plane
symmetry is the Klein four-group $D_{2}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$,
with precisely the inversion and reflection symmetries described above. An
example of such a surface is the $D_{S^{1}}E_{5}$ curve with gauge theory
parameters $\lambda=1$ and $M_{j}=i$ for all $j=1,\dots,4$. Alternatively,
this curve is found by setting the $E_{5}$ characters to
$\bm{\chi}=(-2,-3,0,8,0)$, being a modular surface, with monodromy group
$\Gamma^{0}(4)\cap\Gamma(2)$ Closset:2021lhd . Rescaling the CB parameter
$U=4u$, the $\mathcal{J}$-invariant becomes

$\mathcal{J}(u)=\frac{4\left(u^{4}+u^{2}+1\right)^{3}}{27u^{4}\left(u^{2}+1\right)^{2}}\leavevmode\nobreak\
.$ (3.43)

It is straightforward to check that it is invariant under $r:u\mapsto-u$ and
$s:u\mapsto 1/u$. The singular fibres are $(I_{4}^{\infty};I_{4},2I_{2})$,
where $s$ interchanges $I_{4}^{\infty}$ with the bulk $I_{4}$, while
exchanging the two $I_{2}$ factors. We plot the $u$-plane with this
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry in Figure 4(a). In Appendix C,
we explore the possibility of a quotient of the surface by this non-cyclic
abelian group.

(a)

(b)

Figure 4: (a) $D_{S^{1}}E_{5}$ configuration with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. The blue lines constitute the
so-called _partitioning_ of the $U$-plane, that is, the preimage of the
boundary pieces of the $\text{SL}(2,\mathbb{Z})$ fundamental domain inside the
fundamental domain for this configuration under the map $u=u(\tau)$, as put
forward in Aspman:2021vhs . It can be found directly from the
$\mathcal{J}$-invariant, and makes the symmetry therefore manifest. (b)
$D_{S^{1}}E_{7}$ configuration with tetrahedral symmetry $A_{4}$.

Tetrahedral symmetry. Except for the cyclic groups and
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, the groups
$\text{Aut}_{\mathcal{S}}(\mathbb{P}^{1})$ are non-abelian – either the
dihedral groups $D_{n}$ for $n=3,4,6$, or the tetrahedral group $A_{4}$. The
‘largest’ symmetry groups are therefore the dihedral group $D_{6}$ and the
tetrahedral group $A_{4}$. Rational elliptic surfaces with tetrahedral
symmetry are highly restricted: the possible singular configurations are
$(4I_{3})$ and $(12I_{1})$. The latter, $(12I_{1})$, is the generic
configuration of the 6d $E$-string curve curve Eguchi:2002fc ; Eguchi:2002nx .
Meanwhile, the configuration $(I_{3};3I_{3})$ can be realised for the
$D_{S^{1}}E_{6}$ curve with characters $\bm{\chi}=(0,0,-3,9,0,0)$. It is a
modular elliptic surface, with monodromy group $\Gamma(3)$. In order to make
the $A_{4}$ symmetry manifest, let us rescale $U=4u$. Then we find that

$\mathcal{J}(u)=\frac{\left(u^{4}+8u\right)^{3}}{2^{6}\left(u^{3}-1\right)^{3}}\leavevmode\nobreak\
$ (3.44)

is invariant under the Möbius transformations $g_{1}:u\mapsto\omega_{3}u$ and
$g_{2}:u\mapsto\frac{u+2}{u-1}$, with $\omega_{3}=e^{2\pi i/3}$. These indeed
generate the tetrahedral group $A_{4}$ (see also (KARAYAYLA20121, , Section
5.3.5)).

The action of $A_{4}$ on $\mathbb{P}^{1}$ can be thought of as orientation-
preserving transformations of a tetrahedron inside the Riemann sphere
$\mathbb{P}^{1}$, where the four vertices are the $u$-plane singularities
$P=(1,\omega_{3},\omega_{3}^{2},\infty)$. In terms of the monodromy group
$\Gamma(3)$, the $A_{4}$ symmetry is carried by the cosets, as can be seen
from the fact that $\text{SL}(2,\mathbb{Z})/\Gamma(3)\cong A_{4}$. We plot the
$u$-plane with $A_{4}$ symmetry in Figure 4(b).

Non-induced symmetries. In order to shed some light on the origin of such non-
cyclic symmetries, it is useful to consider all Möbius transformations
preserving the $\mathcal{J}$-invariant. It turns out that not all such
transformations are induced by $\text{Aut}(\mathcal{S})$. An example is 4d
SU(2) SQCD with $N_{f}=2$ massless hypermultiplets. Here, in some
normalisation, the SW curve is given by

$\mathcal{J}(u)=\frac{\left(u^{2}+3\right)^{3}}{27\left(1-u^{2}\right)^{2}}\leavevmode\nobreak\
,$ (3.45)

which has a configuration $(I_{2}^{*};2I_{2})$ and is modular with monodromy
group $\Gamma(2)$. This curve is invariant under the Möbius transformations
$m:u\mapsto\frac{u-3}{u+1}$ and $r:u\mapsto-u$. We have
$m^{3}=r^{2}=(mr)^{2}=1$, giving a presentation of the dihedral group $D_{3}$
of order 6. In this case, the $m$-transformation is induced by a modular
transformation $ST:\tau\mapsto-\frac{1}{\tau+1}$, while the $r$-transformation
is induced by $T:\tau\mapsto\tau+1$. Similar to the $\Gamma(3)$ modular
surface described above, here the symmetry is recovered as well by a quotient
with respect to the monodromy group, $\Gamma/\Gamma(2)\cong S_{3}\cong D_{3}$.
Thus in this case again the group of Möbius maps preserving the
$\mathcal{J}$-invariant is given by the $\text{SL}(2,\mathbb{Z})$ duality
group. That is, in these cases, all $\text{SL}(2,\mathbb{Z})$ electric-
magnetic duality transformations act on the Coulomb branch by Möbius
transformations. For the $\Gamma(2)$ surface, this $D_{3}$ symmetry is however
not an induced symmetry, since it exchanges the fibre $F_{\infty}=I_{2}^{*}$
with a bulk $I_{2}$, and therefore does not originate from an automorphism. As
was remarked in Klemm:1995wp ; Furrer:2022nio , duality transformations
generally do _not_ act as Möbius transformations on the base, but rather as
rational radical functions. We elaborate further on these aspects in Appendix
C.

##  4 Folding across dimensions

Supersymmetric quantum field theories exhibit a rich structure whenever
considered on spacetimes containing an $S^{1}$ factor. Specifically, in the
small radius limit, a supersymmetric $d$-dimensional theory can reduce to a
disjoint sum of multiple $(d-1)$-dimensional theories differing by their
holonomies. In Hwang:2018riu ; Hwang:2017nop , these $(d-1)$-dimensional
theories were dubbed _holonomy saddles_. For 5d $\mathcal{N}=1$ theories on a
circle, such phenomena manifest directly on the SW geometry as multiple
decoupling limits of 4d SW theories Jia:2022dra ; Jia:2021ikh . The SW
geometry of the 5d pure $SU(N)$ theory, for instance, comprises $N$ copies of
locally indistinguishable 4-dimensional pure $SU(N)$ theories. Each such
holonomy saddle comes with its own 4d BPS quiver, which gives a natural
prediction for the 5d BPS quivers in terms of 4d BPS quivers.

In this section, we present a simple argument for detecting holonomy saddles
in the formalism of rational elliptic surfaces. This extends the analysis of
Jia:2022dra ; Jia:2021ikh to theories with fundamental matter, as well as to
6d theories. For rank-one theories, holonomy saddles appear from
$\mathbb{Z}_{2}$-foldings of the $U$-plane, which we discuss in Section 4.1.
Meanwhile, in Sections 4.2 and 4.3, we discuss all examples included in this
class of theories, which corroborate with massless and massive 4d SQCD
theories, respectively.

###  4.1 The 5d–4d map

In Section 2 we have argued that a $\mathbb{Z}_{k}$-folding of the $U$-plane
is given by a map:

$U\mapsto z=U^{k}\leavevmode\nobreak\ .$ (4.1)

This map corresponds to a base-change at the level of the Seiberg–Witten
geometry. The discrete gaugings previously discussed in this context were
realised as mappings $I_{n}^{\infty}\longrightarrow I_{n/k}^{\infty}$ between
the fibres at infinity for the KK theories. These mappings required a
quadratic twist on top of the base-change (4.1), in order to preserve the type
of fibre at infinity. If such quadratic twists were not added, we would
instead have the mapping

$I_{n}^{\infty}\longrightarrow\left(I_{n/k}^{*}\right)^{\infty}\leavevmode\nobreak\
.$ (4.2)

Following (2.7), this appears as a relation between 5d KK geometries and 4d
SQCD theories. In fact, we will show that this link has highly non-trivial
implications, and could be used to determine the spectra of BPS states of 5d
theories based on that of 4d SQCD theories.141414In some particular chamber,
and up to a tower of KK states. Having this in mind, the map (4.2) can be
argued as follows.

##### Foldings across dimensions.

The BPS quiver of a 4d $\mathcal{N}=2$ theory can be determined directly from
the singular fibres of the SW geometry, especially when the CB is _modular_
Magureanu:2022qym . This map between $U$-plane singularities and nodes of BPS
quivers can be directly realised when these CB singularities are of
multiplicative type, i.e. of $I_{n}$ type. We would like to give a new
interpretation to CB foldings, where the folded and unfolded geometries
correspond to seemingly distinct theories. Namely, these should not be related
by a discrete gauging. The utility of such an interpretation will be the BPS
quiver description.

Accordingly, we are seeking foldings that do not introduce elliptic points,
and only involve multiplicative fibres in the bulk. Recall first from
(LABEL:possible_foldings) that the _width_ $n$ of the cusp at infinity
$F_{\infty}=I_{n}$ reduces by a factor of $k$ under a
$\mathbb{Z}_{k}$-folding. At the same time, $k$ identical bulk fibres get
identified under this operation. Thus, the only way to preserve the Euler
number (2.10) of the RES while also avoiding the introduction of elliptic
points, is to introduce $I_{m}^{*}$ singular fibres in the folded geometry.
Nevertheless, in order to have a clear BPS quiver description, these fibres
should be the fibres at infinity, describing thus 4d SQCD theories
Closset:2021lhd ; Caorsi:2018ahl ; Magureanu:2022qym .

Let us first assume that all bulk fibres are of $I_{1}$ type and are related
by the $\mathbb{Z}_{k}$ symmetry. Schematically, we are then looking at the
following folding:

$\big{(}I_{n}^{\infty};\,(12-n)I_{1}\big{)}\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{k}\leavevmode\nobreak\ \leavevmode\nobreak\
}\big{(}(I_{n/k}^{*})^{\infty};\,\frac{12-n}{k}I_{1}\big{)}\leavevmode\nobreak\
.$ (4.3)

The Euler number constraint (2.10) for the new surface reads:

$\frac{12-n}{k}+\frac{n}{k}=6\leavevmode\nobreak\ ,$ (4.4)

which thus restricts $k$ to the unique value $k=2$. In turn, this implies that
such foldings exist only for even $n$. As the starting point is a field theory
with fibre $I_{n}^{\infty}$ at infinity with $n$ even, the obvious candidates
are the $D_{S^{1}}E_{2N_{f}+1}$ theories with $N_{f}=0,\dots,3$. Thus, the
folding realizes the map:

${\rm 5d}\leavevmode\nobreak\ {\rm SU}(2)+2N_{f}\text{ fund
}\xrightarrow[]{\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\ }2\text{ copies of 4d
}{\rm SU}(2)+N_{f}\text{ fund}\leavevmode\nobreak\ .$ (4.5)

In fact, these foldings are related to the notion of holonomy saddles
Hwang:2018riu ; Jia:2022dra , and correspond to decompositions of the 5d BPS
quivers in $k=2$ copies of 4d quivers. Schematically, the 5d quivers take the
following form, where $Q$ is a given 4d quiver:

$Q$$Q$ (4.6)

##### Flavour symmetry.

Before we make the folding explicit, let us comment briefly on the relation
between the flavour symmetries of the theories. For this, we can consider a
more general setup than (4.3) by allowing other types $I_{m_{j}}$ of
singularities, with $m_{j}\geq 1$. The singular configurations of the
$E_{2N_{f}+1}$ theory that we can $\mathbb{Z}_{2}$-fold are then necessarily
of the form $(I_{8-2N_{f}};\\{2n_{j}I_{m_{j}}\\})$, where $n_{j}$ and $m_{j}$
are some integers satisfying

$\sum_{j}n_{j}\,m_{j}-N_{f}=2\leavevmode\nobreak\ .$ (4.7)

This identity simply follows from the Euler number constraint (2.10). As
before, the corresponding 4d configurations will be
$(I_{4-N_{f}}^{*};\\{n_{j}I_{m_{j}}\\})$. Since the folding from 5d to 4d
pairwise identifies two $I_{m_{j}}$ singularities, the action of the
$\mathbb{Z}_{2}$-folding is a reduction of the flavour symmetry algebra
$\mathfrak{g}_{F}$,

$\mathfrak{u}(1)^{\text{rk}(\Phi)}\oplus\bigoplus_{j}A_{m_{j}-1}^{\oplus_{2n_{j}}}\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}\mathfrak{u}(1)^{\frac{\text{rk}(\Phi)-1}{2}}\oplus\bigoplus_{j}A_{m_{j}-1}^{\oplus_{n_{j}}}\leavevmode\nobreak\
.$ (4.8)

Clearly, the rank of the MW group also changes across the dimensions. Indeed,
for generic masses, the MW group of the $E_{2N_{f}+1}$ curve has rank
$2N_{f}+1$, while the corresponding 4d SW surface with $N_{f}$ generic masses
has MW rank $N_{f}$.

In the following, we characterise the geometry changing
$\mathbb{Z}_{2}$-foldings by quadratic relations between the CB parameters in
4d and 5d.151515An intricate issue regarding the CB parameters in 5d is their
normalisation, see (Magureanu:2022qym, , Section 3.3) for a discussion. As is
clear from the explicit expressions, the normalisation does not affect the
structure of the results. First, we discuss the case of massless SQCD, after
which we lay out the structure of the massive theories.

###  4.2 The massless cases

##### The $\bm{D_{S^{1}}E_{1}}$ theory.

We start by considering the $D_{S^{1}}E_{1}$ theory, with the mirror curve
given explicitly in Closset:2021lhd . We find for generic values of $\lambda$
(see also Nekrasov:1996cz ; Gottsche:2006bm ; kim2023 ), that

$U(\tau,\lambda)=\sqrt{8\sqrt{\lambda}\tfrac{u_{0}(\tau)}{\Lambda_{0}^{2}}+4(\lambda+1)}\leavevmode\nobreak\
,$ (4.9)

where $u_{0}$ is the CB parameter of the pure 4d $N_{f}=0$ theory, being a
Hauptmodul of $\Gamma^{0}(4)$. This offers an alternative proof to the
decomposition of the $U$-plane of the $D_{S^{1}}E_{1}$ theory in two identical
copies of the $u$-plane for the 4d ${\rm SU}(2)$ theory. This relationship was
also recently shown in Jia:2022dra and holds true for generic values of the
deformation parameter $\lambda$. In particular, the $U$-plane has a
$\mathbb{Z}_{2}$ symmetry for any values of the gauge coupling, which is also
the centre symmetry of the 5d gauge group ${\rm SU}(2)$. Thus, the folding
that we are considering is:

$E_{1}:\quad(I_{8};\,4I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{4}^{*};\,2I_{1})\leavevmode\nobreak\ .$ (4.10)

In order to see the decomposition of the BPS quiver, we consider the specific
value $\lambda=-1$ of the gauge coupling, where the $\mathbb{Z}_{2}$ symmetry
enhances to $\mathbb{Z}_{4}$. As a result, it is not difficult to compute the
$(a,a_{D})$ periods on the $w=U^{4}$ plane, which lead to the following basis
of BPS states (as computed explicitly in Closset:2021lhd ; Jia:2022dra ):

$\displaystyle\gamma_{1}\,:$ $\displaystyle\quad(1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{2}\,:$
$\displaystyle\quad(-1,2)\leavevmode\nobreak\ ,$ (4.11)
$\displaystyle\gamma_{3}\,:$ $\displaystyle\quad(-1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{4}\,:$
$\displaystyle\quad(1,-2)\leavevmode\nobreak\ ,$

Thus, the associated BPS quiver becomes:

$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma_{1}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma_{4}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma_{3}$
(4.12)

The two colours used in the above figure indicate the two BPS quivers of the
4d pure ${\rm SU}(2)$ gauge theory. The full spectrum of this quiver was
recently computed in DelMonte:2021ytz ; Closset:2019juk for some particular
tame chamber. These works have already indicated that the spectrum of the
$D_{S^{1}}E_{1}$ theory can be organised as two copies of the maximal chamber
for the Kronecker quiver.

##### The $\bm{D_{S^{1}}E_{3}}$ theory.

From the previous arguments, another example of (4.3) is:

$E_{3}:\quad(I_{6};\,6I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{3}^{*};\,3I_{1})\leavevmode\nobreak\ .$ (4.13)

To realise the above map, we need $E_{3}$ configurations that have a
$\mathbb{Z}_{2}$ symmetry. This can be achieved by setting $M_{1}=-M_{2}=M$ in
the SW curve.

Let us consider the massless 4d $N_{f}=1$ theory, whose $u$-plane has a
$\mathbb{Z}_{3}$ symmetry, which is a remnant of the ${\rm U}(1)_{\mathtt{r}}$
R-symmetry. Then, if the above map were to exist, we would need a
configuration of $D_{S^{1}}E_{3}$ exhibiting a
$\mathbb{Z}_{6}=\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ symmetry. As discussed
before, such configurations do exist, but only appear for isolated values of
$(\lambda,M)$. For instance, for $(\lambda,M)=(e^{4\pi i\over 3},e^{7\pi
i\over 6})$, we find the following identity:

$U_{E_{3}}(\tau)=\sqrt{-16\tfrac{u_{1}(\tau,0)}{\Lambda_{1}^{2}}}\leavevmode\nobreak\
,$ (4.14)

where $u_{1}$ is the order parameter for the massless 4d ${\rm SU}(2)$
$N_{f}=1$ theory. For these values of the mass parameters it is again not
difficult to compute a basis of light BPS states (see Closset:2021lhd ),
leading to:

$\displaystyle\gamma_{1}\,:$ $\displaystyle\quad(1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{2}\,:$
$\displaystyle\quad(1,-1)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{3}\,:$
$\displaystyle\quad(0,-1)\leavevmode\nobreak\ ,$ (4.15)
$\displaystyle\gamma_{4}\,:$ $\displaystyle\quad(-1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{5}\,:$
$\displaystyle\quad(-1,1)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{6}\,:$
$\displaystyle\quad(0,1)\leavevmode\nobreak\ ,$

The associated quiver is given below:

$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma_{6}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma_{5}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\gamma_{4}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\gamma_{3}$
(4.16)

Fortunately, it was shown in DelMonte:2021ytz that for this particular quiver
there exists a chamber where the BPS spectrum decomposes into two copies of
the 4d ${\rm SU}(2)$ massless $N_{f}=1$ theory, together with a tower of KK
states. The corresponding 4d sub-quivers are coloured in the above diagram.

We can also find the explicit holonomy saddles from the $D_{S^{1}}E_{3}$
theory to _massive_ $N_{f}=1$ SQCD with a generic mass. For $M_{1}=-M_{2}=M$,
we find agreement of the curves for

$U(\tau)=\sqrt{16(M\lambda)^{\frac{2}{3}}\,\tfrac{u_{1}(\tau,m)}{\Lambda_{1}^{2}}+4\lambda(1-M^{2})+4},$
(4.17)

where $u_{1}(\tau,m)$ corresponds to the massive 4d $N_{f}=1$ curve, and we
relate $\Lambda_{1}=4m(M^{4}\lambda)^{\frac{1}{3}}/(1-M^{2}(1+\lambda))$. This
agrees with (4.14) in the massless case. We will extend this example to other
massive cases below.

##### The $\bm{D_{S^{1}}E_{5}}$ theory.

Consider next the case of $D_{S^{1}}E_{5}$, where (4.3) specializes to:

$\displaystyle E_{5}:\quad(I_{4};\,8I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{2}^{*};\,4I_{1})\leavevmode\nobreak\ .$ (4.18)

We immediately note that the RHS is a configuration of the 4d ${\rm SU}(2)$
$N_{f}=2$ theory. To realise this folding explicitly, we are again looking for
$E_{5}$ configurations that have at least a $\mathbb{Z}_{2}$ symmetry.

Consider the massless 4d theory, in which case the $\mathbb{Z}_{2}$-folding
corresponds to:

$\displaystyle E_{5}:\quad(I_{4};\,4I_{2})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{2}^{*};\,2I_{2})\leavevmode\nobreak\ .$ (4.19)

This $E_{5}$ configuration was discussed in detail in Closset:2021lhd , and
can be found, for instance, for the following values of the mass parameters:
$M_{1}=M_{2}=M$, $M_{3}=M_{4}=-{1\over M}$, with $\lambda=1$. In this case the
$U$-plane has a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry, which turns
into $\mathbb{Z}_{4}$ for the fixed value $M=e^{\pi i\over 4}$. Note that
these $\mathbb{Z}_{2}$ factors are extensions of the $\mathbb{Z}_{2}$ residual
R-symmetry present on the Coulomb branch of the massless 4d $N_{f}=2$ theory.
When the $\mathbb{Z}_{4}$ symmetry is present, the light BPS states are given
by Closset:2021lhd :

$\displaystyle\gamma_{1},\gamma_{5}\,:$
$\displaystyle\quad(1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{2},\gamma_{6}\,:$
$\displaystyle\quad(-1,1)\leavevmode\nobreak\ ,$ (4.20)
$\displaystyle\gamma_{3},\gamma_{7}\,:$
$\displaystyle\quad(-1,0)\leavevmode\nobreak\ ,$
$\displaystyle\qquad\qquad\gamma_{4},\gamma_{8}\,:$
$\displaystyle\quad(1,-1)\leavevmode\nobreak\ .$

The associated quiver is depicted below:

$\gamma_{1}$$\gamma_{5}$$\gamma_{4}$$\gamma_{8}$$\gamma_{3}$$\gamma_{7}$$\gamma_{2}$$\gamma_{6}$
(4.21)

with the massless 4d ${\rm SU}(2)$ $N_{f}=2$ subquivers coloured. Here we
introduce the _block_ notation for a BPS quiver Wijnholt:2002qz ;
Hanany:2012hi . Namely, we group together in a single block the nodes that
share the same incidence with all the other nodes in the quiver. For instance,
there are four morphisms involving node 1: $X_{12},X_{16}$, as well as
$X_{41},X_{81}$. We also find:

$U_{E_{5}}(\tau,M)=\sqrt{64\tfrac{u_{2}(\tau,0)}{\Lambda_{2}^{2}}+4\tfrac{1+M^{4}}{M^{2}}}\leavevmode\nobreak\
,$ (4.22)

with $u_{2}(\tau,0)$ the order parameter of the massless 4d ${\rm SU}(2)$
$N_{f}=2$ theory. As for the $E_{1}$ and $E_{3}$ quivers discussed above, it
was also shown in DelMonte:2021ytz that there exists a chamber of the above
quiver where the BPS spectrum organises as two distinct copies of spectra of
the massless 4d ${\rm SU}(2)$ $N_{f}=2$ quivers, as explained around (4.12).

##### The $\bm{D_{S^{1}}E_{7}}$ theory.

Another example of the folding (4.3) involves the $D_{S^{1}}E_{7}$ and the 4d
${\rm SU}(2)$ $N_{f}=3$ theories. For the massless 4d theory, this reads:

$\displaystyle E_{7}:$
$\displaystyle(I_{2};\,2I_{4},2I_{1})\xrightarrow[]{\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{Z}_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
}(I_{1}^{*};\,I_{4},I_{1})\leavevmode\nobreak\ .$ (4.23)

To find the above $E_{7}$ configuration, we first express the characters in
terms of gauge theory parameters. Then, we set the masses to $M_{i}=1$ for
$i=1,2,3$ and $M_{j}=-1$ for $j=4,5,6$, while the $\lambda$ parameter does not
need to be fixed. In this case, we find:

$U_{E_{7}}(\tau,\lambda)=\sqrt{-2^{12}\tfrac{u_{3}(\tau,0)}{\Lambda_{3}^{2}}-4\tfrac{(\lambda^{2}-1)^{2}}{\lambda^{2}}}\leavevmode\nobreak\
.$ (4.24)

The relevant $\mathbb{Z}_{2}$ symmetric $D_{S^{1}}E_{7}$ quiver is given
by:161616This is mutation equivalent to other known $dP_{7}$ quivers
Wijnholt:2002qz .

$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\gamma_{4}$$\gamma_{5}$$\gamma_{6}$$\gamma_{7}$$\gamma_{8}$$\gamma_{9}$$\gamma_{10}$
(4.25)

As before, we use the block notation. We again expect that there is a chamber
where the BPS spectrum of the KK theory decomposes into two copies of the
massless $N_{f}=3$ sub-quivers. To the best of our knowledge, this has not
been discussed in the literature.

##### Other examples.

It was recently argued in Closset:2023pmc that the global forms of a theory
can be detected directly from the Seiberg–Witten geometry. More specifically,
while the SW geometry for the 4d pure ${\rm SU}(2)$ theory is known to be
described by the configuration $(I_{4}^{*};2I_{1})$, the ${\rm SO}(3)_{\pm}$
global forms were found to be $(I_{1}^{*};I_{4},I_{1})$, where the $I_{4}$
singularity is undeformable.

In a similar fashion, as the $D_{S^{1}}E_{1}$ theory has both a 0 and a 1-form
symmetry, there are multiple global forms allowed for this theory. The
geometry that we have mostly dealt with, $(I_{8};4I_{1})$, was denoted in
Closset:2023pmc as $E_{1}[{\rm SU}(2)]^{[0,1]}$, emphasising the choice of
global form for the gauge group. Meanwhile, the $E_{1}[{\rm
SO}(3)_{\pm}]^{[0,1]}$ forms are described by the configurations
$(I_{2};2I_{4},2I_{1})$. The Coulomb branches of these models can be shown to
consist of two distinct copies of the 4d ${\rm SO}(3)_{\pm}$ theories.

###  4.3 Massive holonomy saddles

The correspondence between 4d and 5d theories can be refined in the presence
of mass deformations, which allows to test the proposal (4.5) in a large part
of the parameter space. While on the 5d side we are able to probe roughly half
of the parameter space, in 4d this map characterises the whole parameter
space.

When turning on arbitrary masses, the Coulomb branch geometry becomes rather
involved, and we have to proceed more systematically in finding the explicit
mapping between the CB order parameters.

####  4.3.1 The 5d–4d dictionary

In order to find a 5d ${\rm SU}(2)+2N_{f}$ phase which allows a
$\mathbb{Z}_{2}$-folding, we need to find the $E_{2N_{f}+1}$ curves whose
functional invariant $\mathcal{J}(U)$ is symmetric, i.e.
$\mathcal{J}(-U)=\mathcal{J}(U)$. Let us first focus on $N_{f}=0,1,2,3$, and