♯♯institutetext: Department of Mathematical Sciences
Durham University, Durham, DH1 3LE, United Kingdom††institutetext: Department
of Mathematics and Department of Physics and Astronomy
Uppsala University, Uppsala, Sweden

# Global Structures from the Infrared

Michele Del Zotto ♯ and Iñaki García Etxebarria<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum field theories with identical local dynamics can admit different
choices of global structure, leading to different partition functions and
spectra of extended operators. Such choices can be reformulated in terms of a
topological field theory in one dimension higher, the symmetry TFT. In this
paper we show that this TFT can be reconstructed from a careful analysis of
the infrared Coulomb-like phases. In particular, the TFT matches between the
UV and the IR. This provides a purely field theoretical counterpart of several
recent results obtained via geometric engineering in various string/M/F theory
setups for theories in four and five dimensions that we confirm and extend.

##  1 Introduction

Among the most important questions about the dynamics of quantum fields is the
task of characterising features that are robust under renormalization flow.
One such feature is the anomalous behaviour of global symmetries
tHooft:1979rat ,111 In this paper by anomalies we always mean ’t Hooft
anomalies for global symmetries. which are often captured via inflow
CALLAN1985427  from an invertible field theory in one dimension higher, known
as the _anomaly theory_ Freed:2014iua .

This inflow picture can be enriched whenever the QFT has local dynamics
compatible with inequivalent global structures (these can be detected by
analysing multiple related properties of the theory: the spectra of non-local
operators, the spectrum of generalised symmetries, or the partition functions
on compact curved spacetimes). When that happens we can split off the choice
of global form from the behaviour of the local degrees of freedom, by viewing
the local degrees of freedom of the QFT as a theory relative Freed:2012bs  to
a non-invertible theory in one dimension higher. A proper QFT, with a fully
specified global form, can then be interpreted as the compactification of the
bulk non-invertible theory on an interval: on one end of the interval the bulk
theory couples to a gapped TFT responsible for choosing the global structure,
and on the other end of the interval the bulk theory couples to the local
degrees of freedom.222 It is not always the case that diffeomorphism invariant
choices of gapped boundary conditions exist. The 6d $(2,0)$ theories
associated to the algebra $\mathfrak{g}$ are examples of consistent local
dynamics whose associated bulk theories are expected not to admit such choices
for generic $\mathfrak{g}$. (See Garcia-Etxebarria:2019cnb ; Gukov:2020btk
for an analysis of which $\mathfrak{g}$ are expected to admit diffeomorphism
invariant gapped boundary conditions.) This is one way to understand
statements in the literature that such theories are not “genuine QFTs”. We
expect that the bulk theories associated to the 4d $\mathcal{N}=2$ theories
studied in this paper always admit such choices. This structure is well-known
in the context of Lagrangian theories — for an enlightening discussion about
the $SU(N)$ versus $PSU(N)$ cases we refer our readers to section 6 of
Gaiotto:2014kfa .

It is generically the case that the resulting QFT after reduction on the
interface is anomalous, in which case the picture above needs to be further
refined, as elaborated on in Apruzzi:2021nmk  (see also Gukov:2020btk ): we
consider a non-invertible theory on an interval, where a gapped interface on
the left connects it to the anomaly theory for the QFT of interest (different
global forms have different symmetries and therefore different anomaly
theories, which are connected via suitable interfaces to the same non-
invertible theory), and the gapless boundary on the right encodes the local
degrees of freedom of the QFT. So in this case it is the gapped interface that
encodes the choice of global form. As in Apruzzi:2021nmk , we refer to the
non-invertible theory inside the interval as the _symmetry theory_.

|
---|---
Figure 1: Left: Inflow picture of anomaly matching for intrinsic QFTs: the
anomaly is an invertible TFT $\bm{A}$ in one dimension higher. Right: TFT
matching for QFTs with global structure. The choices of global structure are
encoded by non-invertible TFT on an interval with a boundary condition
interface $\bm{B}$ to an anomaly theory. This symmetry TFT must match for the
theories in the UV and the IR, thus generalizing the anomaly matching
procedure.

In the presence of anomalies for continuous symmetries the symmetry theory
includes sectors of Chern-Simons type, or more generally $\eta$-invariants. In
this note we focus on the choice of global form, which involves discrete
symmetries only, so we do not need to worry about such sectors, and we can
restrict ourselves to a part of the symmetry theory which is a proper
topological field theory (TFT). By an abuse of language, we will refer to this
TFT sector of the symmetry theory as _symmetry TFT_. It is natural to expect
this symmetry TFT to be invariant under RG flows triggered by deformations
invariant under the symmetries. Equivalently, we expect that the set of
choices for the global form of the theory persists at all energy scales — see
figure 1.

We will test this expectation on four dimensional QFTs in four space time
dimensions that admit a Coulomb phase, i.e. an infrared regime with an
effective description in terms of $r$ independent Maxwell fields. Let us
denote such theory $\mathcal{T}$. In this paper we are interested in
determining the global structure of $\mathcal{T}$ from the perspective of such
infrared regime. All the examples we will consider in this note are
$\mathcal{N}=2$ supersymmetric, but we stress that supersymmetry in itself is
not a necessary requirement for the discussion below: the two assumptions we
are making on $\mathcal{T}$ are that it has an infrared regime where an
effective $U(1)^{r}$ gauge theory description is valid, and that the structure
of the massive spectrum in this IR regime is sufficiently well understood. The
restriction to $\mathcal{N}=2$ comes from this second assumption, which in the
$\mathcal{N}=2$ context we take to mean the (likely weaker) assumption that
the BPS spectrum, which we understand well enough for our needs, is
representative of the full spectrum.333We emphasize that this as an
assumption. A known example where the non-BPS spectrum includes states that
would invalidate an analysis based on the BPS spectrum only is the type I
string, with gauge group $\mathrm{Spin}(32)/\mathbb{Z}_{2}$. In this case the
spinor representation appears as a non-BPS brane with discrete K-theory
charge; the BPS spectrum appears in vector representations only. In all
examples without torsional charges we know the BPS spectrum is representative.
It is tempting to conjecture that this is a general fact.

The class of $\mathcal{N}=2$ theories that we study naturally includes
geometrically engineered $\mathcal{N}=2$ SCFTs. In the geometric engineering
program Katz:1996fh  one aims to establish a dictionary between the properties
of some version of string theory, denoted by $\mathscr{S}$, on a singular
background444 Here we let $\mathscr{S}$ denote also M-theory or F-theory, and
by $\mathcal{X}$ we denote schematically the whole data needed to prescribe a
BPS background for $\mathscr{S}$ of the form $\mathbb{R}^{d}\times\mathcal{X}$
giving rise to a QFT in $d$ spacetime dimensions. $\mathcal{X}$ and a quantum
field theory $\mathcal{T}_{\mathscr{S}/\mathcal{X}}$. From this perspective
the symmetry TFT can be recovered via the analysis of the effective theory
arising after compactification on $\partial\mathcal{X}$, which captures the
anomalies and other features of $\mathcal{T}_{\mathscr{S}/\mathcal{X}}$,
including its higher form symmetries in terms of defect groups
DelZotto:2015isa ; Heckman:2017uxe ; Eckhard:2019jgg ;
GarciaEtxebarria:2019caf ; Albertini:2020mdx ; Morrison:2020ool ;
Dierigl:2020myk ; Closset:2020scj ; DelZotto:2020esg ; Apruzzi:2020zot ;
Bhardwaj:2020phs ; Closset:2020afy ; Heidenreich:2020pkc ; DelZotto:2020sop ;
Gukov:2020btk ; Bah:2020uev ; Bhardwaj:2021pfz ; Apruzzi:2021mlh ;
Apruzzi:2021phx ; Hosseini:2021ged ; Apruzzi:2021vcu ; Bhardwaj:2021wif ;
Bhardwaj:2021zrt ; Closset:2021lwy ; Heidenreich:2021xpr ; Buican:2021xhs ;
Cvetic:2021maf ; Debray:2021vob ; Apruzzi:2021nmk ; Braun:2021sex ;
Bah:2021brs ; Bhardwaj:2021mzl ; Cvetic:2020kuw ; Cvetic:2021sxm ;
Apruzzi:2022dlm ; Hubner:2022kxr ; DelZotto:2022joo ; Cvetic:2022imb . In
particular, if the theory $\mathcal{T}_{\mathscr{S}/\mathcal{X}}$ has non-
trivial choices of global structure, in the simplest cases this has been
understood Garcia-Etxebarria:2019cnb ; Albertini:2020mdx ; DelZotto:2020esg ;
Bhardwaj:2021mzl  in terms of a Heisenberg algebra of non-commuting FMS fluxes
on $\partial\mathcal{X}$ Freed:2006ya ; Freed:2006yc . Below we will give an
alternative purely field theoretical derivation of this same Heisenberg
algebra, showing that it also arises from the infrared perspective on theories
with a Coulomb phase. The advantage of this formulation is that it extends
field theoretically the results about theories with known Lagrangian
formulations to arbitrary SCFTs with mutually non-local massless excitations.
In particular, we recover field theoretically the results on global structures
of four-dimensional Argyres-Douglas theories that have been obtained via
geometric string theory techniques in the literature, and also some results
that have no geometric understanding.

Our main result can be derived closely following Aharony:2013hda . In the pure
$\mathfrak{u}(1)^{r}$ gauge theory, before choosing a global form, one can in
principle consider Wilson and ’t Hooft lines with arbitrary rational dyonic
charges. Once we include massive states, two things happen: the set of
allowable charges for the lines reduces to those mutually local with respect
to the charges of the dynamical states, and some of the line operators get
screened. Depending on the structure of the charge lattice of the theory non-
trivial lines might remain after screening. Generically not all remaining
lines will be mutually local, so an specification of a global form will
consist, as in Aharony:2013hda , on a specification of a maximal subset of
lines that will be genuine line operators in our theory. The rest of the lines
should then be viewed as open surface operators.

We emphasise that from this point of view the usual choice of having line
operators with arbitrary integer dyonic charge is a possible choice for the
maximally commuting set of line operators, but as shown below this is not the
only possible choice given any fixed lattice of charged states. (In making
this statement we assume that we have normalised our charges so that the
lattice of charged states is a sublattice of $\mathbb{Z}^{2r}$, which will be
the case throughout the paper.)

The argument in terms of screening of lines given above can be recast as the
derivation of a symmetry TFT for the theory of the form555Throughout the paper
we will use conventions where the $b$ fields are periodic with period $1$,
instead of $2\pi$.

$S_{5d}=2\pi i\sum_{j=1}^{r}{n_{j}}\int_{\mathcal{M}_{5}}b^{(2)}_{e,j}\wedge
db^{(2)}_{m,j}\,$ (1)

where $n_{j}\in\mathbb{Z}_{\geq 1}$. Below we will give an explicit
prescription for how to extract the integers $n_{j}$ from the Coulomb phase of
the theory. When some of these integers differ from 1 we recover the
Heisenberg algebra of non-commuting fluxes we found from geometric engineering
in terms of this bulk TFT.

All conventional 4d $\mathcal{N}=2$ gauge theories can be analysed with our
methods: giving generic vevs in the Cartan of the gauge group to the adjoint
scalars in the $\mathcal{N}=2$ vector multiplets, breaks the gauge group to
$U(1)^{r}$, giving us a plethora of consistency checks. Moreover, this feature
is also shared by all non-conventional 4d $\mathcal{N}=2$ SQFTs that have a
Coulomb branch of complex dimension $r$. In this case the integer $r$ is known
as the ‘rank’ of the corresponding non-conventional SQFT. From this
perspective our results extend and generalize results that have appeared
previously in the literature by giving a common ground for many computations
of defect groups in various dimensions.

We conclude this paper by extending our formulation to other theories with IR
phases under control, namely 6d SCFTs with a tensor branch, and 5d SCFT with a
Coulomb branch. Also for those systems we find a symmetry TFT that has the
structure of a BF theory in 7 ad 6 dimensions respectively, which is
responsible for the global structure of the theories in question.

The structure of this paper is as follows. In section 2 we review some well-
known features of Maxwell type theories and their higher form symmetries to
set up our notation and conventions. We proceed by revisiting the field
theoretical origin of global structures for 4d theories from an infrared
perspective. In section 3 we summarise the main features of BF theories we
will need, and derive our main result. Results in these two sections only
assume the theory under scrutiny has a Coulomb phase, and are independent from
supersymmetry. In section 4 however, we apply this result in the context of
$\mathcal{N}=2$ theories where the charge lattice of the theories can be
explicitly calculated, from this we recover the well-known center symmetries
and global structures of conventional gauge theories, and we also reproduce
and extend the results obtained about non-conventional SCFTs via geometric
engineering methods. In section 5 we give a generalization of our findings to
theories in various dimensions. In section 6 we present our conclusions and
comment on future directions and applications.

##  2 Genuine and non-genuine lines from the infrared

###  2.1 Line operators from the Coulomb phase

In what follows we explore the constraints on the line operators of Maxwell
theory that arise from the presence of charge dyonic BPS states in the
spectrum. Many important aspects of the analysis below can be found in section
4.1 of Gaiotto:2014kfa , see also appendix C in Cordova:2018cvg .

Figure 2: Aharonv-Bohm effect for lines: monodromies can give rise to phases
proportional to the corresponding Dirac pairings. Only lines which have
charges satisfying the Dirac quantization condition can be simultaneously
genuine. The presence of non-genuine lines is the hallmark of a relative QFT –
see e.g. section 2 of Bhardwaj:2021mzl  for a nice review.

The dynamics of a theory in a Coulomb phase can be described in terms of $r$
copies of the $\mathfrak{u}(1)$ Maxwell theory coupled to massive states.
(There can be non-generic points where some of these massive states can become
massless.) Ignoring the massive states for a moment, at a generic point we
have a higher 1-form global symmetry of the form

$\big{(}U(1)^{(1)}_{e}\times U(1)^{(1)}_{m}\big{)}^{r}$ (2)

where $U(1)^{(1)}_{e}\times U(1)^{(1)}_{m}$ are the 1-form global symmetries
of Maxwell theory. The operators charged under these symmetries are dyonic
lines. We denote the electric and magnetic charges of one such line by a
$2r$-component vector $\bm{\alpha}$. As in Gaiotto:2010be ; Aharony:2013hda ,
two lines with charges $\bm{\alpha}_{1}$ and $\bm{\alpha}_{2}$ can only be
genuine operators in the Maxwell theory (as opposed to open Gukov-Witten
surface operators Gukov:2006jk ; Gukov:2008sn ) if their Dirac pairing
$\bigl{\langle}\bm{\alpha}_{1},\bm{\alpha}_{2}\bigr{\rangle}_{D}\coloneqq\sum_{i=1}^{r}(\bm{\alpha}_{1}^{2i-1}\bm{\alpha}_{2}^{2i}-\bm{\alpha}_{1}^{2i}\bm{\alpha}_{2}^{2i-1})$
is an integer.666Relatedly, we can determine the commutation relations of the
line operators in a spatial slice using the exponentiated form of the
commutators found in Ashtekar:1997rg , which leads to a commutator $\exp(2\pi
i\bigl{\langle}\bm{\alpha}_{1},\bm{\alpha}_{2}\bigr{\rangle}_{D})$. (See
figure 2.) This is often accomplished by requiring
$\bm{\alpha}_{1},\bm{\alpha}_{2}\in\mathbb{Z}^{2r}$, but in the absence of a
dynamical spectrum of point particles this is not necessary for consistency of
the theory.

Figure 3: Top: action of a $U(1)^{(1)}$-form symmetry on a line operator.
Bottom: Constraint on the higher form symmetry in presence of charged
particles (red dot at the end of the line). The presence of a particle of
charge $q\in\mathbb{Z}$ enforces $e^{i\theta q}=1$ which in turns entail that
only rotations with phase $\theta=2\pi k/q$ are allowed, thus breaking
$U(1)^{(1)}$ down to $\mathbb{Z}^{(1)}_{q}$.

This structure gets simplified once we couple the pure Maxwell theory to
charged dynamical states. We will refer to this theory as the “UV theory”, in
contrast to the “IR theory” by which we mean pure Maxwell with no dynamical
states. The electromagnetic charges of the dynamical states live on a lattice
$\Gamma\subseteq\mathbb{Z}^{2r}$. In practice, it is convenient to choose a
basis $\gamma_{i}$ of generators for $\Gamma$, and this gives rise to an
explicit expression for the Dirac pairing in terms of
$\mathcal{Q}_{ij}=\langle\gamma_{i},\gamma_{j}\rangle_{D}\in\mathbb{Z}$.

The first effect of coupling the theory to charged dynamical states is that
the spectrum of admissible lines is reduced. Given a line with charges
$\bm{\alpha}$, it is only admissible if

$\bigl{\langle}\bm{\alpha},\gamma\bigr{\rangle}_{D}\in\mathbb{Z}\quad\text{for
all }\gamma\in\Gamma\,.$ (3)

In other words, $\bm{\alpha}\in\Gamma^{*}$, the dual lattice. The lines in
$\Gamma^{*}$ do not need to commute — see figure 2. Integrally charged line
operators always commute, as pointed out in Freed:2006yc , but $\Gamma^{*}$ is
not necessarily an integral lattice. This is the Coulomb branch counterpart of
the effect discussed at length by Gaiotto:2010be ; Aharony:2013hda : the same
local dynamics can be compatible with inequivalent global structures, and
different global forms of the theory can be detected from the spectrum of line
defects in four-dimensions.

Moreover, the spectrum of allowed charges for genuine line defects is also
constrained as a sublattice of $\Gamma_{L}\subset\Gamma^{*}$ consisting of a
maximally mutually local collection of defect charges satisfying the Dirac
quantization condition
$\langle\bm{\alpha},\widetilde{\bm{\alpha}}\rangle_{D}\in\mathbb{Z}$.
Inequivalent global forms of the theory correspond to inequivalent choices of
sublattices $\Gamma_{L}$ after screening.

Indeed, a second effect is that the higher form symmetry group (2) is
explicitly broken to a subgroup via screening. This can be understood as
follows: the line defects can have endpoints corresponding to charged
operators, and this constrains the electric and magnetic symmetry operators in
the UV theory to be those under which lines which can end are neutral. For an
example see figure 3 where we illustrate the breaking of a $U(1)^{(1)}$ higher
symmetry to ${\mathbb{Z}_{q}}^{(1)}$ in presence of a particle of charge
$q\in\mathbb{Z}$. From this perspective, the symmetry (2) is an emergent
symmetry in the IR, which is broken by massive states that at sufficiently
high energies become dynamical. For this reason the UV theory has a much
smaller 1-form global symmetry, which oftentimes is completely trivial
(corresponding to the cases when the spectrum of the theory is complete,
meaning that all defect charges can be screened).777 For a recent discussion
about this point and applications beyond the scope of the present work, we
refer our readers to the nice work Rudelius:2020orz . A related effect is that
the potential background fluxes that the theory admits get reduced: a
background for $U(1)_{e}$ can be understood as a modification for the
quantisation condition for the magnetic flux (see e.g. section 5 of
Hsin:2019fhf  for a discussion), so in the presence of electrically charged
states only those backgrounds that result in quantisation conditions
compatible with the charges of the dynamical matter are allowed. For instance,
if all of our electrically charged dynamical particles have even charge, we
can only have background magnetic fluxes with holonomies 0 and $\frac{1}{2}$.
In general, the holonomies of the electric and magnetic fluxes must be such
that their Dirac pairing with all dynamical states is integral.

The screening of $\Gamma^{*}$ with respect to $\Gamma$ gives rise to 1-form
factor of the defect group, $\mathbb{D}^{(1)}=\Gamma^{*}/\Gamma$
DelZotto:2015isa . The actual higher form symmetry for a given theory
corresponding to the sublattice $\Gamma_{L}$ is $\Gamma_{L}/\Gamma$
Gaiotto:2010be ; Aharony:2013hda .

###  2.2 An example: the $\mathcal{N}=2$ $\mathfrak{su}(2)$ theory on the
Coulomb branch

As a simple illustration of the previous discussion, here we briefly review
the results of Gaiotto:2010be  about the global form of the $\mathfrak{su}(2)$
gauge theories. From the Seiberg-Witten solution it is known that the BPS
spectrum of the $\mathfrak{su}(2)$ theory can be generated as bound states of
two mutually non-local excitations with charges $\gamma_{1}$ and $\gamma_{2}$
such that $\langle\gamma_{1},\gamma_{2}\rangle_{D}=-2$. In the choice of
electromagnetic duality frame by Seiberg and Witten one can view $\gamma_{1}$
as a monopole of charge $(0,1)$ and $\gamma_{2}$ as a dyon of charge $(2,-1)$.
To determine the dual lattice $\Gamma^{*}$ we can proceed as follows

$\langle\bm{\alpha},\gamma_{1}\rangle_{D}=\alpha_{1}\qquad\langle\bm{\alpha},\gamma_{2}\rangle_{D}=-\alpha_{1}-2\alpha_{2}$
(4)

Then the first equation implies that $\alpha_{1}\in\mathbb{Z}$, while the
second implies that $\alpha_{2}\in{1\over 2}\mathbb{Z}$. Let us consider the
possible mutually local sublattices of $\Gamma^{*}$ from this perspective,
with respect to the corresponding screenings. There are three minimal defect
charges which get nontrivial monodromies with respect to one another,
corresponding to non-integer quantized Dirac pairings, namely

$(1,0)\qquad(0,\tfrac{1}{2})\qquad\text{and}\qquad(1,\tfrac{1}{2})\,.$ (5)

Let us first consider the sublattice $\Gamma^{*}_{(1,0)}\subset\Gamma^{*}$
that contains the line defect with charge $(1,0)$. The requirement of maximal
mutual locality then implies that

$\langle(1,0),\bm{\alpha}\rangle_{D}=\alpha_{2}\in\mathbb{Z}$ (6)

Hence the charges of the line defects in $\Gamma^{*}_{(1,0)}$ have the form
$(n,m)$ where $n,m$ are both integers. Considering the screening by the
charges $\gamma_{1}$ and $\gamma_{2}$ corresponds to identifying

$(n,m)\sim(n^{\prime},m^{\prime})+k_{1}\gamma_{1}+k_{2}\gamma_{2}$ (7)

where $k_{i}\in\mathbb{Z}$. We see that we are left only with two equivalence
classes in $\Gamma^{*}_{(1,0)}$, namely $[(1,0)]$ and $[(0,0)]$, hence we
obtain an electric 1-form symmetry $(\mathbb{Z}_{2})^{(1)}_{e}$ corresponding
(with a natural choice of duality frame) to the gauge group $SU(2)$. Now
consider the lattice $\Gamma^{*}_{(0,\tfrac{1}{2})}$ which contains the line
with charge $(0,\tfrac{1}{2})$. The requirement of maximal mutual locality for
this class of charges is

$\langle(0,\tfrac{1}{2}),\bm{\alpha}\rangle_{D}=-{\alpha_{1}\over
2}\in\mathbb{Z}\,\Rightarrow\,\alpha_{1}\in 2\mathbb{Z}$ (8)

Hence the charges of the line defects in $\Gamma^{*}_{(0,\tfrac{1}{2})}$ have
the form $(2n,m/2)$ where $n,m$ are both integers. The screening equivalence
relation is again

$(2n,m/2)\sim(2n^{\prime},m^{\prime}/2)+k_{1}\gamma_{1}+k_{2}\gamma_{2}.$ (9)

We see that we are left with only two equivalence classes again $[(0,0)]$ and
$[(0,{1\over 2})]$ corresponding to a magnetic 1-form symmetry
$(\mathbb{Z}_{2})^{(1)}_{m}$ which gives a gauge group $SO(3)_{+}$. Now,
consider $\Gamma^{*}_{(1,\tfrac{1}{2})}$: procceeding analogously, we obtain
that

$\displaystyle\langle(1,\tfrac{1}{2}),\bm{\alpha}\rangle_{D}$
$\displaystyle=\alpha_{2}-{\alpha_{1}\over 2}\in\mathbb{Z}$ (10)
$\displaystyle\Rightarrow\,(\alpha_{1},\alpha_{2})\in\\{((2n+1),(2m+1)/2)\\}\text{
or }\\{(2n,m)\\}\qquad n,m\in\mathbb{Z}$

Again by screening we see that we obtain only two equivalence classes:
$[(0,0)]$ and $[(1,\tfrac{1}{2})]$, corresponding to a
$(\mathbb{Z}_{2})^{(1)}_{diag}$ 1-form symmetry, which corresponds to the
gauge group $SO(3)_{-}$.

We stress here that the above result is independent of the choice of
electromagnetic frame: we can choose to work with any different basis as long
as we are preserving the Dirac paring. For instance, one could do the analysis
working with $\gamma_{1}=(1,1)$ and $\gamma_{2}=(1,-1)$. In this case the
charges in $\Gamma^{*}$ have the form

$\langle\bm{\alpha},\gamma_{1}\rangle_{D}=\alpha_{1}-\alpha_{2}\qquad\langle\bm{\alpha},\gamma_{2}\rangle_{D}=-\alpha_{1}-\alpha_{2}$
(11)

And therefore one obtains

$(\alpha_{1},\alpha_{2})\in\\{(\tfrac{2m+1}{2},\tfrac{2n+1}{2})\\}\text{ or
}\\{(m,n)\\}\qquad m,n\in\mathbb{Z}.$ (12)

With this choice of duality frame we have the following defect charges that
would violate Dirac quantization

$(\tfrac{1}{2},\tfrac{1}{2})\qquad(\tfrac{1}{2},-\tfrac{1}{2})\qquad\text{and}\qquad(0,1)\,.$
(13)

These correspond to the three choices of lattices above, if we identify the
direction $(1,1)$ with the magnetic charge and the direction $(0,1)$ with the
electric one.

##  3 Symmetry TFT and global structure

We have just argued that whenever the lattice $\Gamma$ of charges particles in
Maxwell theory is not unimodular we have the possibility of having choices of
global structure, encoded as choices of maximal sets $\Gamma_{L}$ of commuting
elements in $\Gamma^{*}/\Gamma$. This is perhaps a surprising statement, as we
typically don’t think of Maxwell theory as admitting different global forms.
The key difference between our analysis and the usual analysis is that we do
not impose a priori integral quantization for the electric and magnetic fluxes
in the theory, but rather accept as valid any flux quantisation structure
compatible with the dynamical matter content. For instance, if all
electrically charged states have even charge we include half-integrally
quantised fluxes in the path integral for Maxwell theory. If we are
considering theories with only electrically charged states (in some duality
frame) then this is purely a matter of convention, and the half-integrality
can be rescaled away. In contrast, the theories of interest to us are richer,
and include dyonic states, which lead to genuinely different prescriptions for
which fluxes to sum over.

Our task is therefore classifying all the possibilities for flux quantisation
conditions compatible with the local dynamics. As mentioned above, the choice
of quantisation for the electric and magnetic fields can be understood as a
choice of background fields for the electric and magnetic 1-form symmetries.
So our problem may be recast as the determination of which choices of
background fluxes are allowed in a given quantum theory, given the dynamical
matter content. This kind of problem has a familiar solution (see for instance
Witten:1998wy ): the possible flux choices can be understood as states in the
Hilbert space of a (generically non-invertible) BF theory in one dimension
higher.

###  3.1 A quick review of BF theory in $D+1$ dimensions

In this section we review some basic details of the BF theory, following the
discussion in Kapustin:2014gua . These details are well-known, and can be
skipped by cognoscenti. A BF theory is a TFT in $D+1$ dimensions with action

$\displaystyle S={2\pi in}\int_{\mathcal{M}^{D+1}}b^{(q+1)}\wedge
db^{(D-q-1)}$ (14)

where $b^{(q+1)}$ and $b^{(D-q-1)}$ are a $(q+1)$-form and a $(D-q-1)$-form.
We stress that the coefficient $n$ which multiplies the action must be an
integer for $\text{exp}(-S)$ to be well defined and compatible with the local
$U(1)$ gauge transformations

$b^{(q+1)}\to b^{(q+1)}+d\lambda^{(q)}\qquad b^{(D-q-1)}\to
b^{(D-q-1)}+d\lambda^{(D-q-2)}\,.$ (15)

Here we are being naive and focusing only on the local structure of these
higher gauge transformations. More refined experts might look into the full
fledged gerby behavior of these Deligne-Beilinson cocylces. For our purposes
in this paper the above description will suffice. Notice that upon a gauge
variations we obtain

$S\to S+{2\pi in}\int_{\partial\mathcal{M}^{D+1}}\lambda^{(q)}\wedge
db^{(D-q-1)}\,.$ (16)

This boundary term is crucial for applications to inflow and generalizations
that we are after in this paper.

The theory is topological for a simple reason: the equation of motion for this
theory are

$f^{(q+2)}=db^{(q+1)}=0\qquad\qquad f^{(D-q)}=db^{(D-q-1)}=0$ (17)

and forbid any local propagating degree of freedom. The theory has
nevertheless interesting non-local gauge invariant operators corresponding to
closed $(q+1)$-dimensional and $(D-q-1)$-dimensional hypersurfaces of
$\mathcal{M}^{D+1}$, that generalize the familiar Wilson lines:

$\displaystyle\mathcal{W}_{\Sigma^{(q+1)}}=\exp 2\pi
i\int_{\Sigma^{(q+1)}}b^{(q+1)}$ (18)
$\displaystyle\mathcal{W}_{\Sigma^{(D-q-1)}}=\exp 2\pi
i\int_{\Sigma^{(D-q-1)}}b^{(D-q-1)}$

To determine the algebra of these operators notice that the insertion of
$\mathcal{W}_{\Sigma^{(q+1)}}$ in a correlator can be absorbed in the action
introducing a source term in the equations of motion (17)

$nf^{(D-q)}=\delta_{\Sigma^{q+1}}$ (19)

where $\delta_{\Sigma^{(q+1)}}$ is the Poincaré dual to the cycle
$\Sigma^{(q+1)}$ in $\mathcal{M}^{D+1}$. As a result we obtain that

$\langle\mathcal{W}_{\Sigma^{(q+1)}}\mathcal{W}_{\Sigma^{(D-q-1)}}\rangle=\exp\left({2\pi
i\over n}\ell(\Sigma^{(q+1)},\Sigma^{(D-q-1)})\right)$ (20)

where $\ell(\Sigma^{(q+1)},\Sigma^{(D-q-1)})$ is the linking number of
$\Sigma^{(q+1)}$ and $\Sigma^{(D-q-1)}$ in $\mathcal{M}^{D+1}$. Equivalently,
restricting everything along a spatial slice and considering a Hamiltonian
quantization, these generalized Wilson lines form a Heisenberg algebra

$\mathcal{W}_{\Sigma^{(q+1)}}\mathcal{W}_{\Sigma^{(D-q-1)}}=\exp{2\pi i\over
n}\left(\Sigma^{(q+1)}\cdot\Sigma^{(D-q-1)}\right)\mathcal{W}_{\Sigma^{(D-q-1)}}\mathcal{W}_{\Sigma^{(q+1)}}$
(21)

where $\cdot$ is the intersection pairing along the spatial slice for the
Hamiltonian quantization. Since this Heisenberg algebra is nontrivial the
Hilbert space associated to a generic codimension one submanifold
$\mathcal{M}^{D}$ will have dimension greater than one.

###  3.2 Global structures from the infrared

We now specialize the discussion in the previous section to the case of
interest for this paper in which $D=4$ and $q=1$. Our main claim is that the
introduction of the dynamical states on Maxwell theory leads to a 4d theory
relative to the 5d theory888We stress that this is not the whole action of the
symmetry TFT, just the part from which the choice of global structures will
follow.

$\mathcal{S}={2\pi i\over
2}\int_{\mathcal{M}^{5}}\mathcal{Q}^{\alpha\beta}\,b^{(2)}_{\alpha}\wedge
db^{(2)}_{\beta}=\frac{2\pi
i}{2}\int_{\mathcal{M}^{5}}\bigl{\langle}b^{(2)},db^{(2)}\bigr{\rangle}_{D}$
(22)

where the matrix $\mathcal{Q}^{\alpha\beta}$ is the $2r\times 2r$ Dirac
pairing for the BPS states in 4d, and $b^{(2)}$ is a $2r$ component 2-form (or
more precisely, a $2r$-dimensional vector of degree 3 differential characters,
but as mentioned above we will not need to worry about such topological
subtleties here).

This action can be justified as follows. Note first that since $\mathcal{Q}$
is an antisymmetric matrix we can do an invertible integral change of basis to
bring it into a block diagonal form

$\widetilde{\mathcal{Q}}=\left(\begin{array}[]{cccccccc|ccc}0&n_{1}&&&\cdots&&&\\\
-n_{1}&0&&&&&&\\\ &&\ddots&&&&&\vdots\\\ \vdots&&&0&n_{k}&&&\\\
&&&-n_{k}&0&&&&\\\ &&&&&\ddots&&\\\ &&&&&&0&n_{r}\\\ &&&\cdots&&&-n_{r}&0\\\
\hline\cr&&&&&&&&\\\ &&&&&&&&&\mathbf{0}_{f\times f}\\\
&&&&&&&&&&\end{array}\right)$ (23)

with the $n_{i}$ integral and non-negative (see theorem IV.1 of
newman1972integral  as well as the discussion in Caorsi:2017bnp ). In practice
the $n_{i}$ can be obtained easily by going to the Smith normal form of
$\mathcal{Q}$. If we name the components of
$b^{(2)}=(b^{(2)}_{e,1},b^{(2)}_{m,1},b^{2}_{e,2},\ldots)$ we see that our 5d
theory becomes (up to boundary counterterms that we are neglecting)

$\mathcal{S}_{5}={2\pi
i}\sum_{i=1}^{r}n_{i}\int_{\mathcal{M}^{5}}b^{(2)}_{e,i}\wedge
db^{(2)}_{m,i}\,.$ (24)

So our 5d theory decomposes into a sum of decoupled sectors.

In order to justify (24), consider the effect of going to the basis giving
(23) on the boundary field theory. The generators $\gamma_{i}$ in this basis
satisfy
$\bigl{\langle}\gamma_{2i-1},\gamma_{2i}\bigr{\rangle}_{D}=-\bigl{\langle}\gamma_{2i},\gamma_{2i-1}\bigr{\rangle}_{D}=n_{i}$
and zero otherwise. So the problem reduces to a situation similar to what we
found in the $\mathfrak{su}(2)$ case in §2.2. For simplicity we will
henceforth focus on the first block.

By an $SL(2,\mathbb{Z})$ transformation we can further choose
$\gamma_{1}=(0,1)$, $\gamma_{2}=(n_{1},k)$, with $k\in\mathbb{Z}$. We see that
all states in this basis have electric charge divisible by $n_{1}$, so we can
consistently introduce ’t Hooft lines of charge $1/n_{1}$. Wilson lines, on
the other hand, necessarily have integral charge. More generally, the spectrum
of allowed lines has charges $(p,q/n_{1})$, with $p,q\in\mathbb{Z}$. This
implies, in turn, that the electric background fields have periodicity 1, but
magnetic background fields have periodicity $n_{1}$. This means that the
allowed flux operators on the 5d BF theory are of the form
$\mathcal{W}_{e}^{p}\mathcal{W}_{m}^{q/n_{1}}$, with $p,q\in\mathbb{Z}$ (but
$n_{1}$ not necessarily dividing $q$). Equivalently, we can think of the line
operators as being the boundary of Gukov-Witten open surface operators, which
when pulled to the 5d bulk become the operators in the BF theory.

A BF theory with action

$\mathcal{S}_{5}=2\pi i\int_{\mathcal{M}^{5}}b_{e}^{(2)}\wedge db_{m}^{(2)}$
(25)

where $b_{m}^{(2)}$ has periodicity $n_{1}$ (so $\mathcal{W}_{m}^{1/n_{1}}$ is
allowed) and $b_{e}^{(2)}$ periodicity 1 is the same as a BF theory with
action

$\mathcal{S}_{5}=2\pi i\,n_{1}\int_{\mathcal{M}^{5}}b_{e}^{(2)}\wedge
d\tilde{b}_{m}^{(2)}$ (26)

with both $b_{e}^{(2)}$ and $b_{m}^{(2)}$ of periodicity 1 (via
$b_{m}^{(2)}\coloneqq n_{1}\tilde{b}_{m}^{(2)}$). Unwinding the choices of
basis, this proves that (22) is indeed the bulk theory for Maxwell theory with
pairing $\mathcal{Q}$.

From our review in section 3.1 it follows that this TFT has an algebra of
generalized Wilson lines of the form

$\mathcal{W}^{e,i}_{\Sigma^{2}}=\exp 2\pi
i\int_{\Sigma_{2}}b^{(2)}_{e,i}\qquad\mathcal{W}^{m,i}_{\Sigma^{2}}=\exp 2\pi
i\int_{\Sigma_{2}}b^{(2)}_{m,i}$ (27)

which along a spatial slice satisfy an Heisenberg algebra

$\mathcal{W}^{e,i}_{\Sigma^{2}}\mathcal{W}^{m,i}_{\widehat{\Sigma}^{2}}=\exp\left({2\pi
i\over
n_{i}}(\Sigma^{2}\cdot\widehat{\Sigma}^{2})\right)\mathcal{W}^{m,i}_{\widehat{\Sigma}^{2}}\mathcal{W}^{e,i}_{\Sigma^{2}}\,.$
(28)

This entails that whenever one of the $n_{i}$’s is different from one, we
obtain a Hilbert space of dimension greater than one for the theory on the
boundary. From the point of view of the four dimensional theory, these
generalised Wilson lines are the operators measuring background flux for the
1-form symmetries, so the fact that they do not commute implies that we cannot
choose Dirichlet boundary conditions for all fluxes simultaneously, as argued
originally in the holographic context in Witten:1998wy . In the specific case
of geometrically engineered four dimensional theories our field theory
discussion will reproduce the results obtained from geometry previously in
Closset:2020scj ; DelZotto:2020esg ; Hosseini:2021ged .

##  4 Examples and consistency checks from $\mathcal{N}=2$ theories

In order to compute the Dirac pairing $\mathcal{Q}$ we need to know the charge
lattice of the theory, meaning the electromagnetic charges of the particles in
the spectrum of the theory. In this paper, for concreteness, we focus on
examples arising in the context of 4d $\mathcal{N}=2$ theories where we can
easily extract this information exploiting BPS quivers Alim:2011kw . Here we
are assuming that the BPS spectrum faithfully reproduces the charge lattice,
meaning that we are assuming that in any charge sector populated by states we
can always find a BPS representative. By definition of BPS quiver the Dirac
pairing is captured by the quiver exchange matrix

$\mathcal{Q}_{ij}=\\#(\text{ arrows }i\to j)-\\#(\text{ arrows }j\to i)$ (29)

where one works with an extended charge lattice with a number of generators
that equals the number of electric, magnetic and flavour charges (where the
latter are taken in the Cartan of the flavour symmetry $F$ of the theory). The
generators of the charge lattice are in one to one correspondence with the
nodes of the BPS quiver and all other stable BPS states have charges that can
be expressed as linear combinations of the form
$\gamma=\sum_{i=1}^{2r+f}M_{i}\gamma_{i}$ where $M_{i}\in\mathbb{Z}_{\geq 0}$
for all $i$ (particles) or $M_{i}\in\mathbb{Z}_{\leq 0}$ for all $i$
(antiparticles). In particular, we are granted that the $\gamma_{i}$ are in
the spectrum and hence our screening argument is faithfully representing the
1-form symmetry.

###  4.1 Pure $\mathcal{N}=2$ SYM theories with algebra $\mathfrak{g}$

We begin considering the center symmetries of pure SYM theories with gauge
algebra $\mathfrak{g}$. The BPS quivers for all the ADE cases are described in
Cecotti:2010fi ; Alim:2011kw  and those for the non-simple laced cases where
given in Cecotti:2012gh . We emphasize that in the latter case we have no IIB
geometric engineering construction for the BPS quiver, so the agreement that
we will find between the field theory expectation and the result from the
analysis in terms of the BPS quiver can be taken as evidence that our
discussion above remains valid for those cases for which the string theory
analysis is not available.

$\Big{|}$$\mathfrak{g}$ | $\mathbb{D}^{(1)}$ |
---|---|---
$\Big{|}$ $A_{n}$ | $\mathbb{Z}_{n+1}\oplus\mathbb{Z}_{n+1}$ |
$D_{n}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ | $\text{if }\phantom{n}2\mid n$
$\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}$ | $\text{if }\phantom{n}2\nmid n$
$\Big{|}$$E_{6}$ | $\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}$ |
$\Big{|}$$E_{7}\,,\;B_{n}\,,\;C_{n}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ |
$\Big{|}$$E_{8}\,,\;G_{2}\,,\;F_{4}$ | $0$ |
Table 1: Defect groups for the pure $\mathcal{N}=2$ SYM theories with algebra
$\mathfrak{g}$.

We find that for all these semi-simple Lie algebras, the Dirac pairing
$\mathcal{Q}$ can be written in the block diagonal form and the corresponding
defect groups are given in 1. This is clear from the structure of the Dirac
pairing for such models, which is given by

$\mathcal{Q}_{\mathfrak{g}}=\left(\begin{array}[]{c|c}C_{\mathfrak{g}}-C_{\mathfrak{g}}^{t}\phantom{\Big{|}}&\phantom{\Big{|}}C_{\mathfrak{g}}^{t}\\\
\hline\cr\phantom{\Big{|}}-C_{\mathfrak{g}}&0\phantom{\Big{|}}\end{array}\right)$
(30)

where $C_{\mathfrak{g}}$ is the $r\times r$ Cartan matrix for the Lie algebra
$\mathfrak{g}$. For example, the BPS quivers for gauge groups $SU(N+1)$ and
$USp(2N)$ are

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
177.90005pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\circ_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
188.35501pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
183.55508pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
177.90005pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(31)

respectively. From these we obtain the block diagonal forms

$\widetilde{\mathcal{Q}}_{A_{N}}=\left(\begin{array}[]{ccccc}0&N+1&&&\\\
-(N+1)&0&&&\\\ &&0&1&\\\ &&-1&0&\\\
&&&&\ddots\end{array}\right)\qquad\qquad\widetilde{\mathcal{Q}}_{C_{N}}=\left(\begin{array}[]{ccccc}0&2&&&\\\
-2&0&&&\\\ &&0&1&\\\ &&-1&0&\\\ &&&&\ddots\end{array}\right)\,,$ (32)

respectively. Thus, it is clear that,

$\mathbb{D}^{(1)}_{A_{N}}=(\mathbb{Z}_{N+1})^{(1)}_{e}\oplus(\mathbb{Z}_{N+1})^{(1)}_{m}\,,\quad\mathbb{D}^{(1)}_{C_{N}}=(\mathbb{Z}_{2})^{(1)}_{e}\oplus(\mathbb{Z}_{2})^{(1)}_{m}\,.$
(33)

Moreover, the resulting Heisenberg algebra precisely reproduces the expected
global forms for the corresponding Lie groups Garcia-Etxebarria:2019cnb . The
check for all other groups is carried in a similar fashion and we consistently
recover all results in table 1.

###  4.2 Example of $\mathcal{N}=2^{*}$ theories with algebra $\mathfrak{g}$

In order to obtain the $\mathcal{N}=2^{*}$ quivers for a simple gauge theory
with gauge algebra of ADE type, one can start from the affine quiver
$A(1,1)\square\widehat{G}$ and replace one of the Kronecker subquivers that
correspond to a node with Dynkin weight 1 in the affine $\widehat{G}$ diagram
with a single node $\ast$, giving a ‘specialization’ of the corresponding
quiver in the language of Cecotti:2012gh . Graphically such operation
corresponds to

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern 30.75pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\bullet}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-3.0pt\raise-34.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.75pt\raise-69.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\dots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
33.85004pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.64996pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.75pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\dots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
41.75002pt\raise-69.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{gathered}\quad\longrightarrow\qquad\begin{gathered}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern 33.25pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 65.75pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-3.0pt\raise-36.82639pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
30.75pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
67.57813pt\raise-5.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
4.91699pt\raise-5.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
69.5pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.75pt\raise-73.65277pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\dots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
33.29688pt\raise-39.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
33.25pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
65.75pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\dots}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
39.2002pt\raise-39.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(34)

Using this trick, for instance, the BPS quiver for $SU(N+1)$
$\mathcal{N}=2^{*}$ is

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 29.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
5.50002pt\raise-31.26192pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.30002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.30002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{2}}$}}}}}}}{\hbox{\kern
105.10004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.10004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
142.60004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
118.60004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
182.71002pt\raise-31.43918pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
29.5pt\raise-66.67865pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
33.40001pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
71.20003pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
108.85004pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
147.65503pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
182.71002pt\raise-36.82639pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
158.24486pt\raise-66.20833pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
29.5pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\circ_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.79997pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.00005pt\raise-4.80444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.30002pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
76.59999pt\raise-66.20833pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
71.80006pt\raise-66.20833pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
43.30002pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.10004pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
105.10004pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
142.60004pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\circ_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
153.055pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
148.25507pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
142.60004pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(35)

where the nodes $\ast$ appearing on the left and on the right of the above
equation need to be identified. The resulting $\mathcal{Q}_{ij}$ can be read
off straightforwardly from equation (29). Similarly, the BPS quiver for
$SO(2N)$ $\mathcal{N}=2^{*}$ is

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 29.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{1}}$}}}}}}}{\hbox{\kern 67.30002pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
1.31721pt\raise-29.17426pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.30002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
105.10004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
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81.10004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
157.37668pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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197.48666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bullet_{N-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
133.36919pt\raise 9.64857pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
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197.48666pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise
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210.27495pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
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205.47502pt\raise-4.91333pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
201.33983pt\raise-78.56996pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-108.15277pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(36)

Also in these examples we obtain perfect agreement with the defect groups we
expect from gauge theory. For instance starting from the quiver of $SU(N)$
$\mathcal{N}=2^{*}$ we obtain

$\widetilde{\mathcal{Q}}_{SU(N)\,\mathcal{N}=2^{*}}=\left(\begin{array}[]{ccccc}0&N&&&\\\
-N&0&&&\\\ &&0&1&\\\ &&-1&0&\\\ &&&&\ddots\end{array}\right)$ (37)

as expected.

###  4.3 Adding matter in various representations

The BPS quiver for a gauge theory with gauge group $\mathfrak{g}$ and matter
in an principal representation $R$ is easily obtained Alim:2011kw . There is a
one-to-one correspondence between principal representations and nodes of the
Dynkin diagram of $\mathfrak{g}$. We can schematically summarize it as
follows: the nodes of the Dynkin diagram are in one-to-one correspondence with
the basis elements of the weight lattice, and principal representations are
such that their highest weight $w(R_{i})$ is $w(R_{i})=\delta_{ij}\omega_{j}$,
where $\omega_{j}$ is the weight basis. Graphically, the $i$-th node in the
Dynkin diagram

$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
(38)

correspond to the principal representation $R_{i}$. For the pure SYM BPS
quiver of type $\mathfrak{g}$ there is a one-to-one correspondence between the
nodes of the Dynkin diagram and full Kronecker subquivers. To obtain the
quiver for SYM coupled to the $i$-th principal representation one adds a node
to the BPS quiver connected to the $i$-th Kronecker subquiver as follows

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65.75pt\raise-73.65277pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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(39)

Notice that this prescription is compatible with the prescription discussed in
the previous section about $\mathcal{N}=2^{*}$ theories whenever the adjoint
is a principal representation (e.g. this is the case for $SO(2N)$ above).

In general, if the extra matter corresponds to a tensor product of principal
representations $R_{i}\otimes R_{j}$ the corresponding BPS quiver is obtained
by connecting the extra node to the rest of the quiver with an oriented
triangle for each of the corresponding Kroneckers. An example for this
prescription is the BPS quiver for the adjoint of $\mathfrak{su}(N)$ in the
previous section. The quiver in equation (35) is the BPS quiver for the
representation

$\mathbf{N}\otimes\overline{\mathbf{N}}=1\oplus\text{Adj}.$ (40)

and it corresponds to the tensor product of the fundamental
$\textstyle{\bullet_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
and the antifundamental
$\textstyle{\cdots}$$\textstyle{\bullet_{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
representations of $SU(N)$.

It is amusing to check explicitly that the breaking of the center symmetry by
the $N$-ality of the corresponding representation is respected, thus giving
further consistency checks to our general result. As an explicit example of
how this works in practice let us discuss here the case of the BPS quiver for
a Lagrangian theory with gauge group $SU(4)$ and matter in the direct sum of a
symmetric two-index representation of $SU(4)$ and an anti-symmetric two-index
representation. Since both the $\mathbf{6}$ and the $\mathbf{10}$ of $SU(4)$
have quadrality 2, we expect the center symmetry of $SU(4)$ to be broken down
to $\mathbb{Z}_{2}$ in this example. The resulting quiver is in Figure 4. We
indeed find a defect group $\mathbb{D}^{(1)}$ given by two copies of
$\mathbb{Z}_{2}^{(1)}$ and a single non-trivial BF coupling $n_{1}=2$,
compatible with the field theory expectations.

|  |
---|---|---
$\textstyle{\circ_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ_{2}}$$\textstyle{\ast_{\mathbf{10}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast_{\mathbf{6}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\circ_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 4: The BPS quiver for $SU(4)$ with matter in the
$\mathbf{6}\oplus\mathbf{10}$

###  4.4 Non-Lagrangian theories

The results in this paper confirm field theoretically all the results we
obtained from studying the 4d $\mathcal{N}=2$ theories arising from IIB on
isolated hypersurface singularities. This follows from a simple remark: in the
IIB geometric engineering of both Lagrangian and non-Lagrangian theories the
Dirac pairing is captured by the intersection form among the special
Lagrangian vanishing 3-cycles of the corresponding CY (see e.g. Shapere:1999xr
). In that context the charge lattice of the theory is a sublattice of
$H_{3}(\mathcal{X},\mathbb{Z})$ given by stable collections of wrapped D3
branes. The quiver captures precisely this information, and the nodes of the
quiver give rise to a collection of 3-cycles that are always stable, in
regions of the moduli space that are compatible with that quiver descriptions.
The intersection pairing is therefore identified with the Dirac pairing, and
that is precisely the quantity which enters in all the computations we carried
out in our previous paper on the subject and that determines the structure of
the Heisenberg algebra of FMS fluxes. For all these examples, therefore, our
field theoretical results and the ones obtained from geometric engineering
agree by construction. In particular, this confirms previous results
Closset:2020scj ; DelZotto:2020esg ; Hosseini:2021ged ; Closset:2021lwy ;
Buican:2021xhs ; Carta:2022spy ; Bhardwaj:2021mzl  about the Argyres-Douglas
theories of type $(G,G^{\prime})$ constructed by Cecotti, Neitzke, and Vafa
Cecotti:2010fi , the various Arnol’d SCFTs Cecotti:2011gu ; DelZotto:2011an
and other theories originating from singularity theory Xie:2015rpa , as well
as the SCFTs of type $D_{p}(G)$ Cecotti:2012jx ; Cecotti:2013lda .

We stress that our methods extend straightforwardly to all other theories with
a known BPS quiver.

As a further example of an application, we present in the rest of this section
an analysis of the rank one theories with known BPS quivers, namely theories
in the $\mathcal{I}_{1}$ series and in the $\mathcal{I}_{4}$ series in Table 1
of Argyres:2016xmc . Therefore we obtain examples in all possible
characteristic dimensions Cecotti:2021ouq . The BPS quivers for these theories
have been obtained in CDZ  — see section 4 of Caorsi:2019vex  for a review
(see also Alim:2011kw ; Cecotti:2013lda ; Cecotti:2013sza  for previous
results on the topic). The resulting quivers have the form

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10.24359pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise-67.03331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
29.5pt\raise-37.23888pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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59.5pt\raise-37.23888pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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(41)

where:

  * •

$f$ is the rank of the flavor symmetry of the rank 1 theory of interest;

  * •

$q_{\mathcal{K}}$ is a positive integer denoting the multiplicity of the
arrows $\circ\to\bullet$ determined as follows:

$q_{\mathcal{K}}=\begin{cases}3&\text{for
}\mathcal{K}=II^{*},III^{*},IV^{*}\\\ 2&\text{for }\mathcal{K}=I^{*}_{0}\\\
1&\text{for }\mathcal{K}=II,III,IV\end{cases}$ (42)

  * •

$a_{1},a_{2},...,a_{f}$ are positive integers denoting the multiplicities of
the arrows $\bullet\to\ast_{i}\to\circ$, determined from the decomposition of
the Kodaira fiber

$\mathcal{K}\to
I_{1},I_{1},I_{(a_{1})^{2}},\,I_{(a_{2})^{2}},\,\dots,\,I_{(a_{f})^{2}}\,.$
(43)

As an example for the $E_{8}$ Minhan-Nemeshansky theory Minahan:1996cj  we
have $\mathcal{K}=II^{*}$ and $II^{*}\to(I_{1})^{10}$, hence
$q_{\mathcal{K}}=3$, $f=8$ and $a_{i}=1$ for all $i=1,...,8$. As another
example, for the Argyres-Wittig theory with a flavor symmetry with Lie algebra
$C_{5}$ we have $II^{*}\to(I_{1})^{6},I_{4}$, hence $q_{\mathcal{K}}=3$, $f=5$
and $a_{1}=2$ while $a_{2,3,4,5}=1$.

Exploiting the data of Table 1 of Argyres:2016xmc  it is straightforward to
read off the corresponding BPS quivers for these theories, then by (29) the
resulting BF levels follow by our method. The result we obtain is that for all
these theories $n_{1}=1$, but for the case $\mathcal{K}=I_{0}^{*}$ with
decomposition $I_{0}^{*}\to{I_{1}}^{2},I_{4}$ which corresponds to $SU(2)$
$\mathcal{N}=2^{*}$. In this case we obtain $n_{1}=2$, as we already discussed
above.

The result we obtain for the global forms of these theories can be also
recovered exploiting the fact that these models arise as fixed points of
supersymmetry enhancing RG flows, starting from $\mathcal{N}=1$ Lagrangians
Gadde:2015xta . Another interesting class of susy enhancing RGs are those of
Maruyoshi-Song type which give results for the theories in the
$\mathcal{I}_{1}$ series Maruyoshi:2016tqk ; Maruyoshi:2016aim ;
Agarwal:2018ejn . For the theory in the $\mathcal{I}_{4}$ series with global
symmetry $USp(4)\times U(1)$ as well as for the $E_{6}$ MN theory, UV
$\mathcal{N}=1$ theories have been obtained by brane bending and deconfinement
Etxebarria:2021lmq . In all these examples we reproduce easily the fact that
the UV theory has a trivial defect group, thus confirming our findings.

##  5 Generalization to other dimensions

In this section we quickly comment about the generalization of the argument
above to some other theories in higher dimensions. In general we expect the
$D+1$ TFT action will contain terms of the following form (with an additional
factor of 2 in the self-dual case)

$\displaystyle\mathcal{S}^{TFT}_{D+1}\supseteq 2\pi
i\sum_{q=0}^{D}\mathcal{Q}^{\alpha,\beta}_{q}\int_{D+1}b^{(q+1)}_{\alpha}\wedge
db^{(D-q-1)}_{\beta}\,.$ (44)

The latter are relevant to probe more general global forms of QFTs in
$D$-dimensions with different kinds of higher $q$-form symmetries
corresponding to defects with non-trivial generalized Dirac strings. We stress
that other couplings can be allowed, which in this paper we are omitting. In
these cases the symmetry properties of the matrices
$\mathcal{Q}^{\alpha,\beta}_{q}$ depend crucially on $D$ and on $q$. For
instance for $D=6$ and $q=2$, we have a symmetric pairing that was explored in
the context of defect groups and global structures of 6d (2,0) and (1,0)
theories DelZotto:2015isa ; GarciaEtxebarria:2019caf . In what follows we
quickly address the IR origin of the global structures for the cases of 6d
(2,0) SCFTs and of 5d SCFTs.

###  5.1 The case of 6d (2,0) theories

In the case of 6d (2,0) theories we have an analogue of the Coulomb branch,
where the nonabelian string dynamics reduces to an abelian one, the so-called
tensor branch. Along the tensor branch we have a $(U(1)_{e}^{(2)})^{r}$ higher
2-form symmetry, which has 3-form currents $J^{(3)}=h^{(3)}_{i}$ corresponding
to the anti-self-dual 3-form curvatures $h_{i}^{(3)}=db_{i}^{(2)}$ where
$b_{i}^{(2)}$ are the 2-form fields in the 6d tensormultiples. We can couple
the latter to background fluxes $B^{(3)}_{i}$, which have 3-form background
gauge transformations analogous to the discussion we had for the Maxwell
theory. Also in this case, when we excite the (2,0) BPS strings, the current
conservation equation $d\ast J^{(3)}=0$ is broken by the presence of sources
for the $h^{({3})}_{i}$ fluxes, represented by the string charges.

The effect of such breaking is again detected by the inflow mechanism which
associates to the tensor branch a 7d BF like theory of the form

$\mathcal{S}_{7}\supseteq{2\pi i\over
2}C^{\alpha\beta}_{\mathfrak{g}}\int_{7}b^{(3)}_{\alpha}\wedge
db^{(3)}_{\beta}$ (45)

where $C^{\alpha\beta}_{\mathfrak{g}}$ is the Cartan matrix of the Lie algebra
of type $\mathfrak{g}$, which gives the BPS string Dirac pairing for the (2,0)
theory of type $\mathfrak{g}$ in 6d, again accounting for the ’t Hooft
screening DelZotto:2015isa .

There is a crucial difference between six dimensions and four dimensions: in
six dimensions the Dirac pairing in 6d is a symmetric matrix, which is
compatible with the symmetry properties of the 7-form in the action 45. For
this reason, 6d strings can be non-mutually local with respect to themselves.
As examples one can consider e.g. the rank one non Higgsable cluster theories,
6d SCFTs with tensor branch of the form $\overset{\mathfrak{g}}{n}$. For these
models we obtain

$\mathcal{S}_{7}\supseteq{2\pi in\over 2}\int_{7}b^{(3)}\wedge db^{(3)}$ (46)

This physical distinction comes together with a very important mathematical
distinction: while the theory of antisymmetric integral bilinear forms
relevant to the four dimensional case is very simple (and in particular
implies that a change of basis to the simple block diagonal form (23) always
exists), the theory of symmetric bilinear forms over the integers, relevant in
the six dimensional case, is significantly more complicated. For instance, one
can show that the Cartan matrix arising in the $\mathfrak{su}(2)$ case cannot
be taken to diagonal form via an integral congruence Conway1999 .

Interestingly, it is the diagonal form that appears in the holographic result
arising from the reduction of the Chern-Simons couplings of M-theory
Witten:1998wy ; Aharony:1998qu

$\mathcal{S}_{7}(A_{N-1})\supseteq{2\pi i\over 2}N\int_{7}c^{(3)}\wedge
dc^{(3)}+...$ (47)

We do not understand the physical meaning (if any) of this discrepancy, we
hope to return to it in future work.

###  5.2 The case of 5d theories

Another class of theories with interesting Coulomb phases is provided by the
5d SCFTs with non-trivial ranks. Along a 5d Coulomb branch we have an emergent
$(U(1)^{(1)}_{e}\times U(1)^{(2)}_{m})^{r}$ higher form symmetry. The latter
is similarly broken to subgroups by the spectrum and, mutatis mutandis, the
same logic applies. The resulting BF-theory in this case has the form

$\mathcal{S}_{6}\supseteq{2\pi
i}\mathcal{Q}^{\alpha\beta}\int_{6}b^{(2)}_{e,\alpha}\wedge
db^{(3)}_{m,\beta}$ (48)

where $\mathcal{Q}^{\alpha\beta}$ is the Dirac pairing among the BPS electric
states and the 5d BPS monopole strings. This latter quantity determines the
structure of the Heisenberg algebras of non-commuting fluxes via a field
redefinition to its Smith normal form, thus reproducing all the results
obtained via M-theory in this context (see e.g. Albertini:2020mdx ;
Morrison:2020ool ; Bhardwaj:2020phs ; DelZotto:2022fnw ; DelZotto:2022joo ;
Cvetic:2022imb ).999The Dirac pairing $\mathcal{Q}^{\alpha\beta}$ coincides
with the intersection form between 2-cycles and 4-cycles relevant for the
computation of the global forms from M-theory geometric engineering. For this
class of theories, it is known there might be further terms in the bulk TFT
Apruzzi:2021nmk : it should be possible to recover the latter in terms of the
infrared as well, however this would go beyond the modest scope of this short
note.

As a consistency check of the above formula, one can consider the 4d KK theory
of the corresponding SCFT, obtained from circle reduction. These systems have
5d BPS quivers Closset:2019juk . The latter can be exploited in a way
analogous to the one we discussed above to capture the global structure of the
4d KK theory so obtained. See reference DelZotto:2022fnw  for an application
of this idea in the context of 5d orbifold SCFTs discussed e.g. in
Tian:2021cif .

##  6 Conclusions

In this paper we have begun exploring a mechanism to recover the global
structure of a given QFT from an infrared phase which is under perturbative
control. Our main result is to recover, from Coulomb-like phases, the
Heisenberg algebra of non-commuting fluxes that was found in the geometric
engineering analysis in purely field theoretical terms.

An interesting question that we leave for future analysis is to characterize
the full structure of the symmetry TFT from an IR perspective. We expect this
to be possible by a suitable extension of the ’t Hooft anomaly matching
argument: while the theory on the boundary flows, the symmetry TFT in the bulk
must match along the flow. In this short note, we recovered the term
responsible for the possible choices of global structures, but we stress that,
for instance, we expect mixing terms between the various higher form fields in
the symmetry TFT. The latter are not captured by the argument presented here,
but are known to arise from a geometric engineering perspective
Apruzzi:2021nmk ; Apruzzi:2022dlm .

In this context a direction that we find particularly interesting is the
question of recovering higher group structures or more general non-invertible
symmetries from the IR. For two-groups, evidence that this is indeed possible
in some cases was given in the context of little string theories in the papers
Cordova:2018cvg ; Cordova:2020tij ; DelZotto:2020sop . There it was shown that
higher group structure constants are related to specific terms in the anomaly
polynomial of the corresponding little strings. Based on that analysis we
conjecture the anomaly theories on the worldvolumes of the various BPS degrees
of freedom of the boundary QFTs of interest must know about these finer
details of the symmetry TFT. A similar effect was recently exploited to
unravel certain non-invertible symmetries in Kaidi:2021xfk ; Choi:2021kmx .

###### Acknowledgements.

We thank P. Argyres and M. Martone for illuminating discussions. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 851931). I.G.-E. is partially supported by STFC consolidated
grant ST/T000708/1. We also both acknowledge support from the Simons
Foundation via the Simons Collaboration on Global Categorical Symmetries.

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