Global symmetry-breaking and generalized
theta-terms in Type IIB EFTs

Thomas W. Grimm, Stefano Lanza, Thomas van Vuren

_Institute for Theoretical Physics, Utrecht University_

_Princetonplein 5, 3584 CE Utrecht, The Netherlands_

###### Abstract

A longstanding conjecture states that global symmetries should be absent in
quantum gravity. By investigating large classes of Type IIB four-dimensional
$\mathcal{N}=2$ effective field theories, we enlist the potential generalized
global symmetries that could be present and explore how they are avoided.
Crucial ingredients that arise in such effective field theories are
generalized $\theta$-terms. These introduce non-linear couplings between axion
fields and topological terms quadratic in the gauge field strengths which
break a large subset of the global symmetries. Additional residual global
symmetries may further be broken by assuming the existence of some charged
states. However, we illustrate that the presence of generalized $\theta$-terms
leads to a generalized Witten effect, which implies that the spectrum of
charged states is constituted by an infinitely populated lattice. We further
show that such a lattice is generated by the action of the monodromy
transformation that characterizes the moduli space boundary near which the
effective theory is defined.

###### Contents

  1. 1 Introduction
  2. 2 Global symmetries and the Witten effect
    1. 2.1 Generalized global symmetries, and how to avoid them
    2. 2.2 The Witten effect and an infinitely populated charge lattice
  3. 3 Global symmetries and generalized $\theta$-terms in Type IIB EFTs
    1. 3.1 $\mathcal{N}=2$ Type IIB EFTs
    2. 3.2 The symplectic structure of Type IIB EFTs
    3. 3.3 Generalized global symmetries of Type IIB EFTs
    4. 3.4 A generalized Witten effect for Type IIB EFTs
  4. 4 Global symmetries in the rigid case and one-modulus asymptotic limits
    1. 4.1 The rigid case
    2. 4.2 The one-modulus case: the large complex structure point
    3. 4.3 The one-modulus case: Tyurin degeneration
    4. 4.4 An example of finite distance boundary: the conifold singularity
  5. 5 Global symmetries in general asymptotic limits
    1. 5.1 The generalized Witten Effect and global symmetries towards a generic boundary
    2. 5.2 Avoiding global symmetries along boundary enhancement chains
  6. 6 Conclusions
  7. A Charge quantization and dualization
  8. B Gauge interactions in asymptotic limits
    1. B.1 A brief overview of asymptotic Hodge Theory
    2. B.2 The asymptotic form of the gauge-kinetic matrix and the generalized $\theta$-terms
    3. B.3 Outline of the procedure
  9. C Reconstructing the Hodge-star in the one-modulus cases
    1. C.1 The Large Complex Structure Point
    2. C.2 Type $\text{II}_{0}$
    3. C.3 Type $\text{I}_{1}$

##  1 Introduction

In recent years, our understanding of global symmetries has improved greatly
following [1], where the notion of _generalized global symmetries_ was
introduced. Such global symmetries play a critical role in determining the
consistency of field theories that originate from fully-fledged quantum
gravity theories. In fact, quantum gravity is expected to be devoid of any
global symmetries [2, 3, 4, 5, 6], as asserted by the _No Global Symmetry
Conjecture_. The latter is a foundational part of the _Swampland program_
which seeks to identify the criteria that any effective field theory (EFT)
ought to satisfy in order to be consistent with a quantum gravity UV
completion (see, for instance, [7, 8] for reviews on the subject), and many of
the subsequently formulated conjectures rely upon it. The conjecture can be
phrased differently as stating that, within any EFT that is valid below an
energy scale $\Lambda_{\text{\tiny EFT}}$ and which admits a UV completion
within quantum gravity, global symmetries are at most approximate or emergent,
meaning that they are removed by mechanisms that occur at energies above
$\Lambda_{\text{\tiny EFT}}$. For instance, in [9] it was shown that all
generalized global symmetries of the ten-dimensional string theory EFTs, or
the eleven-dimensional M-theory can be removed, either because the eventual
global symmetries are gauged, or because they are broken by brane states.
However, given a lower-dimensional EFT that ought to be consistent with a
quantum gravity completion, it is highly nontrivial to generically show that
every generalized global symmetry is removed by invoking conventional string
theory ingredients.

The aim of this work is to investigate how global symmetries are avoided in a
large family of concrete string theory EFTs. In particular, we will examine
the four-dimensional EFTs that stem from compactifications of Type IIB string
theory over Calabi-Yau three-folds. Within the resulting $\mathcal{N}=2$
supergravity EFTs, our attention will be focused on the generalized global
symmetries associated to the bosonic fields of the vector multiplets, namely
the complex scalar fields, acting as moduli fields which describe the complex
structure deformations of the internal Calabi-Yau, and the supersymmetrically
paired one-form gauge fields. Although the vector multiplets only constitute a
subsector of the fields populating the more general $\mathcal{N}=2$ Type IIB
EFTs, the effective description capturing their interactions is already quite
rich. Indeed, the vector multiplet sector could contain some emergent
generalized global symmetries, which ought to be removed in the UV-complete
theory.

We will examine such Type IIB EFTs in the near-boundary regions of the moduli
space. Remarkably, such parts of the moduli space can be investigated in
detail by means of asymptotic Hodge theory [10]. The latter has proven to be a
powerful tool for studying the near-boundary physics in full generality. For
instance, in [11, 12, 13, 14, 15] Hodge theory has been employed to show how
the Distance Conjecture is realized in generic, concrete string theory EFTs,
or in [16, 17, 18, 19] to infer general features of the scalar potential and
the structure of stringy vacua, or to analyze the properties of BPS objects
[20, 21, 22, 23]. In this work, we will illustrate how Hodge theory can be
used in order to generically study the behavior of the interactions of the
gauge fields in terms of the complex structure moduli fields. In particular,
in Section 4 we will perform a detailed computation of the near-boundary
effective action describing the interaction of the vector multiplets in EFTs
containing one dynamical complex structure modulus. For each of these models,
we will enlist the possible global symmetries and present how these can be
removed using stringy ingredients. These findings will be generalized in
Section 5 to include generic multi-moduli cases.

As we will show, a common feature of the near-boundary effective descriptions
of the Type IIB EFTs is the presence of several ‘ _generalized
$\theta$-terms_’. These terms correspond to couplings between topological
terms that are quadratic in the field strengths of the gauge fields and the
axions of the theory. We will refer to them as ‘ _generalized_ ’ because,
unlike the more standard ‘$\theta$-terms’, of the form $\theta F\wedge F$,
introduced in [24, 25] which are linear in the so-called _Peccei-Quinn axion_
$\theta$, the ‘$\theta$-terms’ encountered in Type IIB EFTs may be non-linear
in the axionic fields of the model. These couplings have profound
phenomenological consequences. In fact, as is well known in the literature,
the standard $\theta$-terms are responsible for the Witten effect [26]:
namely, the $\theta$-term guarantees that, if a magnetic monopole is present
in the spectrum, then so too is an infinite tower of dyons. In this work, we
will illustrate that a similar mechanism is realized in Type IIB EFTs with
generalized $\theta$-terms. Indeed, we will show that the generalized
$\theta$-terms break some of the potential generalized global symmetries,
while also leading to a realization of a generalized version of the Witten
effect.

The work is articulated as follows. In Section 2 we introduce the notion of
generalized global symmetry and review some of the mechanisms that quantum
gravity is expected to be endowed with that prevent the presence of global
symmetries. We further show how the presence of $\theta$-terms implies that
the breaking of global symmetries may lead to an infinitely populated charge
lattice. In Section 3 we examine the potential generalized global symmetries
that the vector multiplet sector of four-dimensional $\mathcal{N}=2$ Type EFTs
may display. Therein, we additionally illustrate how a generalized version of
the Witten effect is realized in such EFTs, and how this effect influences the
definition of gauge-invariant conserved currents that could deliver
generalized global symmetries. In Section 4 we explicitly compute the gauge
interactions of Type IIB EFTs with no moduli or in the near-boundary region of
EFTs with one complex structure modulus. For each of the models therein
presented we display the potential emerging generalized global symmetries, and
how these can be broken. In Section 5 we will then show how the Hodge theory
techniques can be employed to count the global symmetries that can emerge
towards any arbitrary field space boundary.

Furthermore, the Appendices contain several additional technical details that
have been used throughout the main text. In particular, Appendix A collects
some conventions on gauge fields used throughout the work. Appendix B is
devoted to the introduction of asymptotic Hodge theory, the technology that is
foundational for Sections 4 and 5. Therein, we will show how the latter offers
the tools necessary to obtain the interactions of the bosonic components of
the $\mathcal{N}=2$ vector multiplets in the near-boundary region of any
Calabi-Yau singularities in full generality. Finally, in Appendix C we
illustrate how to compute the periods and the gauge interactions towards
generic boundaries in EFTs with a single complex structure modulus by means of
the asymptotic Hodge theory.

##  2 Global symmetries and the Witten effect

Recently, our knowledge about how global symmetries may be realized has been
greatly expanded. Indeed, the recent [1] has shown that the notion of global
symmetry is wider than was previously thought, leading to the notion of
_generalized global symmetries_ , and illustrated under which conditions these
can be avoided, via their breaking or gauging.

The complete identification of global symmetries within a theory plays a
crucial role in Quantum Gravity. Indeed, as captured by the _No Global
Symmetry Conjecture_ , one of the pillars of the Swampland program, any
consistent theory of quantum gravity ought to be devoid of any global
symmetry. The foundations of the conjecture are grounded in the black hole
physics [2, 3, 4, 5, 6]. Loosely speaking, if there exist black holes carrying
global charges, the Hawking evaporation process would produce an infinite
number of states, with masses and sizes of the order of the Planck scale and
each characterized by arbitrary values of global charges. The coexistence of
an infinite number of species of such _remnants_ is believed to be
pathological [27], and in conflict with the Covariant Entropy Bound [6].

However, as noticed in [9, 28] and further explained in the following
sections, EFTs stemming from string theory are _apparently_ plagued by such
generalized global symmetries. Therefore, in order to comply with the No
Global Symmetry conjecture, their UV stringy origin has to deliver the
appropriate mechanisms that avoid them.

In this section, we set the stage for the forthcoming discussion by briefly
recalling the notion of generalized global symmetries in Quantum Field Theory,
and how these can be avoided. In addition, we show that the breaking of some
global symmetries may lead to the realization of a _Witten effect_ , which
predicts properties about the composition of the allowed spectrum of states
that the theory has to support.

###  2.1 Generalized global symmetries, and how to avoid them

In a $D$-dimensional spacetime, a _$p$ -form global symmetry_ is a symmetry
whose action is implemented via operators supported on a $(p-1)$-codimension
manifold, $\mathcal{M}^{(D-p-1)}$. Such global symmetries form a group, which
we denote with $G$. In general, the group $G$ may be either abelian or non-
abelian; however, for $p>0$, the symmetry group $G$ needs to be abelian.
Moreover, the group $G$ can be either discrete or continuous. Given an element
$g\in G$, one can then associate an operator $U_{g}(\mathcal{M}^{(D-p-1)})$.
The operator $U_{g}(\mathcal{M}^{(D-p-1)})$ is assumed to be topological,
namely, it is invariant under continuous deformations of the supporting
manifold $\mathcal{M}^{(D-p-1)}$, provided that no charged spacetime defect is
crossed. $G$ being a group, it follows that for any two group elements
$g_{1},g_{2}\in G$, also $g=g_{1}g_{2}\in G$. Such a composition law is
realized at the operator level as

$U_{g_{1}}(\mathcal{M}^{(D-p-1)})\times
U_{g_{2}}(\mathcal{M}^{(D-p-1)})=U_{g}(\mathcal{M}^{(D-p-1)}).$ (2.1)

Observables – such as charged operators – can be understood as operators
$V(\mathcal{C}^{(p)})$ supported on $p$-dimensional mainfolds
$\mathcal{C}^{(p)}$. Then, a symmetry operator $U_{g}(\mathcal{M}^{(D-p-1)})$,
with the manifold $\mathcal{M}^{(D-p-1)}$ encircling $\mathcal{C}^{(p)}$, acts
on $G$ as follows111Recall that, in $D$ dimensions, a $p$-dimensional object
can be surrounded by a $(D-p-1)$-dimensional sphere.

$U_{g}(\mathcal{M}^{(D-p-1)})V(\mathcal{C}^{(p)})=g(V)V(\mathcal{C}^{(p)})\,,$
(2.2)

where $g(V)$ denotes the action of $g$ in some convenient representation.

For concreteness, and to set the ground for the upcoming sections, let us now
focus on the case where the symmetry group $G$ is the continuous, abelian
$U(1)$ group. In such a case, one can single out a $(D-p-1)$-form Noether
current $J_{D-p-1}$ that is conserved on-shell:222We refer to Appendix A for
the conventions on the gauge fields that we use throughout the work.

${\rm d}J_{D-p-1}=0\,.$ (2.3)

In turn, given an element $g=e^{\mathbf{i}\alpha}$, with
$\alpha\in\mathbb{R}$, one can define the following topological operator

$U_{g=e^{\mathbf{i}\alpha}}(\mathcal{M}^{(D-p-1)})=\exp\left(\mathbf{i}\alpha
Q(\mathcal{M}^{(D-p-1)})\right)=\exp\left(\mathbf{i}\alpha\int_{\mathcal{M}^{(D-p-1)}}J_{D-p-1}\right)\,,$
(2.4)

where $Q$ denotes the conserved charge. The operator (2.4) is indeed
topological due to the conservation law (2.3). The ‘ordinary’ global $U(1)$
symmetries correspond to the case $p=0$, for which the charge operators are
supported on codimension-one manifolds and the charged observables can be
viewed as localized, zero-dimensional defects.

Let us now assume that some generalized global symmetries, constituting the
group $G_{\text{\tiny EFT}}$, can be identified within an EFT that is valid up
to the energy scale cutoff $\Lambda_{\text{\tiny EFT}}$. Then, according to
the No Global Symmetry Conjecture, these symmetries ought to be ultimately
removed in a full quantum gravity theory, eventually by effects emerging at
energy scales above $\Lambda_{\text{\tiny EFT}}$. The two most common
mechanisms that could remove the global symmetries are the following:

Breaking the symmetries

The global symmetries can be _explicitly_ broken by some dynamical operators
that transform non-trivially under the symmetry group $G_{\text{\tiny EFT}}$.
As a result, the group $G_{\text{\tiny EFT}}$ is not a symmetry group of the
theory. For $G_{\text{\tiny EFT}}=U(1)$, breaking a symmetry can be achieved
via the explicit breaking of the conservation law (2.3), thus rendering the
operator (2.4) no longer topological. Examples of such operators include heavy
objects that carry a nontrivial $U(1)$ charge. It is worth noticing, however,
that the inclusion of charged operators typically reduces $G_{\text{\tiny
EFT}}$ to a discrete subgroup thereof, $G_{\text{\tiny
EFT}}^{\text{\tiny{discrete}}}$. By appropriately choosing the charge of said
operator, the global discrete subgroup can also be broken.

Gauging the symmetries

Alternatively, the global symmetry group may be gauged in the UV-completed
theory. Namely, the global symmetry group $G_{\text{\tiny EFT}}$ is promoted
to a wider local symmetry group $G_{\rm gauge}$. In the case $G_{\text{\tiny
EFT}}=U(1)$, gauging the global symmetry can be achieved by rendering the
current $J_{D-p-1}$ exact so that the operator (2.4) becomes trivial.

A simple example: the free Maxwell theory

Before closing this brief overview of generalized global symmetries, let us
consider a simple, illustrative example, namely the free Maxwell theory. We
consider the case where only a single $U(1)_{\rm g,e}$ gauge field is
dynamical. The latter is represented by a gauge one-form $A_{1}$, subjected to
the gauge transformation $A_{1}\to A_{1}+{\rm d}\Lambda_{0}$, for an arbitrary
zero-form $\Lambda_{0}$. The free Maxwell action is

$S=-\frac{1}{2g^{2}}\int F_{2}\wedge*F_{2}\,,$ (2.5)

with $F_{2}={\rm d}A_{1}$ being the field strength, and we will refer to $g$
as the electric gauge coupling.

The action (2.5) is endowed with a $U(1)_{\rm e}$ one-form global symmetry.
Such a symmetry is realized as a shift of the gauge one-form $A_{1}$ by a flat
connection, as $A_{1}\to A_{1}+\Lambda_{1}$ with ${\rm d}\Lambda_{1}=0$.
Applying Noether’s theorem yields the following two-form current and
topological operator:

$J_{2}^{\rm e}=-\frac{1}{g^{2}}*F_{2}\,,\qquad
U_{\alpha}(\Sigma_{2})=e^{\mathbf{i}\alpha Q^{\rm
e}(\Sigma_{2})}=e^{-\mathbf{i}\frac{\alpha}{g^{2}}\int_{\Sigma_{2}}*F_{2}}\,,$
(2.6)

where $\alpha\in\mathbb{R}$ and $\Sigma_{2}$ is a closed two-dimensional
manifold. The charged operators under such electric one-form global symmetry
are the _Wilson lines_

$W_{n}(\Gamma_{1})=\exp\left(\mathbf{i}n\oint_{\Gamma_{1}}A_{1}\right)\,,$
(2.7)

with $n\in\mathbb{N}$ denoting the elementary charge of the Wilson line, and
$\Gamma_{1}$ the one-dimensonal closed manifold supporting the operator. The
Wilson line transforms non-trivially under the one-form shift $A_{1}\to
A_{1}+\Lambda_{1}$ for closed $\Lambda_{1}$. In fact, $W_{n}(\Gamma_{1})$ gets
mapped to
$e^{\mathbf{i}\alpha\oint_{\Gamma_{1}}\Lambda_{1}}W_{n}(\Gamma_{1})$, with the
phase operator $e^{\mathbf{i}\alpha\oint_{\Gamma_{1}}\Lambda_{1}}$ depending
only on the homology class of $\Gamma_{1}$. The elementary charge of the
Wilson line can be measured via the relation (2.6), which reads

$U_{\alpha}(\Sigma_{2})W_{n}(\Gamma_{1})=e^{\mathbf{i}n\alpha}W_{n}(\Gamma_{1})\,,$
(2.8)

for a closed two-manifold $\Sigma_{2}$ encircling $\Gamma_{1}$.

However, the $U(1)_{\rm e}$ group is not the only one to deliver a generalized
global symmetry. In fact, the free Maxwell theory (2.5) admits a dual,
magnetic description in which the degrees of freedom are described in terms of
new gauge one-form $V_{1}$, the electromagnetic dual form of $A_{1}$. By
following the dualization procedure outlined in Section A, one can show that
the action magnetic dual to (2.5) is

$S_{\rm dual}=-\frac{g^{2}}{2}\int G_{2}\wedge*G_{2}\,,$ (2.9)

with $G_{2}={\rm d}V_{1}$. The action (2.9) is endowed with an additional
$U(1)_{\rm m}$ one-form global symmetry that is realized via shifts of the
magnetic gauge one-form $V_{1}$ by flat connections as $V_{1}\to
V_{1}+\tilde{\Lambda}_{1}$, with ${\rm d}\tilde{\Lambda}_{1}=0$. The
associated conserved two-form current and charge operator are

$J_{2}^{\rm m}=-g^{2}*G_{2}\,,\qquad U_{\beta}(\Sigma_{2})=e^{\mathbf{i}\beta
Q^{\rm m}(\Sigma_{2})}=e^{-\mathbf{i}\beta g^{2}\int_{\Sigma_{2}}*G_{2}}\,,$
(2.10)

with $\beta\in\mathbb{R}$. As outlined in Appendix A, the conservation of the
magnetic current $J_{2}^{\rm m}$ can be equivalently read from the Bianchi
identities of the electric frame, since the dualization procedure enforces the
on-shell relation $F_{2}=-g^{2}*G_{2}$. An example of observables that can be
detected in the dual frame (2.9) are the _‘t Hooft lines_

$T_{m}(\Gamma_{1})=\exp\left(\mathbf{i}m\oint_{\Gamma_{1}}V_{1}\right)\,,$
(2.11)

with $m$ denoting the elementary, magnetic charge of the operator.

Therefore, in summary, the free Maxwell theory is characterized by the global
symmetry group $G_{\text{\tiny EFT}}=U(1)_{\rm e}\times U(1)_{\rm m}$. But how
can such symmetries be removed? The simplest possibility is that they are
broken explicitly. This can be achieved via an explicit violation of the
conservation laws of both the electric and magnetic two-form currents in (2.6)
and (2.10) as

${\rm d}J^{\rm e}_{2}=-j^{\rm e}_{3}\,,\qquad{\rm d}J^{\rm m}_{2}=-j^{\rm
m}_{3}\,,$ (2.12)

where $j^{\rm e}$( $j^{\rm m}$) denotes a localized electric (magnetic) charge
distribution. For instance, in the presence of a single charged fundamental
object, such currents are of the form

$j^{\rm e}=q^{\rm e}\delta_{3}(\Gamma_{1})\,,\qquad j^{\rm m}=q^{\rm
m}\delta_{3}(\tilde{\Gamma}_{1})$ (2.13)

with $q^{\rm e}$ ($q^{\rm m}$) the electric (magnetic) elementary charge of
the object and $\delta_{3}(\Gamma_{1})$ ($\delta_{3}(\tilde{\Gamma}_{1})$) the
three-form Dirac delta distribution localized on the object’s worldline
$\Gamma_{1}$($\tilde{\Gamma}_{1}$). At the EFT level, these charged,
fundamental objects can be understood as (heavy) particles that manifest
themselves via the couplings

$S^{\rm e}=q^{\rm e}\int_{\Gamma_{1}}A_{1}=\int A_{1}\wedge j^{\rm e}\,,\qquad
S^{\rm m}=q^{\rm m}\int_{\tilde{\Gamma}_{1}}V_{1}=\int V_{1}\wedge j^{\rm m}$
(2.14)

to the actions (2.5) and (2.9) respectively.

Alternatively to their explicit breaking, the electric and magnetic one-form
global symmetries can be removed if they are gauged in the UV-completed
theory. The gauging is performed by assuming that the one-form shifts,
$A_{1}\to A_{1}+\Lambda_{1}$ and $V_{1}\to V_{1}+\tilde{\Lambda}_{1}$, leave
the theory invariant, even for non-flat one-forms $\Lambda_{1}$,
$\tilde{\Lambda}_{1}$. Clearly, under such shifts, neither of the field
strengths $F_{2}$ and $G_{2}$ are invariant. In fact, constructing a gauge
invariant EFT requires the presence of additional degrees of freedom that may
stem from the UV completion. Such degrees of freedom can be, for instance,
associated to some gauge two-forms $B_{2}$ or $\tilde{B}_{2}$, with gauge
transformations $B_{2}\to B_{2}+{\rm d}\Lambda_{1}$ and
$\tilde{B}_{2}\to\tilde{B}_{2}+{\rm d}\tilde{\Lambda}_{1}$. These would allow
for defining the gauge invariant field strengths $\hat{F}_{2}=F_{2}-B_{2}$ and
$\hat{G}_{2}=G_{2}-\tilde{B}_{2}$, which can be employed to conveniently
define the one-form kinetic terms. An example of such a gauging occurs in the
so-called $BF$-theory – see, for instance, [6].

###  2.2 The Witten effect and an infinitely populated charge lattice

As we have illustrated, breaking certain global symmetries may require the
existence of particular extended objects in the spectrum. However, whenever
the theory is endowed with additional couplings, the mere existence of said
states may give important information about the composition of the allowed
_(sub)lattice_ of states that the theory must support. In particular, if the
theory is endowed with some ‘ _$\theta$ -terms_’, whereby an axion couples to
topological terms of the form $F\wedge F$, the _Witten effect_ [26] (see also
[29, 30, 31] for later reviews) predicts the existence of an infinitely
populated lattice of states, generated by a _seed_ state.

In order to illustrate how the Witten effect is realized, let us consider a
simple model, namely the four-dimensional free Maxwell theory (2.5) with the
addition of a $\theta$-term:

$S=\int\left(-\frac{1}{2}F_{2}\wedge*F_{2}+\frac{\theta}{2}F_{2}\wedge
F_{2}\right)$ (2.15)

Owing to the quantization condition for the curvature $F_{2}$, integral
‘monodromy’ transformations $\theta\to\theta+n$ only shift the action by an
integer, as $S\to S+n$. As such, the monodromy transformation
$\theta\to\theta+n$ ought to be regarded as a perturbative symmetry of the
model, under which any correlation function is left invariant.

Clearly, if the field strength $F_{2}$ obeys the Bianchi identity ${\rm
d}F_{2}=0$, the last term in (2.15) is a topological term. Moreover, as
discussed in the previous section, $J_{2}^{\rm m}=F_{2}$ also plays the role
of a conserved current associated to a magnetic one-form global symmetry. In
order to remove such a global symmetry without introducing additional gauge
fields, one can assume that a magnetic monopole with elementary charge $q_{\rm
m}$ is present in the spectrum. As a result, the Bianchi identity for $F_{2}$
gets modified to ${\rm d}F_{2}=-q_{\rm m}\delta^{(3)}(\Gamma)$, with $\Gamma$
the monopole worldline.

Here, the Witten effect comes into play. The Witten effect [26] asserts that
the single assumption of the existence of a magnetic monopole in the spectrum
is enough to guarantee that the spectrum of states also allows for the
presence of dyons, namely particles both electrically and magnetically
charged. There are several ways to see the emergence of dyons in the spectrum.
The simplest is to notice that the action (2.15) can be recast as [32]

$S=-\frac{1}{2}\int F_{2}\wedge*F_{2}-q_{\rm m}\theta\int_{\Gamma}A_{1}\,.$
(2.16)

The last term displays the coupling of the one-form gauge field $A_{1}$ to an
electric particle with charge $q_{\rm e}=-q_{\rm m}\theta$, tracing out the
same worldline as the magnetic monopole introduced above. Moreover, due to the
perturbative symmetries of the action, _all_ the electric charges that are
obtained from the one explicitly appearing in (2.16) via integral shifts of
the parameter $\theta$ should be present. As a result, the presence of a
magnetic monopole implies an _infinite_ tower of dyons with elementary charges

${\bf q}^{T}=(q_{\rm m},-q_{\rm e})=\left(q_{\rm m},(\theta+n)q_{\rm
m}\right)\,.$ (2.17)

Moreover, due to the explicit dependence of the electric charges on the real
parameter $\theta$, in general, the magnetic and electric charges are not
mutually quantized.

Alternatively, one can see the emergence of an infinite tower of dyonic
particles by examining the currents and charges of the model. In the absence
of objects, the action (2.15) exhibits a magnetic one-form global symmetry.
Thus, there exists a conserved, gauge-invariant two-form current $J_{2}^{\rm
m}=F_{2}$. This current is however not conserved in a monopole background, for
${\rm d}J_{2}^{\rm m}=-q_{\rm m}\delta^{(3)}(\Gamma)$. It is then convenient
to introduce the modified current and charge:333In the language of the
upcoming Section 3.3, these can be identified as the _Page current_ and the
_Page charge_ of the model.

$(J_{2}^{\rm e})^{{\rm mod}}=-*F_{2}+\theta F_{2}\,,\qquad(Q^{\rm e})^{{\rm
mod}}=\int_{\Sigma_{2}}(J_{2}^{\rm e})^{{\rm mod}}\,.$ (2.18)

In the absence of electric sources, i.e. $j^{\rm e}=0$, we have ${\rm
d}(J_{2}^{\rm e})^{{\rm mod}}=0$; in turn, this leads to ${\rm
d}*F_{2}=-q_{\rm m}\theta\delta^{(3)}(\Gamma)$, consistently with (2.16).
Still, the current and charge (2.18) are not single-valued: under shifts
$\theta\to\theta+n$, they shift as

$(J_{2}^{\rm e})^{{\rm mod}}\to(J_{2}^{\rm e})^{{\rm
mod}}+nF_{2}\,,\qquad(Q^{\rm e})^{{\rm mod}}\to(Q^{\rm e})^{{\rm mod}}-nq_{\rm
m}\,.$ (2.19)

Such shifts can only be consistently justified by assuming that the
conservation of the currents is broken due to the appearance of an infinite
tower of electric states sourcing the current $j^{\rm e}$, defined in such a
way that

${\rm d}(J_{2}^{\rm e})^{{\rm mod}}=nq_{\rm m}\delta^{(3)}(\Gamma)=:-j^{\rm
e}\,.$ (2.20)

Therefore, although the avoidance of the magnetic one-form global symmetry
requires the existence of only a single magnetic monopole of elementary
charge, the Witten effect predicts that the monopole is not the sole charge
that has to be present in the spectrum. Rather, an _infinite_ tower of dyonic
states has to be supported by the theory.

##  3 Global symmetries and generalized $\theta$-terms in Type IIB EFTs

In this section we present the class of EFTs that will be the main object of
study: namely, the four-dimensional EFTs that arise from the compactification
of the ten-dimensional Type IIB string theory over a Calabi-Yau three-fold.
After a brief overview of the salient features of such EFTs and of the
multiplet sector that we focus on, we will initiate the study of the
generalized global symmetries that the 4D Type IIB EFTs may be endowed with,
and how these may be broken. We further show that the breaking of the global
symmetry triggers a generalized version of the Witten effect, and we
characterize the spectrum of states that this effect predicts. The analysis
carried out in this section is general, relying only on the $\mathcal{N}=2$
local supersymmetry that the Type IIB EFTs are equipped with.

###  3.1 $\mathcal{N}=2$ Type IIB EFTs

In this section we review some basic aspects of $\mathcal{N}=2$ EFTs
originating after compactifying Type IIB string theory over a Calabi-Yau
three-fold $Y$ [33, 34]. We will be interested in the vector multiplet sector
coupled to gravity, and we will focus on the bosonic sector only. The fields
entering the EFT are the graviton $g_{\mu\nu}$, the graviphoton gauge one-form
$A^{0}$ and $h^{2,1}$ gauge one-forms $A^{i}$, $i=1,\ldots,h^{2,1}$, which we
collectively denote $A^{I}$, $I=0,1,\ldots,h^{2,1}$, and $h^{2,1}$ complex
scalar fields $t^{i}$.

The complex scalar fields $t^{i}$ parametrize deformations of the complex
structure of $Y$. Indeed, by introducing a real integral basis of three-forms
$\gamma_{\mathcal{I}}$, $\mathcal{I}=1,\ldots,2h^{2,1}+2$ of $H^{3}(Y)$, we
can expand the Calabi-Yau holomorphic three-form as follows

$\Omega=\Pi^{\mathcal{I}}(t)\gamma_{\mathcal{I}}={\bm{\Pi}}^{T}(t){\bm{\gamma}}\,.$
(3.1)

where the functions $\Pi^{\mathcal{I}}(t)$, that are holomorphic in $t^{i}$,
are the ‘periods’ of the holomorphic three-form $\Omega$. The periods can be
further computed as $\Pi^{\mathcal{I}}(t)=\int_{\Gamma_{\mathcal{I}}}\Omega$,
given a basis $\Gamma_{\mathcal{I}}$ of three-cycles of $H_{3}$ such that
$\int_{\Gamma_{\mathcal{I}}}\gamma_{\mathcal{J}}=\delta_{\mathcal{J}}^{\mathcal{I}}$.

For the discussion in the upcoming sections, it is convenient to further
specialize the choice of basis of cycles $\gamma_{\mathcal{I}}$. We introduce
a symplectic basis of three-cycles $\Gamma^{\mathcal{I}}=(A^{I},B_{J})$ of $Y$
and, accordingly, a basis of dual three-forms
$\gamma_{\mathcal{I}}=(\alpha_{I},\beta^{J})$, with $I,J=1,\ldots,h^{2,1}+1$
defined in such a way that the intersection matrix $\eta$ has the standard
form

$\eta=\begin{pmatrix}\int\alpha_{I}\wedge\alpha_{J}&\int\alpha_{I}\wedge\beta^{J}\\\
\int\beta^{I}\wedge\alpha_{J}&\int\beta^{I}\wedge\beta^{J}\end{pmatrix}=\begin{pmatrix}0&\mathds{1}\\\
-\mathds{1}&0\end{pmatrix}$ (3.2)

expressed in $(h^{2,1}+1)\times(h^{2,1}+1)$ blocks.

We can then expand the holomorphic three-form $\Omega$ in the symplectic basis
as

${\bm{\Pi}}(t)=\begin{pmatrix}\int_{B_{I}}\Omega\\\
-\int_{A^{I}}\Omega\end{pmatrix}=\begin{pmatrix}X^{I}(t)\\\
-\mathcal{F}_{I}(t)\end{pmatrix}\,,$ (3.3)

with $X^{I}(t)$ and $\mathcal{F}_{I}(t)$ holomorphic functions of the complex
structure moduli $t^{i}$.

The four-dimensional $\mathcal{N}=2$ effective action describing the
interactions among the bosonic components of the gravity multiplet and the
$h^{2,1}$ vector multiplets is

$S=\int\left(\frac{1}{2}M^{2}_{\rm P}R*1-M^{2}_{\rm P}K_{i\bar{\jmath}}^{\rm
cs}{\rm d}t^{i}\wedge*{\rm d}\bar{t}^{\bar{\jmath}}+\frac{1}{2}{\rm
Im}\mathcal{N}_{IJ}F^{I}\wedge*F^{J}+\frac{1}{2}{\rm
Re}\mathcal{N}_{IJ}F^{I}\wedge F^{J}\right)\,.$ (3.4)

Here we have denoted with $R$ the Ricci scalar and $K_{i\bar{\jmath}}^{\rm
cs}=\partial_{t^{i}}\partial_{\bar{t}^{\bar{\jmath}}}K^{\rm cs}$, with
$\partial_{t^{i}}:=\frac{\partial}{\partial t^{i}}$, is the Kähler metric that
can be obtained out of the Kähler potential $K^{\rm cs}$. The latter is fully
specified by the periods as

$K^{\rm
cs}(t,\bar{t})=-\log\left(\mathbf{i}\int_{Y}\Omega\wedge\bar{\Omega}\right)=-\log\mathbf{i}\,{\bm{\Pi}}^{T}\eta\bar{\bm{\Pi}}=-\log\mathbf{i}(\bar{X}^{I}\mathcal{F}_{I}-X^{I}\bar{\mathcal{F}}_{I})\,.$
(3.5)

The interactions of the gauge one-forms $A^{I}$ are dictated by the complex
matrix $\mathcal{N}_{IJ}$ computed as follows

$\mathcal{N}_{IJ}=(\mathcal{F}_{I}\quad
D_{\bar{\imath}}\bar{\mathcal{F}}_{\bar{I}})(X^{J}\quad
D_{\bar{\imath}}\bar{X}^{\bar{J}})^{-1}\,.$ (3.6)

Whenever the matrix $(D_{i}X^{I},X^{I})$ is invertible, the action (3.4) is
fully specified by a _prepotential_ $\mathcal{F}(X)$, which is a homogeneous
function of degree two in the projective coordinates $X^{I}$. Then, the
quantities $\mathcal{F}_{I}$ that appear in (3.3) can be regarded as
derivatives of the prepotential
$\mathcal{F}_{I}=\frac{\partial\mathcal{F}}{\partial X^{I}}$. Moreover, when a
prepotential exists, one can recast the matrix $\mathcal{N}_{IJ}$ in (3.6) in
the more explicit form

$\mathcal{N}_{IJ}=\bar{\mathcal{F}}_{IJ}+2\mathbf{i}\frac{{\rm
Im}\mathcal{F}_{IK}X^{K}{\rm Im}\mathcal{F}_{JL}X^{L}}{{\rm
Im}\mathcal{F}_{MN}X^{M}X^{N}}\,.$ (3.7)

A key ingredient of the analysis that follows is given by the examination of
the equations of motion of the fields entering the effective description
(3.4). The equations of motion for the gauge fields $A^{I}$ are:

${\rm d}G_{I}=0\,,$ (3.8)

where we have defined

$G_{I}\equiv\frac{\delta S}{\delta F_{\mu\nu}^{I}}{\rm d}x^{\mu}\wedge{\rm
d}x^{\nu}={\rm Im}\mathcal{N}_{IJ}*\\!F^{J}+{\rm Re}\mathcal{N}_{IJ}F^{J}\,,$
(3.9)

to which we will oftentimes refer as the field strengths _dual_ to $F^{I}$.
The equations of motion (3.7) are additionally supported by the Bianchi
identities ${\rm d}F^{I}=0$.

The equations of motion for the complex structure moduli $t^{i}$ are

$M_{\rm P}^{2}\left[{\rm d}(K_{i\bar{\jmath}}^{\rm cs}*{\rm
d}\bar{t}^{\bar{\jmath}})-K_{il\bar{\jmath}}^{\rm cs}{\rm d}t^{l}\wedge*{\rm
d}\bar{t}^{\bar{\jmath}}\right]=-\frac{1}{2}\frac{\partial}{\partial
t^{i}}{\rm
Im}\mathcal{N}_{IJ}F^{I}\wedge*F^{J}-\frac{1}{2}\frac{\partial}{\partial
t^{i}}{\rm Re}\mathcal{N}_{IJ}F^{I}\wedge F^{J}\,.$ (3.10)

Additionally, for the following discussion, it will be convenient to consider
the special case in which the Kähler metric is real, and it is only saxion
dependent; namely $K_{i\bar{\jmath}}(t,\bar{t})=:K_{ij}(s)$, so that the
general action (3.4) reduces to

$\displaystyle S$ $\displaystyle=\int\Big{(}\frac{1}{2}M^{2}_{\rm
P}R*1-M^{2}_{\rm P}K_{ij}^{\rm cs}({\rm d}a^{i}\wedge*{\rm d}a^{j}+{\rm
d}s^{i}\wedge*{\rm d}s^{j})+\frac{1}{2}G_{I}\wedge F^{I}\Big{)}\,,$ (3.11)

and the equations of motion for the saxions $s^{i}$ and the axions $a^{i}$
are, respectively

$\displaystyle M_{\rm P}^{2}\left[{\rm d}(K_{ij}^{\rm cs}*{\rm
d}\bar{t}^{\bar{\jmath}})-\frac{\partial}{\partial s^{i}}K_{lj}^{\rm cs}({\rm
d}a^{l}\wedge*{\rm d}a^{j}+{\rm d}s^{l}\wedge*{\rm
d}s^{j})\right]=-\frac{1}{2}\frac{\partial}{\partial s^{i}}G_{I}\wedge
F^{I}\,,$ (3.12a) $\displaystyle M_{\rm P}^{2}\,{\rm d}(K_{ij}^{\rm cs}*{\rm
d}a^{j})=-\frac{1}{2}\frac{\partial}{\partial a^{i}}G_{I}\wedge F^{I}\,.$
(3.12b)

As of now, the analysis of the Type IIB EFTs has been focused on the closed
string sector only. However, the four-dimensional Type IIB EFTs can be
populated by a plethora of objects that originate from higher-dimensional
$p$-branes. For the upcoming discussion, we will be interested in the coupling
of the vector multiplet sector to BPS particles that are electrically charged
under the one-form gauge fields $A^{I}$ or their dual gauge fields. Such
particles stem from D3-branes that wrap internal special Lagrangian three-
cycles. The mass of the a D3-particle obtained wrapping the D3-brane on the
special Lagrangian $\Sigma_{\bf q}\in H^{3}(Y,\mathbb{Z})$ is given by the
central charge $\mathcal{Z}_{\bf q}$ as [35, 36]:

$M_{\bf q}=|\mathcal{Z}_{\bf q}|=e^{\frac{K^{\rm
cs}}{2}}\left|\int_{\Sigma_{\bf q}}\Omega\right|=e^{\frac{K^{\rm
cs}}{2}}\left|\int_{Y}q\wedge\Omega\right|\,.$ (3.13)

Here we have denoted with ${\bf q}$ the _elementary charges_ of the particle,
whereas $q$ denotes the three-form Poincaré dual to the three-cycle
$\Gamma_{\bf q}$. It is convenient to expand
$q=\alpha_{I}p^{I}-q_{I}\beta^{I}$, or equivalently the charge vector ${\bf
q}$ as

${\bf q}=\begin{pmatrix}p^{I}\\\ -q_{I}\end{pmatrix}\,,\qquad
q_{I},p^{I}\in\mathbb{Z}\,.$ (3.14)

We will refer to $q_{I}$ as the elementary _electric_ charges and $p^{I}$ as
the elementary _magnetic_ charges of the D3-brane. With such a decomposition,
the D3-particle mass (3.13) can be rewritten as

$M_{\bf q}=e^{\frac{K^{\rm cs}}{2}}|{\bf q}^{T}\,\eta\,{\bm{\Pi}}(t)|\,.$
(3.15)

In addition to the elementary charges, one can define the _physical charge_ of
the D3-particle as follows

$\mathcal{Q}^{2}_{\bf q}=\frac{1}{2}\int_{Y}q\wedge\star q=-\frac{1}{2}{\bf
q}^{T}\mathcal{T}{\bf q}\,,$ (3.16)

where

$\mathcal{T}=\begin{pmatrix}{\rm Im}\mathcal{N}+{\rm Re}\mathcal{N}({\rm
Im}\mathcal{N})^{-1}{\rm Re}\mathcal{N}&{\rm Re}\mathcal{N}({\rm
Im}\mathcal{N})^{-1}\\\ ({\rm Im}\mathcal{N})^{-1}{\rm Re}\mathcal{N}&({\rm
Im}\mathcal{N})^{-1}\end{pmatrix}\,,$ (3.17)

that is expressed in the symplectic basis $\\{\alpha_{I},\beta^{I}\\}$. The
physical charge in (3.16) carries information about the gauge couplings
associated to the $U(1)_{\rm g}$ abelian gauge one-forms $A^{I}$.

In the following, we will regard the D3-particles just introduced as
_fundamental_ objects. Namely, we shall assume that the EFT cutoff
$\Lambda_{\text{\tiny EFT}}$ is chosen in such a way that
$\Lambda_{\text{\tiny EFT}}\lesssim M_{\bf q}$, thus that additional brane
worldvolume degrees of freedom can be neglected within the effective
description. Under such an assumption, electrically charged D3-particles with
elementary charges $q_{I}$ can be coupled to the bulk action (3.4) via the BPS
action

$S_{\text{D3},{\bf q}_{\rm e}}=-\int_{\Gamma}{\rm d}\tau\,M_{\bf
q}+q_{I}\int_{\Gamma}\,A^{I}$ (3.18)

where $\tau$ the proper time parametrizing the particle worldline $\Gamma$.
Magnetically charged particles with elementary charges $p^{I}$ cannot be
directly coupled to the EFT action (3.4), but rather to its electromagnetic
dual action that is computed below (see (3.29)).

###  3.2 The symplectic structure of Type IIB EFTs

Within the $\mathcal{N}=2$ supergravity EFTs introduced in the previous
section, the complex structure fields $t^{i}$ span a special Kähler manifold
that is endowed with a flat symplectic bundle (see, for instance, [37, 38]).
This section is devoted to the review of some basic consequences of the
symplectic structure of the $\mathcal{N}=2$ supergravity EFTs. The knowledge
of how the ingredients of $\mathcal{N}=2$ EFTs transform under the symplectic
group will be crucial for the upcoming sections.

Preliminarily, let us recall that a matrix $\mathcal{S}\in{\rm
GL}(2h^{2,1}+2,\mathbb{R})$ is symplectic if
$\mathcal{S}^{T}\eta\mathcal{S}=\eta$, with $\eta$ defined in (3.2). In order
to better characterize the structure of any symplectic $\mathcal{S}$ matrices,
it is convenient to split the matrix $\mathcal{S}$ into
$(h^{2,1}+1)\times(h^{2,1}+1)$ square matrices as follows

$\mathcal{S}=\begin{pmatrix}A^{I}{}_{J}&B^{IJ}\\\
C_{IJ}&D_{I}{}^{J}\end{pmatrix}\,,$ (3.19)

Then, a symplectic matrix $\mathcal{S}$ obey the conditions

$A^{T}D-C^{T}B=AD^{T}-BC^{T}=\mathds{1}\,,\qquad
AB^{T},CD^{T},A^{T}C,B^{T}D\quad\text{symmetric}\,,$ (3.20)

where here, and in the following, we suppress the index structure. Moreover,
the inverse of any symplectic matrix $\mathcal{S}$ is given by
$\mathcal{S}^{-1}=-\eta\mathcal{S}^{T}\eta$.

Let us now focus on the scalar sector of the Type IIB EFTs. The periods (3.3)
transform as sections over the symplectic bundle

${\bm{\Pi}}(t)\rightarrow{\bm{\Pi}}^{\prime}(t)=\mathcal{S}{\bm{\Pi}}(t)\,.$
(3.21)

It can be easily checked that any symplectic transformation of the periods
leaves the complex structure Kähler potential (3.5) invariant. In fact,
consistently transforming the symplectic basis
$\gamma_{\mathcal{I}}=(\alpha_{I},\beta^{J})$ leaves (3.1)
invariant.444Specifically, $(-\beta^{I},\alpha_{I})$ transforms as
$(-\beta^{I},\alpha_{I})\to(-\beta^{I},\alpha_{I})\mathcal{S}^{T}$. It is
straightforward to check that such a transformation leaves also (3.2)
invariant.

However, the symplectic structure of the $\mathcal{N}=2$ supergravity plays a
crucial role in the study of the gauge sector. As shown by Gaillard and Zumino
in the seminal work [39] (see also [37, 38, 40] for later application to
$\mathcal{N}=2$ supergravities), symplectic transformations can be understood
as _dualities_ of the action (3.4). With _duality_ we understand a linear
transformation of the field strengths and the matrix $\mathcal{N}$ determining
the gauge kinetic matrix and Chern-Simons couplings such that the first
relation in (3.9) is left invariant _in form_. In order to see this, let us
first collect the field strengths $F^{I}$ and their duals $G_{I}$ defined in
(3.9) into the single vector

${\bf F}=\begin{pmatrix}F^{I}\\\ -G_{I}\end{pmatrix}\,.$ (3.22)

Under a duality transformation, the field strengths ${\bf F}$ transform as a
section of a flat symplectic bundle that the special Kähler geometry supports
as ${\bf F}\to{\bf F}^{\prime}=\mathcal{S}{\bf F}$. Additionally, the matrix
$\mathcal{N}_{IJ}$ in (3.6) accordingly transforms as

$\mathcal{N}\rightarrow(C+D\mathcal{N})(A+B\mathcal{N})^{-1}\,.$ (3.23)

The general action (3.4) is not invariant under general duality
transformations; indeed, it gets transformed as

$S\rightarrow S+\frac{1}{2}\int\left(2GB^{T}CF-FC^{T}AF-GD^{T}BG\right)\,.$
(3.24)

As of now, we have considered the symplectic group as defined over the real
field. However, as we will see momentarily, dualities do also act on the
D3-particle charge vectors (3.14). This feature can be employed as a
constraint on the allowed symplectic transformations. In fact, let us assume
that the D3-brane charges are quantized in a given frame. Then, requiring that
the D3-brane charges are mutually quantized after performing a duality
transformation requires that the symplectic matrix that generates the duality
displays integral entries. Therefore, from now on, we shall always consider
dualities as induced by matrices $\mathcal{S}\in{\rm
Sp}(2h^{2,1}+2,\mathbb{Z})$.

In the following, we will be largely interested in two families of symplectic
transformations:

Monodromies. Monodromy transformations constitute the subset of the symplectic
transformations that are induced by an integral shift of the axionic fields.
These can be most readily computed by looking at how the period vectors (3.3)
transform under such shifts. Namely, we identify the monodromy matrix
$T_{i}\in{\rm Sp}(2h^{2,1}+2,\mathbb{Z})$ by how the periods (3.3) transform
under the sole shift $a^{i}\to a^{i}+1$ as:555As we shall see below, such
discrete shifts can be seen as the gauge subgroup of the general shift
symmetries $a^{i}\to a^{i}+c$, with $c\in\mathbb{R}$.

${\bm{\Pi}}(a,s)\xrightarrow{a^{i}\to
a^{i}+1}{\bm{\Pi}}^{\prime}(a,s)=:T_{i}{\bm{\Pi}}(a,s)\,.$ (3.25)

Clearly, under $k$ shifts of the axions $a^{i}$, the periods get mapped as
${\bm{\Pi}}(a,s)\to T_{i}^{k}{\bm{\Pi}}(a,s)$. Although we shall give a
detailed characterization of the properties of the monodromy matrices that
appear in the near-boundary EFTs in Section 4, for the following discussion it
suffices to introduce the following matrix

$N_{i}=\log T_{i}\,,$ (3.26)

to which we will refer as _log-monodromy matrix_. In the near-boundary EFTs
studied in Section 4 such a matrix is always well-defined. Assuming that the
periods are expressed in special coordinates, i.e.
${\bm{\Pi}}(t)=(1,t^{i},\mathcal{F}_{I}(t))$, any monodromy matrix $T_{i}$
acquires the form

$T_{i}=\begin{pmatrix}A&0\\\ C&D\end{pmatrix}\,,\qquad\text{with}\quad
D=(A^{T})^{-1}\,.$ (3.27)

As inferred from (3.24), under a generic monodromy transformation (3.27), the
action (3.4) is not invariant. However, the lack of invariance is due to the
integral shift of the action

$S\to S-\frac{1}{2}\int FC^{T}AF\,,\qquad\frac{1}{2}\int
FC^{T}AF\in\mathbb{Z}\,.$ (3.28)

Consequently, any partition function is expected to be invariant under generic
monodromy transformations. Thus, despite not being classical symmetries of the
action (3.4), monodromies can be regarded as _pertubative_ symmetries of the
model.

Electromagnetic duality. We regard as _electromagnetic duality_ the symplectic
transformation induced by the matrix $\eta$ defined in (3.2). Such a duality
transformation exchanges the role of field strengths, mapping $F^{I}\to-
G_{I}$, $G_{I}\to F^{I}$, and consistently mapping
$\mathcal{N}\to-\mathcal{N}^{-1}$. Consequently, the action (3.4) gets mapped
to

$\displaystyle S^{\rm dual}=\int\Big{(}$ $\displaystyle\frac{1}{2}M^{2}_{\rm
P}R*1-M^{2}_{\rm P}K_{i\bar{\jmath}}^{\rm cs}{\rm d}t^{i}\wedge*{\rm
d}\bar{t}^{\bar{\jmath}}$ (3.29) $\displaystyle-\frac{1}{2}{\rm
Im}(\mathcal{N}^{-1})^{IJ}G_{I}\wedge*G_{J}-\frac{1}{2}{\rm
Re}(\mathcal{N}^{-1})^{IJ}G_{I}\wedge G_{J}\Big{)}\,,$

with

$\displaystyle\mathcal{N}^{-1}$ $\displaystyle=[({\rm
Re}\,\mathcal{N})^{-1}{\rm Im}\,\mathcal{N}+({\rm Im}\,\mathcal{N})^{-1}{\rm
Re}\,\mathcal{N}]^{-1}[({\rm Im}\,\mathcal{N})^{-1}-\mathbf{i}({\rm
Re}\,\mathcal{N})^{-1}]$ (3.30) $\displaystyle=[\mathds{1}+({\rm
Im}\,\mathcal{N})^{-1}{\rm Re}\,\mathcal{N}({\rm Im}\,\mathcal{N})^{-1}{\rm
Re}\,\mathcal{N}]^{-1}[({\rm Im}\,\mathcal{N})^{-1}{\rm Re}\,\mathcal{N}({\rm
Im}\,\mathcal{N})^{-1}-\mathbf{i}({\rm Im}\,\mathcal{N})^{-1}]$

In this frame, the elementary degrees of freedom may be understood as being
carried by the field strengths $G_{I}$. We can then set $G_{I}={\rm d}V_{I}$
for some gauge one-forms $V_{I}$. Moreover, as is clear from (3.29), the role
of electric and magnetic gauge couplings is reversed in passing from (3.4) to
(3.29). For this reason, we will refer to (3.29) as the _electromagnetic
action dual_ to (3.4). 666Such dualities that relate strongly and weakly
coupled regimes of a theory were dubbed $U$-dualities in [41] in analogy to
those studied in [35]. In the following, we will refer to the frame where the
Type IIB EFTs action acquires the form (3.4) as the _electric frame_ ,
understanding that the electric gauge couplings $g^{2}_{{\rm
e},I}(t)=\mathcal{Q}^{2}_{{\bf q}_{\rm e}^{(I)}}$ are small, namely
$g^{2}_{{\rm el},I}(t)\lesssim 1$; instead, we will refer to the frame where
the action is (3.29) as the _magnetic_ one, with gauge couplings $g^{2}_{{\rm
m},I}(t)\gtrsim 1$.

###  3.3 Generalized global symmetries of Type IIB EFTs

In this section, we examine the generalized global symmetries that
characterize the $\mathcal{N}=2$ Type IIB EFTs reviewed in Section 3.1. The
philosophy that we follow is the one outlined earlier in Section 2.1: within
the Type IIB EFTs, we first single out the conserved currents associated to
the global symmetries that the EFTs exhibit and then inquire whether one can
identify well-defined topological operators – i.e. the charge operators – for
said global symmetries.

To begin with, we investigate the global symmetries that are associated to the
gauge sector of $\mathcal{N}=2$ Type IIB EFTs.

Electromagnetic one-form global symmetries. As for the simple free Maxwell
theory examined in Section 2.1, the $\mathcal{N}=2$ Type IIB EFTs may be
endowed with abelian $U(1)$ one-form global symmetries. Indeed, the electric-
frame action (3.4) is trivially invariant under shifts of the gauge one-forms
$A^{I}$ by flat connections, namely $A^{I}\to A^{I}+\Lambda^{I}_{\rm e}$ with
${\rm d}\Lambda^{I}_{\rm e}=0$. Analogously, the magnetic-frame action (3.29)
is invariant under shifts of the magnetic gauge one-forms $V_{I}$, as
$V_{I}\to V_{I}+\Lambda_{I}^{\rm m}$ with ${\rm d}\Lambda_{I}^{\rm m}=0$.
Therefore, the (maximal) global one-form symmetry group is

$G_{1}=U(1)^{h^{2,1}+1}_{\rm e}\times U(1)^{h^{2,1}+1}_{\rm m}\,,$ (3.31)

where, as in Section 2.1, we have introduced the subscripts to distinguish
whether the symmetry subgroup is associated to an electric or a magnetic one-
form symmetry.

In order to tell whether the group (3.31) is broken to a subgroup within the
Type IIB EFTs, we need to examine the structure of the associated currents.
From the general discussion of Section 2, the maximal global symmetry group
(3.31) may lead to, at most, $b_{3}(Y)=2h^{2,1}+2$ two-form currents. However,
the simultaneous presence of multiple $p$-forms, namely the gauge one-forms
$A^{I}$ and the axion zero-forms $a^{i}$, renders the identification of well-
defined currents rather subtle. Indeed, as recognized in [42], one can
identify _three_ different kinds of currents that in [42] were referred to as
the _Page_ , the _Maxwell_ and the _brane currents_. These definitions of
currents were further employed in [9, 28], since they provide a neat
classification of the available currents when the theory is endowed with
couplings that involve several field strengths of the gauge fields. In the
following, we show how such three notions of charges appear in the Type IIB
EFTs under examination, and illustrate their key properties.

The simplest choice of currents is the one obtained by the straightforward
application of Noether’s first theorem for the group (3.31). Applied to the
electric one-form symmetries, starting from the action (3.4), Noether’s first
theorem leads to the $h^{2,1}+1$ electric two-form currents

$(J_{I}^{\text{e}})^{(P)}=G_{I}={\rm Im}\mathcal{N}_{IJ}*\\!F^{J}+{\rm
Re}\mathcal{N}_{IJ}F^{J}\,.$ (3.32)

Instead, applying Noether’s theorem to the magnetic subgroup of (3.31)
delivers the magnetic two-form currents

$(J^{I}_{\text{m}})^{(P)}=F^{I}\,,$ (3.33)

as can be straightforwardly checked from the dual action (3.29), with the help
of (3.30). We can collect the two-form currents in the vector
${\bm{J}}^{(P)}=((J^{I}_{\text{m}})^{(P)},-(J_{I}^{\text{e}})^{(P)})^{T}$.
From the equations of motion for the electric one-forms $A^{I}$, and their
Bianchi identities, the currents so defined are conserved on-shell:

${\rm d}{\bm{J}}^{(P)}=0\,.$ (3.34)

Are the currents ${\bm{J}}^{(P)}$ _good_ currents that lead to well-defined
charge operators? In general, the currents ${\bm{J}}^{(P)}$ suffer from a
pathology: they are not single-valued. In fact, recalling the discussion in
Section 3.2, the currents ${\bm{J}}^{(P)}$ coincides with the field strength
vector (3.22). Therefore, the current vector ${\bm{J}}^{(P)}$ is a section of
the flat symplectic bundle $Sp(2h^{2,1}+2,\mathbb{Z})$, and thus it does
transform under general symplectic transformations. In particular, the
currents change also under isometries of the scalar manifold, such as the
monodromy transformations introduced in Section 3.2 as ${\bm{J}}^{(P)}\to
T{\bm{J}}^{(P)}$. Therefore, though conserved, the currents ${\bm{J}}^{(P)}$,
are _not_ generically gauge invariant. For these reasons, following [42], we
will refer to the currents ${\bm{J}}^{(P)}$ as the _Page currents_. In the
concrete examples that we will consider in Sections 4 we will see that this is
partly due to the explicit axion dependence of the currents (3.32) on the
axion fields entering the EFT.

Clearly, such an ambiguity in the definition of the currents is also reflected
in the charge operators. Given a closed two-dimensional surface $\Sigma_{2}$,
one can define the Page charge operators

${\bm{Q}}^{(P)}=\int_{\Sigma_{2}}{\bm{J}}^{(P)}\,.$ (3.35)

As we will motivate shortly, the Page charges are localized charges and are
expected to be quantized [42]. For fixed scalar fields $t^{i}=t^{i}_{0}$ each
charge operator within ${\bm{Q}}^{(P)}$ is topological. However, after the
axionic shifts $a^{i}\to a^{i}+1$, the charge operators nontrivially transform
as ${\bm{Q}}^{(P)}\to T{\bm{Q}}^{(P)}$.

It is worth remarking that whether _all_ or _some_ of the Page currents
${\bm{J}}^{(P)}$ (or the corresponding Page charges ${\bm{Q}}^{(P)}$) are not
single-valued depends on the explicit form of the monodromy transformation
$T$. Hence, among the Page currents or charges, there could be some that are
both conserved and single-valued. These ought to be considered as stemming
from genuine global symmetries of the model and, by the No Global Symmetry
Conjecture, should be removed in a full quantum gravity theory. We will
comment on how to consistently break these symmetries later in this section.

Since the Page currents and charges are generically ill-defined, one could
wonder whether it is possible to define some alternative single-valued
conserved currents. Single-valued currents and charges can be most readily
defined out of the Page currents and charges as

${\bm{J}}^{(M)}=e^{-a^{i}N_{i}}{\bm{J}}^{(P)}\,,\qquad{\bm{Q}}^{(M)}=\int_{\Sigma_{2}}{\bm{J}}^{(M)}\,,$
(3.36)

where $N_{i}$ are the matrices defined in (3.26). We will refer to (3.36) as
_Maxwell currents_ and _Maxwell charges_ , respectively. In fact, under the
integral axion shift $a^{i}\to a^{i}+1$, the prefactor in the definition of
the above currents gets mapped according to $e^{-a^{i}N_{i}}\to
e^{-a^{i}N_{i}}T^{-1}$, undoing the monodromy transformation of the Page
currents ${\bm{J}}^{(P)}$ and rendering (3.36) single-valued. However, the
Maxwell currents do suffer from a novel issue: in a background with dynamical
axions, they are not generically conserved, for in general ${\rm
d}{\bm{J}}^{(M)}\neq 0$ and the Maxwell charges ${\bm{Q}}^{(M)}$ are not
topological operators.777Notice that in [42], Maxwell currents are defined as
non-localized, gauge-invariant conserved currents. Here we adopt the weaker
definition of Maxwell charges used in [9], where Maxwell currents are gauge-
invariant but not generically conserved.

To summarize, in a dynamical axion background, the sole well-defined global
symmetries are those which deliver currents that are simultaneously conserved,
gauge-invariant and single-valued. In order to break the global symmetries at
the EFT level, we can employ the last type of currents, the _brane currents_.
Within the $\mathcal{N}=2$ Type IIB EFTs described by the action (3.4), the
brane currents can originate from two sources. The first kind of brane current
stems from the direct coupling of the Type IIB EFTs (3.4) with the electric
D3-particles effectively described by the action (3.18). In the presence of
$N_{\rm e}$ electrically charged D3-particles, the on-shell current
conservation rules for the Page electric currents (3.32) get modified to

${\rm d}(J_{I}^{\text{e}})^{(P)}=-\sum\limits_{\alpha=1}^{N_{\rm
e}}q_{I}^{\alpha}\delta^{(3)}(\Gamma_{\alpha})=:-j_{I}^{\text{e}}$ (3.37)

where $q_{I}^{\alpha}$ denotes the elementary electric charges of the
D3-particle drawing the worldline $\Gamma_{\alpha}$. Additional localized
sources can be included in order to break the Bianchi identities for the field
strengths $F^{I}$ as

${\rm d}F^{I}={\rm d}(J^{I}_{\text{m}})^{(P)}=-\sum\limits_{\beta=1}^{N_{\rm
m}}p^{I}_{\beta}\delta^{(3)}(\Gamma^{\beta})=:-j^{I}_{\text{m}}$ (3.38)

These localized sources can be understood as stemming from $N_{\rm m}$
magnetic D3-particles, with elementary charges $p^{I}_{\beta}$ and worldlines
$\Gamma^{\beta}$, coupled to the dual effective action (3.29). The relations
(3.37) and (3.38) can be more compactly written as

${\rm
d}{\bm{J}}^{(P)}=-{\bm{j}}\,,\qquad{\bm{j}}:=(j^{I}_{\text{m}},-j_{I}^{\text{e}})\,,$
(3.39)

and we will refer to ${\bm{j}}$ as _brane currents_. These currents are
localized by assumption, and consequently, the Page charges in (3.35) are
localized. Breaking the conservation rule for the currents explicitly, the
brane currents are the prime candidates that break any residual
electromagnetic one-form global symmetry. It is worth noting, however, that
generic brane currents may break the continuous global symmetries down to a
discrete subgroup thereof; choosing the brane currents as composed of unit
charges breaks the continuous group entirely.

One could then inquire what is the minimal choice of brane currents that could
lead to the breaking of a global symmetry. Specifically, let us assume that a
brane with unit charge is included so as to break the conservation rule of one
of the Page currents: is this configuration consistent? As mentioned earlier,
the Page current ${\bm{J}}^{(P)}$ is not single-valued under monodromy
transformations. For instance, shifting every axion as $a^{i}\to a^{i}+1$,
(3.39) maps to ${\rm d}{\bm{J}}^{(P)}=-T{\bm{j}}$. In other words, the brane
currents are also required to transform under generic monodromies. This hints
at the fact that the inclusion of single brane current ${\bm{j}}$ is
generically inconsistent, for also the currents $T{\bm{j}}$, $T^{2}{\bm{j}}$,
…should be present in the EFT. This phenomenon has profound phenomenological
which will be examined in the following section.

Having exhausted the discussion of the electromagnetic one-form global
symmetries, let us pass to the global symmetries that can stem from the scalar
sector of the $\mathcal{N}=2$ Type IIB EFTs.

Axionic zero-form global symmetries. Let us assume that the Kähler metric is
axion-independent, so that the $\mathcal{N}=2$ Type IIB EFTs acquires the form
(3.11). The scalar kinetic terms of the action (3.11) are then ostensibly
invariant under any shift of the axion fields $a^{i}\to a^{i}+c^{i}$, with
$c_{i}\in\mathbb{R}$. We further assume that the axions $a^{i}$ span a compact
domain, namely circles of unit circumference, so that we may regard
$c_{i}\in\mathbb{R}/\mathbb{Z}$. Then, the axion fields can be understood as
zero-form gauge fields, and they can be responsible for zero-form global
symmetries, that act on the axion via said shift symmetry. The maximal abelian
global symmetry group that can be associated to such zero-form global
symmetries is then:

$G_{0}=U(1)^{h^{2,1}}_{a}\,.$ (3.40)

The following, gauge-invariant Maxwell currents can be defined:

$J_{a,i}^{(M)}=M_{\rm P}^{2}K_{ij}^{\rm cs}*{\rm d}a^{j}\,.$ (3.41)

However, in general, these currents are not conserved due to the explicit
dependence of the gauge interactions on the axion fields $a^{i}$:

${\rm d}J_{a,i}^{(M)}=-\frac{1}{2}\frac{\partial}{\partial a^{i}}G_{I}\wedge
F^{I}\,.$ (3.42)

If ${\rm d}J_{a,i}^{(M)}=0$ for some $i$, then the zero-form global symmetry
is a genuine global symmetry of the EFT. As we show in details in Appendix B,
we should expect that these continuous axion shift symmetries are at most
approximate in the EFT, for they may be broken to a discrete subgroup by some
‘essential’ instanton effects. Alternatively, if those essential instantons
are absent, the axion zero-form global symmetry can be explicitly broken by
including some brane currents composed by $n_{i}$ fundamental instantons as:

${\rm d}J_{a,i}^{(M)}=-j_{a,i}\,,\qquad
j_{a,i}=\sum\limits_{\alpha=1}^{n_{i}}m_{\alpha}\delta^{(4)}(x^{\alpha}_{0})\,,$
(3.43)

where we have denoted with $m_{\alpha}$ the elementary charges of the
fundamental instanton localized at the spacetime point $x^{\alpha}_{0}$.

Two-form global symmetries. Before concluding this section, let us briefly
comment on one last kind of generalized global symmetries that the Type IIB
EFTs may be endowed with and should be thus removed. These are the two-form
generalized global symmetries that can be understood as the electromagnetic
duals of the axionic zero-form global symmetries discussed above. They are
associated to the one-form currents $J_{B}^{i}={\rm d}a^{i}$, which are
trivially conserved. In order to understand why such symmetries may be also
present in the models at hand, let us first consider the simple case in which
the EFT is endowed with an axionic zero-form global symmetry that shifts the
axion $a^{i}\to a^{i}+c$, for some $c\in\mathbb{R}$ – namely, the Kähler
metric is independent of the axion $a^{i}$ and the axion $a^{i}$ does not
enter any generalized $\theta$-term. Then, one could perform a dualization
procedure as outlined in Appendix A, replacing the axion $a^{i}$ with its
dual, two-form gauge field $B_{2,i}$ (see also [22, 43, 44] for details about
such a dualization procedure, and how this can be performed in a manifestly
supersymmetric way). Then, proceeding as in Section 2.1 for the one-form
global symmetries, one could obtain the conserved one-form currents
$J_{B}^{i}={\rm d}a^{i}$ by applying the standard Noether procedure.

However, in the more general Type IIB EFTs axions participate in the
generalized $\theta$-terms, and the associated continuous shift symmetries are
broken down to discrete ones. Consequently, as also highlighted in the recent
[45], the standard dualization procedure summarized in Appendix A cannot be
generally performed, and the Noether procedure cannot be straightforwardly
applied in order to compute the eventual one-form currents. Nevertheless, it
is worth remarking that the one-form currents $J_{B}^{i}={\rm d}a^{i}$ still
appear to be conserved, and well-defined – see [46, 47]. This could signal
that the Type IIB EFTs that we are considering are endowed with two-form
global symmetries, which might be broken by including strings with elementary
charges. However, in the presence of $\theta$-terms, the inclusion of
fundamental strings comes at a price: as pointed out in [47], fundamental
strings can only be consistently coupled provided that they carry some
additional string worldsheet degrees of freedom, whose existence is hard to
justify from a top-down perspective (see also [23] for a recent discussion in
the context of $\mathcal{N}=2$ Type IIA EFTs). We leave a better understanding
of how these two-form global symmetries are realized in Type IIB EFTs, and how
they can be avoided for future investigation.

###  3.4 A generalized Witten effect for Type IIB EFTs

The four-dimensional $\mathcal{N}=2$ Type IIB EFTs are plagued by the presence
of various topological Chern-Simons terms, of the form $F^{I}\wedge F^{J}$,
for some $I,J$, that are nontrivially coupled to the axions $a^{i}$, residing
in ${\rm Re}\mathcal{N}_{IJ}$, via the Lagrangian term

$\frac{1}{2}{\rm Re}\mathcal{N}_{IJ}(a)F^{I}\wedge F^{J}$ (3.44)

as appears in (3.4). Such terms are reminiscent of the more commonly known QCD
_$\theta$ -terms_, of the form $\theta{\rm Tr}(F\wedge F)$, for some non-
abelian field strength $F$ and $\theta\in\mathbb{R}$, that were proposed to
help solve the strong $CP$-problem [24, 25]. For this reason, we will refer to
the more general couplings (3.44) as _generalized $\theta$-terms_. As
mentioned in the previous section, such couplings do not allow for a correct
identification of some of the conserved currents, _de facto_ breaking them.

However, this is not the only role of generalized $\theta$-terms (3.44). In
fact, a phenomenon similar to the Witten effect reviewed in Section 5.1 is
triggered by the generalized $\theta$-terms (3.44) and, for this reason, we
will refer to this as _generalized Witten effect_. In order to see in detail
how this generalized Witten effect takes place in the Type IIB EFTs under
examination, let us consider the equations of motion (3.37)-(3.38) in the
presence of sources. After a monodromy transformation as in (3.27), these get
mapped to

$\displaystyle A^{I}{}_{J}{\rm d}F^{J}=-j^{I}_{\text{m}}\,,$ (3.45a)
$\displaystyle D_{I}{}^{J}{\rm d}\left({\rm Im}\mathcal{N}_{JL}*\\!F^{L}+{\rm
Re}\mathcal{N}_{JL}F^{L}+(D^{-1}C)_{JL}F^{L}\right)=-j_{I}^{\text{e}}\,.$
(3.45b)

Let us first assume that there are no magnetic sources, i.e.
$j^{I}_{\text{m}}=0$. Then, (3.45b) signals that, whenever the electric brane
source $j_{I}^{\text{e}}$ is present in the spectrum, then also the brane
source $(D^{-1})_{I}{}^{J}j_{I}^{\text{e}}$ has to be present. However, one
can perform multiple monodromy transformations. As a result, due to
monodromies induced by the Witten effect, a _tower_ of brane sources, namely
$j_{I}^{\text{e}}$, $(D^{-1}j^{\text{e}})_{I}$,
$(D^{-1}D^{-1}j^{\text{e}})_{I}$, $\ldots$ has to be allowed by the EFT
spectrum by solely assuming that one of these sources exists.

In the presence of magnetic sources, a similar argument can be performed: from
the transformation law of the magnetic equations of motion (3.45a), one finds
that, if the magnetic sources $j^{I}_{\text{m}}$ are allowed by the spectrum,
then also $(A^{-1}j_{\text{m}})$, $(A^{-1}A^{-1}j_{\text{m}})$, etc. are
allowed. On the other hand, the electric brane sources are turned into dyons
after a monodromy transformation, for they get mapped to

$j_{I}^{\text{e}}\to(D^{-1})_{I}{}^{J}j_{J}^{\text{e}}-(D^{-1})_{I}{}^{J}C_{JL}(A^{-1})^{L}{}_{K}j^{K}_{\text{m}}\,,$
(3.46)

thus delivering an infinite tower of dyonic brane sources.

The combined interplay of the Witten effect and nontrivial axion monodromies,
therefore, has important consequences on the composition of the spectrum of
any EFT. Let us assume that a localized, dyonic brane ‘ _seed current_ ’
${\bm{j}}_{\rm seed}$ is present in the spectrum:

${\bm{j}}_{\rm seed}^{T}={\bf q}_{\rm
seed}^{T}\delta^{(3)}(\Sigma)=(q^{I}_{\rm m,seed},-q^{\rm e}_{I,{\rm
seed}})\delta^{(3)}(\Sigma)\,.$ (3.47)

Such a seed charge may be given, for instance, by one of the charges that are
required to break the one-form global symmetries. For instance, the seed
current may be induced by a dyonic BPS D3-particle following the worldline
$\Sigma$, with elementary charge ${\bf q}_{\rm seed}$ and mass given by
(3.15). We will assume that such a D3-particle does not form a black hole and
cannot be resolved within the EFT; this is achieved by requiring that

$M_{{\bf q}_{\rm seed}}\gtrsim\Lambda_{\text{\tiny EFT}}\,.$ (3.48)

As shown in the previous section, due to the Witten effect, the axion-
monodromy induced currents $T^{-1}{\bm{j}}_{\rm seed}$,
$T^{-1}T^{-1}{\bm{j}}_{\rm seed}$, $\ldots$ also ought to be allowed by the
effective description. In turn, this implies that BPS D3-particles with
charges

${\bf q}^{(k)}=(T^{-1})^{k}{\bf q}_{\rm seed}$ (3.49)

should also be allowed by the spectrum. Namely, the EFT charge lattice
$\Gamma_{\text{\tiny EFT}}$ of D3-brane charges ought to contain _at least_
the monodromy-induced sublattice

$\Gamma_{T}=\\{{\bf q}\in\mathbb{Z}^{b_{3}(Y)}\;|\;{\bf q}=(T^{-1})^{k}{\bf
q}_{\rm seed}\;\text{for some $k\in\mathbb{N}$}\,\\}\subset\Gamma_{\text{\tiny
EFT}}\,.$ (3.50)

In particular, if the monodromy matrix $T$ is of infinite order, the
sublattice $\Gamma_{T}$ is of infinite cardinality. The masses of the BPS
D3-particles constituting the monodromy-induced sublattice (3.50) can be
computed from (3.15) as

$M_{{\bf q}^{(k)}}=e^{\frac{K^{\rm cs}}{2}}|((T^{-1})^{k}{\bf
q})^{T}\,\eta\,{\bm{\Pi}}(t)|\,.$ (3.51)

##  4 Global symmetries in the rigid case and one-modulus asymptotic limits

The analysis of the Type IIB EFTs global symmetries carried out in the
previous section was general, and mostly based on the features shared by the
underlying $\mathcal{N}=2$ local supersymmetry. However, we would like to be
more specific about the global symmetries that can emerge in concrete stringy
EFTs, and how these are avoided. We will be interested in the Type IIB EFTs
defined in field space regions close to singularities in the complex structure
moduli space $\mathcal{M}_{\rm cs}$. Within such regions, the EFTs are
expected to be weakly-coupled and well-controlled. The physics of these near-
boundary EFTs can be studied in detail by means of _asymptotic Hodge Theory_.
Established in [10], asymptotic Hodge Theory has proven to be a powerful tool
for examining the structure of string vacua and characteristics of the BPS
spectrum in the asymptotic regions of moduli space [11, 12, 15, 16, 17, 18,
19, 48, 49, 50, 51, 52]. A brief overview of asymptotic Hodge Theory is
contained in Appendix B, where we further highlight how the couplings of the
vector multiplet sector can be studied in full generality exploiting the tools
that the Hodge theory provides.

For ease of exposition, in this section, we limit ourselves to the
investigation of the generalized global symmetries in some simple Type IIB
EFTs defined in the near-boundary regime. Specifically, we will first examine
the global symmetries of the gravity multiplet sector of $\mathcal{N}=2$ EFTs
only, without any moduli, after which we will consider the global symmetries
emerging in the $\mathcal{N}=2$ EFTs characterized by $h^{2,1}=1$ towards any
field space boundary. The computations in this section rely on the general
framework of asymptotic Hodge Theory reviewed in Appendix B, and on the
general expressions of the one-modulus-dependent gauge interactions that are
computed in Appendix C.

###  4.1 The rigid case

As a warm-up for the upcoming sections, let us first consider the simplest EFT
that can be described by the action (3.4): namely, a theory _without_ moduli,
only describing the interaction among the components of the $\mathcal{N}=2$
gravity multiplet. The zero-moduli case can be considered as originating from
the compactification of Type IIB string theory over a rigid Calabi-Yau $Y$.
Formally, this case can be thought of as stemming from the prepotential
$\mathcal{F}=\frac{c}{2}(X^{0})^{2}$, for some $c\in\mathbb{C}$. Without loss
of generality, we will take $c=\theta-\mathbf{i}$, with $\theta\in\mathbb{R}$.
Then, the action (3.4) reduces to

$\displaystyle S$ $\displaystyle=\int\Big{(}\frac{1}{2}M^{2}_{\rm
P}R*1-\frac{1}{2}F^{0}\wedge*F^{0}+\frac{\theta}{2}F^{0}\wedge*F^{0}\Big{)}\,,$
(4.1)

where $F^{0}={\rm d}A^{0}$, with $A^{0}$ being the graviphoton gauge one-form.
The action (4.1) might be understood as the deep infrared limit of (3.4), in
the case that the latter is endowed with a mechanism that renders the
$h^{2,1}$ vector multiplets massive. The sole equation of motion for the
graviphoton field now reads ${\rm d}*\\!F^{0}=0$.

The gauge sector contributions to the action (4.1) are the same as those
appearing in (2.15), and thus the analysis of the global symmetries of the
model closely follows that of Section 2.2. Indeed, the action (4.1) admits two
one-form global symmetries: an electric one, associated to the two-form
Maxwell current $J_{2}^{\rm e}=*F^{0}$ that is conserved by the graviphoton
equations of motion; and a magnetic one, with Maxwell current $J_{2}^{\rm
m}=F^{0}$. In order to break both these global symmetries, it is enough to
assume that there exists a D3-brane, wrapping a rigid three-cycle in $Y$, that
leads to a particle magnetically charged under the graviphoton field in the
external space. Namely, assuming the D3-particle has elementary magnetic
charge $q_{\rm m}$ and spans a worldline $\Gamma$, we find that it breaks the
conservation of the magnetic two-form current as ${\rm d}J_{2}^{\rm m}=-q_{\rm
m}\delta^{(3)}(\Gamma)$. However, as stressed in Section 2.1, the generic
inclusion of a particle with charge $q_{\rm m}$ would break the magnetic
$U(1)_{\rm m}$ to a discrete $\mathbb{Z}_{q}$ subgroup. In order to ensure
that the $U(1)_{\rm m}$ global symmetry group is broken completely, a particle
with unit magnetic charge is required.

As illustrated in Section 2.2, the Witten effect induced by the $\theta$-term
implies that an infinite tower of electric states, with brane charges $q_{\rm
e}=-nq_{\rm m}$ with $n\in\mathbb{Z}$, has to exist. Indeed, such a tower of
electric D3-particles breaks the conservation law of the Page two-form current
as

${\rm d}(J_{2}^{\rm e})^{(P)}={\rm d}(-*F^{0}+\theta
F^{0})=\sum\limits_{n\in\mathbb{Z}}nq_{\rm m}\delta^{(3)}(\Gamma)\,.$ (4.2)

###  4.2 The one-modulus case: the large complex structure point

We now investigate the global symmetries emerging in EFTs equipped with a
complex one-dimensional moduli space. The first class of EFTs that we examine
is the familiar case of EFTs defined at large complex structure (LCS). In the
following, after first determining the most general EFT action defined at any
large one-modulus complex structure point, we examine which global symmetries
such EFT is endowed with, and how these can be broken.

The LCS EFT. In the language of Appendix B.1, the LCS point corresponds to a
singularity of Type $\text{IV}_{1}$ within the moduli space. Given the single
complex structure modulus $t=a+\mathbf{i}s$ of the internal Calabi-Yau $Y$,
the LCS point is reached as $s\to\infty$, and we are interested in the EFT
defined in regions of very large saxion, $s\gg 1$. In the mirror-dual Type IIA
setting, the saxion $s$ can be alternatively understood as parametrizing the
volume of the mirror-dual Calabi-Yau $\hat{Y}$, and the region of interest is
the one where the internal volume is very large.

As outlined in Appendix B.1, any near-boundary EFT can be determined by
specifying a handful of boundary data. In this section, we only outline the
main results, and we refer to Appendix C.1 for details. The first ingredient
that we need is the nilpotent operator $N$ that realizes the weight filtration
(B.8). It can be shown that the most general operator $N$ for a Type IV
singularity is [53]

$N=\begin{pmatrix}0&0&0&0\\\ m&0&0&0\\\ c&b&0&-m\\\
b&n&0&0\end{pmatrix},\quad\text{with }\begin{cases}m,n\in\mathbb{Z},\\\
b+mn/2\in\mathbb{Z},\\\ c-m^{2}n/6\in\mathbb{Z},\\\ m\neq 0,\,n>0.\end{cases}$
(4.3)

The above constraints on the parameters $b$, $c$, $m$, $n$ guarantee that
${\rm rk}N=3$, as required by a Type IV singularity, and that the monodromy
matrix $T=e^{N}$ displays integral entries. The matrix $N$ can be most readily
understood as the lowering operator $N_{-}=N$ of the sole ${\rm
sl}_{2}$-triple (B.14) associated to the Type IV boundary. Secondly, we need
to characterize the $F_{0}^{3}$ filtration, which allows for reconstructing
the whole filtration. The most general representative thereof is [53]

${\bf a}_{0}=\left(1,0,\xi,\frac{c}{2m}\right)^{T}\,,$ (4.4)

where $\xi$ is, in general, a complex parameter. For the sake of generality,
we shall leave the parameters $b$, $c$, $m$, $n$, $\xi$ unspecified. However,
in concrete stringy EFTs they are determined by the geometric data of the
internal Calabi-Yau manifold $Y$. Equipped with (4.3) and (4.4), it is
immediate to show that the nilpotent orbit approximation of the periods (B.7)
leads to

${\bm{\Pi}}_{\rm nil}=e^{tN}\bf{a}_{0}=\begin{pmatrix}1\\\ mt\\\
-\frac{1}{6}m^{2}nt^{3}+\frac{1}{2}ct+\xi\\\
\frac{1}{2}mnt^{2}+bt+\frac{c}{2m}\end{pmatrix}.$ (4.5)

It is worth noticing that the periods (4.5) can be regarded as originating
from the following prepotential

$\mathcal{F}_{\text{\tiny{LCS}}}=-\frac{n}{6m}\frac{(X^{1})^{3}}{X^{0}}-\frac{b}{2m}(X^{1})^{2}-\frac{c}{2m}X^{1}X^{0}-\frac{1}{2}\xi(X^{0})^{2}$
(4.6)

upon gauge-fixing $X^{0}=1$, $X^{1}=mt$.

Following the procedure outlined in Appendix C.1, it can be shown that the
boundary data (4.3) and (4.4) yield the following expression for matrix (3.7)
that dictates the interactions of the gauge sector:

$\mathcal{N}_{\text{\tiny{LCS}}}=\begin{pmatrix}-\frac{1}{3}m^{2}na^{3}-\text{Re}\,\xi&\frac{1}{2}mna^{2}-\frac{c}{2m}\\\
\frac{1}{2}mna^{2}-\frac{c}{2m}&-na-\frac{b}{m}\end{pmatrix}+\begin{pmatrix}-\frac{1}{6}m^{2}ns(3a^{2}+s^{2})&\frac{1}{2}mnas\\\
\frac{1}{2}mnas&-\frac{n}{2}s\end{pmatrix}.$ (4.7)

Therefore, the explicit expressions (4.5) and (4.7) lead to the following EFT
action defined towards any arbitrary LCS point:

$\displaystyle S_{\text{\tiny{LCS}}}$
$\displaystyle=\int\Big{(}\frac{1}{2}M^{2}_{\rm P}R*1-M^{2}_{\rm
P}\frac{3}{4s^{2}}\frac{1-\frac{3{\rm
Im}\xi}{m^{2}ns^{3}}}{\left(1-\frac{3{\rm
Im}\xi}{2m^{2}ns^{3}}\right)^{2}}({\rm d}a\wedge*{\rm d}a+{\rm d}s\wedge*{\rm
d}s)$ (4.8)
$\displaystyle\qquad\quad-\frac{1}{12}m^{2}ns(3a^{2}+s^{2})F^{0}\wedge*F^{0}+\frac{1}{2}mnasF^{0}\wedge*F^{1}-\frac{1}{4}nsF^{1}\wedge*F^{1}$
$\displaystyle\qquad\quad-\frac{1}{6}(m^{2}na^{3}+3{\rm Re}\xi)F^{0}\wedge
F^{0}+\frac{1}{2m}\left(m^{2}na^{2}-c\right)F^{0}\wedge
F^{1}-\frac{1}{2m}(b+mna)F^{1}\wedge F^{1}\Big{)}\,.$

In the following, we assume that the saxion $s$ is not dynamical and fixed at
a given vev $s\gg 1$.

The LCS one-form global symmetries. The LCS effective action (4.8) is endowed
with, at most, four one-form global symmetries. These are associated to the
invariance of the effective action (4.8) under transformations of the electric
gauge one-forms as $A^{I}\to A^{I}+\Lambda^{I}$, with ${\rm d}\Lambda^{I}=0$,
and its magnetic counterpart. The corresponding two-form Page currents are the
following

$\displaystyle(J^{0}_{\rm m})^{(P)}$ $\displaystyle=F^{0}\,,$ (4.9)
$\displaystyle(J^{1}_{\rm m})^{(P)}$ $\displaystyle=F^{1}\,,$
$\displaystyle(J_{0}^{\rm e})^{(P)}$
$\displaystyle=G_{0}:=-\frac{m^{2}ns}{6}(3a^{2}+s^{2})*F^{0}+\frac{mnas}{2}*F^{1}-\frac{m^{2}na^{3}+3{\rm
Re}\xi}{3}F^{0}+\frac{m^{2}na^{2}-c}{2m}F^{1}\,,$ $\displaystyle(J_{1}^{\rm
e})^{(P)}$
$\displaystyle=G_{1}:=\frac{mnas}{2}*F^{0}-\frac{ns}{2}*F^{1}+\frac{m^{2}na^{2}-c}{2m}F^{0}-\frac{b+mna}{m}F^{1}\,,$

which are conserved by virtue of the Bianchi identities and the equations of
motion of the gauge fields $A^{I}$.

Indeed, as shown in Section 2.2, the Page currents (4.9) are not in general
single-valued, for they transform as

$\displaystyle\begin{pmatrix}(J^{I}_{\rm m})^{(P)}\\\ -(J_{I}^{\rm
e})^{(P)}\end{pmatrix}\quad\xrightarrow{a\to a+1}\quad
e^{N}\begin{pmatrix}(J^{I}_{\rm m})^{(P)}\\\ -(J_{I}^{\rm
e})^{(P)}\end{pmatrix}\,,$ (4.10)

under a unitary axion shift $a\to a+1$. In particular, from the expression
(4.3) of the nilpotent matrix $N$, one can show that the Page currents
$(J^{1}_{\rm m})^{(P)}$ and $(J_{I}^{\rm e})^{(P)}$ are not invariant under
axion shifts and they are therefore ill-defined. The sole Page current that is
single-valued is $(J^{0}_{\rm m})^{(P)}=F^{0}$,888Notice that this is the sole
Page current among those in (4.9) that can also be regarded as a Maxwell
current. and can thus lead to a genuine, well-defined one-form $U(1)_{\rm m}$
global symmetry. Such a symmetry can be broken explicitly by including
D3-particles magnetically charged under the graviphoton field $A^{0}$.
Minimally, it is enough to include a single of such particles, so that the
Bianchi identity for the graviphoton breaks down to ${\rm
d}F^{0}=p^{0}\delta^{3}(\Gamma)$, with $\Gamma$ the worldline the D3-particle
describes and $p^{0}$ its charge. As shown in detail in Appendix C.1, in the
vector basis in which the nilpotent operator is as in (4.3), one can choose
$p^{0}\in\mathbb{Z}$. For a general $p^{0}$, the $U(1)_{\rm m}$ group is
broken to a discrete subgroup $\mathbb{Z}_{p^{0}}$; breaking the symmetry
completely requires that the charge $p^{0}$ has to be the unit charge
$p^{0}=1$.

The generalized Witten effect in LCS EFTs. The effective action (4.8) displays
generalized $\theta$-terms that mix the graviphoton gauge one-form $A^{0}$
with the vector multiplet gauge one-form $A^{1}$. It is worth remarking that
such generalized $\theta$-terms appearing in (4.8) are _unavoidable_ in the
field space region where the action (4.8) is valid. In fact, it can be shown
that, after canonically normalizing both the axion $a$ and the gauge fields
$A^{0}$ and $A^{1}$, the gauge kinetic terms for the one-form gauge fields
$A^{I}$ and the generalized $\theta$-terms scale in the same fashion with the
saxion $s$. Thus, as explained in Section 3.4, a generalized Witten effect
takes place, also towards the limit $s\to\infty$.

Indeed, as we have seen above, the avoidance of the one-form global symmetries
requires the existence of (at least) a ‘ _seed charge_ ’ carrying elementary
charge ${\bf q}_{\rm seed}=(1,0,0,0)^{T}$. According to the generalized Witten
effect, the spectrum has to additionally contain all the states characterized
by the elementary charges

${\bf q}^{(k)}=e^{-kN}{\bf q}_{\rm
seed}=\left(1,-km,\frac{1}{6}(k^{3}nm^{2}-6kc),-\frac{1}{2}k(2b-kmn)\right)^{T},\qquad\text{with
$k\in\mathbb{Z}$}\,.$ (4.11)

These are dyonic states and, in particular, BPS D3-particle states with
elementary charges (4.11) can be constructed. As can be checked from the
general expression for the masses and physical charges (3.15) and (3.16), the
associated BPS D3-particles are heavy and strongly coupled, with both masses
and physical charges diverging as $s\to\infty$.

The LCS axion global symmetries. Before concluding the analysis of the LCS
EFTs, let us briefly comment on the axion zero-form global symmetries. The
Maxwell three-form current

$J^{(M)}_{a}=-M_{\rm P}^{2}\frac{3}{4s^{2}}\frac{1-\frac{3{\rm
Im}\xi}{m^{2}ns^{3}}}{\left(1-\frac{3{\rm
Im}\xi}{2m^{2}ns^{3}}\right)^{2}}*{\rm d}a\,,$ (4.12)

associated to the continuous axion shift symmetry $a\to a+\chi$,
$\chi\in\mathbb{R}$, is not conserved. The lack of conservation of (4.12) is
due to the axion-dependent gauge interactions appearing in the effective
action (4.8). As such, the global zero-form axion $U(1)_{\rm a}$ is naturally
broken to a discrete subgroup $\mathbb{Z}_{N}$, for some $N$. Fully breaking
the axion shift symmetry can be achieved by including an instanton of unit
charge.

###  4.3 The one-modulus case: Tyurin degeneration

The LCS point is not the only infinite-distance boundary that can be reached
within a complex one-dimensional moduli space. Indeed, in the language of
Appendix B.1, one finds that type II boundaries are similarly located at
infinite geodesic distance. In analogy with [50], we shall call this class of
boundaries ‘ _Tyurin degeneration_ ’ because of their analogy with the Calabi-
Yau degenerations first studied in [54] (see also [55, 56, 57] for later works
that consider such classes of field space boundaries). Following in the
footsteps of the previous section, we illustrate which global symmetries
emerge in the EFTs defined close to any Tyurin degeneration, and how these can
be avoided.

The Tyurin degeneration EFT. We first determine the EFT describing the vector
multiplets interactions close to a Tyurin degeneration, by means of the
boundary data characterizing the field space boundary. We will again be brief
and refer to Appendix C.2 for details. As in the previous section, we split
the single complex structure modulus as $t=a+\mathbf{i}s$, and we assume that
the Tyurin degeneration point is reached as $s\to\infty$. The first ingredient
that we need to construct the near-boundary EFT is the nilpotent operator
associated to the Tyurin degeneration; the nilpotent operator that we shall
consider is [53]

$N=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ m&0&0&0\\\ 0&n&0&0\\\
\end{pmatrix},\quad\text{with }m,n\in\mathbb{N}\,.$ (4.13)

Secondly, the most general representative ${\bf a}_{0}$ of the set $F_{0}^{3}$
is [53]

${\bf a}_{0}=\left(1,0,\mathbf{i}\,\alpha,\mathbf{i}\,\alpha
c+d\right)^{T}\,,\quad\text{with }\alpha=\sqrt{m/n}\quad\text{and}\quad
c,d\in\mathbb{R}.$ (4.14)

Then, Schmid’s nilpotent orbit theorem (B.7) yields the following approximated
periods

${\bm{\Pi}}_{\rm nil}=e^{tN}\bf{a}_{0}=\begin{pmatrix}1\\\
\mathbf{i}\,\alpha\\\ mt\\\ d+\mathbf{i}\,c\alpha+\mathbf{i}\,n\alpha
t\end{pmatrix}.$ (4.15)

The periods (4.15) do not have a clear origin in terms of a prepotential.
However, performing the symplectic transformation

$\mathcal{S}=\begin{pmatrix}1&0&0&0\\\ -\alpha d&-\alpha c&0&\alpha\\\
0&-d&1&0\\\ 0&-\alpha&0&0\end{pmatrix},$ (4.16)

the periods (4.15) get mapped to

$\widetilde{\bm{\Pi}}_{\rm nil}=\mathcal{S}{\bm{\Pi}}_{\rm
nil}=\begin{pmatrix}1\\\ \mathbf{i}\,mt\\\ mt-\mathbf{i}\alpha d\\\
-\mathbf{i}\end{pmatrix}.$ (4.17)

The latter periods can be most readily computed out of the prepotential

$\mathcal{F}_{\text{\tiny{Tyurin}}}=\mathbf{i}X^{0}X^{1}+\mathbf{i}\frac{\alpha
d}{2}(X^{0})^{2}\,,$ (4.18)

upon gauge-fixing $X^{0}=1$, $X^{1}=\mathbf{i}\,mt$.

Following the steps described in Appendix C.2, one can show that the
$\mathfrak{sl}(2)$-approximated matrix $\mathcal{N}$ is

$\mathcal{N}_{\text{\tiny{Tyurin}}}=-\begin{pmatrix}ma&\frac{d}{2}\\\
\frac{d}{2}&na+c\end{pmatrix}-\mathbf{i}\begin{pmatrix}ms&0\\\
0&ns\end{pmatrix}.$ (4.19)

Thus, the EFT defined for large saxionic vevs, near the Tyurin degeneration
is:

$\displaystyle S_{\text{\tiny{Tyurin}}}$
$\displaystyle=\int\Big{(}\frac{1}{2}M^{2}_{\rm P}R*1-M^{2}_{\rm
P}\frac{m^{2}}{(\alpha d-2ms)^{2}}({\rm d}a\wedge*{\rm d}a+{\rm d}s\wedge*{\rm
d}s)$ (4.20)
$\displaystyle\qquad\quad-\frac{1}{2}msF^{0}\wedge*F^{0}-\frac{1}{2}nsF^{1}\wedge*F^{1}$
$\displaystyle\qquad\quad-\frac{1}{2}maF^{0}\wedge
F^{0}-\frac{d}{2}F^{0}\wedge F^{1}-\frac{1}{2}(na+c)F^{1}\wedge
F^{1}\Big{)}\,.$

The Tyurin degeneration one-form global symmetries. As in Section 4.2, the
gauge sector can lead to four one-form global symmetries, associated to shifts
of the one-forms $A^{I}$ by flat connections and their magnetic counterparts.
Employing the Bianchi identities for the one-form gauge fields $A^{I}$ and the
equations of motions stemming from the EFT action (4.20), one can show that
the putative one-form global symmetries are related to the following,
conserved two-form Page currents:

$\displaystyle(J^{0}_{\rm m})^{(P)}$ $\displaystyle=F^{0}\,,$ (4.21)
$\displaystyle(J^{1}_{\rm m})^{(P)}$ $\displaystyle=F^{1}\,,$
$\displaystyle(J_{0}^{\rm e})^{(P)}$
$\displaystyle=G_{0}:=-ms*F^{0}-maF^{0}-\frac{d}{2}F^{1}\,,$
$\displaystyle(J_{1}^{\rm e})^{(P)}$
$\displaystyle=G_{1}:=-ns*F^{1}-\frac{d}{2}F^{0}-(na+c)F^{1}\,.$

Recalling that the Page currents ${\bm{J}}^{(P)}=((J^{I}_{\rm
m})^{(P)},-(J_{I}^{\rm e})^{(P)})^{T}$ transform as ${\bm{J}}^{(P)}\to
e^{N}{\bm{J}}^{(P)}$ under a unit axion shift $a\to a+1$, it can be shown that
the electric Page currents nontrivially transform as

$\begin{pmatrix}-(J_{0}^{\rm e})^{(P)}\\\ -(J_{1}^{\rm
e})^{(P)}\end{pmatrix}\quad\xrightarrow{a\to
a+1}\quad\begin{pmatrix}-(J_{0}^{\rm e})^{(P)}+m(J^{0}_{\rm m})^{(P)}\\\
-(J_{1}^{\rm e})^{(P)}+n(J^{1}_{\rm m})^{(P)}\end{pmatrix}\,.$ (4.22)

As such, the electric currents $(J_{I}^{\rm e})^{(P)}$ are ill-defined. On the
other hand, the magnetic Page currents $(J^{I}_{\rm m})^{(P)}$ are single-
valued and thus deliver genuine one-form global symmetries. These global
symmetries can be minimally broken by assuming the presence of a D3-particle
with nontrivial magnetic charges $p^{I}=(p^{0},p^{1})$ so that the two-form
Page currents $(J^{I}_{\rm m})^{(P)}$ are not conserved as

${\rm d}(J^{I}_{\rm m})^{(P)}={\rm d}F^{I}=-p^{I}\delta^{(3)}(\Gamma)\,.$
(4.23)

A generic choice of magnetic charges $p^{I}$ breaks the global group
$(U(1)_{\rm m})^{2}$ to the global discrete subgroup
$\mathbb{Z}_{p^{0}}\times\mathbb{Z}_{p^{1}}$. Choosing the magnetic charges
$p^{I}$ as unit-charges, the one-form global symmetries are fully broken.

The generalized Witten effect in EFTs defined close to the Tyurin
degeneration. As for the EFTs defined close to an LCS point, also EFTs defined
towards a Tyurin degeneration point are characterized by generalized
$\theta$-terms and, as such, are subjected to a nontrivial Witten effect. In
fact, as we have seen, requiring the breaking of the genuine one-form global
symmetries $(U(1)_{\rm m})^{2}$ requires the presence of a D3-particle with
the seed charge

${\bf q}_{\rm seed}=(p^{0},p^{1},0,0)^{T}\,.$ (4.24)

Then, the generalized Witten effect then predicts the following, infinite
tower of dyonic states

${\bf q}^{(k)}_{\rm
tower}=(p^{0},p^{1},-kmp^{0},-knp^{1})^{T}\,,\quad\text{with}\quad
k\in\mathbb{Z}.$ (4.25)

It can be checked that any state belonging to the tower (4.25) has super-
Planckian mass towards the Tyurin degeneration, as $M_{{\bf q}_{\rm
tower}}\to\infty$ as $s\to\infty$.

The Tyurin degeneration axion global symmetries. Finally, let us consider the
zero-form axion global symmetry. The three-form Maxwell current

$J^{(M)}_{a}=-M_{\rm P}^{2}\frac{m^{2}}{(\alpha d-2ms)^{2}}*{\rm d}a$ (4.26)

is not conserved, as can be checked from the equations of motion for the
axion, and the lack of conservation is due to the generalized $\theta$-terms
appearing in (4.20). Therefore, the continuous axion shift symmetry is
unavoidably broken to a discrete subgroup.

###  4.4 An example of finite distance boundary: the conifold singularity

The final class of EFTs endowed with a single dynamical complex structure
modulus are those encountered close to a conifold singularity. In contrast
with the LCS point examined in Section 4.2 and the Tyurin degeneration studied
in Section 4.3, the conifold singularity is reached at _finite_ distance in
field space. As such, we will see that EFTs defined close to conifold point
exhibit crucial differences compared to the EFTs investigated in the previous
sections.

The conifold EFT. We first determine the general form of the $\mathcal{N}=2$
Type IIB EFTs defined close to a conifold singularity. In the terminology of
Appendix B.1, a conifold singularity corresponds to a Type I field space
boundary. The nilpotent operator $N=N_{-}$ that describes a Type I boundary
and delivers the lowering operator of the single ${\rm sl}_{2}$-triple is
given by [53]

$N=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ n&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix},\quad\text{with }0>n\in\mathbb{Z}\,,$ (4.27)

such that $N^{2}=0$. A general representative ${\bf a}_{0}\in F_{0}^{3}$ is
[53]

${\bf a}_{0}=\left(0,1,\beta-\gamma\tau,\tau\right)^{T}\,,\quad\text{with
}\quad\beta,\gamma\in\mathbb{R},\quad\tau\in\mathbb{C}.$ (4.28)

Moreover, we shall additionally assume $d>0$. The nilpotent operator (4.13)
acts trivially on $F_{0}^{3}$ as $N{\bf a}_{0}=0$. Therefore, the leading term
in the nilpotent orbit theorem (B.7) is moduli-independent, and leads to a
constant Kähler potential (3.5). In order to get a nontrivial Kähler potential
one needs to consider subleading terms in the nilpotent orbit (B.7), which are
generated by the instanton map. Its general form is given in [50], which,
transformed to our basis, leads to the following expression:

${\bf a}_{1}=\Gamma^{\prime}(0){\bf
a}_{0}=\alpha\left(1,\gamma,0,\beta\right)^{T},\qquad\alpha\in\mathbb{C},$
(4.29)

with $\alpha$ a model-dependent parameter appearing in the instanton map.
Then, employing (4.28)-(4.29) along with the explicit expression for the
nilpotent operator $N$ (4.27), the nilpotent orbit theorem (B.7) yields the
following approximated periods:

${\bm{\Pi}}=e^{tN}({\bf a}_{0}+e^{2\pi\mathbf{i}\,t}{\bf
a}_{1}+\ldots)\simeq\begin{pmatrix}\alpha e^{2\pi\mathbf{i}\,t}\\\
1+\alpha\gamma e^{2\pi\mathbf{i}\,t}\\\ \beta-\gamma\tau+\alpha
ne^{2\pi\mathbf{i}\,t}t\\\ \tau+\alpha\beta
e^{2\pi\mathbf{i}\,t}\end{pmatrix}.$ (4.30)

The approximate Kähler potential (3.5) computed out of the periods (4.30) is

$e^{-K^{\rm cs}_{\text{\tiny{conifold}}}}\simeq 2d+2\alpha^{2}nse^{-4\pi
s}\,.$ (4.31)

It is worth mentioning that, by performing the symplectic transformation

$\mathcal{S}=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&-c&0&1\\\
0&0&1&0\end{pmatrix},$ (4.32)

the periods (4.30) can be recast as

$\widetilde{\bm{\Pi}}_{\rm nil}\simeq\mathcal{S}{\bm{\Pi}}_{\rm
nil}=\begin{pmatrix}1\\\ e^{2\pi\mathbf{i}t}\\\ \mathbf{i}d\\\
\beta-\gamma\tau+\alpha nte^{2\pi\mathbf{i}t}\\\ \end{pmatrix},$ (4.33)

where we have neglected subleading corrections. In turn, the periods (4.33)
can be regarded as originating from the superpotential

$\mathcal{F}_{\text{\tiny{conifold}}}=-\frac{\mathbf{i}d}{2}(X^{0})^{2}-(\beta-\gamma\tau)X^{0}X^{1}+\frac{\mathbf{i}n}{4\pi}(X^{1})^{2}\log\frac{X^{1}}{X^{0}}\,,$
(4.34)

upon gauge-fixing $X^{0}=1$, $X^{1}=e^{2\pi\mathbf{i}t}$. Indeed, it can be
checked that computing the periods (3.3) via the prepotential (4.34) and
plugging them into the general (3.5) deliver the same Kähler potential as
(4.31).

In Appendix C.3 the following $\mathfrak{sl}(2)$-approximated matrix
$\mathcal{N}$ is obtained:999Notice that, for $s\gg 1$ and $n<0$, the
eigenvalues of ${\rm Im}\mathcal{N}_{\text{\tiny{conifold}}}$ are both
negative.

$\mathcal{N}_{\text{\tiny{conifold}}}\simeq-\begin{pmatrix}na-\gamma(\beta-\gamma
c)&\beta-\gamma c\\\ \beta-\gamma
c&c\end{pmatrix}+\mathbf{i}\begin{pmatrix}ns&\gamma d\\\ \gamma
d&-d\end{pmatrix},$ (4.35)

where we have defined

$\tau=c+\mathbf{i}\,d,\qquad c,d\in\mathbb{R}.$ (4.36)

Hence, using the general periods (4.30) and the matrix (4.35), the EFT defined
close to a conifold singularity, for large value of saxion $s\gg 1$,
is101010In order for the kinetic terms to get the correct signs, one has to
set $n<0$, $d<0$ – see also footnote 9.

$\displaystyle S_{\text{\tiny{conifold}}}$
$\displaystyle\simeq\int\Big{(}\frac{1}{2}M^{2}_{\rm P}R*1-M^{2}_{\rm
P}\frac{2\pi n}{d}\alpha^{2}e^{-4\pi s}(2\pi s-1)({\rm d}a\wedge*{\rm d}a+{\rm
d}s\wedge*{\rm d}s)$ (4.37)
$\displaystyle\qquad\quad+\frac{1}{2}nsF^{0}\wedge*F^{0}+\gamma
dF^{0}\wedge*F^{1}-\frac{d}{2}F^{1}\wedge*F^{1}$
$\displaystyle\qquad\quad-\frac{1}{2}(\gamma\beta-c\gamma^{2}-an)F^{0}\wedge
F^{0}-(\beta-\gamma c)F^{0}\wedge F^{1}-\frac{c}{2}F^{1}\wedge
F^{1}\Big{)}\,.$

The conifold one-form global symmetries. The four one-form global symmetries
that the conifold EFT may at most support are associated to the following two-
form Page currents

$\displaystyle(J^{0}_{\rm m})^{(P)}$ $\displaystyle=F^{0}\,,$ (4.38)
$\displaystyle(J^{1}_{\rm m})^{(P)}$ $\displaystyle=F^{1}\,,$
$\displaystyle(J_{0}^{\rm e})^{(P)}$
$\displaystyle=G_{0}:=(ns-d\gamma^{2})*F^{0}+\gamma
d*F^{1}+(\gamma\beta-c\gamma^{2}-an)F^{0}-(\beta-\gamma c)F^{1}\,,$
$\displaystyle(J_{1}^{\rm e})^{(P)}$ $\displaystyle=G_{1}:=\gamma
d*F^{0}-d*F^{1}-(\beta-\gamma c)F^{0}-cF^{1}\,,$

which are conserved due to the Bianchi identities and the equations of motion
for the one-forms $A^{I}$. The Page currents ${\bm{J}}^{(P)}=((J^{I}_{\rm
m})^{(P)},-(J_{I}^{\rm e})^{(P)})^{T}$ transform as ${\bm{J}}^{(P)}\to
e^{N}{\bm{J}}^{(P)}$ under a unit-axion shift. From the explicit expression of
the nilpotent matrix (4.27), one recognizes that the electric Page current
$(J_{0}^{\rm e})^{(P)}=G_{0}$ is subjected to the nontrivial shift

$G_{0}\quad\xrightarrow{a\to a+1}\quad G_{0}-nF^{0}$ (4.39)

under a monodromy transformation. Thus, the Page current $(J_{0}^{\rm
e})^{(P)}$ is ill-defined. On the other hand, the remaining Page currents,
$(J^{I}_{\rm m})^{(P)}$ and $(J_{1}^{\rm e})^{(P)}$ are single-valued and lead
to unbroken one-form global symmetries. In order to break them, it is enough
to assume the existence of a dyonic D3-particle with charge ${\bf
q}=(p^{0},p^{1},0,-q_{1})^{T}$, that breaks the conservation laws of the well-
defined Page currents as

${\rm d}(J^{I}_{\rm m})^{(P)}={\rm
d}F^{I}=-p^{I}\delta^{(3)}(\Gamma)\,,\qquad{\rm d}(J_{1}^{\rm e})^{(P)}={\rm
d}G_{1}=-q_{1}\delta^{(3)}(\Gamma)\,.$ (4.40)

The generalized Witten effect in EFTs defined close to a conifold singularity.
The $\theta$-terms populating the near-conifold EFT (4.37) induce a nontrivial
Witten effect. Here, the Witten effect manifest in the generation of an
infinite tower of dyonic, massive states. In fact, consider a seed charge of
the form ${\bf q}_{\rm seed}=(p^{0},p^{1},0,-q_{1})^{T}$, as is required in
order to break the three genuine one-form global symmetries as in (4.40).
Then, an infinite tower of states with charge

${\bf q}^{(k)}_{\rm tower}=e^{-kN}{\bf q}_{\rm
seed}=(p^{0},p^{1},-kmp^{0},-q_{1})^{T}\,,\quad\text{with}\quad
k\in\mathbb{Z}\,,$ (4.41)

is also present in the spectrum. Thus, as can be inferred from the BPS mass
formula (3.15) employing the periods (4.30), the tower of states (4.41) is
constituted by massive, super-Planckian states.

The conifold axion global symmetries. Finally, let us comment on the axion
shift symmetries. The conservation of the Maxwell three-form current

$J^{(M)}_{a}=M_{\rm P}^{2}\frac{2\pi n}{d}e^{-4\pi s}(2\pi s-1)*{\rm d}a$
(4.42)

is broken by the $\theta$-term appearing in (4.37) involving the graviphoton
one-form as

${\rm d}J^{(M)}_{a}=-\frac{n}{2}F^{0}\wedge F^{0}\,.$ (4.43)

Therefore, the continuous axion shift symmetry is broken in the field space
region close to a conifold singularity.

##  5 Global symmetries in general asymptotic limits

Inferring the global symmetries that emerge towards a generic field space
boundary might appear to be a daunting task. Indeed, it is far from trivial to
extend the analysis carried in Section 4 in the one-modulus limits to multi-
moduli limits. However, although the explicit derivation of the most general
EFTs that describe the dynamics towards general field space boundaries remains
out of reach, the Hodge theory technology reviewed in Section B delivers
valuable information about the asymptotic symmetries that emerge in such
limits.

In order to set the stage for the upcoming discussion, let us state the setup
on which we focus. For notational simplicity, we consider generic field space
boundaries that are reached when the first $k$ saxions acquire large values,
as $s^{1},\ldots,s^{k}\to\infty$. Limits that involve a different set of
saxions can be described analogously by a relabeling of the coordinates. Let
us assume that such a boundary is reached by sending the saxions to large
values one by one. For instance, let us first consider the limit in which only
the first saxion $s^{1}$ acquires large vev, as $s^{1}\to\infty$. The locus
towards which $s^{1}\to\infty$ is a boundary on its own, and following the
discussion in Appendix B.1 it is characterized by a type, that we generically
denote ‘Type ${\sf A}_{1}$’, with ${\sf A}_{1}={\rm I}_{a},{\rm II}_{b},{\rm
III}_{c},{\rm IV}_{d}$. We then further send $s^{2}\to\infty$, thus reaching a
new boundary, with type ${\sf A}_{2}$, and analogously proceed so that we
ultimately reach a Type ${\sf A}_{k}$ boundary towards which all the first $k$
saxions acquire large vev. This chain of successive ‘ _enhancements_ ’ of the
boundary type can be schematically written as

$\text{Type ${\sf A}_{1}$}\rightarrow\text{Type ${\sf
A}_{2}$}\rightarrow\ldots\rightarrow\text{Type ${\sf A}_{k}$}\,.$ (5.1)

In the following, employing the general considerations of Appendix B, we will
investigate which global symmetries may emerge at each step of the enhancement
chain (5.1), and how these are inter-related.

###  5.1 The generalized Witten Effect and global symmetries towards a generic
boundary

First, we consider the generalized global symmetries that can emerge at a
given step along the enhancement chain (5.1). Without loss of generality, we
shall focus on the last step of the chain represented in (5.1). Additionally,
we shall regard the residual $(n-k)$-complex moduli fields $t^{i}$ as non-
dynamical, fixed at given vevs.

Our general investigation starts with the analysis of the allowed
electromagnetic one-form global symmetries that the EFT defined close to the
Type ${\sf A}_{k}$ may be equipped with. As illustrated in Section 3.3 on
general grounds, the maximal $2(h^{2,1}+1)$ one-form global symmetries may be
ill-defined, since the conserved, gauge-invariant Page currents (3.32)–(3.33)
non-trivially transform as ${\bm{J}}^{(P)}\to T{\bm{J}}^{(P)}$ under the axion
monodromies $a^{i}\to a^{i}+1$ for all $i=1,\ldots,k$. The monodromy matrix
$T$ is here determined by the nilpotent matrix $N_{(k)}$ defined above (B.8)
as $T=e^{N_{(k)}}$. Therefore, the knowledge of the generic properties of the
operator $N_{(k)}$ deliver pivotal information about the transformation that
the Page currents are subjected to. The relevant properties of the nilpotent
matrix $N_{(k)}$ are indeed related to the boundary type close to which the
EFT is defined, and are summarized in Table B.1.

In particular, the Page currents that get nontrivially transformed are those
which lie in the image of the operator $N_{(k)}$. Therefore, the currents that
can deliver genuine, well-defined Page charges are those which lie in the
complement space $H^{3}(Y)\smallsetminus{\rm Im}\,N_{(k)}$. Interestingly, one
can give an estimation on the number of independent Page currents towards any
field space boundary based on the general considerations of Appendix B.1. In
fact, the independent, well-defined Page currents are counted by the dimension
of the space $H^{3}(Y)\smallsetminus{\rm Im}\,N_{(k)}$, and the latter can be
most readily inferred from the rank of the operator $N_{(k)}$, that depends on
the boundary type as given in Table B.1. Thus, the number of independent,
electromagnetic one-form global symmetries $N_{\rm em}$ depends on the
boundary type as follows:

$N_{\rm em}={\rm dim}_{\mathbb{R}}\left(H^{3}(Y)\smallsetminus{\rm
Im}\,N_{(k)}\right)=\begin{cases}2h^{2,1}+2-a&\text{Type I${}_{a}$}\,,\\\
2h^{2,1}-b&\text{Type II${}_{b}$}\,,\\\ 2h^{2,1}-2-c&\text{Type
III${}_{c}$}\,,\\\ 2h^{2,1}-d&\text{Type IV${}_{d}$}\,.\end{cases}$ (5.2)

Employing the bounds on the parameters $a$, $b$, $c$, $d$ that distinguish the
boundary type listed in Table B.1, the following bounds on the number $N_{\rm
em}$ of genuine, conserved one-form global symmetries can be
obtained:111111Recall that boundaries of Type IIIc can appear only for
$h^{2,1}\geq 2$.

$N_{\rm em}=\begin{cases}2\leq N_{\rm em}\leq 2h^{2,1}+2&\text{Type
I${}_{a}$}\,,\\\ h^{2,1}+1\leq N_{\rm em}\leq 2h^{2,1}&\text{Type
II${}_{b}$}\,,\\\ h^{2,1}\leq N_{\rm em}\leq 2h^{2,1}-2&\text{Type
III${}_{c}$}\,,\\\ h^{2,1}\leq N_{\rm em}\leq 2h^{2,1}-1&\text{Type
IV${}_{d}$}\,.\end{cases}$ (5.3)

Clearly, the bounds in (5.3) are very lax, and their range widens as the
number of moduli increases. However, it is interesting to notice that _for
every moduli space boundary, there is always at least one well-defined one-
form global symmetry_ , leading to a global abelian $U(1)_{\rm em}^{N_{\rm
em}}$ continuous symmetry group. Such a feature is just a rephrasing of the
fact that a nilpotent matrix $N_{(k)}$ can never be full-rank.

The immediate consequence is that, regardless of the boundary the EFT
describes, one needs to assume the existence of at least one fundamental
object that breaks the residual continuous one-form global symmetries. With a
convenient choice of the basis, the fundamental charges ${\bf q}$ of the
required D3-particles can be picked in the set $(H^{3}(Y)\smallsetminus{\rm
Im}\,N_{(k)})\cap\mathbb{Z}^{N_{\rm em}}$. A generic choice of charges would
break the continuous group $U(1)_{\rm em}^{N_{\rm em}}$ down to a discrete
subgroup thereof. However, choosing the charges as being unit charges, the
$U(1)_{\rm em}^{N_{\rm em}}$ group is entirely broken.

In general, the properties of the fundamental objects that ought to be
included to break the residual one-form global symmetries depend on the
specific geometrical data of the compactification. Nevertheless, some of their
crucial features can be understood in general terms. First, notice that the
elementary charges ${\bf q}$ cannot be generated as the image of any of the
operators $N_{(i)}$, $i=1,\ldots,k$. Along each of the enhancement steps in
(5.1) that leads to the Type ${\sf A}_{k}$ boundary that is of interest to us,
one can draw a Deligne diagram as the ones in Table B.1. The requirement that
${\bf q}\notin{\rm Im}\,N_{(i)}$, for every $i=1,\ldots,k$, can then be
understood as the fact that ${\bf q}$ has to lie in the upper-part of each of
these Deligne diagrams, namely ${\bf q}\in I^{p,q}_{(i)}$, with $p+q\geq 3$.
Recalling that the physical charge of the D3-particles (3.16) are Hodge norms,
this implies that $\mathcal{Q}_{\bf q}\gtrsim 1$. Namely, the objects required
to break the residual one-form global symmetries are not expected to be weakly
coupled, as observed in one-modulus examples of Section 4.

Finally, let us comment on the zero-form axion global symmetries that the EFT
defined towards a generic boundary can be endowed with. A necessary condition
in order to have a continuous axion shift symmetry is that the EFT action
(3.4) is invariant under arbitrary shifts of the axions $a^{i}\to
a^{i}+c^{i}$, for some $c^{i}\in\mathbb{R}$. In particular, the Kähler metric
ought to be invariant under such shifts. However, recall that the periods are
generically subjected to corrections of order
$\mathcal{O}(e^{2\pi\mathbf{i}t})$ as captured by the nilpotent orbit theorem
(B.7). Consequently, the Kähler potential (3.5) acquires the general form

$e^{-K}=\mathcal{P}(s)+\mathcal{O}(e^{-2\pi s},\cos(2\pi a),\sin(2\pi a))\,,$
(5.4)

where $\mathcal{P}(s)$ is a polynomial in the saxions and
$\mathcal{O}(e^{-2\pi s},\cos(2\pi a),\sin(2\pi a))$ denotes additional
corrections that may be either axion-dependent or exponentially suppressed in
the saxions. As noticed in [50, 51], the axion-dependent corrections that
appear in (5.4) may be necessary for consistency of the Hodge decomposition
near boundaries different from the Type IV ones, namely the large complex
structure boundaries. Hence, in general, only EFTs defined close to a Type IV
boundaries may lead to an axion-independent Kähler potential, and thus they
might exhibit the axion-shift symmetry. However, the analysis of the Kähler
metric is not enough to conclude that some continuous axionic zero-form
symmetries are unbroken. In fact, Type IV boundaries are associated to
nilpotent matrices of the highest rank, and the generalized $\theta$-terms
necessarily depend non-trivially on the axions, which appear within either
linear, quadratic or cubic couplings with the topological four-form terms
$F^{I}\wedge F^{J}$, which explicitly break the axion shift symmetries.
Whether the continuous axion zero-form symmetry is entirely broken or broken
down to a discrete subgroup towards any boundary depends on the specific EFT
data. Requiring that the discrete shift symmetry is broken completely may
additionally require the presence of instantons with unit charges.

###  5.2 Avoiding global symmetries along boundary enhancement chains

Starting Type ${\sf A}_{k}$ boundary | Enhanced Type ${\sf A}_{k+1}$ boundary
---|---
Type Ia | Type I${}_{\hat{a}}$ for $a\leq\hat{a}$
Type II${}_{\hat{b}}$ for $a\leq\hat{b}$, $a<h^{2,1}$
Type III${}_{\hat{c}}$ for $a\leq\hat{c}$, $a<h^{2,1}$
Type IV${}_{\hat{d}}$ for $a\leq\hat{d}$, $a<h^{2,1}$
Type IIb | Type II${}_{\hat{b}}$ for $b\leq\hat{b}$
Type III${}_{\hat{c}}$ for $2\leq b\leq\hat{c}+2$
Type IV${}_{\hat{d}}$ for $1\leq b\leq\hat{d}-1$
Type IIIc | Type III${}_{\hat{c}}$ for $c\leq\hat{c}$
Type IV${}_{\hat{d}}$ for $c+2\leq\hat{d}$
Type IVd | Type IV${}_{\hat{d}}$ for $d\leq\hat{d}$
Table 5.1: The allowed enhancements from a boundary of Type ${\sf A}_{k}$
reached as the first $k$ saxions $s^{1},\ldots,s^{k}\to\infty$ to a boundary
of Type ${\sf A}_{k+1}$ reached when also the $(k+1)$-th saxion acquires large
vev as $s^{k+1}\to\infty$ [58, 12].

Effective theories defined close to different boundaries display a different
number of global symmetries, as the nilpotent operator that determines the
nontrivial transformation of the Page currents changes. The general framework
of Hodge theory allows one to keep track of how the number of one-form global
symmetries changes as the boundary is enhanced along the chain (5.1). For
concreteness, consider a boundary of Type ${\sf A}_{k}$, which – as in the
previous section – is reached as the first $k$ saxions
$s^{1},\ldots,s^{k}\to\infty$. Then, we consider the boundary that is reached
when one of the residual $(n-k)$-saxions is very large – say, when the
$(k+1)$-th saxion $s^{k+1}\to\infty$. The boundary type is then enhanced to
the different type, that we generically denote ${\sf A}_{k+1}$.

Crucially, there exist restrictions on the Type ${\sf A}_{k+1}$ a Type ${\sf
A}_{k}$ boundary can enhance to, as is explained in [58, 12]. For any boundary
of Type ${\sf A}_{k}$, the allowed types ${\sf A}_{k+1}$ of the enhanced
boundary are collected in Table 5.1.

Starting boundary | Starting $N_{\rm em}^{\rm st}$ | Enhanced boundary | Enhanced $N_{\rm em}^{\rm enh}$
---|---|---|---
|  | Type I${}_{\hat{a}}$ | $2h^{2,1}+2-{\hat{a}}\leq 2h^{2,1}+2-a$
|  | Type II${}_{\hat{b}}$ | $2h^{2,1}-\hat{b}\leq 2h^{2,1}-a$
|  | Type III${}_{\hat{c}}$ | $2h^{2,1}-2-\hat{c}\leq 2h^{2,1}-2-a$
Type Ia | $2h^{2,1}+2-a$ | Type IV${}_{\hat{d}}$ | $2h^{2,1}-\hat{d}\leq 2h^{2,1}-a$
|  | Type II${}_{\hat{b}}$ | $2h^{2,1}-\hat{b}\leq 2h^{2,1}-b$
|  | Type III${}_{\hat{c}}$ | $2h^{2,1}-2-\hat{c}\leq 2h^{2,1}-b$
Type IIb | $2h^{2,1}-b$ | Type IV${}_{\hat{d}}$ | $2h^{2,1}-\hat{d}\leq 2h^{2,1}-1-b$
|  | Type III${}_{\hat{c}}$ | $2h^{2,1}-2-\hat{c}\leq 2h^{2,1}-2-c$
Type IIIc | $2h^{2,1}-2-c$ | Type IV${}_{\hat{d}}$ | $2h^{2,1}-\hat{d}\leq 2h^{2,1}-2-c$
Type IVd | $2h^{2,1}-d$ | Type IV${}_{\hat{d}}$ | $2h^{2,1}-\hat{d}\leq 2h^{2,1}-d$
Table 5.2: The one-form global symmetries o a boundary of Type ${\sf A}_{k+1}$
reached when also the $(k+1)$-th saxion acquires large vev as
$s^{k+1}\to\infty$.

We denote by $N_{\rm em}^{\rm st}$ the maximal number of independent,
continuous one-form global symmetries of the EFT towards the starting boundary
of Type ${\sf A}_{k}$, as given by (5.2). Likewise, we let $N_{\rm em}^{\rm
enh}$ denote the maximal number of symmetries towards the enhanced Type ${\sf
A}_{k+1}$ boundary. Table 5.2 lists these numbers for each of the allowed
enhancements from Table 5.1, and one finds that

$N_{\rm em}^{\rm enh}\leq N_{\rm em}^{\rm st}\qquad\text{for every allowed
boundary enhancement}.$ (5.5)

Therefore, enhancing the boundary type, by sending additional saxions toards
large vevs, generically decreases the number of generalized global symmetries
that one has to avoid.

##  6 Conclusions

In this work we have investigated which generalized global symmetries appear
in the near-boundary regimes of four-dimensional $\mathcal{N}=2$ EFTs obtained
from compactifying Type IIB string theory on a Calabi-Yau three-fold. We have
focused on the global symmetries associated to the vector multiplet sector,
specifically on the electromagnetic one-form and the axion zero-form global
symmetries. As in [9] we have analyzed whether these global symmetries are
broken by studying the associated conserved currents, and inquiring whether
they are are well-defined.

A key role in the analysis is played by certain ‘generalized $\theta$-terms’
that non-linearly couple the axions to the topological $F\wedge F$-terms of
the theory. In particular, for the $\mathcal{N}=2$ Type IIB EFTs examined in
this work, these couplings are at most cubic in the axions. However, although
a more standard linear coupling of the axion to the topological $F\wedge
F$-term has been extensively studied in the physics literature, these
generalized $\theta$-terms deserve further study, specifically in light of
their effect on the symmetry properties of the theory. In fact, our
observations illustrate that these generalized $\theta$-terms are ubiquitous
and unavoidable in stringy EFTs. Indeed, in the EFTs studied in this work,
their presence is due to the underlying $\mathcal{N}=2$ supersymmetry: the
vector multiplets contain one-form gauge fields and axions among their
components and a supergravity action describing their interactions mixes such
components.

Moreover, the generalized $\theta$-terms may lead to a rich phenomenology.
Indeed we have shown that, on the one hand, such terms break the conservation
laws of some generalized global symmetries; on the other hand, they trigger a
generalized Witten effect, that renders the charge lattice infinitely
populated provided that a handful of seed charges exist. However, as we have
seen in the one-modulus cases scrutinized in Section 4, the infinite towers of
states that the generalized Witten effect delivers are constituted by _dyonic_
states, with masses becoming superplanckian as the field space boundary is
reached.

Thus, such towers of states – that are obtained via the combined effect of
breaking the generalized global symmetries and the generalized Witten effect –
are different from the infinite towers of massless states that realize the
Distance Conjecture. Indeed, as showed in [11, 12, 13], towers of states that
are candidates for realizing the Distance Conjecture in Type IIB EFTs should
rather be constituted by _electric_ D3-particles. It is then tantalizing to
relate the infinite tower of states found in this work with those of [11, 12,
13], and an investigation of a tight relationship between these two kinds of
towers is left for future work. Indeed, if the generalized Witten effect
studied in this work could also justify the existence of the infinite towers
of massless states found in [11, 12, 13], this could give a neat bottom-up
argument for the realization of the Distance Conjecture.

It is worth stressing that the analysis carried out here does not cover _all_
the possible generalized global symmetries related to the vector multiplet
sector. In particular, as stressed in Section 3.3, alongside the zero-form
axion symmetries, their electromagnetic dual, two-form global symmetries may
be present. The difficulties in examining the two-form global symmetries in
Type IIB EFTs are rooted in the obstructions to applying the standard
dualization procedure of Appendix A in a straightforward way, due to the
presence of generalized $\theta$-terms. As outlined in [47], understanding how
these two-form global symmetries are avoided may deliver valuable information
about the worldsheet theory of the eventual axion strings that ought to be
present in the spectrum. Moreover, higher-form generalized global symmetries
associated to the Chern-Weil currents studied in [9, 28] might be present. We
leave the systematic analysis of these additional global symmetries for future
work.

It is also tempting to extend our findings to EFTs endowed with higher
supersymmetry, eventually in dimension greater than four. In the
$\mathcal{N}=2$ Type IIB EFTs examined in this work, the monodromy group,
which is part of the isometry group of the scalar manifold, determines the
structure of the generalized $\theta$-terms and dictates the realization of
the Witten effect. In theories with higher supersymmetry or diverse dimension,
the role here played by the monodromy group is expected to be played by a
proper subgroup of the isometries of the scalar manifold. A detailed
investigation along this direction is left for future work.

Acknowledgments

We would like to thank Rigers Aliaj, Luca Martucci, Irene Valenzuela for
insightful discussions and we are especially grateful to Brice Bastian for
fruitful, intensive discussions throughout the completion of this work. This
research is partly supported by the Dutch Research Council (NWO) via a Start-
Up grant and a Vici grant.

##  Appendix A Charge quantization and dualization

In this section we collect the main conventions that we have employed in this
work on gauge fields and their quantization properties.

For concreteness, let us focus on four-dimensional theories. Consider a free
abelian $U_{\rm g}(1)$ gauge theory, whose gauge field is represented by a
$(p-1)$-form $A_{p-1}$. We regard $A_{p-1}$ as the ‘electric’ gauge field. The
kinetic terms of such an $A_{p-1}$ gauge field are 121212In our conventions,
in $D$-dimensions, the Hodge dual of a $p$-form $\omega_{p}$ is the
$(D-p)$-form  $\omega_{p}=\frac{1}{p!}\omega_{M_{1}\ldots M_{p}}{\rm
d}x^{M_{1}}\wedge\ldots\wedge{\rm d}x^{M_{p}}\,,$ (A.1) with
$\varepsilon^{12\ldots D}=1$. This implies that
$*_{D}(*_{D}\omega_{p})=\sigma(-)^{p(D-p)}\omega_{p}$ with $\sigma=+1$
($\sigma=-1$) if the space has Euclidean (Minkowskian) signature.

$L=-\frac{1}{2g^{2}}F_{p}\wedge*F_{p}$ (A.2)

with $F_{p}={\rm d}A_{p-1}$ and $g$ is the gauge coupling. The $p$-form field
strength is assumed to obey the quantization rule

$\int_{\Sigma_{p}}F_{p}\in\mathbb{Z}\,,$ (A.3)

for any closed $p$-dimensional surface $\Sigma_{p}$.

As recalled in Section 2.1, the pure Maxwell theory exhibits two $U(1)$ one-
form global symmetries. The first, electric $U_{\rm e}(1)$ one-form global
symmetry is manifest in the electric frame. Indeed, $U_{\rm e}(1)$ acts by
shifting $A_{p-1}$ by a closed connection, namely $A_{p-1}\to
A_{p-1}+\Lambda_{p-1}$ with ${\rm d}\Lambda_{p-1}=0$, under which the
Lagrangian (A.2) is invariant.

The conserved current associated to the $U_{\rm e}(1)$ global symmetry can be
computed via the standard Noether’s procedure, leading to

$J^{\rm e}_{D-p}=\frac{(-1)^{p-1}}{g^{2}}*F_{p}\,,\qquad{\rm d}J^{\rm
e}_{D-p}=0\,.$ (A.4)

Given a closed $(D-p)$-surface $\Sigma_{D-p}$, we identify the quantized
charges

$Q^{\rm e}=\frac{1}{g^{2}}\int_{\Sigma_{D-p}}*F_{p}\in\mathbb{Z}\,.$ (A.5)

However, the free Maxwell theory described by the Lagrangian (A.2) is endowed
with an additional, ‘magnetic’ $U_{\rm m}(1)$ global symmetry, which we call
$U_{\rm m}(1)$. Such a symmetry is made manifest once the electric gauge field
$A_{p-1}$ is dualized to its magnetic dual gauge field. The dualization can be
performed by considering a ‘master Lagrangian’ that is obtained from (A.2) by
promoting $F_{p}$ to an arbitrary $p$-form $\Theta_{p}$ and adding a dualizing
term that contains an generic $(D-p-1)$-form $V_{D-p-1}$:

$L_{\text{master}}=-\frac{1}{2g^{2}}\Theta_{p}\wedge*\Theta_{p}+\Theta_{p}\wedge{\rm
d}V_{D-p-1}\,.$ (A.6)

Starting from (A.6), one can follow two paths. First, integrating out
$V_{D-p-1}$ from (A.6) yields ${\rm d}\Theta_{p}=0$. The latter can be locally
solved by setting $\Theta_{p}={\rm d}A_{p-1}$ for a generic $(p-1)$-form
$A_{p-1}$. Plugging such a solution in (A.6) gives back the electric-frame
Lagrangian (A.2). Alternatively, one can integrate out $\Theta_{p}$ from
(A.6), obtaining

$\frac{1}{g^{2}}*\Theta_{p}={\rm d}V_{D-p-1}\equiv G_{D-p}\,.$ (A.7)

Plugging this solution in the Lagragian (A.6) delivers the ‘dual’ Lagrangian

$\displaystyle L_{\rm dual}$
$\displaystyle=-\frac{g^{2}}{2}G_{D-p}\wedge*G_{D-p}\,,$ (A.8)

that solely depends on the field strength of the magnetic field $V_{D-p-1}$,
dual to $A_{p-1}$.

Analogously to the electric $U_{\rm e}(1)$, here the action of the global
$U_{\rm m}(1)$ is manifest, for it acts by shifting $V_{D-p-1}\to
V_{D-p-1}+\Lambda_{D-p-1}$, with ${\rm d}\Lambda_{D-p-1}=0$. Accordingly,
Noether’s first theorem delivers the conserved $p$-form current

$J^{\rm m}_{p}=(-1)^{D-p-1}g^{2}*G_{D-p}\,,\qquad{\rm d}J^{\rm m}_{p}=0\,.$
(A.9)

Thus, taking a closed $p$-surface $\Sigma_{p}$, we the magnetic charges obey
the quantization condition

$Q^{\rm
m}=(-1)^{D-p-1}g^{2}\int_{\Sigma_{p}}*G_{D-p}=(-1)^{(D-p)(p+1)}\int_{\Sigma_{p}}F_{p}\in\mathbb{Z}\,.$
(A.10)

##  Appendix B Gauge interactions in asymptotic limits

In this section, we review some basic facts about asymptotic Hodge theory and
we show how the tools that such a framework provides may be helpful to study
the vector multiplet gauge interactions of any near-boundary Type
$\mathcal{N}=2$ IIB EFT. The technology reviewed here is used extensively
throughout Sections 4 and 5 of the main text.

###  B.1 A brief overview of asymptotic Hodge Theory

We start by collecting some of the basic concepts of asymptotic Hodge Theory.
Our exposition will be brief, and we refer to [12, 18] for detailed reviews on
the subject. In general, the complex structure moduli space $\mathcal{M}_{\rm
cs}$ is noncompact, and it may be characterized by singularities that serve as
_boundaries_ for the moduli space. These singularities are distributed along
intersecting divisors. We can choose local coordinates $z^{\alpha}$ such that
each boundary component, corresponding to $k\leq h^{2,1}$ intersecting
divisors, is described as the vanishing locus $z^{\alpha}=0$, with
$\alpha=1,\ldots,k$. Each boundary component is further characterized by a set
of monodromy transformations $T_{\alpha}\in{\rm Sp}(2h^{2,1}+2,\mathbb{R})$ of
the kind introduced in Section 3.2. Upon encircling the singularity
$z^{\alpha}=0$ along one of the $m$ transverse directions by drawing a path
that sends $z^{\alpha}\to e^{2\pi\mathbf{i}}z^{\alpha}$, some quantities may
non-trivially transform, and their transformation laws are dictated by
appropriate representation of the monodromy group.

Given the singularity $z^{\alpha}=0$, we will focus on the near-boundary
region $\mathcal{E}$ that we characterize as the punctured polydisk
$\mathcal{E}=(\Delta^{*})^{k}\times(\Delta)^{h^{2,1}-k}$ where,
$\Delta\coloneqq\\{|z|<1\\}$ and $\Delta^{*}\coloneqq\\{0<|z|<1\\}$. To
emphasize this split we similarly decompose our local coordinates as
$(z^{\alpha},\zeta^{i})$. The coordinates $\zeta^{i}$ of the disk $\Delta$ are
referred to as spectator moduli. They will not play an important role in the
following and we suppress any explicit dependence on these spectator moduli.
It is convenient to introduce the following parametrization for the $k$
punctured components

$t^{\alpha}\equiv a^{\alpha}+\mathbf{i}s^{\alpha}=\frac{1}{2\pi\mathbf{i}}\log
z^{\alpha}\,,$ (B.1)

in which the singularities are reached towards large values of the saxions
$s^{\alpha}$ and encircling the locus $z^{\alpha}=0$ once corresponds to
shifting the axions $a^{\alpha}$ by one period as $a^{\alpha}\to
a^{\alpha}+1$. The near boundary region of interest is then described by the
upper-half plane
$\mathcal{H}^{m}=\\{t^{\alpha}=a^{\alpha}+\mathbf{i}s^{\alpha}\mid 0\leq
a^{\alpha}\leq 1,s^{\alpha}>0\\}$, with the identification $a^{\alpha}\simeq
a^{\alpha}+1$.

Let us now better characterize the properties of the monodromy matrices
$T_{\alpha}$. Around each singularity, one can decompose each of the monodromy
matrices $T_{\alpha}$ into a _semi-simple_ , finite-order part
$T_{\alpha}^{(s)}$ and into a _unipotent_ , infinite-order $T_{\alpha}^{(u)}$
as $T_{\alpha}=T_{\alpha}^{(s)}T_{\alpha}^{(u)}$. These satisfy the following
properties [11]:

$\displaystyle(T_{\alpha}^{(s)})^{m_{\alpha}-1}\neq\mathds{1}\,,$
$\displaystyle(T_{\alpha}^{(s)})^{m_{\alpha}}=\mathds{1}\,,$ (B.2)
$\displaystyle(T_{\alpha}^{(u)}-\mathds{1})^{n_{\alpha}}\neq 0\,,$
$\displaystyle(T_{\alpha}^{(u)}-\mathds{1})^{n_{\alpha}}=0\,,$

for some $m_{\alpha},n_{\alpha}\in\mathbb{N}$. For the upcoming discussion, it
will be convenient to extract the unipotent part of the monodromy matrices by
defining the _log-monodromy matrices_

$N_{\alpha}:=\frac{1}{m_{\alpha}}\log T^{m_{\alpha}}=\log T_{\alpha}^{(u)}\,,$
(B.3)

which are nilpotent matrices, obeying $N_{\alpha}^{n_{\alpha}}\neq 0$,
$N_{\alpha}^{n_{\alpha}+1}=0$, for some $n_{\alpha}\in\mathbb{N}$. In
particular, for Calabi-Yau three-folds, $0\leq n_{\alpha}\leq 3$. Moreover,
the log-monodromy matrices associated to different axion shifts around the
same boundary component commute, as $[N_{\alpha},N_{\beta}]=0$. The log-
monodromy matrices just introduced will play a critical role in the analysis
that follows, and in the following sections we shall see that some physical
quantities are closely related to their properties.

One of the core points of asymptotic Hodge Theory resides in the
identification of the structures that characterize the near-boundary regime of
the complex structure moduli space. To begin with, let us recall that the
middle cohomology of a Calabi-Yau three-fold splits as

$H^{3}(Y,\mathbb{C})=\bigoplus\limits_{p=0}^{3}H^{3-p,p}=H^{3,0}\oplus
H^{2,1}\oplus H^{1,2}\oplus H^{0,3}\,,$ (B.4)

such that $\overline{H^{p,q}}=H^{q,p}$. Such a decomposition identifies a
_pure_ Hodge structure, namely it exhibits a fixed weight, given by $w=p+q=3$.
The complex structure moduli space is then characterized by how the Calabi-Yau
holomorphic three-form $\Omega$ varies inside $H^{3}(Y,\mathbb{C})$ as a
function of the moduli. In order to describe such variations, it will prove
more convenient to introduce the _Hodge filtration_ :

$\displaystyle F^{3}=H^{3,0}\,,$ $\displaystyle F^{2}=H^{3,0}\oplus
H^{2,1}\,,$ (B.5) $\displaystyle F^{1}=H^{3,0}\oplus H^{2,1}\oplus H^{1,2}\,,$
$\displaystyle F^{0}=H^{3,0}\oplus H^{2,1}\oplus H^{1,2}\oplus H^{0,3}\,,$

from which one retrieves the Hodge decomposition (B.4) by setting
$H^{p,q}=F^{p}\cap\overline{F^{q}}$. Given the Gauss-Manin connection
$\nabla_{\alpha}\equiv\nabla_{\partial/\partial z^{\alpha}}$, one has
$\nabla_{\alpha}F^{p}\subset F^{p-1}$. For our purposes, the advantage of the
Hodge filtration (B.5) is that it can be fully spanned by the Calabi-Yau
three-form and its derivatives.

In general, it is very hard to construct the filtration (B.5) at arbitrary
points in the moduli space. One should indeed compute the periods (3.3)
explicitly, which can be achieved by solving a set of Picard-Fuchs equations
(see, for instance, [59]), computing its Gauss-Manin derivatives and then
reconstructing the filtration (B.5). Nevertheless, in the near-boundary region
$s^{\alpha}\gg 1$, the problem greatly simplifies. Near any moduli space
boundary, the filtration $F^{p}$ can be approximated by means of _Schimd’s
nilpotent orbit theorem_ [60], which states that, in the asymptotic region
characterized by large values of the saxions, the Hodge filtration takes the
following asymptotic form

$F^{p}=e^{t^{\alpha}N_{\alpha}}e^{\Gamma(z)}F^{p}_{0}\,.$ (B.6)

Here, $F^{p}_{0}$ denotes the (moduli-independent) _limiting Hodge filtration_
, defined via the relation
$F^{p}_{0}:=\lim\limits_{t^{\alpha}\to\mathbf{i}\infty}e^{-t^{\alpha}N_{\alpha}}F^{p}$;
moreover, $\Gamma(z)$ is a holomorphic $\mathfrak{sp}(2h^{2,1}+1)$-valued map,
such that $\Gamma(0)=0$, to which we will refer as _instanton map_. Any
singular behavior of the Hodge filtration near the singularity is thus
completely characterized by the log-monodromy matrices.

It is worth stressing that equation (B.6) is a vector space relation. However,
evaluating it for $p=3$ leads to an asymptotic form for the period vector
${\bm{\Pi}}$, appearing in the expansion of the Calabi-Yau three-form in
(3.1), as

${\bm{\Pi}}=e^{t^{\alpha}N_{\alpha}}e^{\Gamma(z)}{\bf
a}_{0}=e^{t^{\alpha}N_{\alpha}}\left({\bf a}_{0}+z^{\alpha}{\bf
a}_{\alpha}+z^{\alpha}z^{\beta}{\bf a}_{\alpha\beta}+\ldots\right)\simeq
e^{t^{\alpha}N_{\alpha}}{\bf a}_{0}\equiv{\bm{\Pi}}_{\rm nil}.$ (B.7)

Here, ${\bf a}_{0}\in F^{p}_{0}$ does not carry any dependence on the moduli.
Remarkably, due to the nilpotency of the matrices $N_{\alpha}$, the leading
contribution to the periods (B.7) is polynomial in the complex structure
moduli $t^{\alpha}$. Eventual corrections to the leading contributions are
suppressed exponentially in the saxionic fields $s^{\alpha}$.

As is clear from (B.7), the approximation of the periods achieved via the
nilpotent orbit theorem relies on two data: the nilpotent matrices
$N_{\alpha}$ and the limiting filtration $F^{p}_{0}$. This data can succinctly
be captured by associating to it a _mixed Hodge structure_ whose properties
are particularly well-understood. We consider a boundary reached as
$s^{\alpha}\to\infty$, for $\alpha=1,\ldots,k$ and set $N_{(k)}:=N_{1}+\ldots
N_{k}$ and, in order to keep track of the saxions that acquire large vevs
towards such a boundary, we denote as $F^{p}_{(k)}$ the limiting filtration
$F^{p}_{0}$. Then, we first define the _monodromy weight filtration_ :

$W^{(k)}_{\ell}(N_{(k)})=\sum\limits_{j\geq{\rm max}(-1,\ell-3)}\ker
N_{(k)}^{j+1}\cap{\rm Im}N_{(k)}^{j-\ell+3}\,.$ (B.8)

While the filtration $W^{(k)}_{\ell}(N_{(k)})$ involves only the nilpotent
matrices $N_{(k)}$, we can use them to encode the limiting filtration
$F_{0}^{p}$ in a _Deligne splitting_ :

$I^{p,q}_{(k)}=F^{p}_{(k)}\cap W^{(k)}_{p+q}\cap\Big{(}\bar{F}^{q}_{(k)}\cap
W^{(k)}_{p+q}+\sum\limits_{j\geq 1}\bar{F}^{q-j}_{(k)}\cap
W^{(k)}_{p+q-j-1}\Big{)}\,.$ (B.9)

This splitting identifies a _mixed_ Hodge structure, namely a Hodge structure
involving sub-spaces $I^{p,q}$ of various weights $w=p+q$. Moreover, the
limiting filtration $F_{(k)}^{p}$ and the monodromy weight filtration
$W_{\ell}^{(k)}$ can be recovered from the Deligne splitting (B.9) as follows:

$F_{(k)}^{p}=\bigoplus_{r\geq p}\bigoplus_{s}I^{r,s}_{(k)},\qquad
W^{(k)}_{\ell}=\bigoplus_{p+q=\ell}I^{p,q}_{(k)}\,.$ (B.10)

It will be convenient to organize the subspaces $I^{p,q}_{(k)}$ in a diagram
as in Figure 1, on which the action of the nilpotent matrices $N_{(k)}$ can