IFT-UAM/CSIC-23-77

Torsion in cohomology
and dimensional reduction
Gonzalo F. Casas,1 Fernando Marchesano 1 and Matteo Zatti 1,2

1 Instituto de Física Teórica UAM-CSIC, c/ Nicolás Cabrera 13-15, 28049
Madrid, Spain
2 Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049
Madrid, Spain
Abstract

Conventional wisdom dictates that $\mathbb{Z}_{N}$ factors in the integral
cohomology group $H^{p}(X_{n},\mathbb{Z})$ of a compact manifold $X_{n}$
cannot be computed via smooth $p$-forms. We revisit this lore in light of the
dimensional reduction of string theory on $X_{n}$, endowed with a
$G$-structure metric that leads to a supersymmetric EFT. If massive $p$-form
eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to
them will have a non-trivial smeared delta form, that is an EFT long-
wavelength description of $p$-form currents of the $(n-p)$-cycles of $X_{n}$.
We conjecture that, whenever torsion cycles are calibrated, their linking
number can be computed via their smeared delta forms. From the EFT viewpoint,
a torsion factor in cohomology corresponds to a $\mathbb{Z}_{N}$ gauge
symmetry realised by a Stückelberg-like action, and calibrated torsion cycles
to BPS objects that source the massive fields involved in it.

###### Contents

  1. 1 Introduction
  2. 2 The proposal
  3. 3 Localised sources and dimensional reduction
    1. 3.1 Localised sources in ten and four dimensions
    2. 3.2 The supersymmetric case
    3. 3.3 Generalisations
  4. 4 A simple example
    1. 4.1 The 10d background
    2. 4.2 EFT description
    3. 4.3 Hitchin flow equations
  5. 5 Direct computation
  6. 6 More general ${\cal N}=2$ compactifications
  7. 7 Torsion D-branes in ${\cal N}=1$ vacua
  8. 8 Beyond the BPS case
  9. 9 Conclusions
  10. A NS5-branes and domain wall solutions
  11. B Massive $p$-from spectra in twisted tori

##  1 Introduction

String theory compactifications provide a remarkable connection between the
geometry of extra dimensions and the physics of Effective Field Theories
(EFTs) [1, 2, 3, 4, 5]. An early lesson that one obtains upon exploring this
link is that the more an EFT quantity is protected against quantum
corrections, the simpler is its description in geometric terms. Typical
examples arise in the context of supergravity and supersymmetric gauge
theories, where protection mechanisms involve both gauge invariance and
renormalisation effects constrained by supersymmetry.

In several instances, discrete EFT data protected by gauge invariance are
described in terms of the topology of the compact manifold $X_{n}$, while
quantities protected by supersymmetry enjoy a simple description in terms of
differential and/or algebraic geometry. A clear example of the second is BPS
states or extended objects of the EFT, which can be obtained from, e.g.,
D-branes in type II compactifications at weak coupling. In that case, the
BPSness condition requires that the D-brane extra dimensions wrap a $p$-cycle
of $X_{n}$ that is calibrated, in the sense of [6]. This condition not only
has a neat differential geometric description for compactification manifolds
$X_{n}$ with special holonomy, but it can be generalised whenever $X_{n}$ has
a $G$-structure metric and a flux background that leads to a supersymmetric
EFT [7, 8, 9]. The central charge of the BPS object at tree-level is then
determined by the integral over the $p$-cycle of the suitable $p$-form
calibration, or generalisations that allow us to calibrate D-brane bound
states. This picture also applies to space-time filling D-branes that are part
of the background in type II orientifold compactifications, as well as to
Euclidean D-branes that play the role of BPS instantons.

An example of discrete EFT data with a topological higher-dimensional origin
is the presence of discrete gauge symmetries. In the Abelian case, a
$\mathbb{Z}_{N}$ gauge symmetry of a $d$-dimensional EFT can be described by a
Lagrangian coupling of the form [10]

$N\,B_{d-2}\wedge F_{2},$ (1.1)

where $B_{d-2}$ is a $(d-2)$-form of the EFT dual to an axion $C_{0}$, and
$F_{2}=dA_{1}$ is the field strength of the $U(1)$ boson gauged by $C_{0}$ à
la Stückelberg. Finally, $N\in\mathbb{Z}$ is the quantity that is described in
terms of the topology of $X_{n}$. For instance, in type II orientifold
compactifications, couplings of this form are specified by the homology
classes of the $p$-cycles wrapped by space-time filling D-branes, which in
turn determine the discrete gauge symmetries acting on the open string sector
of the theory [11]. This case is particularly interesting because the discrete
symmetry acts on the massless chiral spectrum of the EFT. However, it has the
feature that the axion and gauge boson masses induced by (1.1) are usually of
the order of the string scale. This implies that the Stückelberg terms that
complete (1.1) are not part of the EFT Lagrangian.

A different setup where the coupling (1.1) is realised is by threading the
compact manifold $X_{n}$ with quantised background fluxes [12]. In this case,
the coupling $N$ is determined by the flux quanta, or equivalently by an
integral cohomology class in $X_{n}$. Here the interplay with the EFT cutoff
is reversed with respect to the previous one. The Stückelberg-induced masses
for axions and gauge bosons can lie below the EFT cutoff, but now the
resulting discrete gauge symmetry acts on strings and particles that typically
do not correspond to light states of the EFT.

In this work, we are interested in yet another realisation of discrete gauge
symmetries, namely those that arise from torsion factors in the integral
cohomology groups $H^{p}(X_{n},\mathbb{Z})$. That such $\mathbb{Z}_{N}$
factors correspond to $\mathbb{Z}_{N}$ gauge symmetries can be seen in the
AdS/CFT context by following the reasoning in [13, 14], applied to type II
orientifold compactifications in [15], and with subsequent work in similar
setups in [16, 17, 12, 18, 19, 20]. As stressed in [12], the realisation of
discrete gauge symmetries via torsion in cohomology is related to the setting
with background fluxes by dualities such as mirror symmetry. This implies that
the same EFT features should be realised, namely: i) Stückelberg couplings
that are part of the EFT Lagrangian and ii) charged objects that lie above the
EFT cut-off. Indeed, as discussed in [15] such charged objects are given by
D-branes wrapping torsion cycles of $X_{n}$, which from the EFT perspective
look like particles and $(d-3)$-branes coupling to $A_{1}$ and $B_{d-2}$,
respectively.

From this simple observation, an apparent puzzle follows. If in this case
(1.1) and its Stückelberg completion appears in the lower-dimensional EFT, is
because torsion in cohomology is detected by the standard procedure of
$p$-form dimensional reduction. This is rather counter-intuitive, in the sense
that torsion cohomology groups are trivial in de Rham cohomology, or in other
words their elements can only be represented by exact $p$-forms. Since the EFT
data captured upon dimensional reduction typically involves integrals of
$p$-forms over $p$-cycles, it is a priori not clear how torsion cohomology
factors can translate into a Stückelberg Lagrangian term in the EFT. This
naive picture agrees with the standard lore that torsion in cohomology cannot
be detected via smooth $p$-forms, and that one should resort to more advanced
geometric techniques, like the computation of spectral sequences [21] or to
differential cohomology [22].

This paper addresses this puzzle and proposes a prescription to capture
torsion in cohomology via the standard procedure of dimensional reduction. The
basic idea is to use smeared delta forms to construct the integral basis in
which ten-dimensional fields are expanded in order to perform the reduction.
Here a delta $p$-form stands for the $p$-current $\delta_{p}(\Pi_{n-p})$ with
legs transverse to an $(n-p)$-cycle $\Pi_{n-p}\subset X_{n}$, while its
smeared version $\delta_{p}^{\rm sm}(\Pi_{n-p})$ corresponds to the projection
into the light eigen-$p$-forms of the Laplacian. If one projects
$\delta_{p}(\Pi_{n-p})$ to the zero-mode sector of the spectrum one simply
obtains a harmonic $p$-form which is the de Rham Poincaré dual of $\Pi_{n-p}$,
and torsion cycles are projected out. If however, one includes in the
projection those non-vanishing eigenmodes that correspond to massive $p$-form
fields entering the EFT, then torsion cycles can have a non-trivial smeared
delta form, and translate into quantities of the EFT Lagrangian.

More precisely, we propose that one should consider smeared delta forms of
calibrated cycles in order to build the basis for the dimensional reduction.
The physical intuition behind this proposal is that D-branes wrapping
calibrated cycles correspond to BPS objects with a controlled backreaction,
that one can use together with the picture developed in [23, 24, 25] to see
their smeared delta function as an EFT long-wavelength description of the
corresponding object. This can then be used to extract information from the
EFT, as done in [26, 27, 28] in the context of 4d ${\cal N}=1$
compactifications. In particular, D-branes wrapping calibrated torsion cycles
can be seen as BPS operators that gather information on the massive sector of
the EFT Lagrangian, like the kinetic terms of the fields that appear in (1.1).
As a direct consequence of our proposal, the linking number between two
calibrated torsion cycles can be computed using EFT data, or equivalently by
defining a smeared version of the torsion linking number, as summarised in
Conjecture 1.

The notion of calibrated torsion cycle or BPS operator with a $\mathbb{Z}_{N}$
charge may seem puzzling. From a geometric viewpoint, calibrations in special
holonomy manifolds are closed $p$-forms, and therefore they can never
calibrate a torsion $p$-cycle. This obstruction is however absent in the more
general set of manifolds with $G$-structure metrics, since there the exterior
derivative of a calibration does not need to vanish, and one can indeed
construct explicit examples with torsion $p$-cycles that are calibrated. From
a physics viewpoint, due to the no-force condition between mutually BPS
objects, one should always be able to stack an arbitrary number of them on top
of each other without any binding energy. This fits naturally with a
$\mathbb{Z}$-valued charge, but not with a $\mathbb{Z}_{N}$-valued one. To
address this issue we construct examples of BPS objects with $\mathbb{Z}_{N}$
charge, in the context of domain-wall solutions of type II string compactified
on half-flat manifolds [29]. We find that the process that annihilates $N$ BPS
D-branes wrapped on a torsion cycle is indeed possible topologically, but not
energetically favoured. As a result, it is energetically stable to stack an
arbitrary number of such objects with $\mathbb{Z}_{N}$ charge, as implied by
the BPS condition.

Expressing a delta form as a sum of eigenforms of the Laplacian is in general
involved, as it requires knowledge of the massive $p$-form spectrum of a
manifold. This difficulty is however less severe for three-dimensional
manifolds with isometries, a fact that we exploit to perform an explicit
computation of a torsion linking number and its smeared version in twisted
tori, in order to verify Conjecture 1. While it seems challenging to extend
such a computation to more general setups, one can provide physical evidence
that our proposal should also be valid in SU(3)-structure manifolds. Indeed,
using smeared delta forms of calibrated cycles as a basis for dimensional
reduction fits perfectly with the framework developed in [30, 31, 32] to
describe 4d ${\cal N}=2$ gauged supergravities as EFTs of type II string
compactifications and, in fact, one may argue that it is necessary for the
consistency of the approach. Similar considerations can be drawn in the
context of 4d ${\cal N}=1$ type II orientifold vacua, where the BPS torsion
objects are given by membranes ending on strings, and by space-time filling
branes ending on membranes.

In most of our examples it seems that an extension of Conjecture 1 is
required. In such a generalisation, the torsion linking number can be computed
not only when elements of ${\rm Tor}H_{n-p}(X_{n},\mathbb{Z})$ contain
calibrated representatives, but also when they can be expressed as linear
combinations of elements of $H_{n-p}(X_{n},\mathbb{Z})$, all of them with
calibrated representatives. This extension could in principle be applied to
compute torsion factors in the cohomology of Calabi–Yau manifolds, whenever
they contain light eigenforms of the Laplacian other than harmonic forms. One
may even speculate that our approach could be useful to compute torsion
linking numbers even in cases where torsion cycles cannot be related to
calibrated submanifolds, by providing an estimate of the associated error in
the linking number computation. In any event, our findings support that one
may compute certain torsion topological invariants in terms of smeared or EFT
data such as masses and kinetic terms, extending the dictionary between
geometry and physics to the more subtle and unexplored sector that is torsion
in cohomology.

The rest of the paper is organised as follows. In section 2 we describe what
is our proposal to compute the torsion linking numbers of a manifold via
smeared delta forms, as well as an extension of such proposal. In section 3 we
motivate the proposal from a physics viewpoint, by interpreting torsion
calibrated cycles as BPS objects of the EFT with a non-trivial backreaction.
In section 4 we analyse our proposal in the context of domain-wall solutions
of 4d ${\cal N}=2$ EFTs obtained from compactifications of type IIA string
theory on half-flat manifolds. The simplest examples of such manifolds are
based on twisted three-tori, for which our conjecture can be verified
explicitly using the techniques of section 5. Section 6 tests our proposal in
the context of general SU(3)-structure compactifications of type IIA string
theory, finding agreement with previous analysis in the literature and giving
a more precise prescription to perform the dimensional reduction in this
context. Section 7 extends our general strategy to 4d ${\cal N}=1$ type II
orientifold vacua, and section 8 contains some speculative remarks on how to
perform a further extension to the case where no EFT BPS objects are available
to detect torsion. We finally draw our conclusions in section 9.

Several technical details have been relegated to the appendices. Appendix A
analyses a mirror dual setup to that of section 4 from a microscopic
viewpoint, in order to classify the relevant set of BPS D-branes in both
backgrounds. Appendix B analyses the massive $p$-form spectrum for the case of
the twisted three-torus, as a necessary step to perform the direct computation
of the torsion linking number of section 5.

##  2 The proposal

Let us consider a compact manifold $X_{n}$ of real dimension $n$, and a
submanifold $\Pi_{p}\subset X_{n}$ which is a $p$-cycle. We can define a bump-
delta $(n-p)$-current or distributional form $\delta(\Pi_{p})$, such that

$\int_{X_{n}}\omega_{p}\wedge\delta(\Pi_{p})=\int_{\Pi_{p}}\omega_{p}\,,$
(2.1)

for any smooth $p$-form $\omega_{p}\in\Omega^{p}(X_{n})$. If $X_{n}$ is
endowed with a smooth metric $ds^{2}_{X^{n}}$ measured in string units
$\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$, one can solve the eigenvalue problem
for $(n-p)$-forms

$\Delta b_{n-p}^{i}=\lambda_{i}^{2}b_{n-p}^{i}\,,$ (2.2)

where $\Delta=d^{\dagger}d+dd^{\dagger}$ is the Laplace-de Rham operator,
$\\{b_{n-p}^{i}\\}_{i}$ is an orthonormal basis of eigenforms with respect to
the Hodge product, and $\\{\lambda_{i}^{2}\\}_{i}$ the corresponding set of
non-negative, dimensionless eigenvalues. Then one can expand the bump-delta
$(n-p)$-form on such a basis

$\delta(\Pi_{p})=\sum_{i}c_{i}\,b^{i}_{n-p}\,,\qquad
c_{i}=\int_{\Pi_{p}}*b^{i}_{n-p}\,,$ (2.3)

and from here define a smeared version of the delta-form, by keeping only
those terms in the expansion that satisfy $\lambda_{i}<\lambda_{\rm max}$, for
some choice of $\lambda_{\rm max}$. This defines a smooth bump $(n-p)$-form
localised within a tubular neighbourhood of radius $\ell_{s}/\lambda_{\rm
max}$ around $\Pi_{p}$. Such a $(n-p)$-form can be identified with the Thom
class of the normal bundle of $\Pi_{p}$, which is known to lie in the de Rham
Poincaré dual to $[\Pi_{p}]\in H_{p}(X_{n})$ [21]. Indeed, notice that all
elements $b^{i}_{n-p}$ of the expansion (2.3) must be exact $(n-p)$-forms
except those with vanishing eigenvalue, which must correspond to the harmonic
representative of the Poincaré dual to $[\Pi_{p}]$. It follows that if
$[\Pi_{p}]$ lies in a torsion class of $H_{p}(X_{p},\mathbb{Z})$ then
$\delta(\Pi_{p})$ must be a sum of exact $(n-p)$-forms.

In string theory compactifications there is a natural choice of metric for
$X_{n}$ that comes from solving the 10d supergravity equations of motion, as
well as a natural choice of $\lambda_{\rm max}$ that one identifies with the
compactification scale $m_{\rm KK}$. One can define $\ell_{s}m_{\rm KK}$ to be
the typical spacing between positive eigenvalues $\lambda_{i}$, oftentimes
estimated by the average radius Vol$(X_{n})^{1/n}$. Physically, we understand
$m_{\rm KK}$ as the energy scale below which we recover a $D$-dimensional EFT
description, with $D=10-n$, that describes all eigenmodes with
$\lambda_{i}\ll\ell_{s}m_{\rm KK}$ as $D$-dimensional fields. The standard
practice in the string literature is to either assume that only harmonic modes
satisfy the requirement $\lambda_{i}\ll\ell_{s}m_{\rm KK}$, such that the
procedure of dimensional reduction simply projects the spectrum of $p$-forms
to the harmonic sector.111Alternatively, one may set the EFT cut-off
$\Lambda_{\rm EFT}$ below any non-vanishing mode. However, it has been shown
that in certain compactification regimes, and in particular in six-dimensional
manifolds with SU(3)-structure [33, 34, 35, 36, 37, 38, 39, 9], one has a non-
vanishing $p$-form eigenvalues well below the compactification scale. This
will be the case of interest in this work, and henceforth our definition of
smeared delta-form will correspond to the following:

$\delta^{\rm sm}(\Pi_{p})=\sum_{\lambda_{i}\ll\ell_{s}m_{\rm
KK}}c_{i}\,b^{i}_{n-p}\,.$ (2.4)

Note that if $[\Pi_{p}]\in{\rm Tor}H_{p}(X_{p},\mathbb{Z})$ then (2.4) may
contain no terms at all and, if it does, it will be a sum of exact
$(n-p)$-forms. This reflects the difficulties in obtaining information from
the torsion (co)homology classes from the viewpoint of the lower dimensional
EFT, as integrals of (2.4) over any $(n-p)$-cycle of $X_{n}$ simply vanish.
There is however a well-defined topological invariant for torsion homology
classes, which is the torsion linking number. Given the torsion classes
$[\Pi_{p}]\in{\rm Tor}H_{p}(X_{n},\mathbb{Z})$ and $[\Pi_{n-p-1}]\in{\rm
Tor}H_{n-p-1}(X_{n},\mathbb{Z})$, one can define their linking number in terms
of the bump delta-forms of two of their representatives as [15]

$L(\Pi_{n-p-1},\Pi_{p})=\int_{X_{n}}d^{-1}\delta(\Pi_{n-p-1})\wedge\delta(\Pi_{p})\quad\mod\
1\,.$ (2.5)

Following [40], we can rewrite this quantity as follows. Notice that
$\\{\lambda_{i}^{-1}d*b^{i}_{n-p}\\}_{i}$ is an orthonormal basis of exact
$(p+1)$-forms, so one can perform the expansion

$\delta(\Pi_{n-p-1})=\sum_{i}\frac{e_{i}}{\lambda_{i}}d*b^{i}_{n-p}\,,\qquad
e_{i}=\frac{(-1)^{n(n-p)}}{\lambda_{i}}\int_{\Pi_{n-p-1}}d^{\dagger}b^{i}_{n-p}\,,$
(2.6)

from where one obtains

$L(\Pi_{n-p-1},\Pi_{p})=\sum_{i}\frac{c_{i}e_{i}}{\lambda_{i}}\quad\mod\ 1\,.$
(2.7)

One can now define a smeared linking number. From $[\Delta,d]=[\Delta,*]=0$ it
follows that $d*b_{n-p}^{i}$ has the same eigenvalue as $b_{n-p}^{i}$, and so
the smeared version of (2.6) corresponds to the same truncation as in (2.3).
Thus, it is natural to define the smeared analogue of (2.7) as

$L^{\rm sm}(\Pi_{n-p-1},\Pi_{p})=\sum_{\lambda_{i}\ll\ell_{s}m_{\rm
KK}}\frac{c_{i}e_{i}}{\lambda_{i}}\quad\mod\ 1\,.$ (2.8)

On the one hand, this quantity is not a topological invariant of $X_{n}$.
Unlike for (2.7), there is no reason for it to remain invariant under a
continuous deformation of either of the representatives $\Pi_{p}$ or
$\Pi_{n-p-1}$. On the other hand, as we argue in section 3, whenever (2.8) is
non-vanishing the massive sector of the $D$-dimensional EFT knows about the
value of (2.7), so there must be some way in which one can find about this
topological invariant in terms of smeared data.

In the following, we propose a solution to this conundrum, namely that one
needs to focus on certain minimal-volume representatives within the torsion
homology class. More precisely, we consider manifolds $X_{n}$ that contain
calibration $p$-forms, and torsion $p$-cycles that are calibrated by them.
Calibration $p$-forms are standard objects in manifolds endowed with metrics
of special holonomy [6], and from the viewpoint of string theory, they are
present whenever $X_{n}$ leads to a $D$-dimensional supersymmetric EFT [7, 8,
9]. In the case of special holonomy metrics calibrations are closed $p$-forms,
and therefore torsion $p$-cycles cannot be calibrated. This is however
different for manifolds with $G$-structure metrics, where several examples of
calibrated torsion cycles exist. An illustrative case for our discussion in
the following sections will be the case of six-dimensional manifolds with
SU(3)-structure, whose metric is specified by the pair of calibrations
$(J,\Omega)$, which can respectively calibrate two- and three-cycles that are
torsion or even trivial in homology. In terms of this language, our proposal
can be expressed as follows:

###### Conjecture 1

A non-trivial smeared linking number between two calibrated torsion cycles
equals their actual linking number.

From a physics viewpoint, D-branes wrapping calibrated cycles correspond to
BPS objects of different dimensions in the lower-dimensional supersymmetric
EFT. In this sense, Conjecture 1 can be understood as the equality between
(2.7) and (2.8) for the case of D-branes that wrap torsion cycles and that at
the same time are mutually BPS, that is, they preserve some common
supercharges in the lower-dimensional EFT. Notice that equating (2.7) to (2.8)
implies a cancellation in the contribution of very massive modes to the
torsion linking number. Physically this suggests that a protection mechanism
against threshold corrections must be in place, which is indeed a
characteristic feature of certain supersymmetric settings.

To make the proposal more precise, a number of comments are in order. First,
some of the compactification manifolds that we will consider correspond to
supersymmetric $D$-dimensional EFTs without vacua in the interior of their
field space. Instead, they describe supersymmetric solutions that probe a
family of metrics of $X_{n}$. For this reason, we require that the torsion
representatives that are BPS/calibrated must be so in a region of the EFT
field space, as opposed to in a single point. In particular, they must remain
calibrated upon local deformations of the metric that either are moduli or
involve energies below the compactification scale. Notice that, in general,
calibrated $p$-cycles in fixed homology classes can cross walls of marginal or
threshold stability when one deforms the metric of the compactification
manifold, so this condition is a significant restriction in the definition of
calibrated submanifolds, that we will dub strict calibration condition. In the
string theory literature, examples of BPS objects with this property are the
EFT strings and membranes defined in [26, 27, 28], so in this sense some the
objects of study in this work can be thought of as their torsion analogues.

Second, notice that Conjecture 1 implies that the smeared linking number
should not vary upon infinitesimal deformations of the embedding of the
torsion representatives that respect the calibration condition, which we will
dub as BPS deformations. In the following sections we argue that this is
indeed the case, by relating the coefficients $c_{i}$, $e_{i}$ with the volume
of the respective $p$-cycles. Now, since we are interested in metrics with
non-closed calibration $p$-forms, one may in principle encounter BPS
deformations that vary the $p$-cycle volume. We have not found instances of
this possibility for our more restrictive definition of calibrated cycle.
However, in case that it occurred to apply Conjecture 1 one should consider
the calibrated representative of $\Pi_{p}$ that locally minimises its volume
for a fixed metric in $X_{n}$.

Finally, using the bilinearity of the linking number one may extend the
conjecture to torsion $p$-cycles that are not calibrated by themselves, but
that are linear combinations of calibrated cycles. For instance, let us
consider a manifold $X_{n}$ with a $G$-structure metric and a pair of
$p$-cycles $\Pi_{p}$ and $\Pi_{p}^{\prime}$ calibrated by the same
calibration, both in the strict sense, that correspond to the same class on
$H_{p}(X_{n},\mathbb{R})$, but such that $[\Pi_{p}^{\rm
tor}]=[\Pi_{p}^{\prime}]-[\Pi_{p}]$ is a non-trivial element of ${\rm
Tor}H_{p}(X_{n},\mathbb{Z})$. Then one may smear out both delta-forms
separately, and define a smeared description of the torsion two-cycle as

$\delta^{\rm sm}(\Pi_{p}^{\prime})-\delta^{\rm sm}(\Pi_{p})\,.$ (2.9)

By construction, this is an exact smooth $(n-p)$-form, from where one can
extract the coefficients $c_{i}$ as in (2.4). A different realisation of
$\Pi_{p}^{\rm tor}$ in terms of BPS cycles, like for instance a representative
$\Pi_{p}^{\rm tor}$ that is BPS by itself, may give rise to different
coefficients $c_{i}$. However, the extension of the conjecture would imply
that all these choices give rise to the same smeared linking number with a
given BPS torsion $(n-p-1)$-cycle. Notice that with this extension one may not
only compute torsion linking numbers via smeared data in $G$-structure
manifolds with non-closed calibrations, but also in manifolds with metrics of
special holonomy.

To sum up, our proposal means that for manifolds endowed with certain metrics,
one can compute some torsion invariants in terms of smeared/massive EFT data.
One only needs i) the eigenforms of the Laplacian that correspond to their
lowest eigenvalues and ii) the projection of torsion, strict-calibrated cycles
into them.

##  3 Localised sources and dimensional reduction

The aim of this section is to motivate the content of Conjecture 1 from a
physics viewpoint, by considering the effect of localised sources in
compactifications of string theory. If these sources wrap torsion cycles in
the compact dimensions and couple to the massive fields present in the lower-
dimensional EFT, then by consistency of the low-energy description there
should be terms in the EFT Lagrangian that know about their torsion linking
number. The reason is that, in this case, the EFT contains localised objects
charged under a discrete gauge symmetry (the torsion cohomology group) with a
non-trivial backreaction at EFT wavelengths. The corresponding EFT Lagrangian
has the form proposed in [15] (see also [16, 17, 12, 18, 19, 20]) to describe
torsion in (co)homology from the viewpoint of dimensional reduction. However,
this does not guarantee that one can compute the torsion linking number from
smeared data. For this, one in addition needs that such localised sources
appear as BPS objects of the EFT.

###  3.1 Localised sources in ten and four dimensions

For concreteness, let us consider a static D4-brane in 10d, with worldvolume
$\Sigma_{5}=\mathbb{R}\times\Sigma_{4}\subset\mathbb{R}^{1,9}$. Its
backreaction sources a RR field strength $F_{4}=dC_{3}$, such that
$dF_{4}=\delta_{5}(\Sigma_{5})$ corresponds to the bump delta 5-form with
support on $\Sigma_{5}$. On a 4-sphere $S^{4}$ surrounding this source, the
pullback of $F_{4}$ is of the form $2\pi\Phi_{S^{4}}$, where $\Phi_{S^{4}}$ is
such that $\int_{S^{4}}\Phi_{S^{4}}=1$. Analogously to the Wu-Yang description
of a 4d monopole, we need to describe the potential $C_{3}|_{S^{4}}$ as a
connection, more precisely as the connection of a 2-gerbe on $S^{4}$, see e.g.
[41].

Let us recall how a probe D2-brane feels this background. In particular, let
us consider the case where its worldvolume $\Sigma_{3}$ sweeps a 3-sphere
$S^{3}$ at the equator of $S^{4}$. In analogy with the Wu-Yang monopole, the
non-trivial pull-back of $F_{4}$ on $S^{4}$ has the effect that
$C_{3}|_{S^{3}}$ is non-trivial in the cohomology of $S^{3}$. However, one can
still globally write it as $d\lambda_{2}$, where $\lambda_{2}$ is not a
globally well-defined smooth two-form, but nevertheless
$e^{i\int_{\Pi_{2}}\lambda_{2}}$ is well-defined for any two-cycle $\Pi_{2}$
inside $S^{3}$. This property amounts to saying that the wavefunction of the
probe D2-brane is well-defined in the backreacted background of its magnetic
dual.

We now consider the particular 10d background $\mathbb{R}^{1,3}\times X_{6}$,
with $X_{6}$ a compact manifold with a given metric. If the D4-brane wraps a
4-cycle $\Pi_{4}\subset X_{6}$, then it will look like a point-like source in
the 4d EFT, very much like a monopole. Its backreaction may be described by a
2-gerbe in the microscopic 10d picture, but its effective description at
wavelengths larger than $1/m_{\rm KK}$ should correspond to a bundle similar
to that of the Wu-Yang monopole. This is indeed the case whenever $[\Pi_{4}]$
is a non-trivial class in $H_{4}(X_{6},\mathbb{R})$. A probe D2-brane wrapping
a two-cycle $\Pi_{2}\subset X_{6}$ with non-trivial transverse intersection
$\Pi_{2}\cdot\Pi_{4}=Q$ looks, from the 4d viewpoint, like a (test) particle
circling around the monopole-like source. The pull-back of $F_{4}$ on
$S^{2}\times\Pi_{2}$ with the two-sphere surrounding the source reads:

$2\pi\left(\Phi_{S^{2}}+d\chi_{1}\right)\wedge\delta_{2}(\Pi_{4})|_{\Pi_{2}}=2\pi
Q\left(\Phi_{S^{2}}+d\chi_{1}\right)\wedge\left(\Phi_{\Pi_{2}}+d\tilde{\chi}_{1}\right),$
(3.1)

where the $\Phi$’s are volume forms normalised to unity, $\chi_{1}$ and
$\tilde{\chi}_{1}$ are globally well-defined one-forms on $S^{2}$ and
$\Pi_{4}$, respectively, and $\delta_{2}(\Pi_{4})$ is the bump delta two-form
of $\Pi_{4}$ in $X_{6}$. The result gives an integral of $2\pi Q$, that
corresponds to the product of electric and magnetic charges. We can now
restrict our attention to $\gamma\times\Pi_{2}$, where $\gamma$ is at the
equator of $S^{2}$. The difference of connections $C_{3}$ on two patches
overlapping over this submanifold can be written as

$C_{3}=d\lambda_{2},\qquad\text{with}\quad\lambda_{2}=\lambda\,Q\left(\Phi_{\Pi_{2}}+d\tilde{\chi}_{1}\right),$
(3.2)

and $\lambda$ a function $\lambda:\gamma\to S^{1}$ with a single winding. When
we describe this system at energies well below the compactification scale, we
simplify the internal profile for $\lambda_{2}$. In the effective description,
one replaces $\delta_{2}(\Pi_{4})$ by a harmonic two-form
$\omega_{2}^{\Pi_{4}}$ in the Poincaré dual class to $[\Pi_{4}]$, which is the
lowest lying mode (the harmonic piece) of the Kaluza–Klein (KK) decomposition
of $\delta_{2}(\Pi_{4})$. Therefore, we write $\lambda_{2}=\lambda
Q\omega_{2}^{\Pi_{4}}$, as a more detailed profile would involve gauge
transformations for massive $U(1)$’s that are beyond our 4d EFT description.

If we now assume that $[\Pi_{4}]\in{\rm Tor}H_{4}(X_{6},\mathbb{Z})$, then
$\omega_{2}^{\Pi_{4}}$ vanishes, and the presence of a D4-brane wrapped on
$\Pi_{4}$ remains undetected by any D2-brane wrapping a two-cycle
$\Pi_{2}\subset X_{6}$. Instead, as discussed in [15], one needs to consider a
D4-brane wrapping a three-cycle $[\Pi_{3}]\in{\rm
Tor}H_{3}(X_{6},\mathbb{Z})$, which is perceived by the low-energy EFT as a 4d
string. Let us for simplicity place this 4d string in
$\mathbb{R}^{1,1}\subset\mathbb{R}^{1,3}$, and take $z=re^{2\pi i\theta}$ to
be the complex coordinate transverse to its worldsheet. This time one can
provide a global description of the RR potential sourced by the D4-brane

$C_{3}=2\pi d\left(\theta\rho_{2}\right),$ (3.3)

where $\rho_{2}$ is a two-form on $X_{6}$ such that
$d\rho_{2}=\delta_{3}(\Pi_{3})$ is the bump delta three-form of $\Pi_{3}$.
This background is detected by a D2-brane with worldvolume
${\gamma}\times\Pi_{2}$, where $[\Pi_{2}]\in{\rm Tor}H_{2}(X_{6},\mathbb{Z})$
is a torsion class with linking number $L\in\mathbb{Q}$ with respect to
$[\Pi_{3}]$, and $\gamma$ is a 4d worldline surrounding once the string
location $\\{z=0\\}$. If we pull back (3.3) into the D2-brane worldvolume we
obtain

$C_{3}|_{{\gamma}\times\Pi_{2}}=2\pi
L\left(d\phi+df(\phi)\right)\wedge\left(\Phi_{\Pi_{2}}+d\tilde{\chi}_{1}\right),$
(3.4)

where $\phi\in\mathbb{R}/\mathbb{Z}$ parametrises $\gamma$ and $f(\phi)$ is a
periodic function in it, and we have used that $\int_{\Pi_{2}}\rho_{2}=L$.
Therefore, we obtain that $C_{3}=d\lambda_{2}$, with $\lambda_{2}$ of the form
(3.2), except for the replacement $Q\to L$. The fact that $L$ is not an
integer number implies that the D2-brane picks a non-trivial phase $e^{2\pi
iL}$ when circling around $\gamma$, which is a trait of 4d Aharanov-Bohm (AB)
strings and signals the presence of a discrete gauge symmetry [10].

Let us describe the discrete gauge symmetry in terms of the gauge
transformations involved in the backreacted D4-brane background. At the
microscopic 10d level, these are of the form

$d\left(\lambda\rho_{2}\right)=d\lambda\wedge\rho_{2}+\lambda\,\delta_{3}(\Pi_{3}),$
(3.5)

with $\lambda$ well-defined on loops on $\mathbb{R}^{1,3}$ but not on
$\mathbb{R}^{1,3}$ itself. It now remains to see what is the long-wavelength
4d EFT description of this transformation. As already discussed
$\delta_{3}(\Pi_{3})$ has no harmonic component, and the same can be assumed
for $\rho_{2}$.222A priori nothing forbids $\rho_{2}$ to have a harmonic
piece, which would even be required if we impose that
$\int_{\Pi_{2}}\rho_{2}\in\mathbb{Z}$ for any two-cycle $\Pi_{2}$ [42].
Following [43], this piece would imply a non-trivial kinetic mixing between
massive and massless $U(1)$’s of the compactification, which could then be
removed by an appropriate change of basis. To simplify the discussion, here we
assume the absence of such a harmonic piece. The question is then if
$\delta_{3}(\Pi_{3})$ has a non-trivial projection into the massive field
content of the 4d EFT spectrum, or in other words if it has a non-trivial 4d
smearing. If it does, the D4-brane backreaction should be seen by the 4d EFT,
in the sense that it sources some of it fields, that pick a non-trivial
profile involving wavelengths above $1/m_{\rm KK}$. So in the following we
will assume that $\delta_{3}(\Pi_{3})$ has a non-trivial 4d smearing, which is
also necessary for Conjecture 1 to provide a non-trivial statement.

For simplicity let us assume that $X_{6}$ is such that ${\rm
Tor}H_{3}(X_{6},\mathbb{Z})=\mathbb{Z}_{N}$. By the Universal Coefficient
Theorem [21] and Poincaré duality this implies that ${\rm
Tor}H_{2}(X_{6},\mathbb{Z})=\mathbb{Z}_{N}$, and also that $LN\in\mathbb{Z}$.
Let us in addition assume that there is a single exact eigen-three-form
$b_{3}$ of the Laplacian with unit norm and a non-vanishing eigenvalue below
the compactification scale. That is, we have a unique solution of the form
$dd^{{\dagger}}b_{3}=\lambda_{\rm st}^{2}b_{3}$, with $m_{\rm
st}=\ell_{s}^{-1}\lambda_{\rm st}\ll m_{\rm KK}$. Then to obtain our 4d EFT
via dimensional reduction we must consider the following set of $p$-forms

$b_{3},\qquad*b_{3},\qquad\lambda_{\rm st}^{-1}d*b_{3},\qquad\lambda_{\rm
st}^{-1}d^{\dagger}b_{3},$ (3.6)

all of them with unit norm and the same eigenvalue, because they are
associated with the same mass scale. The standard dimensional reduction
procedure consists of expanding the 10d $p$-form potentials in a basis of
harmonic forms plus the above, non-harmonic set. For instance, reducing the
type IIA three-form $C_{3}$ to 4d with respect to the above non-harmonic
sector gives

$C_{3}=2\pi\ell_{s}^{3}\left(A_{1}\wedge\omega_{2}+C_{0}\,\beta_{3}\right),$
(3.7)

where $A_{1}$ and $C_{0}$ describe a 1-form and a 0-form in 4d, respectively,
and we have defined

$\beta_{3}=f\,b_{3},\qquad\omega_{2}=\frac{1}{g\lambda_{\rm
st}}d^{\dagger}b_{3},\qquad\text{with}\quad f,g\in\mathbb{R}.$ (3.8)

so $d\omega_{2}=N_{\rm eff}\beta_{3}$ with $N_{\rm eff}=\frac{\lambda_{\rm
st}}{fg}$. The reduction to 4d of the 10d kinetic term $\int F_{4}^{2}$ gives

$(2\pi\hat{f})^{2}\left(dC_{0}-N_{\rm
eff}A_{1}\right)^{2}+\frac{4\pi^{2}}{g^{2}}(dA_{1})^{2},\qquad\hat{f}\coloneqq
fe^{\phi_{4}}M_{\rm P}=\frac{fe^{\phi}}{(4\pi{\rm Vol}_{X_{6}})^{1/2}}M_{\rm
P},$ (3.9)

namely a Stückelberg-like Lagrangian, where $e^{\phi_{4}}$ is the 4d and
$e^{\phi}$ the 10d dilaton, Vol${}_{X_{6}}$ is the volume of $X_{6}$ in string
units, and $M_{\rm P}$ the 4d Planck mass. This is precisely the dimensional
reduction scheme proposed in [15] to describe discrete gauge symmetries from
torsion in cohomology, if one imposes the constraint $N_{\rm eff}=N$ and
treats $C_{0}$ as an axion-like particle of unit periodicity $C_{0}\sim
C_{0}+1$. In this case, the discrete gauge symmetry is generated by the shift

$2\pi C_{0}\to 2\pi C_{0}+\lambda,\qquad 2\pi A_{1}\to 2\pi
A_{1}+\frac{d\lambda}{N},$ (3.10)

with $\lambda\in 2\pi\mathbb{Z}$. A particle with charge $NL$ under $A_{1}$
will pick up a phase $e^{2\pi iL}$ upon (3.10), for instance when circling a
string of unit charge. This is how the 4d EFT reflects the linking number
between torsion cycles on $X_{6}$, and in particular that ${\rm
Tor}H_{3}(X_{6},\mathbb{Z})\simeq$ ${\rm
Tor}H_{2}(X_{6},\mathbb{Z})=\mathbb{Z}_{N}$. So while at this point we have
not determined the parameters $f$ and $g$, consistency of the 4d EFT requires
that they are constrained by $fg=\frac{\lambda_{\rm st}}{N}$. Therefore we
have the relation

$\frac{1}{N}=\frac{fg}{\lambda_{\rm st}}=\frac{\hat{f}g}{m_{\rm st}}.$ (3.11)

Note that the expression in the middle resembles a smeared linking number, as
defined in (2.7), while the rhs corresponds to how the EFT massive sector
encodes this quantity.

The 4d effective Lagrangian (3.9) should be sufficient to give a long-
wavelenght description (more precisely in the range $(m_{\rm KK}^{-1},m_{\rm
st}^{-1}))$ of the backreaction of D4-branes wrapping torsion three-cycles of
$X_{6}$. Recalling our 10d analysis, one may try to provide such a 4d
description by directly smearing the 10d solution, that is by projecting the
background (3.3) into the massive sector (3.6). However, if one does so the
gauge transformation (3.5) translates into

$2\pi C_{0}\to 2\pi C_{0}+\frac{c}{f}\lambda,\qquad 2\pi A_{1}\to 2\pi
A_{1}+\frac{c}{f}\frac{d\lambda}{N},$ (3.12)

where $\delta_{3}^{\rm sm}(\Pi_{3})=cb_{3}$, and we have imposed that $N_{\rm
eff}=N$. So only when $c=f$ we recover the expected gauge transformation
(3.10). While this may seem surprising, it does not necessarily indicate any
inconsistency of the 4d EFT. Instead, one may interpret it as the fact that
smearing a 10d background is a classical procedure that may be subject to
corrections, like quantum corrections associated to the fields above the
compactification scale that one has truncated. So in principle, it could be
that these or other corrections modify the backreacted 4d background in such a
way that the quotient $c/f$ disappears from (3.12), and one recovers the gauge
transformation (3.10) consistent with (3.9). If this was the case, (3.11)
should be interpreted as a smeared linking number after corrections have been
taken into account.

This proposal to solve the apparent inconsistency in (3.12) has the downside
that it does not give a clear geometric prescription to compute the parameters
$f$ and $g$ which, together with $\lambda_{\rm st}$, are the 4d EFT data that
allow us to compute $N$. However, it gives us the guideline that one should
try to consider D4-branes whose smeared backreaction does not suffer important
corrections upon dimensional reduction. From a physics viewpoint, the best
candidates to display this feature are D4-branes that preserve some
supersymmetry of the background, namely BPS objects of the EFT, as the results
of [44] also suggest. In the next subsection we will argue why this is the
right answer.

Finally, it is instructive to perform the dimensional reduction of the RR
potential $C_{5}$, dual to $C_{3}$ in 10d. An expansion in the relevant non-
harmonic $p$-forms (3.6) gives

$C_{5}=2\pi\ell_{s}^{5}\left(V_{1}\wedge\tilde{\omega}_{4}+B_{2}\wedge\alpha_{3}\right),$
(3.13)

where $V_{1}$ and $B_{2}$ are a 4d 1-form and 2-form in 4d, and we have
defined

$\tilde{\omega}_{4}=\frac{g}{\lambda_{\rm
st}}d*b_{3},\qquad\alpha_{3}=f^{-1}*b_{3}.$ (3.14)

such that
$\int_{X_{6}}\omega_{2}\wedge\tilde{\omega}_{4}=\int_{X_{6}}\alpha_{3}\wedge\beta_{3}=1$,
as in [29, 30, 31, 32]. This is required for the fields $(V_{1},B_{2})$ to be
quantised 4d duals to $(A_{1},C_{0})$. It also implies that
$d\alpha_{3}=N_{\rm eff}\tilde{\omega}_{4}=N\tilde{\omega}_{4}$, so upon
dimensional reduction one obtains

$(2\pi
g)^{2}\left(dV_{1}+NB_{2}\right)^{2}+\frac{4\pi^{2}}{\hat{f}^{2}}(dB_{2})^{2}.$
(3.15)

This 4d effective Lagrangian should describe the backreaction of D2-branes
wrapping torsion two-cycles $\Pi_{2}$ of $X_{6}$, in the long-wavelength
approximation. In these dual variables the discrete gauge symmetry reads

$2\pi V_{1}\to 2\pi V_{1}-\lambda_{1},\qquad 2\pi B_{2}\to 2\pi
B_{2}+\frac{d\lambda_{1}}{N}.$ (3.16)

A D2-brane wrapping a torsion two-cycle $\Pi_{2}$ such that $\delta_{4}^{\rm
sm}(\Pi_{2})=\frac{e}{\lambda_{\rm st}}d*b_{3}$ will not generate this shift
via its backreaction, unless $e=g$. Again, one could interpret this mismatch
as the result of non-trivial quantum corrections, and argue that the equality
should hold for D2-branes wrapping calibrated two-cycles, as we proceed to
argue.

###  3.2 The supersymmetric case

One may summarise the reasoning of the previous subsection as follows. A
D4-brane wrapping a torsion three-cycle $\Pi_{3}$ of $X_{6}$ becomes, upon
compactification to 4d, an Aharanov-Bohm 4d string that realises ${\rm
Tor}H_{3}(X_{6},\mathbb{Z})=\mathbb{Z}_{N}$ as a discrete gauge symmetry. This
object will be perceived by the 4d EFT if it couples to some massive $p$-form
modes below the compactification scale $m_{\rm KK}$. In that case the
backreaction has size $m_{\rm st}=\lambda_{\rm st}/\ell_{s}\ll m_{\rm KK}$,
where $\lambda_{\rm st}$ is the eigenvalue of such massive eigenmodes, and
there must be a term in the 4d EFT that describes such a backreacted solution
at long wavelengths. This 4d Lagrangian term is (3.9), with $N_{\rm
eff}=\lambda_{\rm st}/fg=N$ encoding the topological information of the
torsion homology group. Knowledge of the massive spectrum and of the
parameters $f,g\in\mathbb{R}$ thus allows us to compute torsion cohomology
groups, and to represent them via smooth $p$-forms (3.8) and (3.14) that are,
from the 4d viewpoint, analogous to the harmonic representatives of de Rham
cohomology groups. The parameters $f$ and $g$ are not determined from the 4d
smearing of the backreaction of a D4-brane wrapping an arbitrary torsion
three-cycle, since in general there can be significant corrections to the
smeared background. Notice that these parameters are intrinsic of the 4d EFT,
and so they only depend on the topology and metric of $X_{6}$.

There is however a particular class of 4d strings for which quantum
corrections should be under control, namely BPS fundamental strings of the
EFT. We are particularly interested in D4-branes that correspond to the 4d EFT
strings of [26, 27, 28], except that they source 4d axions with a mass $m_{\rm
st}$ induced by a Stückelberg coupling. As stressed in [26], near the string
core and at wavelenghts below $m_{\rm st}^{-1}$ one should be able to describe
the 4d backreaction of these objects with a solution similar to that of
standard EFT strings, implying that their tension is determined by the kinetic
terms of the 4d EFT Lagrangian, and in particular by parameters like $f$.

Geometrically, the BPS condition means that the torsion three-cycle $\Pi_{3}$
is calibrated by a complex three-form $\Omega$. This is not possible when
$X_{6}$ is a Calabi–Yau, but it occurs in SU(3)-structure manifolds with a
metric specified by $(J,\Omega)$ and a non-vanishing intrinsic torsion,333The
two meanings of the word torsion should not be confused. By intrinsic torsion
we mean the five torsion classes which enter in the description of manifolds
with SU(3)-structure metrics, and which show up in the derivatives of the
globally well-defined forms $\Omega$ and $J$ [33, 34, 35, 36, 37, 38, 39, 9].
In any other instance, the word torsion refers to torsion classes in
(co)homolgy groups of $X_{6}$, and to their representatives. which we will
assume in the following. Notice that the calibration condition selects a
specific representative within the torsion class $[\Pi_{3}]\in{\rm
Tor}H_{3}(X_{6},\mathbb{Z})$, that directly depends on the metric of $X_{6}$.
Therefore, it is reasonable to assume that $f$, which also depends on the
metric of $X_{6}$, can be computed from $\delta_{3}(\Pi_{3})$ with $\Pi_{3}$
calibrated. More precisely, we will argue that $f$ can be computed from the
smeared delta-form $\delta_{3}^{\rm sm}(\Pi_{3})$.

To see this, let us assume that upon compactification of type IIA string
theory on $X_{6}$ we recover a 4d EFT with ${\cal N}=2$ supersymmetry.
According to [31, 32] this is guaranteed if $X_{6}$ is an SU(3)-structure
manifold with calibrations $(J,\Omega)$. Following their approach, we may
expand $J$ and $\Omega$ in the set of harmonic two- and three-forms,
respectively, plus the non-harmonic set (3.6). Let us first consider $\Omega$
and assume an expansion of the form $\Omega=\Omega^{\rm
harm}+ia\,\alpha_{3}+b\,\beta_{3}$, where $\Omega^{\rm harm}$ is a sum of
harmonic three-forms and $a$, $b$ are real functions of the 4d fields. If we
impose the condition $*\Omega=-i\Omega$ we find that the more precise form

$\Omega=\Omega^{\rm harm}+{\rm
Vol}_{X_{6}}^{1/2}\left(if\alpha_{3}+f^{-1}\beta_{3}\right),$ (3.17)

where we have taken into account that
$\int_{X_{6}}i\bar{\Omega}\wedge\Omega=8{\rm Vol}_{\rm X_{6}}$, and that
$\alpha_{3},\beta_{3}$ are orthogonal to any harmonic form. Notice that here
$f$ is not a fixed number, but depends on the choice of SU(3)-metric or, from
the 4d viewpoint, on the vevs of the 4d scalar fields.

Next, we use that for a 4d BPS string its tension is proportional to the
kinetic term of the axion to which it couples magnetically. In the case at
hand, the orthogonality of the massive modes implies that the string charge-
to-mass ratio equals one, and so for a string inducing a single winding of
$C_{0}$ around its core the tension is determined by the axion decay constant
in (3.9) as ${\sqrt{\pi}}\hat{f}M_{\rm P}$. This quantity should correspond to
the 4d string tension obtained from a D4-brane wrapped on a three-cycle
$\Pi_{3}$ calibrated by $\Omega$, see [27, section 6.4]. We thus find

$\frac{\hat{f}}{M_{\rm P}}=\frac{e^{\phi}}{{\sqrt{4\pi}}{\rm
Vol}_{X_{6}}}\left|\int_{\Pi_{3}}\Omega\right|\implies f={\rm
Vol}_{X_{6}}^{-1/2}\left|\int_{X_{6}}\Omega\wedge\delta_{3}(\Pi_{3})\right|.$
(3.18)

Notice that in the second equation we can replace
$\delta_{3}(\Pi_{3})\to\delta_{3}^{\rm sm}(\Pi_{3})$. Using that $\Pi_{3}$ is
a torsion three-cycle and therefore $\delta_{3}(\Pi_{3})$ is an exact three-
form we finally obtain

$\delta_{3}^{\rm sm}(\Pi_{3})=\beta_{3}.$ (3.19)

That is, $f$ can be found from smearing the bump delta-form of a calibrated
torsion three-cycle.

Similarly, one may consider a D2-brane wrapping a BPS representative of
$[\Pi_{2}]\in{\rm Tor}H_{2}(X_{6},\mathbb{Z})$, or in other words $\Pi_{2}$ is
calibrated by $J$. A BPS particle of unit charge with respect to $A_{1}$ will
have a mass $gM_{\rm P}$, so putting both statements together results in the
equality

$g={\rm
Vol}_{X_{6}}^{-1/2}\left|\int_{X_{6}}J\wedge\delta_{4}(\Pi_{2})\right|.$
(3.20)

Again, expanding $e^{iJ}$ in harmonic and non-harmonic forms and using the
Hodge duality relations translates into the equality

$\delta_{4}^{\rm sm}(\Pi_{2})=\tilde{\omega}_{4}.$ (3.21)

Equivalently, $g$ results from smearing the bump delta-form of a calibrated
torsion two-cycle.

Notice that in this construction the torsion cycles $\Pi_{3}$ and $\Pi_{2}$
that lead to $f$ and $g$ have a minimal 4d charge and tension. Therefore we
expect them to generate ${\rm Tor}H_{3}(X_{6},\mathbb{Z})$ and ${\rm
Tor}H_{2}(X_{6},\mathbb{Z})$, respectively, and to have a linking number
$1/N\mod 1$. When plugging the values of $f$ and $g$ into the smeared linking
number one indeed finds that $L^{\rm sm}(\Pi_{2},\Pi_{3})=1/N$, in agreement
with Conjecture 1. If instead $\Pi_{2}$ corresponds to a particle of charge
$LN$, then repeating the same reasoning its smeared delta-form will have to be
multiplied by $LN$, and we will recover a smeared linking number of $L$, again
supporting the conjecture.

An interesting point is that, when dealing with mutually BPS objects, one
should be able to add up their tensions to compute the energy of the total
system. Geometrically, this amounts to say that even if the topological charge
of a calibrated torsion $p$-cycle or a sum of them lives in $\mathbb{Z}_{N}$,
its central charge lives in a lattice. This does not imply any contradiction
with the $\mathbb{Z}_{N}$ discrete system of the 4d EFT, provided that the
process that reduces the number of BPS objects by $N$ has a non-vanishing
energy which compensates for the loss of $N$ $p$-cycles. To illustrate this,
let us consider the torsion three-cycle class $[\Pi_{3}]$ generating ${\rm
Tor}H_{3}(X_{6},\mathbb{Z})$ in a SU(3)-structure manifold. A set of $N$
D4-branes wrapping calibrated representatives $\Pi_{3,i}$ of this class looks
like $N$ BPS strings in 4d. These can end on a 4d monopole, made up of a
D4-brane wrapping a four-chain $\Sigma_{4}$ whose boundary is given by
${\partial}\Sigma_{4}=\sum_{i}\Pi_{3,i}$ [15]. On the one hand, using Stokes’
theorem one can relate the sum of string tensions with the integral of
$d\Omega$ over $\Sigma_{4}$. On the other hand, the mass of the 4d monopole is
proportional to the volume of $\Sigma_{4}$ which, if the monopole is BPS, is
given by the integral of $\pm\frac{1}{2}J\wedge J$ over $\Sigma_{4}$.
Therefore we find that the marginal stability of $N$ BPS AB strings implies

$\ell\left|\int_{X_{6}}d\Omega\wedge\delta_{2}(\Sigma_{4})\right|+\left|\frac{1}{2}\int_{X_{6}}J\wedge
J\wedge\delta_{2}(\Sigma_{4})\right|={\rm const.}$ (3.22)

where $\ell$ is the length of the AB string in $\ell_{s}$ units. Notice that
this relation can only make sense if the monopole mass depends on $\ell$,
which should then be a feature of backgrounds with BPS AB strings and
particles. We postpone a more precise explanation of this statement to the
next section, where both quantities in (3.22) will be evaluated in a simple
setup based on half-flat manifolds. For the time being, it is worth pointing
out that the above reasoning leads to an interpretation of the non-closed two-
form $\omega_{2}$ in (3.8). Indeed, notice that in (3.22) we can replace
$\delta_{2}(\Sigma_{4})\to\delta_{2}^{\rm sm}(\Sigma_{4})$ and that because
the action of smearing commutes with the exterior derivative,
$d\delta_{2}^{\rm sm}(\Sigma_{4})=N\delta_{3}^{\rm sm}(\Pi_{3})=N\beta_{3}$.
It is thus natural to guess that $\delta_{2}^{\rm sm}(\Sigma_{4})=\omega_{2}$
when $\Sigma_{4}$ is a calibrated four-chain, something that can be verified
by noting that a 4d BPS monopole of unit charge has mass $g^{-1}M_{\rm P}$,
and running a reasoning analogous to the previous ones. Similarly, one can
deduce that $\delta^{\rm sm}_{3}(\Sigma_{3})=\alpha_{3}$, where $\Sigma_{3}$
is a calibrated three-chain ending on $N$ calibrated torsion two-cycles.
Therefore, one concludes that the set of harmonic plus non-harmonic forms in
which one expands $J$, $\Omega$ and the RR potentials to obtain the 4d fields
can be interpreted as smeared delta-forms of a basis of calibrated chains and
cycles. Notice that this fits well with the notion that the set of forms
$\\{\omega_{2},\alpha_{3},\beta_{3},\tilde{\omega}_{4}\\}$ reflect
quantisation features of the 4d EFT, like axions of unit periodicity and
$U(1)$ gauge symmetries. This quantisation also implies that these $p$-forms
generate a lattice just like quantised harmonic $p$-forms do, which seems to
be in contradiction with the fact that these D-brane charges are torsion.
However, as mentioned above when dealing with mutually BPS objects the
mass/tensions are additive, which explains the lattice structure. Finally,
while here we have considered a very simple case, it is reasonable to expect
that this description of the reduction basis of $p$-forms extends to the
general framework of SU(3)-structure manifold dimensional reduction analysed
in [31, 32], as we will further discuss in section 6.

It is also instructive to consider what happens when we slightly depart from a
BPS embedding. In particular, let us take a D4-brane wrapping a calibrated
torsion three-cycle $\Pi_{3}$ and perform a small deformation of its
embedding, such that the torsion linking number with a calibrated torsion two-
cycle $\Pi_{2}$ does not change

$L(\Pi_{2},\Pi_{3})=\sum_{i}\frac{c_{i}e_{i}}{\lambda_{i}}\,.$ (3.23)

Notice that this quantity is not defined mod 1. Geometrically, this means that
upon the deformation $\Pi_{3}$ does not cross $\Pi_{2}$. A simple deformation
of this sort that changes the smeared linking number takes the form

$c_{\rm KK}\to c_{\rm KK}-{\epsilon},\qquad c_{\rm st}\to c_{\rm
st}+\frac{\lambda_{\rm st}}{\lambda_{\rm KK}}{\epsilon},$ (3.24)

where $c_{\rm KK}$ represents the coefficient of a mode above the
compactification scale and $c_{\rm st}$ one below. We thus find that the naive
gauge transformation (3.12) changes with a suppression factor of $m_{\rm
st}/m_{\rm KK}$ with respect to the BPS case. This is indeed the kind of
suppression that one would expect from integrating out massive operators at
the Kaluza-Klein scale, which supports the interpretation that the expected
discrete gauge transformation (3.10) could be restored for the non-BPS case,
once that quantum corrections are taken into account. It would however be
important to perform a more direct test of this proposal.

###  3.3 Generalisations

In our discussion so far we have focused in a type IIA setup, in which D4- and
D2-branes look respectively like strings and particles in the 4d EFT. However,
it is clear that the same reasoning can be applied to any other kind of string
compactifications, as long as the 4d picture is similar. For instance, in type
IIB compactified in a SU(3)-structure manifold, Aharanov-Bohm strings and
particles would be realised by D3-branes wrapping torsion two- and three-
cycles. There are other extended objects that can give rise to 4d AB strings
and particles [15], but in many instances they do not wrap calibrated cycles,
and so the BPS property, which is an important ingredient of our logic, is
missing.

Nevertheless, one may extend our reasoning in yet another direction, since
there are other BPS objects in a 4d EFT that encode torsion in cohomology.
Indeed, a key property of AB strings with $\mathbb{Z}_{N}$ charge is that $N$
of them can end on a monopole, while $N$ AB particles can end on a 4d
instanton [10]. As a general rule, $\mathbb{Z}_{N}$ charges are detected in
the 4d theory by $p$-branes ending on $(p-1)$-branes with $p=0,1,2,3$, and in
certain instances these $\mathbb{Z}_{N}$ charges reflect torsion cohomology
groups of the compactification manifold [12]. In our previous discussion we
have focused on the cases $p=0$ and $p=1$, which are typically represented in
4d EFT language by the Lagrangians (3.9) and (3.15), respectively, and are
dual to each other. The case $p=2$ corresponds to 4d membranes ending on
strings, and it is related to the following EFT Lagrangian piece [45]

$\left(dB_{2}-NC_{3}\right)^{2},$ (3.25)

where $C_{3}$ is a three-form that couples to the membrane and $B_{2}$ is a
two-form coupling to the string. In our previous type IIA setup, these objects
would arise from D4-branes wrapping torsion two-cycles and the three-chain
connecting them, respectively, and signal the presence of a non-trivial
superpotential. The case $p=3$ describes 4d space-time filling branes ending
on membranes, and the corresponding Lagrangian piece reads [46]

$\left(dD_{3}-NA_{4}\right)^{2},$ (3.26)

where $A_{4}$ coupling to the space-time filling branes and $D_{3}$ to the
membranes. In our type IIA setup these 4d objects arise from D6-branes wrapped
on torsion three-cycles and on a four-chain linking them, respectively.444In
most of the literature, these Lagrangians are shown to arise from
compactifications with NS $H$-fluxes. In this case, $N$ represents an $H$-flux
quantum and the feature of $p$-branes ending of $(p-1)$-branes has a
microscopic description in terms of $d_{H}$ cohomology and its dual homology
[47]. By looking at concrete constructions, it is easy to convince oneself
that such a setup is connected by mirror symmetry to the one that we are
considering [48, 49].

The general philosophy of the previous subsection also applies to these
Stückelberg-like couplings. That is, the 4d $p$-forms that appear in (3.25)
and (3.26) should arise from expanding the 10d RR potentials on smeared delta-
forms of calibrated torsion cycles of $X_{6}$. The resulting coefficients that
multiply both expressions are the analogues of $f$ and $g$ in (3.9) and
(3.15), and so together with the relevant Laplace eigenvalue they determine
$N$. Notice that the analogy is not straightforward, because (3.25) and (3.26)
are not dual Lagrangians, which reflects the fact that D4-branes and D6-branes
do not couple to dual 10d RR potentials. However one may consider a gauge
instanton on the space-time filling D6-branes, which amounts to a D2-brane
wrapping the same torsion three-cycle and coupling to a massive 4d axion
$C_{0}^{\prime}$ that arises from reducing $C_{3}$ on a coexact three-form
like $\alpha_{3}$ in (3.14). This EFT object is sensitive to the backreaction
of D4-branes wrapping torsion two-cycles and the three-chain connecting them,
and an analogy with the gauge transformations involving AB strings and
particles can be drawn. The precise statement is that there exists a gauged
$(-1)$-form symmetry that describes the discrete gauge symmetries of the EFT
superpotential [50, 51].

For the purposes of computing torsion in cohomology, to consider these new
terms in the Lagrangian may seem redundant, since in the type IIA
constructions that we have discussed they are related to the same kind of
torsion groups, namely ${\rm Tor}H_{3}(X_{6},\mathbb{Z})\simeq$ ${\rm
Tor}H_{2}(X_{6},\mathbb{Z})$ and their linking number. However, an important
difference is that the terms (3.25) and (3.26) appear in 4d ${\cal N}=1$
string theory vacua, like in type II orientifold compactifications, while
(3.9) and (3.15) typically appear in 4d ${\cal N}=2$ compactifications without
vacua, like the example considered in [29]. In this case the ${\cal N}=2$
supersymmetry of the Lagrangian is realised off-shell, while solutions to the
equations of motion at most preserve a fraction of this supersymmetry, like
the domain-wall solution preserving four supercharges to be discussed in the
next section. In practice this implies that the 10d background is not of the
form $\mathbb{R}^{1,3}\times X_{6}$, but instead a fibration of $X_{6}$ over a
real line or a plane in $\mathbb{R}^{1,3}$. Following the general philosophy
of [29, 31] we are entitled to carry out the usual procedure of dimensional
reduction to 4d – and therefore our discussion above – as long as the
variation of this fibration is very small compared to the compactification
scale. The only additional thing that we need to take into account is that for
Conjecture 1 to apply the objects like 4d strings and particles must be BPS
with respect to the 4d solution, which is a stronger condition that being BPS
in a would-be ${\cal N}=2$ vacuum. In practice, this means that they must be
calibrated also from the point of view of the fibration, as we will illustrate
in the next section.

The fact that AB strings and particles cannot be BPS in ${\cal N}=1$
orientifold compactifications seems to clash with the proposal in [15], in the
sense that the basis of non-harmonic $p$-forms in which one expands the 10d RR
potentials to obtain a Stückelberg Lagrangian (3.9) cannot come from smearing
the delta-forms of calibrated cycles. Nevertheless, one can still make sense
of such a basis of non-harmonic forms if one considers the extension of
Conjecture 1 formulated around (2.9). For instance, one could try to describe
the torsion two- and three-cycles of a Calabi–Yau threefold as the difference
of two calibrated cycles with equal volume, or some other combination of
calibrated cycles. As long as there are some eigenmodes below the
compactification scale that couple differently to these calibrated cycles,
there will be non-harmonic $p$-forms that one builds from smearing their bump
delta-forms. Finally, one should make sure that such harmonic forms have the
appropriate parity under the orientifold action to lead to a Stückelberg term.

##  4 A simple example

The simplest example of SU(3)-structure manifolds with torsion in cohomology
are nilmanifolds or twisted tori, which in the context of type II string
compactifications were initially considered in [29, 37, 52]. Particularly
interesting for our discussion is the setup of [29], in which the simplest
kind of twisted torus is realised as a 4d domain-wall solution. In the
following we will see how the objects defined in the previous sections, in
particular torsion calibrated cycles and their smeared delta sources, are
described in this case.

###  4.1 The 10d background

Let us recall the main idea behind the construction in [29]. One first
considers a toroidal compactification of type IIB string theory to 4d with a
backreacted NS5-brane wrapping a special Lagrangian three-cycle of ${\bf
T}^{6}$ and extended along $\mathbb{R}^{1,2}\subset\mathbb{R}^{1,3}$ in the
non-compact dimensions. The long-wavelength approximation of this backreaction
provides a domain-wall solution in 4d, which upon three T-dualities in ${\bf
T}^{6}$ becomes a type IIA background with constant dilaton and a twisted six-
torus $\tilde{\bf T}^{6}$ fibered over a non-compact direction. A simple
generalisation of this setup results in the following type IIA 10d string
frame background:

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
ds^{2}_{\mathbb{R}^{1,2}}+\ell_{s}^{2}V(d\xi)^{2}+\ell_{s}^{2}ds^{2}_{\tilde{\bf
T}^{6}},$ (4.1) $\displaystyle ds^{2}_{\tilde{\bf T}^{6}}$ $\displaystyle=$
$\displaystyle(2\pi)^{2}\left[\frac{R_{1}^{2}}{V_{1}}(\eta^{1})^{2}+\frac{R_{2}^{2}}{V_{2}}(\eta^{2})^{2}+\frac{R_{3}^{2}}{V_{3}}(\eta^{3})^{2}+\frac{VR_{4}^{2}}{V_{1}}(\eta^{4})^{2}+\frac{VR_{5}^{2}}{V_{2}}(\eta^{5})^{2}+\frac{VR_{6}^{2}}{V_{3}}(\eta^{6})^{2}\right],$
(4.2)

where $\xi$ is the 4d coordinate transverse to the domain-wall, $R_{i}$ are
radii measured in string units, and $\eta^{i}$ are the left-invariant one-
forms of the twisted six-torus, defined as

$\begin{array}[]{ll}\eta^{1}=dx^{1}+M_{1}x^{6}dx^{5}\,,&\qquad\eta^{4}=dx^{4}\,,\\\
\eta^{2}=dx^{2}+M_{2}x^{4}dx^{6}\,,&\qquad\eta^{5}=dx^{5}\,,\\\
\eta^{3}=dx^{3}+M_{3}x^{5}dx^{4}\,,&\qquad\eta^{6}=dx^{6}\,,\end{array}$ (4.3)

with $M_{i}\in\mathbb{N}$. Finally,

$V=V_{1}V_{2}V_{3},\qquad
V_{i}=1-\zeta_{i}\xi,\qquad\zeta_{i}=\frac{M_{i}}{2\pi}\frac{R_{i}R_{i+3}}{R_{4}R_{5}R_{6}}.$
(4.4)

To recover the case of [29] one needs to take $M_{i}=M\in\mathbb{N}$ and
$M_{j}=M_{k}=0$, with $i\neq j\neq k\neq i$. The solution applies to the range
$(0,\xi_{\rm end})$, with $\xi_{\rm end}={\rm min}\\{\zeta_{i}^{-1}\\}_{i}$,
while for $\xi<0$ one should glue a direct product $\mathbb{R}^{1,3}\times{\bf
T}^{6}$, with torus radii $R_{i}$.555Our background differs slightly from the
one in [29], in the sense that therein the choice $V_{i}=\zeta_{i}\xi$ along
the range $\xi\geq 0$ is taken, for a domain wall at $\xi=0$. Both choices are
compatible with the domain-wall analysis of [53, 54, 55], but we find that our
choice also reproduces the scalar flow features of $\frac{1}{2}$BPS domain
walls in ${\cal N}=1$ EFTs (see e.g. [26, section 4.3.2]) and is compatible
with the presence of BPS AB strings as particles, as discussed below.

We refer to [49] for more details on the geometry and topology of this class
of twisted six-tori. As in there, one can impose a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold projection that reduces the
structure of the internal manifold to a genuine SU(3) structure, and which we
will assume in the following. In the conventions $d{\rm
vol}_{X_{6}}=-\frac{1}{6}J^{3}=\frac{i}{8}\bar{\Omega}\wedge\Omega$, the
SU(3)-structure calibrations $(J,\Omega)$ are given by

$\displaystyle J$ $\displaystyle=$ $\displaystyle
4\pi^{2}\left(t^{1}\,\eta^{1}\wedge\eta^{4}+t^{2}\,\eta^{2}\wedge\eta^{5}+t^{3}\,\eta^{3}\wedge\eta^{6}\right),$
(4.5) $\displaystyle\Omega$ $\displaystyle=$
$\displaystyle{i}(2\pi)^{3}V^{-1/2}R_{1}R_{2}R_{3}\left(\eta^{1}+i\tau^{1}\eta^{4}\right)\wedge\left(\eta^{2}+i\tau^{2}\eta^{5}\right)\left(\eta^{3}+i\tau^{3}\eta^{6}\right),$
(4.6)

with

$t^{i}=\frac{V^{1/2}}{V_{i}}R_{i}R_{i+3},\qquad\tau^{i}=V^{1/2}\frac{R_{i+3}}{R_{i}}.$
(4.7)

The calibrated objects of this SU(3)-structure manifold are those $p$-chains
whose volume is computed by integrating $\Omega$ or $e^{iJ}$. Recall, however,
that we are interested in a particular kind of calibrated cycles. First, they
need to be calibrated in a strict sense, meaning that upon varying the values
of the $R_{i}$ they are still calibrated. Second, they need to be mutually BPS
with the domain-wall source, in order to be actual BPS objects of the
background (4.1). This second criterion is more easily analysed in the type
IIB mirror background, as done in Appendix A. In our context, one finds the
following BPS objects that are relevant to our discussion:

  * -

A D4-brane wrapped on $\Pi_{3}^{\rm tor}=\\{x^{4}=x^{5}=x^{6}=0\\}$ in
$\tilde{\bf T}_{6}$ and extended along $\xi$.

  * -

A D4-brane wrapped on $\Sigma_{4}^{i}=\\{x^{i}=x^{i+3}=0\\}$ in $\tilde{\bf
T}_{6}$.

  * -

An Euclidean D2-brane on $\Pi_{2}^{i}=\Sigma_{4}^{j}\cap\Sigma_{4}^{k}$, with
$i\neq j\neq k\neq i$ in $\tilde{\bf T}_{6}$ and extended along $\xi$.

  * -

An Euclidean D2-brane wrapped on $\Sigma_{3}=\\{x^{1}=x^{2}=x^{3}=0\\}$ in
$\tilde{\bf T}_{6}$.

Notice that, when extending a D-brane along $\xi$, it does not make sense to
do it beyond $\xi_{\rm end}$, where the metric degenerates and we enter a
strong coupling region.

The submanifolds $(\Pi_{3}^{\rm tor},\Sigma_{4}^{i},\Pi_{2}^{i},\Sigma_{3})$
and others can be described via group theory techniques, by first writing the
twisted six-torus as a coset $\tilde{\bf T}^{6}=G/\Gamma$, with $G$ a Lie
group of a 2-step nilpotent algebra and $\Gamma$ a co-compact lattice, and
then exponentiating different set of generators of $G$, see [49, Appendix A].
Using this framework and the results of [56, 57], one can see that

${\rm Tor}H_{3}(\tilde{\bf
T}^{6},\mathbb{Z})_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}={\rm
Tor}H_{2}(\tilde{\bf
T}^{6},\mathbb{Z})_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}=\mathbb{Z}_{M},$ (4.8)

where $M={\rm g.c.d.}(M_{1},M_{2},M_{3})$, and the subindex represents those
cycles invariant under the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold
projection.666The generators of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
orbifold group act on the left-invariant one forms as
$\theta_{1}:(\eta^{1},\eta^{2},\eta^{3},\eta^{4},\eta^{5},\eta^{6})\mapsto(\eta^{1},-\eta^{2},-\eta^{3},\eta^{4},-\eta^{5},-\eta^{6})$
and
$\theta_{2}:(\eta^{1},\eta^{2},\eta^{3},\eta^{4},\eta^{5},\eta^{6})\mapsto(-\eta^{1},-\eta^{2},-\eta^{3},-\eta^{4},-\eta^{5},\eta^{6})$
[49]. It is not obvious if the torsion cohomology of the orbifold quotient
$\tilde{\bf T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ corresponds to (4.8) or
if it has further elements. However, in case that some additional torsion
cycles existed, one can show that they are not calibrated and they do not
couple to any light eigenmode. Therefore one can ignore them for the purposes
of this work.  The three-cycle $\Pi_{3}^{\rm tor}$ is the generator of ${\rm
Tor}H_{3}(\tilde{\bf T}^{6},\mathbb{Z})_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$,
while ${\rm Tor}H_{2}(\tilde{\bf
T}^{6},\mathbb{Z})_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$ is generated by
$\Pi_{2}^{\rm tor}=\sum_{i}(M_{i}/M)\Pi_{2}^{i}$. Additionally, $\Sigma_{3}$
is a three-chain with a boundary homotopic to $M\Pi_{2}^{\rm tor}$ and, if
$M_{i}\neq 0$, $\Sigma_{4}^{i}$ is a four-chain with a boundary homotopic to
$M_{i}\Pi_{3}^{\rm tor}$. All these $p$-chains are calibrated by either
$\Omega$ or $e^{iJ}$, with a calibration phase that will depend on their
orientation. The D-branes listed above are $\frac{1}{2}$BPS in the background
(4.1), which means that they preserve two supercharges out of the four
supercharges preserved by the solution. The two supercharges that they
preserve will depend on their orientation. For instance, a D4-brane wrapping
$\Sigma_{4}^{i}$ looks like a $\frac{1}{2}$BPS monopole in 4d, and preserves
two supercharges of the domain-wall solution. Reversing the orientation and
wrapping the D4-brane on $-\Sigma_{4}^{i}$ corresponds to a 4d
$\frac{1}{2}$BPS monopole with opposite charge and preserving the other two
supercharges of the background, an object that we will refer to as anti-BPS
monopole. Here we will not keep track of which objects preserve which
supercharges, because a much more straightforward picture will arise when we
interpret this system in terms of Hitchin flow equations.

A D4-brane wrapping a chain $\Sigma_{4}^{i}$ with a boundary is not consistent
by itself, as it develops a worldvolume anomaly, but one can make it
consistent by attaching D4-branes wrapped on ${\partial}\Sigma_{4}^{i}$. In
the present setup, if the D4-brane wrapping $\Sigma_{4}^{i}$ is located at
$\xi_{0}\in(0,\xi_{\rm end})$, one can cure its worldvolume anomaly by
wrapping $M_{i}$ D4-branes on $\Pi_{3}^{\rm tor}$, and connecting them to
${\partial}\Sigma_{4}^{i}$. These $M_{i}$ D4-branes will look like 4d strings
that extend along the coordinate $\xi$, and either end on an anti-monopole in
a different location, or stretch up until the origin $\xi=0$. From the 4d
perspective, in the first case we have a monopole-anti-monopole pair connected
by $M_{i}$ AB strings, as expected for a 4d EFT with a Lagrangian of the form
(3.15) and a monopole of charge $M_{i}/M$. In the second case, we have a 4d
avatar of a Hanany-Witten brane creating effect [58], mirror dual to a
D3-brane crossing the NS5-brane (the domain wall), with $M_{i}$ D1-branes
stretching along both after the crossing. As stressed in [12], this effects
also signal the presence of a discrete gauge symmetry, encoded either in the
Lagrangian (3.15) or its dual. Similarly, the worldvolume anomaly of an
Euclidean D2-brane in $\Sigma_{3}$ can be cured by $M$ D2-branes wrapped on
$\Pi_{2}^{\rm tor}$ and connected to ${\partial}\Sigma_{3}$. From the 4d
viewpoint this is perceived like $M$ AB Euclidean particles ending on an
instanton [10]. Notice that in this case a configuration made of
$\frac{1}{2}$BPS objects involves $M$ Euclidean AB particles extended along
the coordinate $\xi$, that stretch either between the domain-wall source and
the instanton or between an instanton-anti-instanton pair.

In terms of these 4d objects one can compute the quantities $f$ and $g$ that
feature the discussion of section 3, by using (3.18) and (3.20). Since in our
example there are many axions and gauge bosons, in order to isolate a pair of
them in the Lagrangian, as in (3.9), we must consider the particular case
$M_{i}\neq 0$, while $M_{j}=M_{k}=0$ for $i\neq j\neq k\neq i$. One then finds

$f=\left(\tau^{1}\tau^{2}\tau^{3}\right)^{-1/2},\qquad
g=\frac{1}{2\pi}\sqrt{\frac{t^{i}}{t^{j}t^{k}}}.$ (4.9)

Additionally, via the direct computation of section 5 (see eq.(5.20)) or the
results of Appendix B, one obtains that the smallest non-vanishing eigenvalue
of $\tilde{\bf T}^{6}$ is

$\lambda_{\rm st}=\zeta_{i}V_{i}^{-3/2},$ (4.10)

and so it follows that the first equality in (3.11) is satisfied with
$N=M_{i}$, even if all quantities depend on the coordinate $\xi$.

One can also see that the unit-norm exact three-form eigenmode corresponding
to (4.10) is

$b_{3}=f^{-1}\eta^{4}\wedge\eta^{5}\wedge\eta^{6}=f^{-1}\delta_{3}^{\rm
sm}(\Pi_{3}^{\rm tor}),$ (4.11)

where in the second equality we have again used the results of section 5, cf.
eq.(5.23). We thus find perfect agreement with the discussion of section 3, in
which the definition of $f$ via a smeared delta bump-form coincides with the
value in (3.18). A similar check can be made for $g$, and the combined result
is such that Conjecture 1 is verified. In the following we will discuss how to
extend this result to general $M_{i}\in\mathbb{N}$, using the 4d EFT
description.

###  4.2 EFT description

To obtain the 4d effective description of this system one may follow the
approach in [29], or its extension to more general setups discussed in [31,
32]. One first defines the following basis of three-forms

$\displaystyle\alpha_{0}=\eta^{1}\wedge\eta^{2}\wedge\eta^{3},$
$\displaystyle\qquad\beta^{0}=\eta^{4}\wedge\eta^{5}\wedge\eta^{6},$ (4.12a)
$\displaystyle\alpha_{1}=\eta^{4}\wedge\eta^{2}\wedge\eta^{3},$
$\displaystyle\qquad\beta^{1}=-\eta^{1}\wedge\eta^{5}\wedge\eta^{6},$ (4.12b)
$\displaystyle\alpha_{2}=\eta^{1}\wedge\eta^{5}\wedge\eta^{3},$
$\displaystyle\qquad\beta^{2}=-\eta^{4}\wedge\eta^{2}\wedge\eta^{6},$ (4.12c)
$\displaystyle\alpha_{3}=\eta^{1}\wedge\eta^{2}\wedge\eta^{6},$
$\displaystyle\qquad\beta^{3}=-\eta^{4}\wedge\eta^{5}\wedge\eta^{3},$ (4.12d)

and a basis of two- and four-forms

$\displaystyle\omega_{1}=\eta^{1}\wedge\eta^{4},\qquad\omega_{2}=\eta^{2}\wedge\eta^{5},\qquad\omega_{3}=\eta^{3}\wedge\eta^{6},$
(4.13a)
$\displaystyle\tilde{\omega}^{1}=-\omega_{2}\wedge\omega_{3},\qquad\tilde{\omega}^{2}=-\omega_{3}\wedge\omega_{1},\qquad\tilde{\omega}^{3}=-\omega_{1}\wedge\omega_{2}.$
(4.13b)

This set of forms are those that are invariant under the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ projection that takes us to a genuine
SU(3)-structure. Notice that they satisfy $\int_{\tilde{\bf
T}^{6}}\alpha_{i}\wedge\beta^{j}=\int_{\tilde{\bf
T}^{6}}\omega_{i}\wedge\tilde{\omega}^{j}=\delta_{i}^{j}$, with a specific
normalisation which is crucial for the discussion that follows, since we are
going to expand both the calibrations $(J,\Omega)$ and the 10d RR fields in
these forms, and the latter are going to define the 4d axion periodicities the
global $U(1)$ gauge transformations. While for harmonic $p$-forms one has a
clear prescription to define an integral basis, the same is not true for exact
and co-exact elements of this set, which can be identified thanks to the
relations

$d\omega_{i}=-M_{i}\beta^{0},\qquad d\alpha_{0}=-M_{i}\tilde{\omega}^{i}.$
(4.14)

In the present setup such a normalisation can be fixed by means of mirror
symmetry, just as in [29]. However, in the general setting of [31, 32] it is
simply assumed as an input. In section 6 we will argue that one can fix it by
defining the set $\\{\alpha_{A},\beta^{B},\omega_{a},\tilde{\omega}^{b}\\}$ as
smeared delta-forms.

Following [31, 32] we expand the NS-NS sector in the above basis

$\displaystyle J_{c}$ $\displaystyle=$ $\displaystyle
B+iJ=4\pi^{2}(b^{j}+it^{j})\,\omega_{j},$ (4.15) $\displaystyle\Omega$
$\displaystyle=$
$\displaystyle{i}(2\pi)^{3}\sqrt{\frac{t^{1}t^{2}t^{3}}{\tau^{1}\tau^{2}\tau^{3}}}\left(\alpha_{0}+z^{i}\alpha_{i}-z^{2}z^{3}\beta^{1}-z^{1}z^{3}\beta^{2}-z^{1}z^{2}\beta^{3}+z^{1}z^{2}z^{3}\beta^{0}\right),\quad
z^{j}=a^{j}+i\tau^{j}.$

Similarly, one expands the RR potential $C_{3}$ as

$C_{3}=2\pi\ell_{s}^{3}\left[A_{1}^{i}\wedge\omega_{i}+\theta^{I}\alpha_{I}+\tilde{\theta}_{K}\beta^{K}\right],$
(4.16)

where $(\theta^{I},\tilde{\theta}_{I})$ with $I=(0,i)$ represent axions of
unit periodicity. The dimensional reduction of this term gives

$(2\pi)^{2}\left[g_{ii}(dA_{1}^{i})^{2}+\hat{f}^{00}\left(d\tilde{\theta}_{0}-M_{i}A_{1}^{i}\right)^{2}+\hat{f}^{ii}(d\tilde{\theta}_{i})^{2}+\hat{f}_{II}(d{\theta}^{I})^{2}\right],$
(4.17)

plus a mass term for $\theta^{0}$. Here we have defined

$g_{ii}=(2\pi)^{2}\,\frac{t^{j}t^{k}}{t^{i}},\qquad\hat{f}^{00}=e^{2\phi_{4}}(\tau^{1}\tau^{2}\tau^{3})^{-1}M_{\rm
P}^{2},\qquad\hat{f}^{ii}=e^{2\phi_{4}}\frac{\tau^{i}}{\tau^{j}\tau^{k}}M_{\rm
P}^{2},$ (4.18)

with $i\neq j\neq k\neq i$, and
$\hat{f}_{II}=(\hat{f}^{II})^{-1}e^{4\phi_{4}}M_{\rm P}^{4}$, where
$e^{\phi_{4}}=e^{\phi}/\sqrt{4\pi\,t^{1}t^{2}t^{3}}$ is the 4d dilaton. Notice
that all these couplings depend on the domain-wall transverse coordinate
$\xi$, while the axion vevs remain constant along it. The NS-NS sector of the
compactification varies along $\xi$ via the non-trivial profile of the saxions
$t^{i},\tau^{i}$ along this coordinate, as captured by (4.7), and in agreement
with the results of [55, 59].

The lightest massive $p$-form mode has the following squared mass

$m_{\rm
st}^{2}=V^{-1}\left[\sum_{i}\frac{\zeta_{i}^{2}}{V_{i}^{2}}\right]e^{2\phi_{4}}M_{\rm
P}^{2},$ (4.19)

and so it is a priori not obvious how to compute the smeared linking number
using (3.11). To do so, one must take into account that for generic $M_{i}$’s
the torsion two-cycle $\Pi_{2}^{\rm tor}=\sum_{i}(M_{i}/M)\Pi_{2}^{i}$ is not
a smooth calibrated cycle, but instead a linear combination of them. In this
case, it is the extension of the conjecture made around (2.9) that should be
applied. One obtains

$\delta^{\rm sm}_{4}(\Pi_{2}^{\rm
tor})=\sum_{i}\frac{M_{i}}{M}\tilde{\omega}^{i},$ (4.20)

whose projection into the four-form eigenmode with the smallest non-vanishing
eigenvalue gives

$g=\frac{1}{2\pi}\sqrt{\sum_{i}\frac{M_{i}^{2}}{M^{2}}\frac{t^{i}}{t^{j}t^{k}}}.$
(4.21)

Defining $f=\sqrt{\hat{f}^{00}}$ one reproduces (3.11) with $N=M={\rm
g.c.d.}(M_{1},M_{2},M_{3})$, as expected. From a purely 4d EFT viewpoint, one
can interpret (4.21) as the gauge coupling of the linear combination of
$U(1)$’s that develops a Stückelberg mass.

###  4.3 Hitchin flow equations

As already pointed out in [29], the background (4.1) can be understood
geometrically as a fibration of a half-flat manifold $X_{6}$ over a real
coordinate, that gives a seven-dimensional $G_{2}$-manifold $Y_{7}$. The
general description of this kind of fibrations has been given in [60, 34], and
are known as Hitchin flow equations. In the standard description, the real
coordinate $z$ has a flat metric, and one constructs the $G_{2}$-structure
forms

$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle dz\wedge
J-{\mbox{Re}\,\Omega},$ (4.22) $\displaystyle\varphi$ $\displaystyle=$
$\displaystyle-dz\wedge{\mbox{Im}\,\Omega}-\frac{1}{2}J\wedge J,$ (4.23)

where $J$ and $\Omega$ are the $z$-dependent SU(3)-structure calibrations of
$X_{6}$. Demanding that $Y_{7}$ has $G_{2}$ holonomy amounts to impose that
$\varphi$ is harmonic in $Y_{7}$. If we describe the 7d derivative as

$d_{7}={\partial}_{z}dz\wedge+\,d,$ (4.24)

with $d$ the exterior derivative along the 6d fibre, this requirement reads

$\displaystyle d{\mbox{Im}\,\Omega}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\partial}_{z}\left(J\wedge J\right),$ (4.25)
$\displaystyle dJ$ $\displaystyle=$
$\displaystyle-{\partial}_{z}{\mbox{Re}\,\Omega}.$ (4.26)

In our background the coordinate $\xi$ has a non-trivial metric, more
precisely $dz=-V^{1/2}d\xi$, where the sign choice accounts for the difference
in our background compared to [29] (see footnote 5). The Hitchin flow
equations then take the following form :

$\displaystyle V^{1/2}d{\mbox{Im}\,\Omega}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{\partial}_{\xi}\left(J\wedge J\right),$ (4.27)
$\displaystyle V^{1/2}dJ$ $\displaystyle=$
$\displaystyle{\partial}_{\xi}{\mbox{Re}\,\Omega}.$ (4.28)

Applied to the background (4.1) these equations reduce to

${\partial}_{\xi}V_{i}=-\zeta_{i},$ (4.29)

which is clearly satisfied by (4.4).

The Hitchin flow equations have a nice interpretation when it comes to
D-branes on torsion cycles, that can be illustrated explicitly in the solution
(4.1). Let us consider $M$ D4-branes wrapped on $\Pi_{3}^{\rm tor}$ and
extended along an interval $(0,\xi_{0})\subset(0,\xi_{\rm end})$. At $\xi_{0}$
one places a D4-brane wrapping a four-chain $\Sigma_{4}$, such that its
boundary coincides with the $M$ torsion three-cycles. From the 4d viewpoint,
this represents a 4d monopole in which $M$ AB strings end, with their other
end at the domain-wall source. Since both sets of D4-branes yielding the
monopole and the AB strings are calibrated by $*\varphi$, they must be
mutually BPS, and satisfy the marginal stability condition (3.22).
Additionally, the total energy of the system must be given by its central
charge, which is the integral of $*\varphi$ over the full D4-brane worldvolume
in the $G_{2}$ manifold $Y_{7}$, and it is easy to argue that this central
charge must be independent of the monopole position $\xi_{0}$.

Indeed, notice that shifting the value of $\xi_{0}$ corresponds to add $M$ AB
strings extended along the interval
$(\xi_{0},\xi_{0}^{\prime})\subset(0,\xi_{\rm end})$, with a D4-branes
wrapping $\Sigma_{4}$ at each end, with opposite orientations. From the 4d
viewpoint, this realises a monopole-anti-monopole pair in which AB strings
end. Since this object can annihilate by itself, one expects that its central
charge vanishes. Microscopically, the whole object corresponds to a trivial
four-cycle in the $G_{2}$ manifold $Y_{7}$, and so since $*\varphi$ is closed
its integral must vanish on it. So indeed the monopole-anti-monopole pair
carries no central charge and changing the value of $\xi_{0}$ in the above BPS
configuration should not change the energy of the system. In particular this
energy should match that of a monopole placed at $\xi=\xi_{\rm end}$ which is
equivalent to having $M$ AB strings, and to a monopole at $\xi=0$, which does
not have any AB strings attached to it.

This is indeed what the Hitchin flow equations are telling us, and in
particular (4.25). On the one hand, ${\partial}_{z}J\wedge J$ represents the
variation of the mass of BPS monopoles when we move along $z$. On the other
hand, ${\mbox{Im}\,\Omega}$ integrated along the torsion three-cycle measures
the tension of a BPS 4d AB string, and by Stokes’ theorem, this is equivalent
to integrating ${d\mbox{Im}\,\Omega/M}$ over the four-chain $\Sigma_{4}$
linking $M$ of them. So what (4.25) is saying is that it is the monopoles in
which $M$ AB strings can end the ones whose mass varies along the coordinate
transverse to the domain wall. Moreover, there is a mass scale associated to
the 4d string, which is its tension integrated along the interval
$(0,\xi_{0})$. For BPS objects, this energy increases with $\xi$ at the same
rate as the monopole mass decreases, and that is why the total central charge
and therefore the energy of the system stays constant. In our example (4.1)
one can see that the factors of $V$ cancel for a 4d AB string, so the energy
of $M_{i}$ BPS AB strings is given by $\ell_{s}^{-1}{\rm Vol}(\Pi_{3}^{\rm
tor})M_{i}\xi_{0}\propto\xi_{0}$. Additionally, the mass of the monopole in
which such strings can end is given by $\ell_{s}^{-1}{\rm
Vol}(\Sigma_{4}^{i})\propto V_{i}|_{\xi_{0}}$. Therefore, it decreases
linearly with $\xi_{0}$, precisely compensating the change in the energy of
the AB strings.777One can engineer the BPS configuration of $M$ AB strings
ending on a monopole by a Hanany-Witten brane-creation effect, as one can
check using the mirror type IIB picture, see Appendix A. The interpretation is
then that a Hanany-Witten effect does not change the energy of a BPS object.
The mass of a monopole located at $\xi_{0}\in(-\infty,0)$ and at
$\xi_{0}\in(0,\xi_{\rm end})$ is the same, if in the second case we include
the energy of the extended AB strings. That is, if at both sides we compute
the energy or central charge of the gauge invariant operator.

This example illustrates how (3.22) can be satisfied, and the expectation of
subsection 3.2, that one should be able to add up central charges of BPS
objects in $\mathbb{Z}$, even when their topological charge is
$\mathbb{Z}_{N}$. In the case at hand, $M$ D4-branes wrapping $\Pi_{3}^{\rm
tor}$ can disappear by ending on a monopole, but a monopole nucleation process
costs energy, which is minimised for the case of BPS monopoles. The discussion
above implies that this energy is at least that of $M$ BPS strings extended
along the interval $(\xi_{0},\xi_{\rm end})$. Therefore nucleating a monopole
at $\xi_{0}$ is topologically possible, but not energetically favoured. In
this sense, adding up an arbitrary number of AB strings is well-defined in the
BPS context, as well as considering a cone of 4d AB string charges.

##  5 Direct computation

While the general arguments of section 3 motivate Conjecture 1 for torsion
cycles in SU(3)-structure manifolds, it is instructive to work out in detail
how the conjecture is realised in explicit examples, by a direct comparison of
the torsion linking number and its smeared version. In this section we perform
such a comparison for the SU(3)-structure manifold of section 4, more
precisely for the twisted six-torus with a single metric flux. As we will see,
the direct computation of the torsion linking number displays a series of
cancellations between terms that is reminiscent of those that occur in the
computation of topological indices, and that leaves the smeared torsion
linking number (2.8) as the only non-vanishing contribution. The reader not
interested in these technical details may safely skip to the next section.

#### The setup

Let us consider the twisted six-torus background in (4.2), rewritten as

$ds^{2}_{\tilde{\bf
T}^{6}}=(2\pi)^{2}\sum_{i}\left(\frac{t^{i}}{\tau^{i}}(\eta^{i})^{2}+t^{i}\tau^{i}(\eta^{i+3})^{2}\right),$
(5.1)

and with the definitions (4.7) and (4.3). In particular we consider $M_{i}\neq
0$ and $M_{j}=M_{k}=0$ with $i\neq j\neq k\neq i$. In this case, the metric
background factorises as $\tilde{\bf T}^{6}=\tilde{\bf T}^{3}\times{\bf
T}^{3}$, and all the torsion cycles correspond to a direct product of a
torsion one-cycle in $\tilde{\bf T}^{3}\simeq\langle
x^{i},x^{j+3},x^{k+3}\rangle$ and a non-trivial cycle in ${\bf
T}^{3}\simeq\langle x^{j},x^{k},x^{i+3}\rangle$. As a result, all torsion
linking numbers of $\tilde{\bf T}^{6}$ stem from the torsion linking numbers
between one-cycles in $\tilde{\bf T}^{3}$. Moreover, the calibration condition
in $\tilde{\bf T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ will translate into
a subset of such torsion one-cycles. Therefore our strategy will be to verify
Conjecture 1 for such a subset, then extend the result into calibrated two and
three-cycles of $\tilde{\bf T}^{6}$, and finally check that the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ projection does not modify the statement.
A necessary first step is to describe the set of massive $p$-form modes in
$\tilde{\bf T}^{3}$, which one can accomplish using a general method for
three-manifolds with isometries.

#### Massive spectra of three-manifolds

To describe the massive $p$-form spectrum of a twisted three-torus, one may
use the method of [61], which applies to compact Riemannian three-dimensional
manifolds $X_{3}$ with a continuous isometry. Such a manifold admits a unit-
norm Killing vector $\chi$, and we assume that its dual one-form satisfies

$\star
d\chi=\lambda_{\chi}\,\chi\,,\qquad\Delta_{3}\chi=\lambda_{\chi}^{2}\,\chi\,,\qquad\chi^{2}=1\,,\qquad\lambda_{\chi},\in\mathbb{R}\,,$
(5.2)

and that its integral curves are closed. Here $\star$ and $\Delta_{3}$ stand
for the Hodge star operator and the Laplacian on $X_{3}$, respectively. Then,
let $\\{\phi_{\alpha}\\}$ be an orthonormal basis of complex scalar eigenforms
of the Laplacian such that888Notice that such a basis always exists because
$[\Delta_{3},\mathcal{L}_{\chi}]=0$.

$\Delta_{3}\phi_{\alpha}=\sigma_{\alpha}^{2}\phi_{\alpha}\,,\qquad\mathcal{L}_{\chi}\phi_{\alpha}=i\mu_{\alpha}\phi_{\alpha}\,,\qquad\sigma_{\alpha}\in\mathbb{R}\,,\quad\mu_{\alpha}\in\mathbb{R}\,.$
(5.3)

Solving the second condition of (5.3) we can obtain the explicit dependence of
the $\\{\phi_{\alpha}\\}$ on the isometry coordinate ${\theta}$ associated to
$\chi$

$\phi_{\alpha}=e^{i\mu_{\alpha}\theta}K_{\alpha}\,,\qquad{\theta}\sim{\theta}+2\pi
r\,,$ (5.4)

with $d\theta=\chi$ and $K_{\alpha}$ functions which do not depend on
${\theta}$. Given that ${\theta}$ parameterises a closed integral curve of
radius $r$ in a compact manifold, we obtain the quantisation condition
$\mu_{\alpha}r\in\mathbb{Z}$ .

In this setup, it is possible to give a simple description of non-harmonic
eigen-one-forms of the Laplacian, in terms of the Killing vector $\chi$ and
the scalar eigenforms $\phi_{\alpha}$. We define

$R_{\alpha}=d\phi_{\alpha}\,,\qquad S_{\alpha}=\star
d(\phi_{\alpha}\chi)\,,\qquad T_{\alpha}=\star dS_{\alpha}\,.$ (5.5)

It is easy to see that the set $R_{\alpha}$ forms a complete basis of exact
eigen-one-forms. The set of co-exact one-forms $S_{\alpha}$ and $T_{\alpha}$
is closed under the action of the operator $\star d$

$\star dS_{\alpha}=T_{\alpha}\,,\qquad\star
dT_{\alpha}=\sigma_{\alpha}^{2}S_{\alpha}+\lambda_{\chi}T_{\alpha}\,,$ (5.6)

from where one can find the following eigenforms of $\star d$

$U^{\pm}_{\alpha}=\left(\frac{1}{2}\pm\frac{\lambda_{\chi}}{2\sqrt{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}}\right)T_{\alpha}\pm\frac{\sigma_{\alpha}^{2}}{\sqrt{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}}S_{\alpha}\,.$
(5.7)

Therefore, since the action of the Laplacian $\Delta_{3}$ on co-closed forms
amounts to $\star\,d\star d$, we obtain that the $U^{\pm}$ are eigenforms of
the Laplace operator with eigenvalues

$(\lambda^{\pm}_{\alpha})^{2}=\sigma_{\alpha}^{2}+\frac{\lambda_{\chi}^{2}}{2}\pm\frac{\lambda_{\chi}}{2}\sqrt{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}\,.$
(5.8)

Let us dub the constant eigenmode of the Laplacian as
$\phi_{0}=1/\sqrt{V_{3}}$, with $V_{3}$ the volume of $X_{3}$. Then the
eigenmode $U_{0}^{-}$ identically vanishes, while $U_{0}\equiv U_{0}^{+}$ has
eigenvalue $\lambda_{\chi}^{2}$ with respect to $\Delta_{3}$ and takes form
$U_{0}=\lambda_{\chi}^{2}\,\chi\phi_{0}$. Moreover, the set of co-exact one-
forms $U^{\pm}_{\alpha}$ are normalised to unity by multiplying them by the
following factor

$c_{\alpha}^{\pm}=\left[\frac{(\lambda_{\chi}^{4}+3\lambda_{\chi}^{2}\sigma_{\alpha}^{2}+\sigma_{\alpha}^{4})-(\lambda_{\chi}^{2}+\sigma_{\alpha}^{2})\,\mu_{\alpha}^{2}}{2}\pm\lambda_{\chi}\,\frac{(\lambda_{\chi}^{4}+5\lambda_{\chi}^{2}\sigma_{\alpha}^{2}+5\sigma_{\alpha}^{4})-(\lambda_{\chi}^{2}+3\sigma_{\alpha}^{2})\,\mu_{\alpha}^{2}}{2\sqrt{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}}\right]^{-1/2}\,.$
(5.9)

In the following we will assume that the $U_{\alpha}^{\pm}$ have been
normalised to unit norm. In particular, we have that $c_{0}\equiv
c_{0}^{+}=\lambda_{\chi}^{-2}$, and so $U_{0}=\chi\phi_{0}$.

The set $\\{U_{\alpha}^{\pm}\\}$ is part of the co-exact one-form
eigenspectrum of $X_{3}$, but the above method does not guarantee that it is a
complete set. In the particular case of $\tilde{\bf T}^{3}$ one can check that
the whole co-exact spectrum is of this form, as verified in Appendix B by
using the results of [62].

#### Computing the linking number

Using that $\\{U_{0},U_{\alpha}^{\pm}\\}$ is a complete basis of co-exact
eigen-one-forms of $X_{3}$,999There may be more than one eigenform for a given
eigenvalue, but this will not change our final result. we can expand the bump
delta two-form for a torsion one-cycle $\pi_{1}\subset X_{3}$ as

$\delta^{(2)}(\pi_{1})=K_{0}\star U_{0}+\sum_{\alpha}\left(K_{\alpha}^{+}\star
U^{+}_{\alpha}+K_{\alpha}^{-}\star U^{-}_{\alpha}\right)\,,$ (5.10)

where

$K_{0}=\int_{\pi_{1}}U_{0},\qquad
K_{\alpha}^{\pm}=\int_{\pi_{1}}U_{\alpha}^{\pm}\,.$ (5.11)

In terms of these expressions, the linking number (2.7) between two torsion
one-cycles reads

$L(\pi,\tilde{\pi})=\frac{1}{\lambda_{\chi}}K_{0}\tilde{K}_{0}+\sum_{\alpha}\left[\frac{1}{\lambda_{\alpha}^{+}}K^{+}_{\alpha}\tilde{K}^{+}_{\alpha}+\frac{1}{\lambda_{\alpha}^{-}}K^{-}_{\alpha}\tilde{K}^{-}_{\alpha}\right]\,,$
(5.12)

where the coefficients $\tilde{K}$ arise from integration over a different
torsion one-cycle $\tilde{\pi}_{1}$.

We now impose the calibration condition. One can check that calibrated torsion
two- and three-cycles in $\tilde{\bf T}^{3}\times{\bf
T}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ correspond to torsion one-cycles on
$\tilde{\bf T}^{3}$ that are integral curves of $\chi$. For such one-cycles we
have that $\int_{\pi_{1}}\alpha=\int_{0}^{2\pi
r}\iota_{\chi}\alpha\,d{\theta}$, for any one-form $\alpha$. Therefore

$K_{0}=\int_{0}^{2\pi r}\iota_{\chi}U_{0}\,d{\theta}=2\pi r\phi_{0}\,,\qquad
K_{\alpha}^{\pm}=\int_{0}^{2\pi r}\iota_{\chi}U_{\alpha}^{\pm}\,d{\theta}\,.$
(5.13)

From these expressions, one may compute each of the terms in the torsion
linking number. Indeed, one first notices that

$\iota_{\chi}S_{\alpha}=\lambda_{\chi}\phi_{\alpha}\,,\qquad\iota_{\chi}T_{\alpha}=(\lambda_{\chi}^{2}+\sigma_{\alpha}^{2}-\mu_{\alpha}^{2})\,\phi_{\alpha}\,,$
(5.14)

which imply

$\iota_{\chi}U^{\pm}_{\alpha}=\pm\left\\{\lambda^{\pm}_{\alpha}\bigg{[}\lambda_{\chi}^{2}+\sigma_{\alpha}^{2}-\mu_{\alpha}^{2}\bigg{]}+\sigma_{\alpha}^{2}\lambda_{\chi}\right\\}\frac{c_{\alpha}^{\pm}}{\sqrt{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}}\phi_{\alpha}\,.$
(5.15)

As a result, the massive eigenmodes with $\mu_{\alpha}\neq 0$ have a vanishing
coefficient, since

$K^{\pm}_{\alpha}\propto\int_{0}^{2\pi r}\phi_{\alpha}\,d{\theta}=0\,,$ (5.16)

where we have used (5.4). It remains to check the contribution of the modes
with $\mu_{\alpha}=0$ to (5.12). Recall that those modes with $\alpha\neq 0$
come in pairs, and one can check that they satisfy the following relation:

$\frac{1}{\lambda_{\alpha}^{+}}K^{+}_{\alpha}\tilde{K}^{+}_{\alpha}+\frac{1}{\lambda_{\alpha}^{-}}K^{-}_{\alpha}\tilde{K}^{-}_{\alpha}=\frac{\epsilon_{\alpha}}{\lambda_{\chi}^{2}+4\sigma_{\alpha}^{2}}\int_{0}^{2\pi
r}\phi_{\alpha}\,d{\theta}\int_{0}^{2\pi r}\phi_{\alpha}\,d{\theta}\,,$ (5.17)

where we have defined

$\epsilon_{\alpha}\equiv\frac{(c_{\alpha}^{+})^{2}}{\lambda_{\alpha}^{+}}\left[\lambda_{\alpha}^{+}(\lambda_{\chi}^{2}+\sigma_{\alpha}^{2})+\sigma_{\alpha}^{2}\lambda_{\chi}\right]^{2}+\frac{(c_{\alpha}^{-})^{2}}{\lambda^{-}_{\alpha}}\left[\lambda_{\alpha}^{-}(\lambda_{\chi}^{2}+\sigma_{\alpha}^{2})+\sigma_{\alpha}^{2}\lambda_{\chi}\right]^{2}=0\,.$
(5.18)

That is, those massive eigenmodes with $\mu_{\alpha}=0$ have non-trivial
coefficients $K_{\alpha}^{\pm}$, but for those contributing to the bracket in
(5.12) there is a non-trivial cancellation by pairs, such that the sum cancels
term by term. The surviving term in (5.12) is the smeared linking number

$L^{\rm
sm}(\pi_{1},\tilde{\pi}_{1})\equiv\frac{1}{\lambda_{\chi}}K_{0}\tilde{K}_{0}=\frac{4\pi^{2}r^{2}}{V_{3}\lambda_{\chi}}\,.$
(5.19)

In a twisted three-torus with metric $ds^{2}_{\tilde{\bf
T}^{3}}=(2\pi)^{2}\left[(R_{i}\eta^{i})^{2}+(R_{j+3}\eta^{j+3})^{2}+(R_{k+3}\eta^{k+3})^{2}\right]$
and twist $d\eta^{i}=-N\eta^{j+3}\wedge\eta^{k+3}$ one obtains

$U_{0}=\frac{2\pi
R_{i}}{\sqrt{V_{3}}}\,\eta^{i},\qquad\lambda_{\chi}=\frac{NR_{i}}{2\pi
R_{j+3}R_{k+3}}\,,\qquad r^{2}=R_{i}^{2}\,,\qquad
V_{3}=8\pi^{3}R_{i}R_{j+3}R_{k+3}\,.$ (5.20)

Therefore applying (5.19) one recovers the result $L^{\rm
sm}(\pi_{1},\tilde{\pi}_{1})=1/N$, as expected.

#### Extension to $\tilde{\bf T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$

Let us now see how the above computation extends to the SU(3)-structure
manifold $\tilde{\bf T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. We first
consider the covering space $\tilde{\bf T}^{6}=\tilde{\bf T}^{3}\times{\bf
T}^{3}$ with metric (5.1), where $\tilde{\bf T}^{3}$ is parametrised by the
coordinates $\\{x^{i},x^{j+3},x^{k+3}\\}$ and ${\bf T}^{3}$ by
$\\{x^{i+3},x^{j},x^{k}\\}$, with $i\neq j\neq k\neq i$. Given the
factorisation of the metric, any eigenform of the Laplacian will be a wedge
product of one in $\tilde{\bf T}^{3}$ and one in ${\bf T}^{3}$. We are in
particular interested in those eigenforms in which the bump delta-forms
$\delta(\Pi_{3}^{\rm tor})$ and $\delta(\Pi_{2}^{\rm tor})$ are decomposed. It
is easy to see that these fall in the subset

$\displaystyle\left[\star U_{\alpha}^{\pm}\right]$ $\displaystyle\wedge$
$\displaystyle\left(e^{2\pi in_{i+3}\,x^{i+3}}dx^{i+3}\right),\qquad
n_{i+3}\in\mathbb{Z}\,,$ (5.21) $\displaystyle\left[\star
U_{\alpha}^{\pm}\right]$ $\displaystyle\wedge$
$\displaystyle\left(e^{2\pi(n_{j}x^{j}+n_{k}x^{k})}dx^{j}\wedge
dx^{k}\right),\qquad n_{j},n_{k}\in\mathbb{Z}\,,$ (5.22)

for $\delta(\Pi_{3}^{\rm tor})$ and $\delta(\Pi_{2}^{\rm tor})$, respectively,
where as above $\star$ stands for the Hodge star operator in $\tilde{\bf
T}^{3}$. As a consequence, the expansion of the smeared deltas
$\delta^{\rm{sm}}(\Pi_{3}^{\rm tor})$ and $\delta^{\rm{sm}}(\Pi_{2}^{\rm
tor})$ are given by $\star U_{0}\wedge dx^{i+3}$ and $\star U_{0}\wedge
dx^{j}\wedge dx^{k}$, accordingly. That is, using the metric (5.1) one obtains

$\delta^{\rm{sm}}(\Pi_{3}^{\rm
tor})=\eta^{4}\wedge\eta^{5}\wedge\eta^{6}\qquad\delta^{\rm{sm}}(\Pi_{2}^{\rm
tor})=-\eta^{j}\wedge\eta^{k}\wedge\eta^{j+3}\wedge\eta^{k+3}.$ (5.23)

With regard to the complete expansion, it is easy to see that the wedge of one
of these forms and its antiderivative will give a non-vanishing contribution
only if $n_{j}=n_{k}=n_{i+3}=0$, that is if we select harmonic forms in ${\bf
T}^{3}$. As a result, the computation of the linking number for calibrated
cycles works precisely as outlined for $\tilde{\bf T}^{3}$, with the same
vanishing coefficients and the same cancellations, and we end up again with
the smeared torsion linking number (5.19).

Let us now implement the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold
projection, where each $\mathbb{Z}_{2}$ generator ${\theta}_{1}$ and
${\theta}_{2}$ acts by flipping two coordinates on $\tilde{\bf T}^{3}$ and
other two on $\tilde{\bf T}^{3}$, as follows from footnote 6. Since this is a
product of two involutions, each acting on one submanifold, we can split the
above exact eigenforms into even and odd under such involutions, and take
(odd, odd) or (even, even) products, such that the result is invariant under
the orbifold generators. While one could perform such an analysis explicitly,
given our discussion above it is sufficient to show the action of these
orbifold generators on the two-forms $\star U_{\alpha}^{\pm}$ only depends on
the value of $\sigma_{\alpha}$ and $\mu_{\alpha}$, since then the orbifold
projection will commute with relations that lead to the cancellations (5.18),
and they will also happen for orbifold-invariant massive modes. One can show
the assumption by using that ${\theta}_{1}$ and ${\theta}_{2}$ act as
isometries when restricted to $\tilde{\bf T}^{3}$, as then they commute with
$\Delta$, and ${\cal L}_{\chi}$. It then follows that they have a well-defined
action on the basis of scalar wavefunctions $\\{\phi_{\alpha}\\}$, and act on
the above set of co-exact one forms as

${\theta}_{\alpha}:S_{\alpha}\mapsto\nu_{\alpha}^{\sigma_{\alpha},\mu_{\alpha}}S_{\alpha}\,,\quad{\theta}_{\alpha}:T_{\alpha}\mapsto\nu_{\alpha}^{\sigma_{\alpha},\mu_{\alpha}}T_{\alpha}\
\implies\
{\theta}_{\alpha}:U_{\alpha}^{\pm}\mapsto\nu_{\alpha}^{\sigma_{\alpha},\mu_{\alpha}}U_{\alpha}^{\pm}\,.$
(5.24)

That is, the orbifold group action on the massive modes of interest only
depends on the value of $\sigma_{\alpha}$ and $\mu_{\alpha}$, as assumed.
Finally, by construction, the orbifold projection leaves invariant the
eigenmodes of $\tilde{\bf T}^{3}\times{\bf T}^{3}$ that contribute to the
smeared linking number.

##  6 More general ${\cal N}=2$ compactifications

The extension of the setup in [29] to more general type II string
compactifications leading to 4d ${\cal N}=2$ gauged supergravities has been
performed in [30, 31, 32, 63]. In the following we focus on the framework
developed in [31, 32], which applies to SU(3)-structure manifolds. The results
of [31, 32] imply that, on an SU(3)-structure manifold $X_{6}$, one should be
able to select a set of smooth $p$-forms:

$\\{\omega_{a}\\}\in\Omega^{2}(X_{6})\,,\qquad\\{\alpha_{A},\beta^{B}\\}\in\Omega^{3}(X_{6})\,,\qquad\\{\tilde{\omega}^{a}\\}\in\Omega^{4}(X_{6})\,,$
(6.1)

with $a=1,\dots,n_{K}$, $A,B=1,\dots,n_{\rm c.s.}$, chosen such that

$\int_{X_{6}}\omega_{a}\wedge\tilde{\omega}^{b}=\delta_{a}^{b},\qquad\int_{X_{6}}\alpha_{A}\wedge\beta^{B}=\delta_{A}^{B},$
(6.2)

and satisfying the relations

$\displaystyle d^{\dagger}\omega_{a}$ $\displaystyle=0,$ (6.3a) $\displaystyle
d\omega_{a}$ $\displaystyle=m_{a}{}^{A}\alpha_{A}+e_{aA}\beta^{A},$ (6.3b)
$\displaystyle d\alpha_{A}$ $\displaystyle=e_{aA}\tilde{\omega}^{a},$ (6.3c)
$\displaystyle d\beta^{B}$ $\displaystyle=-m_{a}{}^{B}\tilde{\omega}^{a},$
(6.3d) $\displaystyle d\tilde{\omega}^{a}$ $\displaystyle=0,$ (6.3e)

with $m_{a}{}^{A},e_{aA}\in\mathbb{Z}$ such that
$m_{a}{}^{A}e_{bA}=m_{b}{}^{A}e_{aA}$. Consistency of the dimensional
reduction implies that the set is closed under the Hodge star operator:

$\tilde{\omega}^{a}=g^{ab}*\omega_{b},\qquad*\alpha_{A}=H_{A}^{B}\alpha_{B}+G_{AB}\beta^{B},\qquad*\beta^{A}=F^{AB}\alpha_{B}-H^{A}_{B}\beta^{B},$
(6.4)

mimicking the relations between harmonic forms in Calabi–Yau manifolds.

Given this set of $p$-forms, one expands the SU(3)-structure calibrations in
terms of them:

$\displaystyle J_{c}$ $\displaystyle=$ $\displaystyle
B+iJ=4\pi^{2}(b^{a}+it^{a})\,\omega_{a},$ (6.5) $\displaystyle\Omega$
$\displaystyle=$ $\displaystyle Z^{A}\alpha_{A}-\mathcal{F}_{B}\beta^{B},$

with $\mathcal{F}_{A}={\partial}_{A}\mathcal{F}$ the derivatives of the
complex structure prepotential $\mathcal{F}$. The 4d kinetic terms of the
corresponding fields are governed by the same expressions as in the Calabi–Yau
case, in terms of Kähler potentials
$K_{\rho}=-\log\int_{X_{6}}i\bar{\Omega}\wedge\Omega$ and
$K_{J}=-\log\frac{4}{3}\int_{X_{6}}-J\wedge J\wedge J$ that correspond to
Hitchin functionals [31]. Finally, one should also expand the 10d RR
potentials in this set of $p$-forms. In the case of type IIA compactifications
such an expansion reads

$C_{3}=2\pi\ell_{s}^{3}\left(A_{1}^{a}\wedge\omega_{a}+\tilde{C}_{0}^{A}\alpha_{A}+C_{0\,B}\beta^{B}\right),$
(6.6)

leading to a set of axions and gauge vectors in 4d. The dual degrees of
freedom are obtained from the expansion of $C_{5}$.

In the framework of [31, 32] there is no geometric interpretation for the set
$\\{\omega_{a},\alpha_{A},\beta^{B},\tilde{\omega}^{b}\\}$, nor a clear
prescription on how to build them from the light eigenmodes of the Laplacian.
Notice that a key property of these $p$-forms is that they define the
quantisation features of the 4d EFT, either in terms of axion periodicities or
$U(1)$ gauge transformations. As such, their definition should be connected to
the presence of 4d EFT objects like strings, particles and instantons, which
implement and detect global gauge transformations. We have already seen this
connection in our discussion of section 3, in light of which one may propose
to describe the set $\\{\omega_{a},\alpha_{A},\beta^{B},\tilde{\omega}^{b}\\}$
as smeared delta forms.

Indeed, based on our previous discussion, it is natural to propose that the
smooth $p$-forms $\\{\omega_{a},\alpha_{A},\beta^{B},\tilde{\omega}^{b}\\}$
correspond to smeared delta-forms $\delta^{\rm sm}_{p}(\Sigma_{6-p})$ of a set
of strictly calibrated $(6-p)$-chains $\Sigma_{6-p}\subset X_{6}$, which
encode the presence of BPS objects in the 4d EFT. More precisely, the closed
four-forms $\tilde{\omega}^{b}$ correspond to the smeared bump delta-forms
$\delta^{\rm sm}_{4}(\Pi_{2})$, where $[\Pi_{2}]$ belongs to the free part of
$H_{2}(X_{6},\mathbb{Z})$ if $\tilde{\omega}^{b}$ is harmonic, and to ${\rm
Tor}H_{2}(X_{6},\mathbb{Z})$ if it is de Rham exact. Similarly, the subset of
three-forms in $\\{\alpha_{A},\beta^{B}\\}$ that are closed correspond to the
smeared delta-forms $\delta^{\rm sm}_{3}(\Pi_{3})$ of strictly calibrated
three-cycles. The remaining set of smooth $p$-forms can be constructed by
taking the anti-derivatives of the exact three- and four-forms and normalising
them such that (6.2) is satisfied, which implies that the integers
$m_{a}{}^{A},e_{aA}$ encode the torsion linking numbers of $X_{6}$. Finally,
as in our simple example above, one could also relate the non-closed two- and
three-forms as the smeared version of delta-forms for calibrated four- and
three-chains in $X_{6}$, whose boundary describes the torsional nature of some
calibrated cycles.

To make this picture more precise, let us consider the subcase
$m_{a}{}^{A}=0$, which also resembles the setup considered in [15]. Then, the
rank $r_{\bf e}$ of the matrix $e_{aA}$ should determine the number of
harmonic two- and three forms of $X_{6}$ as $b_{2}(X_{6})=n_{K}-r_{\bf e}$ and
$b_{3}(X_{6})=n_{\rm c.s.}-r_{\bf e}$. Clearly, the rank of $e_{aA}$ counts
massive eigenforms below the compactification scale, more precisely we should
at least have $r_{\bf e}$ times a spectrum of the form (3.6), as this is what
we obtain from dimensionally reducing the RR sector of the theory. Indeed, let
us consider the type IIA expansion (6.6), and for simplicity assume that in
(6.4) $H^{A}_{B}=0$, so that $G_{AB}F^{BC}=-\delta_{A}^{C}$. Then we find

$(2\pi)^{2}\hat{F}^{AB}\left(dC_{0\,A}-e_{aA}A_{1}^{a}\right)\left(dC_{0\,B}-e_{bB}A_{1}^{b}\right)+(2\pi)^{2}g_{ab}\,dA_{1}^{a}\wedge
dA_{1}^{b}\,,$ (6.7)

with $\hat{F}^{AB}=F^{AB}e^{2\phi_{4}}M_{\rm P}^{2}$, plus a mass term for
$r_{\bf e}$ axions $\tilde{C}_{0}^{A}$.101010These massive axions are more
suitably described in terms of a 4d dual two-form $B_{2}$ involved in a
gauging of the form (3.25), see e.g. [45]. The masses that one reads from such
a mass term, the Lagrangian (6.7) and its dual reproduce the action of the
Laplace operator on the set
$\\{\omega_{a},\alpha_{A},\beta^{B},\tilde{\omega}^{b}\\}$ as expected [48,
31, 32]. For instance, the action of the Laplacian on the closed forms
$\beta^{A}$ and $\tilde{\omega}^{a}$ reads

$\displaystyle\Delta\beta^{A}$ $\displaystyle=$ $\displaystyle
F^{AB}e_{bB}g^{bc}e_{cC}\,\beta^{C}\,,$ (6.8)
$\displaystyle\Delta\tilde{\omega}^{a}$ $\displaystyle=$ $\displaystyle
g^{ab}e_{bB}F^{BC}e_{cC}\,\tilde{\omega}^{c}\,.$ (6.9)

The diagonalisation of these mass matrices gives us the set of massless and
light $p$-form eigenmodes. Such a spectrum is by assumption complete, or
otherwise the expansions (6.5) and (6.6) would be missing light modes of the
EFT. Knowledge of these mass matrices and of the kinetic terms $C^{AB}$ and
$g_{ab}$ leads to $e_{aA}$, in a generalisation of the relation (3.11). As
proposed in [15], the matrix $e_{aA}$ is a sort of the inverse of the torsion
linking numbers, and it encodes the torsion cohomology that is sensitive to
the light EFT modes. This topological information is easier to extract if one
performs a unimodular integral change of basis both in
$\\{\omega_{a},\tilde{\omega}^{b}\\}$ and in $\\{\alpha_{A},\beta^{B}\\}$ that
take $e_{aA}$ to its Smith normal form

$e^{\rm Smith}=\begin{pmatrix}k_{1}\\\ &k_{2}\\\ &&\ddots\\\ &&&k_{r_{\bf
e}}&\dots&0\\\ &&&0&\dots&0\\\ &&&\vdots&\ddots&\vdots\\\
&&&0&\dots&0\end{pmatrix}$ (6.10)

with $k_{i},k_{i}/k_{i+1}\in\mathbb{Z},\forall i$. In this basis the
computation of the smeared linking number gives
$\int_{X_{6}}d^{-1}\tilde{\omega}^{i}\wedge\beta^{j}=k_{i}^{-1}\delta^{ij}$,
for $i,j=1,\dots,r_{\bf e}$, suggesting that the torsion cohomology groups are

${\rm Tor}\,H^{3}(X_{6},\mathbb{Z})\simeq{\rm
Tor}\,H^{4}(X_{6},\mathbb{Z})\simeq\mathbb{Z}_{k_{1}}\times\dots\times\mathbb{Z}_{k_{m}}\,,$
(6.11)

where $|k_{i}|>1$ for $i\leq m$ and $|k_{i}|=1$ for $m<i\leq r_{\bf e}$. Those
entries of (6.10) with value $\pm 1$ should correspond to calibrated
$p$-cycles that are trivial in homology, but that nevertheless are detected by
the 4d EFT because they couple to massive modes below the compactification
scale.

The proposal that the closed $p$-forms within
$\\{\omega_{a},\alpha_{A},\beta^{B},\tilde{\omega}^{b}\\}$ correspond to
smeared bump delta-forms of calibrated cycles can be further motivated by
considering the set of BPS objects in the 4d EFT. For instance, let us again
consider type IIA with $m_{a}{}^{A}=0$ and look at the closed three-forms
$\beta^{A}$, which in the basis (6.10) may either be harmonic or exact in de
Rham cohomology. Each of these forms are related to an axion $C_{0\,A}$, and
from the BPS completeness hypothesis [64], or the EFT string completeness
hypothesis [27] applied to ${\cal N}=2$ gauged supergravities, one expects a
BPS string under which such an axion is magnetically charged. Then, the
results of [31, 32] imply that
$K_{\rho}=-\log\int_{X_{6}}i\bar{\Omega}\wedge\Omega$ describes the metric of
the hypermultiplet moduli space, at least at the classical level. Because the
axion kinetic terms only depend on $K_{\rho}$ and this has the same expression
as in the ungauged case, the tension of a BPS string should have the same
general expression as in a Calabi–Yau. That is, we have that

$\frac{{\cal T}^{A}}{M_{\rm P}^{2}}=\frac{e^{\phi}}{{\rm
Vol}_{X_{6}}}\left|\int_{X_{6}}\Omega\wedge\beta^{A}\right|\,.$ (6.12)

Now, in this context BPS means that the D4-brane internal worldvolume
$\Pi_{3}^{A}$ is calibrated by $\Omega$, and dimensionally reducing its DBI
action one obtains

$\frac{{\cal T}^{A}}{M_{\rm P}^{2}}=\frac{e^{\phi}}{{\rm
Vol}_{X_{6}}}\left|\int_{\Pi_{3}^{A}}\Omega\right|=\frac{e^{\phi}}{{\rm
Vol}_{X_{6}}}\left|\int_{X_{6}}\Omega\wedge\delta(\Pi_{3}^{A})\right|,$ (6.13)

which implies that $\beta^{A}$ must be the smeared version of
$\delta(\Pi_{3}^{A})$. Note that this is a standard result when $\beta^{A}$ is
not exact in de Rham cohomology, since then $[\beta^{A}]$ and $[\Pi_{3}^{A}]$
are related by standard Poincaré duality. Similarly, in 4d ${\cal N}=2$ EFTs,
the mass of charged BPS particles in Planck units is specified by their
central charge. A key result of [31, 32] is that the kinetic terms of vector
multiplet sector is encoded in the Kähler potential
$K_{J}=-\log\frac{4}{3}\int_{X_{6}}-J\wedge J\wedge J$ also for gauged
supergravities obtained from compactifications on SU(3)-structure manifolds.
From here it follows that the central charges of BPS particles charged under
the vector multiplets are precisely the periods of $B+iJ$, that is

$Z^{a}=e^{K_{J}/2}\int_{X_{6}}e^{B+iJ}\wedge\tilde{\omega}^{a}.$ (6.14)

As in the Calabi–Yau case, such BPS particles should arise from wrapping
D2-branes on two-cycles $\Pi_{2}^{a}\subset X_{6}$ calibrated by $J$. By
dimensionally reducing their DBI action we obtain

$\frac{m_{a}^{2}}{M_{\rm
P}^{2}}=e^{K_{J}}\left|\int_{X_{6}}(B+iJ)\wedge\delta(\Pi_{2}^{a})\right|^{2},$
(6.15)

which implies that $\tilde{\omega}^{a}$ should be the smeared version of
$\delta(\Pi_{2}^{a})$. Again, this is independent of whether
$\tilde{\omega}^{a}$ is de Rham exact or not. When $\tilde{\omega}^{a}$ is
exact, it has to be that $\Pi_{2}^{a}$ is either a torsion or a trivial class
of $H_{2}(X_{6},\mathbb{Z})$. Finally, the completeness hypothesis implies
that there is a BPS particle per element of the basis
$\\{\tilde{\omega}^{a}\\}$, which again is a standard result in the Calabi–Yau
case.

##  7 Torsion D-branes in ${\cal N}=1$ vacua

Having discussed the physical meaning of the smeared torsion linking number in
4d ${\cal N}=2$ settings, it is natural to wonder how Conjecture 1 can be
physically realised in 4d ${\cal N}=1$ string vacua. In the ${\cal N}=2$ case,
the realisation was based on the existence of BPS AB particles and strings, a
set of objects that will be essentially absent in 4d ${\cal N}=1$ type II
orientifold settings. Indeed, there are no BPS particles in ${\cal N}=1$
vacua, and a 4d string that arises from a D$(p+1)$-brane wrapped on a
$p$-cycle $\Pi_{p}\subset X_{6}$ can only be BPS if $\Pi_{p}$ is calibrated by
a closed $p$-form.111111The precise statement is that in 4d ${\cal N}=1$
Minkowski vacua the calibration for 4d strings is $d_{H}$-closed [7, 8, 9], an
statement that also holds for the ${\cal N}=0$ Minkowski vacua analysed in
[65]. In practice this implies that, even in compactifications with $H$-flux,
torsion cycles cannot be calibrated. As such, torsion $p$-cycles cannot yield
4d BPS strings. The realisation of the smeared linking number must therefore
be more subtle in this case.

As already mentioned in section 3, one possibility is to invoke the extension
of Conjecture 1 formulated around (2.9), and look for torsion $p$-cycles that
are not calibrated by themselves, but nevertheless its homology class can be
seen as a linear combination of calibrated $p$-cycle classes. This would in
principle allow us to describe AB strings and particles and their associated
smeared $p$-forms in 4d ${\cal N}=1$ orientifold vacua, connecting our
previous discussion with the setup of [15].

The most natural realisation of the smeared torsion linking form seems instead
to involve the Stückelberg-like terms that involve 4d BPS objects in ${\cal
N}=1$ vacua, and that correspond to (3.25) and (3.26). These two couplings
represent 4d membranes ending on strings and space-time filling branes ending
on membranes, and in the context of 4d ${\cal N}=1$ supersymmetry they can be
described in the language of three-form multiplets [46]. Notice that in a type
II compactification on a six-dimensional manifold $X_{6}$, a D$p$-brane that
looks like a 4d particle and a D$(p+2)$-brane that looks like a 4d membrane
can wrap the same $p$-cycle $\Pi_{p}\in X_{6}$, and the same holds for a
D$(p+3)$-brane and an Euclidean D$(p-1)$-brane instanton. So essentially we
are trading the role of 4d particles and instantons for that of 4d membranes
and space-time filling branes, in order to probe a similar set of torsion
$p$-cycles with BPS objects. In practice, this implies that the topological
information that was captured by (3.9) and (3.15) in the ${\cal N}=2$ case,
now is encoded in the couplings (3.25) and (3.26).

To see how this works in practice, let us again focus on type IIA string
theory on a compact SU(3)-structure manifold $X_{6}$, but now with an
orientifold projection that introduces O6-planes. Assuming a 4d Minkowski
vacuum leads to the following metric Ansatz

$ds^{2}=e^{2A}ds^{2}_{\mathbb{R}^{1,3}}+\ell_{s}^{2}ds^{2}_{X_{6}},$ (7.1)

with $A$ a warp factor that depends on the coordinates of $X_{6}$, whose
SU(3)-structure metric satisfies the following equations

$d(3A-\phi)=H+idJ=0,\qquad
d(e^{2A-\phi}{\mbox{Re}\,\Omega})=0,\qquad\ell_{s}d(e^{4A-\phi}{\mbox{Im}\,\Omega})=-e^{4A}*F_{2},$
(7.2)

and $F_{0}=F_{4}=F_{6}=0$, with $F_{2p}$ the gauge-invariant RR field
strength. In this setup 4d strings made up of D4-branes wrapping three-cycles
are calibrated by $\pm e^{2A-\phi}{\mbox{Re}\,\Omega}$ which, as advanced, is
a closed three-form. Therefore, there are no BPS strings of this sort that
correspond to torsion homology classes. The same can be said for membranes,
which are calibrated by $e^{3A-\phi}e^{B+iJ}$.

The last equation in (7.2), however, features a non-closed three-form that
calibrates space-time filling D6-branes. As such, it can detect calibrated
torsion three-cycles. That such three-cycles exist in certain SU(3)-structure
vacua can be deduced from the results of [48], which imply that a D$p$-brane
that is point-like in a Calabi–Yau manifold with $H$-flux is mapped by mirror
symmetry to a D$(p+3)$-brane wrapping a torsion three-cycle $\Pi_{3}^{\rm
tor}$. In our type IIA orientifold context, $\Pi_{3}^{\rm tor}$ will host a 4d
BPS object if it is wrapped either by a D6-brane or by an Euclidean D2-brane.
In practice, the DBI action of these objects is easier to analyse if,
following [66], one trades the last equation in (7.2) by an equivalent one not
involving the Hodge star operator. In the case at hand we find [42]

$\ell_{s}d(e^{-\phi}{\mbox{Im}\,\Omega})=-J\wedge F_{2},$ (7.3)

which already hints that part of the torsion data of $X_{6}$ is encoded in the
RR flux $F_{2}$.

In this context, it is illustrative to consider a simple example, like the
twisted six-torus geometries analysed in [49]. These correspond to an
SU(3)-structure of the form (4.5) and (4.6), with the simplification
$V=V_{i}=1$, $\forall i$, and an orientifold action of the form ${\cal
R}:(J,\Omega)\mapsto{-}(J,\bar{\Omega})$. The $p$-chains
$\\{\Pi_{2}^{i},\Pi_{3}^{\rm tor},\Sigma_{3},\Sigma_{4}^{i}\\}$ play the same
role in terms of torsion homology information as in section 4, but from an EFT
viewpoint they should be associated to either 4d membranes or space-time
filling branes. For concreteness, let us consider a $\tilde{\bf T}^{6}$ with
twisting $M_{1}=-M_{2}=N\in\mathbb{N}$ and $M_{3}=0$. This choice fixes the
complexified Kähler moduli as $b^{1}+it^{1}=b^{2}+it^{2}$ in the vacuum and
leads to a background RR flux of the form

$F_{2}=\ell_{s}K\left(\eta^{1}\wedge\eta^{4}-\eta^{2}\wedge\eta^{5}\right),\qquad
K\in\mathbb{N}\,,$ (7.4)

that solves (7.3) in the constant dilaton approximation, by setting
$Kt^{1}=Kt^{2}=Ne^{-\phi}R_{1}R_{2}R_{3}$. This sort of RR flux background is
the one that appears in the type IIA orientifold flux literature, see e.g.
[67, 68], but it is important to realise that the expression is a consequence
of the constant-dilaton/smeared approximation. Indeed, what occurs in this
background is that there are eight O6-planes wrapped on $[\Pi_{3}^{\rm tor}]$.
In a plain toroidal compactification one could cancel this charge by placing
32 D6-branes in the same three-cycle class of the covering space. In the
twisted torus geometry, because $[\Pi_{3}^{\rm tor}]$ is
$\mathbb{Z}_{N}$-torsion, one only needs to place $32-kN$ of such D6-branes,
for some $k\in\mathbb{Z}$, in order to cancel the RR tadpole. The lack of
D6-branes leads to a RR flux background that satisfies

$dF_{2}\simeq-\ell_{s}kN\delta_{3}(\Pi_{3}^{\rm tor}),$ (7.5)

where for simplicity we have assumed all O6-planes and D6-branes on the same
representative (otherwise one is led to more involved delta-source equations,
like the ones solved in [69]). Upon implementing the smearing approximation
one obtains

$F_{2}=-\ell_{s}kN\,d^{-1}\delta_{3}^{\rm sm}(\Pi_{3}^{\rm tor})\,,$ (7.6)

which reproduces (7.4) for $k=2K$, up to a harmonic form. The actual RR flux
is, however, the one that solves (7.5), since it is the only one that can
satisfy Dirac’s quantisation condition, upon the appropriate choice of
harmonic piece [42].

The couplings (3.25) and (3.26) are obtained upon dimensionally reducing the
RR potentials

$\displaystyle C_{5}$ $\displaystyle=$
$\displaystyle\ell_{s}^{5}2\pi\left[B_{2\,0}\wedge\beta^{0}+B_{2}^{i}\wedge\alpha_{i}+C_{3}^{i}\wedge\omega_{i}\right]\,,$
(7.7) $\displaystyle C_{7}$ $\displaystyle=$
$\displaystyle\ell_{s}^{7}2\pi\left[D_{3\,i}\wedge\tilde{\omega}^{i}+A_{4}^{0}\wedge\alpha_{0}+A_{4\,i}\wedge\beta^{i}\right]\,,$
(7.8)

where we have taken into account the orientifold action, and expanded into
$p$-forms that couple with unit charge to the calibrated $p$-chains
$\\{\Pi_{2}^{i},\Pi_{3}^{\rm tor},\Sigma_{3},\Sigma_{4}^{i}\\}$.121212For
simplicity we are using the notation (4.12), which results in an unusual
convention in orientifold compactifications. The more standard one is obtained
by interchanging the basis elements as $\alpha_{i}\leftrightarrow-\beta^{i}$,
cf. [68, eq.(2.6)]. One finds

$(2\pi)^{2}\left[\hat{g}_{ii}(dC_{3}^{i})^{2}+\tilde{F}^{00}\left(dB_{2\,0}+M_{i}C_{3}^{i}\right)^{2}+e^{-4\phi_{4}}M_{P}^{-4}\tilde{F}_{ii}(dB_{2}^{i})^{2}\right]\,,$
(7.9)

and

$(2\pi)^{2}\left[\hat{g}^{ii}\left(dD_{3\,i}-M_{i}A_{4}^{0}\right)^{2}\right]e^{-8\phi_{4}}M_{P}^{-8}\,,$
(7.10)

where we have assumed generic twists $M_{i}$ and have defined

$\hat{g}_{ii}=g_{ii}e^{-4\phi_{4}}M_{P}^{-4},\qquad\tilde{F}^{AA}=F^{AA}e^{-2\phi_{4}}M_{P}^{-2},$
(7.11)

and $\hat{g}^{ii}=1/\hat{g}_{ii}$, $\tilde{F}_{AA}=1/\tilde{F}^{AA}$. Note
that $\hat{F}^{AB}$ defined below (6.7) satisfies
$\hat{F}^{AB}=\tilde{F}^{AB}e^{4\phi_{4}}M_{P}^{4}$, and so (7.9) contains the
same kind of information as (4.17). As a result, the computation of the
smeared linking number works exactly as in section 4.2. The main difference is
the expression of the smeared linking number in terms of 4d EFT quantities,
which now involves the physical charges of membranes and strings ending on
each other. More precisely in this ${\cal N}=1$ setup, one finds that the
relation (3.11) is substituted by

$\frac{m_{\rm
st}}{N}=\sqrt{\frac{\hat{g}^{\alpha\alpha}}{\tilde{F}_{00}}}=e^{\phi_{4}}M_{\rm
P}\sqrt{\frac{{g}^{\alpha\alpha}}{{F}_{00}}}\,,$ (7.12)

where $N=M={\rm g.c.d.}(M_{1},M_{2},M_{3})$ and
$\hat{g}^{\alpha\alpha}=e^{4\phi_{4}}M_{P}^{4}{g}^{\alpha\alpha}=e^{4\phi_{4}}M_{P}^{4}\sum_{i}\frac{M_{i}^{2}}{M^{2}}g^{ii}$.
Here $\hat{g}^{\alpha\alpha}$ represents the squared physical charge ${\cal
Q}^{2}$ of a BPS membrane ending on a BPS string and $\tilde{F}_{00}$ the
squared charge of such a string, as defined in [26], see also (8.1).

##  8 Beyond the BPS case

Conjecture 1 proposes a method to compute the linking number between two
calibrated torsion cycles from smeared/EFT data. However, as already
mentioned, in Calabi–Yau manifolds torsion $p$-cycles cannot be calibrated, or
equivalently D-branes wrapped on them are not BPS objects of the EFT. The
extension of the conjecture around (2.9) allows us to implement the same
method whenever the torsion class $\Pi_{p}^{\rm tor}$ of interest is a linear
combination of $p$-cycle homology classes with calibrated representatives.
This more general setup could in principle occur in Calabi–Yau
compactifications, and then the extended conjecture would imply that one can
compute torsion in cohomology via smeared data, provided there exist massive
eigenforms of the Laplacian below the compactification scale that couple to
torsion $p$-cycles. Including such a set of light fields in the 4d EFT would
presumably take us to a structure of the form (6.2) and (6.3), in which giving
a non-vanishing vev to a massive, light field deforms an SU(3)-holonomy metric
to an SU(3)-structure one.

Nevertheless, in general, one would expect that a torsion class in homology
does not contain any calibrated representative, and neither can it be
understood as a linear combination of homology classes with them. In that
case, our discussion of section 3 suggests that there should be non-trivial
corrections associated to this sector. More precisely, one would expect that
in the EFT description the bump-delta form $\delta_{6-p}(\Pi_{p}^{\rm tor})$
can still be replaced by its smeared version, but only up to a multiplying
constant that could be interpreted as wavefunction renormalisation. That is,
one does not simply project the delta into its lowest eigenmode component, but
also has to multiply the result by some constant (or a field-dependent
function) in order to correctly reproduce the 4d physics. Whenever this
happens the smeared linking number and the exact linking number do not
coincide, and one should rescale the smeared delta to make it so.

It is hard to have an idea of the magnitude of this rescaling without an
example at hand where the computation can be carried out explicitly. The best
we can do is to give an estimate for the error in the smeared linking number,
as follows. In supersymmetric theories, the EFT kinetic terms obtained from
truncating massive modes at zero vev are exact up to corrections of ${\cal
O}(\Lambda_{\rm EFT}/\Lambda_{\rm UV})$, see e.g. [70]. In our case
$\Lambda_{\rm EFT}$ corresponds to the mass $m_{\rm st}$ of the massive modes
that we keep in our EFT, and $\Lambda_{\rm UV}$ to the compactification scale
$m_{\rm KK}$. Notice that this is the suppression that we found in (3.24) when
moving away from the minimal tension representative.

If the torsion cycle is not calibrated, it means that the EFT data is not
computing its minimal volume properly. In some cases, like in Calabi–Yau vacua
the ‘expected’ volume ${\cal V}_{p}^{\rm BPS}$, namely the integral of the
appropriate calibration over $\Pi_{p}^{\rm tor}$, vanishes. In general, the
volume of a non-calibrated cycle should be larger than the integral of its
would-be calibration over the given homology class. One can think of the
mismatch between volumes as how much one needs to deform a would-be calibrated
cycle to match the actual one. Finally, one usually converts differences of
internal volumes to differences of field vevs via the physical 4d charge
${\cal Q}$ of the corresponding EFT object [27], which here we define as

${\cal Q}^{2}(\Pi_{p})=\sum_{\lambda_{i}\ll\ell_{s}m_{\rm KK}}c_{i}^{2},$
(8.1)

with $c_{i}$ the coefficients of the smeared delta-form of $\Pi_{p}$, as in
(2.4).

With these considerations in mind, let us consider the smeared linking number
between a calibrated cycle $\Pi_{6-p-1}^{\rm tor}$ and a non-calibrated one
$\Pi_{p}^{\rm tor}$. One may propose the following upper bound

$\left|L-L^{\rm sm}\right|<\frac{{\cal V}_{p}-{\cal V}_{p}^{\rm BPS}}{{\cal
Q}}\frac{m_{\rm st}}{m_{\rm KK}},$ (8.2)

where ${\cal V}$ is the actual volume of $\Pi_{p}^{\rm tor}$, in string units.
This upper bound estimates the error when computing the smeared linking
number, with respect to the actual one $L$. If the bound is small, it still
makes sense to compute (2.8), because it gives a good estimate of the actual
linking number. That is, one may still use EFT data to characterise torsion in
cohomology.

##  9 Conclusions

In this work we have proposed a method to detect topological invariants of
torsion cohomology groups via smooth $p$-forms. The proposal is based on what
${\rm Tor}H^{p}(X_{n},\mathbb{Z})$ means when performing dimensional reduction
of type II string theory on $X_{n}$ and obtaining a lower-dimensional EFT with
a massive sector, and it can be summarised in two main points:

  * i)

If a D-brane wrapped on a torsion cycle $\Pi_{p}^{\rm tor}$ has a non-trivial
backreaction at EFT wavelengths, it is because there are light massive
eigenmodes of the Laplacian sourced by it. In geometric terms, this means that
$\Pi_{p}$ has a non-trivial smeared delta form $\delta^{\rm
sm}_{n-p}(\Pi_{p})$, which is a necessary requirement to apply our approach.

  * ii)

Whenever $\Pi_{p}^{\rm tor}$ is calibrated, a D-brane wrapped on it is a BPS
object of the theory whose smeared backreaction is protected from dimensional
reduction corrections. As a result one can compute the torsion linking numbers
of $\Pi_{p}^{\rm tor}$ using its smeared delta form.

This second statement, which is the content of Conjecture 1, provides a method
to detect the $\mathbb{Z}_{N}$ factors in $H^{p}(X_{n},\mathbb{Z})$. The
method has a wider application if one assumes the extension of the conjecture
made around (2.9), and it would be really interesting to see if it can be
applied to manifolds with special holonomy metrics.

The use of smooth $p$-forms to compute torsion in cohomology may seem quite
surprising because such $\mathbb{Z}_{N}$ factors are projected out in de Rham