${\cal W}(z;\tau)=\underbrace{e_{A}{\cal V}^{A}(z)}_{\text{generated by three-
forms}}\quad\underbrace{-Uh^{\Lambda}T_{\Lambda}+\hat{\cal
W}(Z;T)}_{\text{`spectator' sector $\to{\cal W}^{\prime}(z;\tau)$}}\,.$ (6.92)

Having set the NS-NS fluxes as fixed, the tadpole cancellation condition
(6.88) becomes linear with respect to the dynamically generated fluxes, which
here is just the Romans mass:

$m^{0}Q^{\Lambda}+{\cal Q}_{\rm bg}^{\Lambda}=0\,,\qquad{\rm with}\quad
Q^{\Lambda}\equiv-h^{\Lambda}\,.$ (6.93)

It is then immediate to see that, since ${\cal I}^{{\cal A}{\cal
B}}_{\Lambda}\to{\cal
I}^{\Sigma}_{0\Lambda}=\delta_{\Sigma}^{\Lambda}\delta_{0}^{\cal A}$, the
subset $e_{A}$ constitute an isotropic sublattice in the sense of (1.52).

The dynamical generation of the sublattice (6.91) requires the introduction of
a set of $2h_{-}^{(1,1)}(X)+2$ gauge three-form potentials, the same amount as
$e_{A}$ in (6.34). Their origin from a ten-dimensional theory may be better
understood switching to a democratic formulation [153, 120]. Along with the
Romans mass $m^{0}$ and $\tilde{G}_{2}$, $\tilde{G}_{4}$, we also consider
$\tilde{G}_{10}$, $\tilde{G}_{8}$ and $\tilde{G}_{6}$. They obey
$*\tilde{G}_{10}=m^{0}$, $*\tilde{G}_{8}=\tilde{G}_{2}$,
$*\tilde{G}_{6}=G_{4}$, which can be imposed at the level of equations of
motion. The presence of the higher-form potentials $\tilde{C}_{5}$,
$\tilde{C}_{7}$, $\tilde{C}_{9}$ allows one to get, by dimensional reduction,
the following sets of gauge three-forms in 4D EFT:

$\displaystyle\tilde{G}_{4}$ $\displaystyle={\rm{d}}A_{3}^{0}+\ldots\,,$
$\displaystyle\qquad\tilde{G}_{6}$
$\displaystyle={\rm{d}}A_{3}^{i}\wedge\omega_{i}+\ldots\,,$ (6.94)
$\displaystyle\tilde{G}_{8}$
$\displaystyle={\rm{d}}\tilde{A}_{3i}\wedge\omega^{i}+\ldots\,,$
$\displaystyle\qquad\tilde{G}_{10}$
$\displaystyle={\rm{d}}\tilde{A}_{30}\wedge{\rm vol}_{X}+\ldots\,,$ (6.95)

where the dots stand for other contributions where we are presently not
interested into. Together, the gauge three-forms
$\\{A_{3}^{a},\tilde{A}_{3a}\\}$ are a set of $2h_{-}^{(1,1)}+2$ gauge three-
forms. This has to be compared with the number of physical complex scalars
$\varphi^{i}$, $h_{-}^{(1,1)}$, which _actively_ participate in the
dynamically generated part of the superpotential. This case falls into the
_maximally nonlinear_ class of examples of Section 4.2.1. The supersymmetric
EFT which appropriately include the gauge three-forms is described in terms
nonlinear double three-form multiplets of the kind of (4.76a)

$S^{a}\equiv-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R}){\cal
P}^{a}\,,\qquad S_{a}={\cal G}_{a}(S)\equiv-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R})\tilde{\cal P}_{a}\,,$ (6.96)

which collect at once the Kähler moduli (plus the super-Weyl compensator)
$z^{a}$ and the gauge three-forms $A_{3}^{A}$ among their components.

The gauge three-forms so included would arbitrarily generate any possible
constant $e_{A}\in\mathbb{Z}$. The tadpole constraint (6.88) is enforced upon
the dynamically generated $e_{A}$ by inserting, in the EFT, a set of gauge
four-forms $C_{4}^{\Lambda}$ gauging the three-forms
$\\{A_{3}^{a},\tilde{A}_{3a}\\}$. Following the recipe of Section 4.3, we
introduce $b_{3}^{-}(X)$ gauge four-forms, one for each of the constraints
imposed by (6.93). Applying (4.96), with the definitions (6.93), it turns out
that the only multiplets that is subjected to the gauging is

$S_{0}={\cal G}_{0}(S^{a})\;\rightarrow\hat{S}_{0}\equiv-\frac{{\rm
i}}{4}(\bar{\cal D}^{2}-8{\cal R})\tilde{\cal P}_{0}+{\rm
i}h^{\Lambda}\Gamma_{\Lambda}\,.$ (6.97)

This results also into the fact that in the four-dimensional theory, rather
than $\tilde{F}_{40}$, its gauged counterpart appears

$\hat{\tilde{F}}_{40}={\rm{d}}\tilde{A}_{30}-h^{\Lambda}C_{4\Lambda}\;.$
(6.98)

To recap, up to to this point we have arrived at the superspace Lagrangian

$\displaystyle\tilde{\cal L}$ $\displaystyle=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}_{\rm k}(\hat{S},\bar{\hat{S}})e^{-\frac{K_{\rm
cs}(T,\bar{T})}{3}}+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
W}^{\prime}(\hat{S};\tau)+{\rm c.c.}\right)\,,$ (6.99)
$\displaystyle\quad\,+\left[{\rm i}\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
Q}_{\rm bg}^{\Lambda}\Gamma_{\Lambda}+{\rm c.c.}\right]+\tilde{\cal L}_{\rm
bd}$

with the remnant superpotential

${\cal W}^{\prime}(S;N)=-S^{0}h^{\Lambda}T_{\Lambda}+\hat{\cal W}(S)\;.$
(6.100)

The components of (6.99), after gauge-fixing the super-Weyl invariance, are
the same as (4.99), with the three-form kinetic parts given by (4.48) and
(4.83) (see also, in comparison, [120, 121, 122, 123, 154]).

This not the end of the story. The complex structure multiplets $T_{\Lambda}$
may be regarded as axionic multiplets for the Lagrangian (6.99) and can then
be dualized to linear multiplets $L^{\Lambda}$ following the procedure of
Section 4.4.1. In this picture, the R-R axions $\xi_{\Lambda}$ are then
replaced with gauge two-forms ${\cal B}_{2}^{\Lambda}$, while the saxions
$n_{\Lambda}$ are replaced by $\ell^{\Lambda}$, the lowest components of the
linear multiplets. Furthermore, comparing the superpotential (6.100) with
(4.116) considered in Section 4.4.2, we immediately recognize that the first
contribution in (6.100), which couples the compensator $S^{0}$ to the complex
structure superfields $T_{\Lambda}$, can be reabsorbed in a gauging of the
newly added linear multiplets. Hence, we trade $L^{\Lambda}$ with their gauged
variants

$L^{\Lambda}\rightarrow\hat{L}^{\Lambda}=L^{\Lambda}+h^{\Lambda}\tilde{\cal
P}_{0}\;,$ (6.101)

which, at the level of components, reflects into the gauging

${\cal H}_{3}^{\Lambda}={\rm{d}}{\cal B}_{2}^{\Lambda}\rightarrow\hat{\cal
H}_{3}^{\Lambda}={\rm{d}}{\cal B}_{2}^{\Lambda}+h^{\Lambda}A_{3}^{0}\;.$
(6.102)

The full spectrum of superfields which build the theory, along with the
bosonic components that they include, is summarized in Table 6.4. At the
superspace level, the Lagrangian (6.99) becomes

$\displaystyle\tilde{\cal L}$
$\displaystyle=-3\int{\rm{d}}^{4}\theta\,E\,{\cal
F}(\hat{S},\bar{\hat{S}};\hat{L})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,\hat{\cal W}(\hat{S})+{\rm c.c.}\right)\,,$ (6.103)
$\displaystyle\quad\,+\left[{\rm i}\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
Q}^{\rm bg}_{\Lambda}\Gamma^{\Lambda}+{\rm c.c.}\right]+\tilde{\cal L}_{\rm
bd}$

where the only remnant piece of the superpotential is $\hat{\cal W}(S)$.

Superfields | Bosonic fields | Number | Data
---|---|---|---
| $z^{a}(\varphi)$ | $h_{-}^{(1,1)}(X)+1$ | Kähler deformations
| R-R axions
$\hat{S}^{a}$ |  $A_{3}^{A}$, $\tilde{A}_{3A}$ | gauge three-forms
| $l^{\Lambda}(t)$ | $h^{(2,1)}(X)+1$ | complex structure deformations
$\hat{L}^{\Lambda}$ | $B_{2}^{\Lambda}$ | NS-NS axions
$\Gamma^{\Lambda}$ | $C_{4}^{\Lambda}$ | $h^{(2,1)}+1$ | gauge four-forms
Table 6.4: How the 4D fields originating form compactification of Type IIA
superstring theory reassemble as lowest components of ${\cal N}=1$ chiral
superfields.

In components, the Lagrangian (6.103) reads

$\begin{split}S|_{\rm bos}&=M^{2}_{\rm
P}\int\left(\frac{1}{2}\,R*1-K_{\text{k}\,i\bar{\jmath}}\,{\rm{d}}\varphi^{i}\wedge*{\rm{d}}\bar{\varphi}^{\bar{\jmath}}+\frac{1}{4}F_{\Lambda\Sigma}{\rm{d}}\ell^{\Lambda}\wedge*{\rm{d}}\ell^{\Sigma}\right)\\\
&\quad\,+\frac{1}{4M_{\rm P}^{2}}\int F_{\Lambda\Sigma}\,\hat{\cal
H}_{3}^{\Lambda}\wedge*\hat{\cal H}^{\Sigma}_{3}+\\\
&\quad\,-\int_{\Sigma}\Big{[}\frac{1}{2}T_{AB}\hat{F}^{A}_{4}*\\!\hat{F}^{B}_{4}+T_{AB}\Upsilon^{A}\hat{F}^{B}_{4}+\Big{(}\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\Big{)}*\\!1\Big{]}\\\
&\quad\,+\int_{\partial\Sigma}T_{AB}(*\hat{F}^{A}_{4}+\Upsilon^{A})A^{B}_{3}\\\
&\quad+{\cal Q}_{\rm bg}^{\Lambda}\int_{\Sigma}C_{4\Lambda}\end{split}$
(6.104)

with the three-form part specified by (4.133) with $\Pi^{A}$ as in (6.85).
Moreover, the Legendre transform of the Kähler potential (6.70) is here given
by

$F(\varphi,\bar{\varphi};\ell)=K+2\ell^{\Lambda}{\rm Im\hskip
1.00006pt}t_{\Lambda}\,,$ (6.105)

where we have introduced the variables

$\ell^{\Lambda}=-\frac{1}{2}\frac{\partial K_{\rm cs}}{\partial{\rm Im\hskip
1.00006pt}t_{\Lambda}}\,,$ (6.106)

and we recall that
$\tilde{F}(\varphi,\bar{\varphi};\ell)=K(\varphi,\bar{\varphi};{\rm Im\hskip
1.00006pt}t(\varphi,\bar{\varphi};\ell))$.

#####  6.3.4 Including non-geometric fluxes

Non-geometric fluxes can be included by extending the NS-NS sector to

$c^{\Lambda}_{A}=(c^{\Lambda}_{0},c^{\Lambda}_{i},c^{\Lambda j},c^{\Lambda
0})\equiv(h^{\Lambda},\omega^{\Lambda}_{i},Q^{\Lambda j},R^{\Lambda})\,,$
(6.107)

where $h^{\Lambda}$ counts the $H_{3}$ flux quanta, while
$\omega^{\Lambda}_{i}$, $R^{\Lambda}$, $Q^{\Lambda j}$ represent the geometric
and non-geometric fluxes [147, 148, 133, 149]. A superpotential which takes
into account the contribution of the non-geometric sector is a slight
extension of (6.82)

${\cal W}_{\text{NS-NS}}(Z;T)=U\int\Omega_{\rm
c}\wedge\mathfrak{D}(e^{J_{c}})\,.$ (6.108)

Here $\mathfrak{D}$ is the twisted differential operator

$\mathfrak{D}={\rm{d}}+\overline{H}_{3}\wedge\;-\,\omega\,\lhd\;+\,Q\rhd\;-R\,\bullet\;\,,$
(6.109)

where the actions of the operators over the cohomology classes is such that
[149, 155]

$\displaystyle\omega\,\lhd$ $\displaystyle:\quad
H^{p}(X,\mathbb{Z})\rightarrow H^{p+1}(X,\mathbb{Z})$ (6.110) $\displaystyle
Q\,\rhd$ $\displaystyle:\quad H^{p}(X,\mathbb{Z})\rightarrow
H^{p-1}(X,\mathbb{Z})$ $\displaystyle R\,\bullet$ $\displaystyle:\quad
H^{p}(X,\mathbb{Z})\rightarrow H^{p-3}(X,\mathbb{Z})$

Expanding the integrand in (6.108) into an harmonic basis over the Calabi-Yau,
gives the superpotential

${\cal W}_{\text{NS-
NS}}(Z;T)=-UT_{\Lambda}\left(h^{\Lambda}+\omega_{i}^{\Lambda}\Phi^{i}+\frac{1}{2}\kappa_{ijk}Q^{\Lambda
i}\Phi^{j}\Phi^{k}-\frac{1}{6}\kappa_{ijk}\Phi^{i}\Phi^{j}\Phi^{k}R^{\Lambda}\right)$
(6.111)

which can be recast in the more compact form

${\cal W}_{\text{NS-NS}}({\cal Z};T)=-c^{\Lambda}_{A}T_{\Lambda}{\cal
V}^{A}(Z;T)\;.$ (6.112)

where we have used the quantities have been defined so as

${\mathfrak{D}}e^{J_{c}}=:-\left(h^{\Lambda}+\omega_{i}^{\Lambda}\Phi^{i}+\frac{1}{2}\kappa_{ijk}Q^{\Lambda
i}\Phi^{j}\Phi^{k}-\frac{1}{6}\kappa_{ijk}\Phi^{i}\Phi^{j}\Phi^{k}R^{\Lambda}\right)\alpha_{\Lambda}\;.$
(6.113)

The four-dimensional tadpole cancellation condition requires that the
contribution$\int C_{7}\wedge\mathfrak{D}\,\overline{\mathbf{G}}$, once we
expand $C_{7}$, is canceled by the contributions of background charges;
namely, the tadpole cancellation condition (6.93) now becomes

${\cal Q}_{\rm
bg}^{\Lambda}-h^{\Lambda}m^{0}+\omega_{i}^{\Lambda}m^{i}-e_{i}Q^{\Lambda
i}-e_{0}R^{\Lambda}=0\,.$ (6.114)

These can be written in the form (6.3) by making the index change
$(\ldots)_{I}\rightarrow(\ldots)^{\Lambda}$, grouping the fluxes into ${\cal
N}_{\cal A}=(N_{A},c^{\Lambda}_{B})\in\Gamma$ and correspondingly decomposing

$({\cal I}^{\Lambda})^{{\cal A}{\cal
B}}=\begin{pmatrix}0&\delta_{\Theta}^{\Lambda}I^{AD}\\\
-\delta_{\Sigma}^{\Lambda}I^{CB}&0\end{pmatrix}$ (6.115)

with

$I^{AB}=\begin{pmatrix}0&0&0&-1\\\ 0&0&-\delta^{i}_{j}&0\\\
0&\delta_{i}^{j}&0&0\\\ -1&0&0&0\end{pmatrix}\,,$ (6.116)

which identifies the charges

$Q^{\Lambda}\equiv-(R^{\Lambda},Q^{\Lambda
i},-\omega^{\Lambda}_{i},h^{\Lambda})\,.$ (6.117)

The path that we followed in the previous section to get the dual formulation
can be applied to the present case as well, taking care of the change of the
tadpole condition, from (6.93) to the more complicated (6.114). The gauging of
the three-form multiplets of (4.96), due to (6.117), now involve all the
multiplets

$\displaystyle\hat{S}^{0}$ $\displaystyle=-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R}){\cal P}^{0}+{\rm i}R^{\Lambda}\Gamma_{\Lambda}\,,$ (6.118)
$\displaystyle\hat{S}^{i}$ $\displaystyle=-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R}){\cal P}^{i}+{\rm i}Q^{\Lambda i}\Gamma_{\Lambda}\,,$
$\displaystyle\hat{S}_{i}$ $\displaystyle=-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R})\tilde{\cal P}_{i}-{\rm
i}\omega_{i}^{\Lambda}\Gamma_{\Lambda}\,,$ $\displaystyle\hat{S}_{0}$
$\displaystyle=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})\tilde{\cal
P}_{0}+{\rm i}h^{\Lambda}\Gamma_{\Lambda}\,,$

resulting into a gauging of _all_ the gauge three-form entering the four
dimensional theory as

$\displaystyle\hat{F}_{4}^{0}$
$\displaystyle={\rm{d}}A_{3}^{0}-R^{\Lambda}C_{4\Lambda}\,,$ (6.119)
$\displaystyle\hat{F}_{4}^{i}$ $\displaystyle={\rm{d}}A_{3}^{i}-Q^{\Lambda
i}C_{4\Lambda}\,,$ $\displaystyle\hat{F}_{4i}$
$\displaystyle={\rm{d}}A_{3i}+\omega_{i}^{\Lambda}C_{4\Lambda}\,,$
$\displaystyle\hat{F}_{40}$
$\displaystyle={\rm{d}}A_{30}-h^{\Lambda}C_{4\Lambda}\,.$

Of course, this reflects also in a change of the gauging of the linear
multiplets (6.101) which now, in order to incorporate all the non-geometric
fluxes, gets modified to

$\hat{L}^{\Lambda}=L^{\Lambda}+c_{A}^{\Lambda}{\cal P}^{A}$ (6.120)

which, in components, implies the gauging of the two-forms as

$\displaystyle\hat{\cal H}_{3}^{\Lambda}$ $\displaystyle={\rm{d}}{\cal
B}_{2}^{\Lambda}+h^{\Lambda}A_{3}^{0}+\omega_{i}^{\Lambda}A_{3}^{i}+Q^{\Lambda
i}\tilde{A}_{3i}+R^{\Lambda}\tilde{A}_{30}\;.$ (6.121)

The four-dimensional bosonic action is the very same as (6.104), provided
however the redefinition of the gauged two- and three-forms according to
(6.121) and (6.119).

#####  6.3.5 Including open string moduli

The _open string_ moduli sector can also be introduced without many
difficulties. In this case, we will just focus on moduli originating from
spacetime filling D6 branes [120]. Let us for simplicity turn off the
geometric and non-geometric fluxes, restricting $\Gamma$ so that

$\omega^{\Lambda}_{i}=R^{\Lambda}=Q^{\Lambda j}=0\,,$ (6.122)

in order to deal with proper Calabi-Yau orientifold compactifications. The
chiral moduli $\chi_{\alpha}\simeq\chi_{\alpha}+1$ of a D6-brane wrapping a
special Lagranian 3-cycle $S$ (including D6 and image D6) combine Wilson lines
and geometric moduli, both labelled by $\alpha=1,\ldots,b^{-}_{1}(S)$. These
couple to the bulk moduli through a contribution to the superpotential [150,
151]. In super-Weyl invariant notation, this contribution takes the form999In
the presence of metric fluxes D6-branes also develop superpotentials quadratic
in $\chi_{\alpha}$, see e.g. [156].

${\cal W}_{\rm D6}=-n^{\alpha}_{a}\,\chi_{\alpha}z^{a}\,.$ (6.123)

Here $n^{\alpha}_{0}\in\mathbb{Z}$ correspond to the D6-brane world-volume
flux quanta, which take values into $H^{2}_{-}(S;\mathbb{Z})$ and are then
also labelled by $\alpha=1,\ldots,b^{-}_{1}(S)$. The remaining quantized
coupling constants are given by the intersection numbers

$n^{\alpha}_{i}=\int_{S}\omega_{i}\wedge\eta^{\alpha}\,,$ (6.124)

where $\omega_{i}$ is the basis of $H^{2}_{-}(X;\mathbb{Z})$ used to identify
the complexified Kähler moduli $\varphi^{i}$, while $\eta^{\alpha}$ is the
basis of $H^{1}_{-}(S)$ used to identify the world-volume moduli
$\chi_{\alpha}$ [150]. Comparing (6.123) with (4.116), it is clear that also
the coupling (6.123) can be interpreted as produced from a two-form gauging of
the linear multiplets dual to the world-volume chiral fields $\chi_{\alpha}$.
To be more precise, let us denote by $\hat{\cal B}_{2}^{\alpha}$ the
corresponding two-form potentials and split $A^{A}_{3}$ into
$(A^{a}_{3},\tilde{A}_{3a})$. Then, (6.123) corresponds to a gauging

$\hat{\cal B}_{2}^{\alpha}\rightarrow\hat{\cal
B}_{2}^{\alpha}+n^{\alpha}_{a}\Lambda^{a}_{2}\,,$ (6.125)

under the transformation $A^{a}_{3}\rightarrow
A^{a}_{3}+{\rm{d}}\Lambda^{a}_{2}$, implying the introduction of the gauge
invariant three-forms

$\hat{\cal H}_{3}^{\alpha}\rightarrow\hat{\cal
H}_{3}^{\alpha}+n^{\alpha}_{a}A_{3}^{a}\,.$ (6.126)

In doing so, we are assuming the fluxes $n^{\alpha}_{a}$ must be treated as
non-dynamical in the EFT. This also fits well with the observation made in
[121] that the matrix $T^{AB}$ becomes non-invertible in the presence of such
fluxes.

Again, the four-dimensional bosonic action acquires the same form as (6.104),
provided however the redefinition of the gauged three-forms (6.119) and the
gauged two-forms given by (6.121). However, the Legendre transform $F$ will
also depend on the (duals of) the moduli $\chi_{\alpha}$ and the associated
two-forms appear via the gauge-invariant field strengths (6.126).

####  6.4 M-theory effective field theories

At very strong coupling, when $g_{s}\gg 1$, the ten-dimensional Type IIA
effective theory is no longer valid and we should rather pass to the eleven-
dimensional M-theory description. Indeed, the eleven-dimensional M-theory
spectrum is naturally endowed with a gauge three-form, which will reflect, as
we will see shortly, in the presence of multiple gauge three-forms in the
four-dimensional spectrum, and thus seems naturally fit to apply the
dualization procedure developed in Chapter 4.

It is known that, once compactified on a circle, M-theory reproduces the
weakly coupled Type IIA supergravity, but M-theory and Type IIA are definitely
not quite the same. For example, no Romans mass is allowed in the M-theory
description [157] and no tadpole cancellation condition constrains the
background fluxes. These peculiarities forces us to treat the M-theory EFTs
separately from the Type IIA ones. In the following we will first consider the
case where the seven-dimensional manifold upon which we compactify the eleven-
dimensional M-theory enjoys a $G_{2}$–holonomy. This request will be later
relaxed and we will also provide connections with the Type IIA case examined
in the previous section.

#####  6.4.1 From 11D to 4D

The eleven-dimensional massless, bosonic spectrum of M-theory is made up just
by the graviton $g_{MN}$ and a gauge three-form $\tilde{C}_{3}$. The low
energy supergravity EFT is [145, 27, 133]

$\displaystyle S_{\text{M-theory}}^{(11)}$
$\displaystyle=\frac{1}{2\kappa_{11}^{2}}\int\left(\tilde{R}*1-\frac{1}{2}\tilde{G}_{4}\wedge*\tilde{G}_{4}-\frac{1}{6}\tilde{C}_{3}\wedge\tilde{G}_{4}\wedge\tilde{G}_{4}\right)\;,$
(6.127)

where we have introduced the four-form field strength
$\tilde{G}_{4}={\rm{d}}\tilde{C}_{3}$.

A four-dimensional ${\cal N}=1$ supergravity theory can be obtained by
compactifying M-theory over a proper seven manifold $Y$. Presently, we shall
assume that the manifold $Y$ is equipped with a $G_{2}$–holonomy. The manifold
$Y$, similarly to a Calabi-Yau manifold, is embedded with a real, covariantly
constant and harmonic three-form $\Phi_{3}$, satisfying
${\rm{d}}_{Y}\Phi_{3}=0$.

The identification of the four-dimensional fields entering the effective
theory proceeds along the same lines as for Type II superstring theories. The
scalar fields have a two-fold origin. Introducing a basis of harmonic forms
$\\{\check{\rho}_{\check{i}}\\}$ of $H^{3}(X)$, with
$\check{\imath}=1,\ldots,b^{3}(X)$, we may decompose the eleven dimensional
three-form $\tilde{C}_{3}$ as

$\tilde{C}_{3}=c^{\check{\imath}}\check{\rho}_{\check{\imath}}\,,$ (6.128)

as well as the covariantly constant three-form $\Phi_{3}$ as

$\Phi_{3}=\phi^{\check{\imath}}\check{\rho}_{\check{\imath}}\,.$ (6.129)

The zero modes $c^{\check{\imath}}$ and $\phi^{\check{\imath}}$ identify the
complex coordinates

$\check{S}^{\check{\imath}}\equiv\int_{\Sigma^{\check{\imath}}}(\tilde{C}_{3}+{\rm
i}\Phi_{3})=c^{\check{\imath}}+{\rm i}\phi^{\check{\imath}}\,.$ (6.130)

where we have introduced a basis $\Sigma^{\check{\imath}}$ of three-cycles in
$H^{3}(X;\mathbb{Z})$, Poicaré dual to $\check{\rho}_{\check{\imath}}$. The
coordinates $\check{S}^{\check{\imath}}$ parametrize a Kähler manifold with
Kähler potential

$K_{G_{2}}=-3\log\left(\frac{1}{7\kappa_{11}^{2}}\int_{X}\Phi_{3}\wedge*\Phi_{3}\right)\,.$
(6.131)

At the superfield level, we may regard $\check{S}^{\check{\imath}}$ as the
lowest components of a set of $b^{3}(X)$ superfields which, with a slight
abuse of notation, we will call $\check{S}^{\check{\imath}}$ as well.

A superpotential for the chiral superfields $\check{S}^{\check{\imath}}$ is
introduced by turning on some background fluxes $\overline{G}_{4}$ and its
dual $\overline{G}_{7}$. Expanding $\overline{G}_{4}$ in an harmonic basis
$\\{\check{\zeta}^{\check{\jmath}}\\}$ of $H^{4}(X;\mathbb{Z})$, with
$\check{\jmath}=1,\ldots,b^{3}(X)$, dual to
$\\{\check{\rho}_{\check{\imath}}\\}$, as

$\overline{G}_{4}=n_{\check{\jmath}}\check{\zeta}^{\check{\jmath}}$ (6.132)

and $\overline{G}_{7}$ as

$\overline{G}_{4}=n_{0}{\rm{d}}{\rm vol}_{Y}$ (6.133)

we recognize that the available number of fluxes is $b^{3}(X)+1$. Then, a
superpotential for the chiral fields $S^{\check{\imath}}$ acquires the form
[158, 132, 159]

${\cal W}_{G_{2}}=U\left[\int_{X}G_{7}+\int_{X}\left(\tilde{C}_{3}+{\rm
i}\Phi_{3}\right)\wedge\tilde{G}_{4}\right]$ (6.134)

where $G_{4}\equiv{\rm{d}}\tilde{C}_{3}+\overline{G}_{4}$ includes also the
contribution of the background four-form fluxes. The contributions from
$G_{7}$ is actually tied to the one of $G_{4}$ via [158]

$\int_{X}\left(G_{7}+\frac{1}{2}C_{3}\wedge G_{4}\right)=0$ (6.135)

so that (6.134) can be also written as

${\cal W}_{G_{2}}=U\int_{X}\left(\frac{1}{2}C_{3}+{\rm i}\Phi\right)\wedge
G_{4}$ (6.136)

As stressed in [158], the factor of $\frac{1}{2}$ is crucial to make the
superpotential holomorphic in $S^{\check{\imath}}$. In fact, now expanding
$C_{3}$, $\Phi$ and $\overline{G}_{4}$ as in (6.128), (6.129) and (6.132),
(6.134) gives

${\cal
W}_{G_{2}}=U\left(n_{0}+n_{\check{\imath}}\check{S}^{\check{\imath}}\right)\,.$
(6.137)

The introduction of $U$, interpreted here as a compensator, is necessary in
the super-Weyl formulation, in order to ensure the superpotential (6.137) to
properly acquire super-Weyl weight $(1,0)$. In the following, it will also
convenient to collect the fluxes as $n_{\cal I}=(n_{0},n_{\check{\imath}})$,
with ${\cal I}=0,1,\ldots,b^{3}(Y)$. Accordingly, we also define the _periods_
${\cal V}^{\cal I}(S)=(U,U\check{S}^{\check{\imath}})$, the superpotential
being rewritten as

${\cal W}_{G_{2}}=n_{{\cal I}}{\cal V}^{\cal I}(U,S)\,.$ (6.138)

#####  6.4.2 Reformulating the 4D EFT

The potential of the four-dimensional theory is characterized by $b^{3}(Y)+1$
background fluxes $(n_{0},n_{\check{\imath}})$, spanning a lattice

$\Gamma=\left\\{(n_{0},n_{\check{\imath}})\;|\;n_{0},n_{\check{\imath}}\in\mathbb{Z}\right\\}=\\{n_{\cal
I}\\}\,.$ (6.139)

In contrast with the Type II EFTs, no tadpole cancellation condition
constrains the admissible values $n_{{\cal I}}$. We may then regard _all_ the
lattice $\Gamma$ as generated by a set of gauge three-forms $A_{3}^{\cal I}$.
Comparing the number of complex scalars upon which the superpotential (6.137)
depends, $b^{3}(Y)$, with the dimensionality of the lattice (6.139), we
immediately understand that the case at hand falls into the linear
superpotential family examined in Section 4.2.1. We then trade all the periods
${\cal V}^{I}$ with single three-form multiplets as in (4.70)

${\cal V}^{\cal I}(z)=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P^{\cal
I}\equiv S^{\cal I}\,,$ (6.140)

each of them including, among its bosonic components, the moduli
$s^{\check{\imath}}$, along with the compensator $u$, via $z^{\cal I}(u,s)$ as
well as the gauge three-forms $A_{3}^{\cal I}$

$S^{\cal I}=\\{z^{\cal I},A_{3}^{\cal I}\\}\,.$ (6.141)

The dual three-form theory is then immediately read off from (4.71) and is
given by

$\displaystyle S|_{\rm bos}$ $\displaystyle=M_{\rm
P}^{2}\int_{\Sigma}\left(\frac{1}{2}R*1-K_{\check{\imath}\bar{\check{\jmath}}}{\rm{d}}s^{\check{\imath}}\wedge*{\rm{d}}\bar{s}^{\bar{\check{\jmath}}}\right)$
(6.142) $\displaystyle\quad\,-\frac{1}{2}\int_{\Sigma}T_{{\cal I}{\cal
J}}F^{\cal I}_{4}*\\!F^{\cal J}_{4}+\int_{\partial\Sigma}T_{{\cal I}{\cal
J}}*F^{\cal I}_{4}A_{3}^{\cal J}$

with $T_{{\cal I}{\cal J}}$ defined from (4.72), with $\alpha=4$. The spectrum
of the dual theory is summarized in Table 6.5.

Superfields | Bosonic fields | Number | Data
---|---|---|---
|  |  |  $\Phi$–deformations
| $z^{{\cal I}}$ |  |  $C_{3}$–axions
$S^{{\cal I}}$ | $A_{3}^{\cal I}$ | $b^{3}(Y)$ | gauge three-forms
Table 6.5: How the 4D fields originating form compactification of M-theory
theory reassemble as lowest components of ${\cal N}=1$ chiral superfields in
the dual picture.

#####  6.4.3 Connection to Type IIA compactifications

In order to connect this picture with the Type IIA orientifold
compactifications of Section (6.3), let us assume that the internal manifold
$Y$ can be decomposed as [132]

$Y=\frac{X\times S^{1}}{\hat{\cal R}}$ (6.143)

where $X$ is a Calabi-Yau three-fold, $S^{1}$ a circle with radius $R$ and
$\hat{\cal R}$ acts on $X$ as an anti-holomorphic involution and on $S^{1}$ by
inverting the coordinate $y\to-y$ (in other words, the only forms along
$S^{1}$ which we should consider belong to the class $H^{1}_{-}(S^{1})$).

To begin with, we split the eleven-dimensional forms $\Phi_{3}$ and $C_{3}$ in
components along the Calabi-Yau $X$ and the circle $S^{1}$. The choice (6.143)
makes the third cohomology class of $Y$ split as

$H^{3}(Y)=H^{3}_{+}(X)\oplus[H^{2}_{-}(X)\wedge H^{1}_{-}(S^{1})]$ (6.144)

and we can then expand $C_{3}$ and $\Phi_{3}$ as

$\displaystyle C_{3}$
$\displaystyle=\hat{B}_{2}\wedge{\rm{d}}y+\sqrt{2}\hat{C}_{3}$ (6.145)
$\displaystyle\Phi_{3}$ $\displaystyle=\hat{J}\wedge{\rm{d}}y+\sqrt{8}{\rm
Re\hskip 1.00006pt}({\cal C}\Omega)$

where the hatted-quantities have only internal legs.

In order to compute the superpotential (6.134), we will explicitly need the
four-form field strength $G_{4}$ with only internal legs

$\displaystyle
G_{4}|_{Y}=\overline{H}_{3}\wedge{\rm{d}}y+\sqrt{2}\overline{G}_{4}$ (6.146)

where $\overline{H}_{3}$ and $\overline{F}_{4}$ are the background fluxes, as
well as $G_{7}|_{Y}$ decomposed as

$\displaystyle G_{7}|_{Y}=\overline{G}_{6}\wedge{\rm{d}}y$ (6.147)

where $\overline{F}_{6}$ is an internal six-form flux. As in Type IIA, we now
define the complexified forms

$J_{c}=\hat{B}_{2}+{\rm i}J\,,\qquad\Omega_{c}=\hat{C}_{3}+2{\rm i}{\rm
Re\hskip 1.00006pt}({\cal C}\Omega)\,.$ (6.148)

Therefore, we immediately recognize that the chiral coordinates (6.130)
emerging from M-theory

$\check{S}^{\check{\imath}}\check{\rho}_{\check{\imath}}=J_{c}\wedge{\rm{d}}y+\sqrt{2}\Omega_{c}$
(6.149)

do actually include at once both the Kähler and the complex structure moduli,
which constituted the scalar spectrum considered in Section 6.3.1 (see Table
6.3).

Then, the superpotential becomes

${\cal
W}_{G_{2}}=U\int_{X}(\overline{G}_{6}+J_{c}\wedge\overline{G}_{4}+\Omega_{c}\wedge\overline{H}_{3})\,.$
(6.150)

Expanding now over an internal basis of $X$, we get

${\cal W}_{G_{2}}=U(e_{0}+e_{i}\Phi^{i}+h^{\Lambda}T_{\Lambda})$ (6.151)

where the normalization of the background fluxes $e_{i}$ and $h^{\Lambda}$ is
chosen so as to match with (6.79)-(6.82).

We get a four-dimensional theory, which is consistent with those obtained from
Type IIA compactifications. This is indeed a subcase of the superpotentials
appearing in Type IIA (6.79)-(6.82). By consistency with the strongly coupled
regime, the Romans mass $m^{0}$ has necessarily to be zero [157]. Here also
$m^{i}=0$, but we will see in the next section how to reintroduce them. Hence,
the full superpotential which come from compactifying on $G_{2}$-holonomy
seven-manifolds, translated in terms of moduli for $X$, is the same as (6.83),
provided that in the latter we set $\hat{\cal W}=0$ and $m^{a}=0$. The three-
form theory (6.142) indeed can be retrieved from the Type IIA case (6.104)
employing these assumptions.

#####  6.4.4 Relaxing the $G_{2}$–holonomy assumption

The assumption of $G_{2}$-holonomy of the internal seven-manifold allows only
for the expansion (6.128)-(6.129) of the three-forms $C_{3}$ and $\Phi_{3}$.
This, in turn, influences the structure of the superpotential (6.134)
generated by the background fluxes. From the Type IIA perpspective, the
superpotential (6.137) cannot contain two-form background fluxes, as the
$m^{i}$ appearing in (6.84). However, we may relax the $G_{2}$–holonomy
request, so that the three-form $\Phi_{3}$ is not closed anymore, that is
${\rm{d}}_{Y}\Phi_{3}\neq 0$. Then, the expansions (6.128) and (6.129) are
performed over a basis which is not harmonic, that is, in particular,
${\rm{d}}_{X}\check{\rho}^{\check{\imath}}\neq 0$.

The generalization of the superpotential (6.136) is [160, 159, 161, 162, 163]

$\begin{split}{\cal W}_{G_{2}}&=U\int_{X}\frac{1}{2}(C_{3}+{\rm
i}\Phi_{3})\wedge{\rm{d}}(C_{3}+{\rm i}\Phi_{3})\\\
&=U\int_{X}\left[\overline{G}_{7}+(\tilde{C}_{3}+{\rm
i}\Phi_{3})\wedge\overline{G}_{4}+\frac{1}{2}(\tilde{C}_{3}+{\rm
i}\Phi_{3})\wedge{\rm{d}}(\tilde{C}_{3}+{\rm i}\Phi_{3})\right]\\\
&=U\left(n_{0}+n_{\check{\imath}}\check{S}^{\check{\imath}}+\frac{1}{2}m_{\check{\imath}\check{\jmath}}\check{S}^{\check{\imath}}\check{S}^{\check{\jmath}}\right)\,,\end{split}$
(6.152)

where we have defined the symmetric matrix

$m_{\check{\imath}\check{\jmath}}\equiv\int_{X}\check{\rho}_{\check{\imath}}\wedge{\rm{d}}\check{\rho}_{\check{\jmath}}\;,$
(6.153)

whose entries play the role of quantized fluxes. The Type IIA superpotential
(6.84) can now be recovered from the compactification over $G_{2}$-structure
manifolds. The expansions (6.145) of $C_{3}$ and $\Phi$ read

$\displaystyle C_{3}$
$\displaystyle=\hat{B}_{2}\wedge\sigma+\sqrt{2}\hat{C}_{3}+[\overline{B}_{2}\wedge{\rm{d}}y+\sqrt{2}\overline{C}_{3}]$
(6.154) $\displaystyle\Phi_{3}$
$\displaystyle=\hat{J}\wedge\sigma+\sqrt{8}{\rm Re\hskip 1.00006pt}(C\Omega)$

where $\sigma={\rm{d}}y+\overline{A}_{1}$ and
${\rm{d}}\sigma=\overline{F}_{2}=m^{i}\omega_{i}$ represents the failure of
the closure of $\Phi$. Hence,

$G_{4}|_{Y}={\rm{d}}_{Y}C_{3}=\hat{B}_{2}\wedge\overline{F}_{2}+\overline{H}_{3}\wedge{\rm{d}}y+\sqrt{2}\overline{F}_{4}$
(6.155)

Substituted in the first line of (6.152) and integrating over the $S^{1}$
gives

${\cal
W}_{G_{2}}=U\int_{X}\left(\overline{F}_{6}+J_{c}\wedge\overline{F}_{4}+\frac{1}{2}J_{c}\wedge
J_{c}\wedge\overline{F}_{2}+\Omega_{c}\wedge\overline{H}_{3}\right)\,.$
(6.156)

A final expansion over a over an internal basis of $X$ gives

${\cal
W}_{G_{2}}=U\left(e_{0}-e_{i}\Phi^{i}+\frac{1}{2}\kappa_{ijk}m^{i}\Phi^{j}\Phi^{k}+h^{\Lambda}T_{\Lambda}\right)$
(6.157)

The choice of which fluxes to generate dynamically then proceeds as in Section
(6.3.3), with the simplification $m^{0}=0$ and no tadpole cancellation
condition. The bosonic theory is the same as the one obtained in (6.104) for
vanishing Romans mass: this, in particular, implies that in (6.104) the gauge
three-forms are _not_ gauged, with no gauge four-form appearing because no
tadpole condition has to be enforced.

The four-dimensional bosonic action in then again as (6.104), but now no gauge
three-form associated to $m^{0}$ appears: namely, in (6.104) we should set
$\tilde{A}_{30}=0$.

####  6.5 The F-theory uplift

Before concluding this chapter, let us briefly comment on the possible uplift
of the previous constructions to the F-theory case. In Section 6.2 we have
seen how, in a weak-coupling regime with only O3-planes and D3-branes, we may
choose an isotropic sublattice $\Gamma_{\rm EFT}\subset\Gamma$. In more
general flux compactifications, the choice of $\Gamma_{\rm EFT}$ is less
obvious. This is indeed what happens if D7-branes wrapping holomorphic
surfaces are present [164, 165] and we include their world-volume fluxes into
$\Gamma$, or we go to a strongly coupled F-theory regime. In order to
illustrate this point let us consider an F-theory compactification on a smooth
elliptically fibred Calabi-Yau four-fold $Y$ – see [140, 166, 167] for
reviews. We can then identify $\Gamma$ with the lattice of ‘transversal’
fluxes $G_{4}\in H^{4}(Y;\mathbb{Z})_{\rm T}$ [168],101010For smooth
elliptically fibered Calabi-Yau four-folds there is no half-integral
correction to the flux quantization [169]. that is, whose Poincaré dual
4-cycle has vanishing intersection number with any pair of divisors of $Y$. By
introducing an appropriate basis of transversal cocycles $\alpha^{\cal A}\in
H^{4}(Y;\mathbb{Z})_{\rm T}$, we can expand

$G_{4}={\cal N}_{\cal A}\,\alpha^{\cal A}\,,$ (6.158)

and introduce the symmetric pairing

${\cal I}^{{\cal A}{\cal B}}\equiv\int_{Y}\alpha^{\cal A}\wedge\alpha^{\cal
B}\,.$ (6.159)

The D3-charge tadpole condition then takes the form (6.38), with the
additional contribution $-\frac{1}{24}\chi(Y)$ to ${\cal Q}_{\rm bg}$, where
$\chi(Y)$ is the Euler characteristic of $Y$. Furthermore, the flux-induced
superpotential is given by

${\cal W}(z)={\cal N}_{\cal A}{\cal V}^{\cal A}(z)\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{with}\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\cal V}^{\cal A}(z)\equiv\frac{\pi}{\ell^{6}_{\rm
M}}\int_{Y}\Omega_{4}\wedge\alpha^{\cal A}\,,$ (6.160)

where $\Omega_{4}$ is the (dimensionless) holomorphic (4,0)-form on $Y$ and
$\ell^{6}_{\rm M}$ is the M-theory Planck length. The chiral fields $z^{a}$
($a=0,\ldots,h^{3,1}$) parametrize $Y$-complex structure moduli and include
the super-Weyl compensator.

Let us now pick an isotropic sublattice $\Gamma_{\rm EFT}\subset\Gamma$. By
splitting ${\cal N}_{\cal A}$ as in (1.38), and defining ${\cal
V}^{A}(z)\equiv v^{A}_{\cal A}{\cal V}^{\cal A}(z)$, we can write the
superpotential (6.160) in the form

${\cal W}(z)=N_{A}\,{\cal V}^{A}(z)+\hat{\cal W}(z)\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{with}\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \hat{\cal W}(z)\equiv{\cal N}^{\rm bg}_{\cal A}\,{\cal
V}^{\cal A}(z)\,.$ (6.161)

By making an Hodge decomposition of $H^{4}(Y;\mathbb{C})_{\rm T}$, one
realises that the pairing (6.159) has signature $(2+h^{2,2}_{\rm
T},2h^{3,1})$. Then the dimension of any maximal isotropic sublattice
$\Gamma_{\rm EFT}\subset\Gamma$ is at most $2h^{3,1}$. This means there are
always enough complex scalars $z^{a}$ to allow for a supersymmetric
dualization of the flux quanta $N_{A}$ spanning $\Gamma_{\rm EFT}$ to three-
form potentials $A^{A}_{3}$. 111111For a general Calabi-Yau four-fold, the
pairing (6.159) has signature $(b^{+}_{4},b_{4}^{-})$, with
$b^{+}_{4}-b^{-}_{4}=chi/3+32$, If $h^{2,1}=0$, from the

As a concrete example, leaving a more detailed discussion of other settings to
the future, we can consider the model spelled out in [170], for which
$\dim\Gamma=23320$, $h^{3,1}=3878$, $h^{2,2}_{\rm T}=h^{2,2}-2=15562$. In this
case, following our general prescription, one would obtain an EFT with
$\dim\Gamma_{\rm EFT}=7756$ three-form potentials $A^{A}_{3}$ and
$\dim(\Gamma/\Gamma_{\rm EFT})=15564$ non-dynamical fluxes. The $7756$ three-
form potentials may be accommodated, together with the Weyl compensator and
$3877=h^{3,1}-1$ of the $Y$-complex structure moduli, into $3878$ double
three-form multiplets. Notice that there does not appear any ‘natural’ choice
of the sublattice $\Gamma_{\rm EFT}$ and of the $Y$-complex structure modulus
which is excluded from these double three-form multiplets. Analogously,
differently from what we observed in the previous subsection, there is no
obvious hierarchy between the energy scales of the potential and the membrane
tensions.

It is instructive to observe how such ‘democracy’ is removed by going to weak
coupling. In this limit, the total amount of fluxes $23320$ splits into
$2\times 300$ bulk R-R plus NS-NS three-form fluxes and $22720$ D7-brane
fluxes, while the moduli $z^{a}$ include the overall Weyl compensator, the
axio-dilaton $\tau$, $149$ bulk complex-structure moduli, and $3728$ D7-brane
geometric moduli. The $149$ complex-structure moduli and the Weyl compensator
can combine with 300 R-R bulk fluxes into double three-form multiplets, as we
did in the previous subsections. The axio-dilaton $\tau$ can again be dualized
to a linear multiplet which is gauged, with charge defined by the $300$ NS-NS
fluxes. The $7756-300=7456$ D7-brane fluxes in $\Gamma_{\rm EFT}$, can then be
accommodated, together with the D7-brane moduli, into other 3728 double three-
form multiplets. The remaining 15264 D7-brane fluxes stand non-dynamical and
contribute to $\hat{\cal W}$ in (6.161).

Leaving a more detailed study of the three-form formulation of the EFT for
general F-theory compactifications – as well as their weak-coupling limits –
to the future, we close this section by briefly discussing the microscopic
origin of the three-form gauging

$A^{A}_{3}\rightarrow A^{A}_{3}-Q^{A}\Lambda_{3}\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{with}\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ Q^{A}={\cal I}^{{\cal A}{\cal B}}v_{{\cal A}}^{A}{\cal
N}_{\cal B}^{\rm bg}\,.$ (6.162)

This can be understood from the perspective of the dual M-theory compactified
to three dimensions on an elliptically fibred Calabi-Yau four-fold $Y$ and the
derivation is similar to the weakly-coupled IIB case discussed in Section
6.2.3. One can start from the eleven-dimensional Bianchi identity

${\rm{d}}F_{7}-\frac{1}{2}F_{4}\wedge F_{4}=(\text{M2-brane charge})\,.$
(6.163)

By appropriately reducing $F_{7}$, $F_{4}$ and the corresponding potentials to
three dimensions one gets a set of gauge-invariant three-forms which are dual
to the F-theory field-strengths $\hat{F}^{A}_{4}={\rm{d}}A^{A}_{3}+Q^{A}C_{4}$
. These are gauge invariant under $C_{4}\rightarrow C_{4}+{\rm{d}}\Lambda_{3}$
provided that $A^{A}$ transform as in (6.162). In the weak-coupling limit,
this generalizes the gauging of three-forms discussed in the previous
subsections by including a sector supported on the D7-branes.

### Consistency conditions of EFTs and the spectrum of BPS–objects

The previous chapter led us to a full reformulation of the four-dimensional
effective theories originating from string and M-theory. The introduction of
the gauge three-forms into the EFTs allowed us to give a dynamical
interpretation of the appearance of the background fluxes. However, not all of
the background fluxes that enter the four-dimensional EFT can be generically
dualized to gauge three-forms. The tadpole cancellation conditions and the
Freed-Witten anomalies force us to treat some of the fluxes as frozen: these
fluxes in fact identify the gauging parameters for the gauge two- and three-
forms, which cannot be rendered dynamical. Accordingly, in the previous
chapter, in both Type IIA and Type IIB cases, we _arbitrarily_ chose which
fluxes can be dualized and those which may not.

We will now fill the gap of the previous chapter and provide some physical
arguments to enforce the choices of dynamical sublattices of Sections 6.2 and
6.3. These arguments rely on the amount and type of extended objects, either
strings, membranes or 3-branes, which can be included into the effective field
theory. The spectrum of objects is indeed strictly tightened to cut-off of the
EFT as well as on the masses of the scalar fields and how these change across
the spacetime: namely, crossing membranes and encircling strings induce
changes in the background fluxes and axions, leading to a change in the mass
spectrum. Further developing the ideas settled in the introductory Section
1.6, we shall see how forbidding some transitions, basically by requiring the
full consistency of the effective description, naturally leads to the
exclusion of some objects in the EFT.

The picture that will emerge at the end of the chapter convey an EFT whose
spectrum of objects is strictly fixed by the perturbative regimes that the EFT
is supposed to scan. The inclusion of other objects in the theory, as well as
an improper dualization of fluxes to gauge three-forms or of axions to two-
forms, may lead the EFT to the _Swampland_ of inconsistent EFTs.

####  7.1 Extended objects from higher dimensions

In Section 5.1 we have introduced supersymmetric 3-branes, membranes and
strings. Writing down a supersymmetric action for them required to enforce
$\kappa$-symmetry, which strictly tightens together the tensions for these
objects to their charges. From a different viewpoint, in ten-dimensional
string theory $p$-branes can be coupled, understood as hypersurfaces spanning
a $(p+1)$–dimensional volume. Once we reduce the ten-dimensional theory down
to 4D, a plethora of objects stem from such $p$-branes and we end up with a
four-dimensional theory which naturally include a hierarchy of BPS–objects, as
those of Section 5.1.

We will briefly recall how the reduction can be performed, showing that the
tension of the reduced objects naturally depends on the moduli of the
compactification. Our aim is to get a grasp of how the BPS–objects of Section
5.1 may be generated from string theory and we will be very schematic,
referring to [133, 171] and references therein for a more detailed discussion.

In ten dimensions and in the string frame111We recall that, in the string
frame, the Einstein-Hilbert term of the Type II effective actions is
normalized as [133] $S_{\rm EH,II}^{(10)}=\frac{1}{2\kappa^{2}_{10}}\int
e^{-2\phi}R*1$ (7.1) with $\phi$ the ten-dimensional dilaton and
$\kappa^{2}_{10}=\ell_{\rm s}^{8}/(4\pi)$, where $\ell_{\rm
s}=2\pi\sqrt{\alpha^{\prime}}$ (with respect to [133], our $\ell_{\rm s}^{\rm
here}=2\pi l_{s}^{\rm there}$). We also recall that reducing (7.1) to four
dimensions, we get the following identification for the four-dimensional
Planck mass  $M_{\rm P}=\frac{\sqrt{4\pi V_{6}}}{\ell_{\rm s}g_{s}}\,,$ (7.2)
where $V_{6}$ is the volume of $X$ as computed in the string frame – see
(7.10).,the action of a D$p$-brane stretching along the cycle
$(p+1)$-dimensional cycle $\Sigma_{p+1}$ is given by

$S_{\text{D}p}=S_{\text{D}p,\rm DBI}+S_{\text{D}p,\rm CS}\,.$ (7.3)

The first, Dirac-Born-Infeld term is

$S_{\text{D}p,\rm DBI}=-\mu_{p}\int_{\Sigma_{p+1}}{\rm{d}}^{p+1}\sigma
e^{-\phi}\sqrt{-\det\left(P[g_{MN}+B_{MN}]\right)}\,,$ (7.4)

where $g_{MN}$ and $B_{MN}$ are, respectively, the ten-dimensional metric and
the components of the NS-NS two-form, with $P$ denoting their pullback to the
brane worldvolume $\Sigma_{p+1}$. Additionally, here and below we have
neglected possible contributions from the worldvolume gauge field. This term
encodes the kinetic terms of the $p$-brane and $\mu_{p}$ denotes its tension
which, in terms of the string length $\ell_{\rm s}$, is given by

$\mu_{p}=\frac{2\pi}{\ell_{\rm s}^{p+1}}\,.$ (7.5)

The second, Chern-Simons term in (7.3) expresses the minimal coupling of the
$p$-brane to a gauge $p$-form $C_{p+1}$

$S_{\text{D}p,\rm CS}=\mu_{p}\int_{\Sigma_{p+1}}P[C_{p+1}e^{-B_{2}}]\,,$ (7.6)

where, again, $P$ denotes the pull-back to the brane worldvolume. The sign and
charge $\mu_{p}$ in (7.6) are fixed by requiring that the full action (7.3)
preserves supersymmetry. Namely, it has to be $\kappa$-symmetric so that, when
(7.3) is evaluated at its ground state, only half of the supersymmetry
generators are spontaneously broken, making the D$p$-branes $1/2$–BPS objects,
in analogy with our four-dimensional membrane counterparts (5.15). We also
recall that the action of the NS5-branes is the same as (7.3), provided
however that the DBI and CS-terms in (7.3) get multiplied by $1/g_{\rm s}$.

Instead, anti-D$p$-branes, denoted with $\overline{\text{D}p}$, spontaneously
break all the supersymmetry: their action is the very same as (7.3), with,
however, an opposite Chern-Simons term. Such a difference of relative sign
does not allow the $\kappa$-symmetry to be realized, making it impossible to
preserve even a portion of the bulk supersymmetry over the worldvolume.222We
stress that the choices of sign depend on the background, namely from the
solutions of the Killing spinor equations that dictate the amount of preserved
supersymmetry (quite in analogy with what we did in Section 5.3 for four-
dimensional domain walls).

Other objects, already introduced in the previous chapter, are O-planes: they
preserve supersymmetry, albeit their DBI-term has an opposite sign with
respect to (7.3)

$S_{{\text{O}p}}=\mu_{p}\int_{\Sigma_{p+1}}{\rm{d}}^{p+1}\sigma
e^{-\phi}\sqrt{-\det\left(P[g_{MN}]\right)}-\mu_{p}\int_{\Sigma_{p+1}}P[C_{p+1}]\,.$
(7.7)

The sign of the Chern-Simons term has been also changed with respect to (7.3)
in order to ensure these objects to preserve supersymmetry.

In the next sections, we will be mostly interested in retrieving membranes and
strings in four-dimensional EFTs. For simplicity, let us focus just on
D$p$-branes. In order to get four-dimensional membranes, we can start with a
D$p$-brane and assume that it wraps an internal $(p-2)$-cycle $\Sigma_{p-2}$,
so that it spans a three-dimensional hypersurface ${\cal M}$ in the external
space. As a simplifying assumption, we shall take the cycle $\Sigma_{p+1}$ as
the product of $\Sigma_{p-2}$ and the external membrane worldvolume. The 4D
tension can be easily computed by dimensionally reducing the DBI–term (7.4)
as:

$S_{\rm memb,NG}=-\int_{{\cal M}}{\rm{d}}^{3}\sigma\,{\cal T}_{\bf
q}(\phi)\sqrt{-{\bf h}}\,,$ (7.8)

with the tension

${\cal T}_{\bf q}(\phi)=\frac{1}{4\sqrt{\pi}}g_{s}^{2}M_{\rm
P}^{3}\frac{V_{p-2}}{V_{6}^{3/2}}$ (7.9)

and we have defined the volumes in the string frame as

$V_{p-2}\equiv\ell_{\rm s}^{2-p}{\rm vol}\,(\Sigma_{p-2})\;.$ (7.10)

It is however important to stress that the volumes of the cycles ${\rm
vol}\,(\Sigma_{p-2})$ are _moduli dependent_ and therefore we expect also the
tension (7.9) to generically depend on the moduli of the compactification.
Furthermore, if we wish the membrane to be a supersymmetric object in four
dimensions, in the sense of Section 5.1.2, one needs to properly choose the
internal cycles $\Sigma_{p-2}$ (for example, in Type IIB, the three-cycles
wrapped by a D5-brane need to be _special Lagrangian_): this is a rather
important requirement, but its discussion goes beyond the scope of this work
and we refer, for example, to [172, 173] for detailed discussions. The
relation presented above is quite general and does not rely on the bulk
sector. We will however show in the examples of the next sections that the
tension (7.9) reduces to precisely to the expression (5.15) that we obtained,
in four dimensions, from arguments that were genuinely based on
$\kappa$–symmetry.

The same reasoning can be applied to get BPS–strings in the four-dimensional
EFT. A supersymmeytric string is obtain by wrapping a D$p$-brane over a
$\Sigma_{p-1}$ internal cycle, spanning a two-dimensional spacetime surface
${\cal S}$ in the external space. A direct reduction of (7.4), in the absence
of worldvolume fluxes, leads to the Nambu-Goto term

$S_{\rm string,NG}=-\int_{{\cal S}}{\rm{d}}^{2}\sigma\,{\cal T}_{\bf
q}(\phi)\sqrt{-{\bm{\gamma}}}\,,$ (7.11)

with the tension

${\cal T}_{\bf q}(\phi)=\frac{1}{2}g_{s}M_{\rm P}^{2}\frac{V_{p-1}}{V_{6}}\,.$
(7.12)

Supersymmetric 3-branes may, in principle, be treated equivalently as
membranes and strings. However, the 3-branes introduced in Section 5.1.1 are
quite peculiar, for the Nambu-Goto term is absent. Still, as we will see
explicitly in the Type IIB case in Section 7.3.3, those 3-branes constitute a
‘degenerate case’ of the extended objects treated here, because a mutual
cancellation between the Nambu-Goto and Chern-Simons part of the reduced (7.3)
will lead to an action which is the sole (5.2).

####  7.2 Perturbative regimes and spectrum of objects

A membrane allows for describing potential with different shapes in two
regions of spacetime, but there are some caveats. Within the same effective
theory, it is possible to pass from a potential $V(\varphi;N_{A})$, depending
on some quantized constants $N_{A}$, to the potential
$V(\varphi;N_{A}+q_{A})$, where the constants get shifted by $q_{A}$, if and
only if the constants $N_{A}$ are dual to a set of gauge three-forms
$A_{3}^{A}$ and there exists a membrane, with charges $q_{A}$ under
$A_{3}^{A}$, which allows for the flux transition $N_{A}\to N_{A}+q_{A}$.
Although generically a domain wall solution which interpolates between the
vacua on the two sides may not exist, the presence of a membrane is still a
necessary condition to connect the vacua of $V(\varphi;N_{A})$ with those of
$V(\varphi;N_{A}+q_{A})$.

As explained in Section 6.1, generically the potential depends on the fluxes
${\cal N}_{\cal A}$, but only a portion thereof can be properly understood as
dynamical. In fact, whenever there are some constraints on the flux lattice
such as (6.2), we are forced to consider some fluxes _fixed_ at a background
value. The fluxes which are ‘frozen’ _cannot_ be subjected to transitions: in
fact, they are not dualized to gauge three-forms and no membrane coupling is
present in the effective theory. In this section, we elaborate more on the
distinction of which fluxes are dynamical and which are not, further
developing the general discussion of Section 1.6. We will give physical
arguments that corroborate the discussion of Section 6.1, which rely on the
spectrum of membranes that is allowed in the effective theory.

Any effective field theory is characterized by a cut-off scale $\Lambda_{\rm
UV}$, below which the theory is valid. We will assume that such a cut-off is
field independent. If we wish to disregard the Kaluza-Klein excitations, we
need to choose

$\Lambda_{\rm UV}\lesssim m_{\rm KK}\,,$ (7.13)

where $m_{\rm KK}$ is the lowest mass of the Kaluza-Klein excitations. We will
always work in the assumption that (7.13) is satisfied, consequently
identifying $m_{\rm KK}$ as the upper bound for the cut-off scale
$\Lambda_{\rm UV}$. In the effective field theory, plenty of moduli are
present, which we generically denote with $\chi$ and the presence of a
potential may render them massive. Their mass, of generic order $m_{\chi}$,
clearly depends on the fluxes ${\cal N}_{{\cal A}}$. Since we want to keep the
moduli within the EFT description, we assume that they are stabilized at
masses $m_{\chi}$ lower than the cut-off $\Lambda_{\rm UV}$:

$m_{\chi}\lesssim\Lambda_{\rm UV}\,.$ (7.14)

Both the Kaluza-Klein scale $m_{\rm KK}$ as well the moduli masses $m_{\chi}$
are moduli dependent. Therefore, the combined conditions (7.13) and (7.14)
identifies a region within the moduli space in which the effective field
theory is valid, for a given choice of the cutoff $\Lambda_{\rm UV}$.

The cut-off $\Lambda_{\rm UV}$ determines, in turn, which objects we are
legitimate to include within the effective theory. Let us first consider
membranes. We would like to treat the membrane semi-classically, neglecting
their quantum fluctuations. Indeed, the quantum spectrum of a supersymmetric
membrane is continuous, with no mass gap [174]: including the quantum
excitations of membranes would then introduce an infinite tower of fields. In
order to avoid such a tower, it is necessary to assume that the tension of the
membrane is greater that the cut-off $\Lambda_{\rm UV}$ in the following
sense:

$\frac{{\cal T}_{\rm memb}}{\Lambda^{3}_{\rm UV}}\gtrsim
1\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\text{\sf{Semi-
classical membranes requirement}}}$ (7.15)

Let us now assume that on the two sides of the membrane we can select two
different vacua, with the moduli acquiring that may acquire masses of order
$m_{\chi}$ depending on the background fluxes. We can now split the fluxes as
in (1.38), namely into a set $N_{A}$ which is dynamical and into another
${\cal N}^{\rm bg}_{\cal A}$ regarded as frozen. A hierarchy of masses is
originated, which is fixed by both the sectors. A membrane can indeed induce
transitions between a vacuum on the left to another on the right and let us
assume that a domain wall solution, interpolating between them, exists. In
short, we are assuming that it is legitimate to consider transitions between
the vacua. Across the flow which leads to a change of the vevs of the fields,
the masses $m_{\chi}$, which do depend on the particular point of the moduli
space and thus on the background fluxes, generically change. We now require
that _the allowed transitions are those which do not significantly change the
hierarchy of masses_. In other words, we assume that the hierarchy is
basically fixed by the frozen sector ${\cal N}^{\rm bg}_{\cal A}$ and that any
change in the dynamical sector $N_{A}$ has not enough energy to completely re-
shuffle the masses $m_{\chi}$.

Furthermore, the energy involved in transitions across the membrane can
roughly be estimated from the difference of the potential at the vacua as

$|\Delta V|^{2}\simeq M_{\rm P}^{2}\left||{\cal W}_{+\infty}|^{2}-|{\cal
W}_{-\infty}|^{2}\right|\simeq{\cal T}^{2}_{\rm memb}M_{\rm P}^{-4}\,.$ (7.16)

Therefore, if a transition is allowed, the energy involved has to be lower
than the cut-off of the theory

$\frac{{\cal T}_{\rm memb}}{M^{2}_{\rm P}}\lesssim\Lambda_{\rm
UV}\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\text{\sf{Compatibility
with UV cut-off}}}$ (7.17)

This relation becomes indeed the criterion that has to be respected by the
split (1.38). Namely, the membranes that we can include in the four-
dimensional EFT must obey (7.17) and the allowed three-forms are those which
couple to such membranes. This indeed determines which fluxes _cannot_ have a
three-form origin and have to be regarded as ${\cal N}_{{\cal A}}^{\rm bg}$,
which determine the gaugings induced by the tadpole cancellation conditions.

It is also important to stress another important feature of the relation
(7.17). All the quantities that appear in (7.17) are moduli dependent, making
(7.17) be related to the particular region of the moduli space that we are
exploring. The relation (7.17) is, in particular, related to the coupling
constants of the theory, leading to the following interpretation: given a
certain perturbative regime that we want to scan within the EFT, (7.17) tells
what membranes can be included into the effective theory. As an example, in
the EFTs originating from string theory, (7.17) depends on the string coupling
constant $g_{\rm s}$ and, as we shall see in the next two sections, choosing
whether we want to explore the regions $g_{\rm s}\lesssim 1$, $g_{\rm s}\sim
1$ or $g_{\rm s}\gtrsim 1$ different kind of membranes may be included.

Figure 7.1: At different points in the field space, the hierarchy of objects
change: some membranes and strings may become lighter than others.
Accordingly, the dynamical sublattice $\Gamma_{\rm EFT}$ changes. An EFT is
then characterized differently according to the perturbative regime that we
want to scan.

As of now, our discussion has been focused on membranes only, but can be also
extended to BPS–strings, with some modifications and additional notes. As we
noticed for membranes, if we wish to treat strings as semi-classical objects,
their tensions has to satisfy

$\frac{{\cal T}_{\rm string}}{\Lambda^{2}_{\rm UV}}\gtrsim
1\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\text{\sf{Semi-
classical strings requirement}}}$ (7.18)

In an EFT where also semiclassical membranes are present, this criterion
combines with (7.15). However, strings develop the so–called Freed-Witten
anomalies as briefly summarized in Appendix E. These are cured by attaching,
in the effective description, a certain amount of membranes to the string,
creating a BPS–junction as that examined in Section 5.1.5. An anomaly–free
theory has then to consistently include the whole BPS–junction. In terms of
the tension of the string, in analogy with (7.17), we require

$\frac{{\cal T}_{\rm string}}{M^{2}_{\rm P}}\lesssim
1\,\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\text{\sf{Compatibility
with Freed-Witten anomaly cancellations}}}$ (7.19)

However, in contrast with (7.17), no $\Lambda_{\rm UV}$ appears. This can be
traced back to the fact that the backreaction of the strings grow only
logarithmically with the distance and is negligible with respect to that of
membranes, which grows linearly. In Type IIB case, we will deliver an example
of cancellation of Freed-Witten anomalies with BPS–objects, relating the
criteria (7.18), (7.19) for strings, (7.15), (7.17) for membranes with the
consistency of the EFT.

To summarize, the region of the moduli space that we would like to explore
selects the perturbative regime of the EFT. Within such region, the
compatibility of the EFT dictates which elements, namely field representations
and extended objects, may be consistently included. As a result, depicted in
Fig. 7.1, the effective descriptions change moving across the field space.

####  7.3 A hierarchy of objects in Type IIB EFTs

In Section 6.2.3 we arbitrarily regarded the R-R fluxes $m_{A}$ as dynamical,
by properly generating them by using a set of $b_{3}^{-}(X)$ gauge three-
forms, and the NS-NS fluxes $h_{A}$ as frozen, entering the gauging of both
gauge two- and three-forms. There, such splitting seemed rather arbitrary, but
we will now justify it in light of the general discussion of Section 7.2. The
first step is the introduction of a proper cut-off $\Lambda_{\rm UV}$, which
dictates what kinds of membranes and string may be included in the EFT.

#####  7.3.1 Identification of the EFT scales

Preliminarily, we may characterize the EFT by introducing a moduli-independent
UV cut-off scale $\Lambda_{\rm UV}$, which has however to be related to other
mass scales entering the EFT. There are two natural scales that may enter the
effective theory: the Kaluza-Klein mass scale $m_{\rm KK}$ and the masses of
the moduli $m_{\chi}$, which are induced by the background fluxes. On the one
hand, the Kaluza-Klein scale $m_{\rm KK}$ sets an upper bound on the cut-off,
so that we are legitimate to discard all the Kaluza-Klein excitation modes
from the effective theory; on the other, we may assume that the induced masses
$m_{\chi}$ for the moduli lie below the cut-off $\Lambda_{\rm UV}$. In
summary, we choose the cut-off in such a way that

$m_{\chi}\lesssim\Lambda_{\rm UV}\lesssim m_{\rm KK}\,.$ (7.20)

The relation (7.20) indeed specifies a region inside the moduli space as
follows. The Kaluza-Klein mass $m_{\rm KK}$ can be easily computed from (6.4),
obtaining

$m_{\rm KK}\simeq\frac{1}{\rho^{2}\,{\rm Im\hskip 1.00006pt}\tau}\,M_{\rm
P}\,,$ (7.21)

where, for the sake of clarity, we have dropped all the $2\pi$s and ${\cal
O}(1)$ factors. We have also introduced the volume modulus

$\rho\equiv V^{\frac{1}{3}}_{\rm s}\,,$ (7.22)

with $V_{\rm s}$ the string-frame volume of the internal space, measured in
string units. In order to get an estimate over the moduli mass scale
$m_{\chi}$, we focus on the induced mass $m_{\phi}$ of the four-dimensional
dilaton $\phi$. Then, from the usual Cremmer et al. potential (4.18), with the
superpotential (6.28), one can obtain the following estimation for the dilaton
mass:

$m_{\phi}\simeq\frac{|h|\|\Pi(\phi)\|}{\rho^{3}\,{\rm Im\hskip
1.00006pt}\tau}\,M_{\rm P}\,,$ (7.23)

where $|h|$ represents the typical $h_{A}$ flux quanta and
$\left\|\Pi(\phi)\right\|^{2}$ schematically denotes a contribution of the
form $e^{K_{\rm cs}}K_{\rm cs}^{i\bar{\jmath}}\Pi^{A}_{i}\Pi^{B}_{j}$,
$e^{K_{\rm cs}}\Pi^{A}\Pi^{B}$, …We assume that these terms are all of the
same finite order. We also employed the relations

$e^{K_{\rm k}}=\frac{e^{4\phi}}{V_{6}^{2}}\,,\qquad e^{K_{\rm
cs}}=\frac{1}{8V_{6}}\,,$ (7.24)

as can be deduced from (6.14) and (6.22). Here we have introduced the internal
volume $V_{6}$ as measured in the string units, that is
$V_{6}\equiv\frac{1}{3!}\int_{X}J^{3}$.Therefore the requirement (7.20)
translates into the following EFT condition on ${\rm Im\hskip 1.00006pt}\tau$
and $\rho$:

$\rho^{2}\,{\rm Im\hskip 1.00006pt}\tau\ \lesssim\ \frac{M_{\rm
P}}{\Lambda_{\rm UV}}\ \lesssim\ \frac{\rho^{3}\,{\rm Im\hskip
1.00006pt}\tau}{|h|\|\Pi\|}\,,$ (7.25)

which specifies the region depicted in Fig. 7.2.

Figure 7.2:  Region of validity of the EFT specified by (7.25).

From these conditions we obtain the following minimal and maximal values of
$\rho$ and ${\rm Im\hskip 1.00006pt}\tau$ respectively, allowed by the EFT
bounds (7.21)

$\rho_{\rm min}\simeq|h|\|\Pi\|\ ,\ \quad{\rm Im\hskip 1.00006pt}\tau_{\rm
max}\simeq\frac{1}{|h|\|\Pi\|}\left(\frac{M_{\rm P}}{\Lambda_{\rm
UV}}\right)\,.$ (7.26)

Hence, if the combination $|h|\|\Pi\|$ entering the estimate of $m_{\phi}$ in
(7.21) is moderately large, the conditions (7.25) guarantee the geometric
regime. Furthermore, by taking $\Lambda_{\rm UV}\ll M_{\rm P}$, ${\rm Im\hskip
1.00006pt}\tau$ can reach very large values. On the other hand, we will take

${\rm Im\hskip 1.00006pt}\tau\geq{\rm Im\hskip 1.00006pt}\tau_{\rm min}\,,$
(7.27)

for any ${\rm Im\hskip 1.00006pt}\tau_{\rm min}$ which should be large but
much smaller than $M_{\rm P}/\Lambda_{\rm UV}$, so that

$\rho^{2}_{\rm max}=\frac{1}{{\rm Im\hskip 1.00006pt}\tau_{\rm
min}}\left(\frac{M_{\rm P}}{\Lambda_{\rm UV}}\right)$ (7.28)

is large too.

#####  7.3.2 A hierarchy of membranes

Let us now pass to the dual formulation in terms of three-forms. For the
moment, let us assume that we can dualize both the R-R and NS-NS fluxes to
gauge three-forms. Then, accordingly, we should couple all the possible
membranes which make both the kinds of fluxes jump.

Indeed, in Type IIB EFTs, four-dimensional membranes may have only two
origins: they can come from either D5-branes or NS5-branes wrapped over
internal special Lagrangian three-cycles $\Sigma_{A}$; we will call the
membranes so obtained as ‘R-R membranes’ and ‘NS-NS membranes’, respectively.
The former, carrying charges $q^{\text{R-R}}_{A}$ under the gauge three-forms,
provide the jumps of the R-R fluxes as $\Delta m_{A}=q^{\text{R-R}}_{A}$; the
latter do instead make the NS-NS fluxes jumps as $\Delta h_{A}=q^{\text{NS-
NS}}_{A}$, with $q^{\text{NS-NS}}_{A}$ denoting their charges. Their tension
can be obtained from the general reduction formula (7.9), with the volume of
the internal three-cycle $\Sigma^{\Lambda}$ given, schematically, by

${\rm vol}(\Sigma^{\Lambda})\sim
q_{A}\left|\int_{\Sigma^{\Lambda}}\Omega\right|\sim|q_{A}{\cal
V}^{A}(\varphi)|$ (7.29)

where $q_{A}$ are some quantized charges, either $q^{\text{R-R}}_{A}$ or
$q^{\text{NS-NS}}_{A}$. Then, the tensions of R-R and NS-NS membranes are

${\cal T}_{\rm memb}^{\text{R-R}}\simeq M_{\rm
P}^{3}e^{\frac{K}{2}}|m_{A}\Pi^{A}|\,,\quad\quad{\cal T}_{\rm memb}^{\text{NS-
NS}}\simeq M_{\rm P}^{3}\frac{e^{\frac{K}{2}}}{g_{\rm s}}|h_{A}\Pi^{A}|\,,$
(7.30)

coherently with the four-dimensional counterpart (5.16) predicted form
$\kappa$-symmetry. We get the following estimations of the tensions of the R-R
and NS-NS membranes wrapped over the very same cycle $\Sigma_{A}$

$\frac{{\cal T}_{\rm memb}^{\text{R-R}}}{M^{2}_{\rm
P}}\simeq\frac{|q^{\text{R-R}}|\|\Pi\|}{\rho^{3}({\rm Im\hskip
1.00006pt}\tau)^{2}}\,M_{\rm P}\,,\quad\quad\frac{{\cal T}_{\rm
memb}^{\text{NS-NS}}}{M^{2}_{\rm P}}\simeq\frac{|q^{\text{NS-
NS}}|\|\Pi\|}{\rho^{3}{\rm Im\hskip 1.00006pt}\tau}\,M_{\rm P}\,.$ (7.31)

In Table 7.1 we have collected all the relevant membrane and flux information.

Dp-brane | wrapped over… | Charge | Flux | ${\cal T}_{\rm memb}/M^{3}_{\rm P}$ | Dynamical
---|---|---|---|---|---
D5 | three-cycles $\Gamma^{A}$ | $q^{\text{R-R}}_{A}$ | $m_{A}=\int_{\Sigma^{A}}\overline{F}_{3}$ | $g_{s}^{2}\rho^{-3}$ | ✓
NS5 | three-cycles $\Gamma^{A}$ | $q_{A}^{\text{NS-NS}}$ | $h_{A}=\int_{\Sigma^{A}}\overline{H}_{3}$ | $g_{s}\rho^{-3}$ | ✗
Table 7.1: The branes originating the membranes which give the jumps over the
RR fluxes $m_{A}$ and the NS-NS fluxes $h_{A}$.

Now we should question whether it makes sense to include both the kinds of
membranes within the cut-off $\Lambda_{\rm UV}$ determined from (7.20). First,
we can easily relate the tensions to the Kaluza-Klein scale (7.21), obtaining

$\frac{{\cal T}_{\rm memb}^{\text{R-R}}}{M^{2}_{\rm
P}}\simeq\frac{|q^{\text{R-R}}|}{|h|{\rm Im\hskip
1.00006pt}\tau}\frac{\rho_{\rm min}}{\rho}\,m_{\rm KK}\,,\quad\quad\frac{{\cal
T}_{\rm memb}^{\text{NS-NS}}}{M^{2}_{\rm P}}\simeq\frac{|q^{\text{NS-
NS}}|}{|h|}\frac{\rho_{\rm min}}{\rho}\,m_{\rm KK}\,.$ (7.32)

Hence, both R-R and NS-NS membranes do satisfy the condition (7.17) for
$\Lambda_{\rm UV}\simeq m_{\rm KK}$ in the large volume, weak coupling region
in which we are working.

However, it is also clear that, in the weak string coupling regime $g_{\rm
s}=({\rm Im\hskip 1.00006pt}\tau)^{-1}\ll 1$, the NS-NS membranes are much
heavier that the R-R membranes wrapped over the same internal three-cycle,
since their tensions are enhanced with a factor $1/g_{\rm s}$:

$\frac{{\cal T}_{\rm memb}^{\text{R-R}}}{{\cal T}_{\rm memb}^{\rm NS-
NS}}\simeq\frac{1}{{\rm Im\hskip 1.00006pt}\tau}=g_{\rm s}\;.$ (7.33)

In other words, at low energy, the only membranes which we can insert into the
EFT are the R-R membranes; the NS-NS membranes are excluded because the jumps
that they induce over the NS-NS fluxes would involve greater energies with
respect to the cut-off $\Lambda_{\rm UV}$. We are then led to the dynamical
flux sublattice (6.34) or, equivalently, to the explorable sublattice (6.36).

We can also be more precise, trying to quantify the cutoff $\Lambda_{\rm UV}$
that realizes such a situation. First, combining (7.31) with (7.23), we can
relate the membrane tensions to the dilaton mass as

$\frac{{\cal T}_{\rm memb}^{\text{R-R}}}{M^{2}_{\rm
P}}\simeq\frac{|q^{\text{R-R}}|}{|h|{\rm Im\hskip
1.00006pt}\tau}\,m_{\phi}\,,\quad\quad\frac{{\cal T}_{\rm memb}^{\text{NS-
NS}}}{M^{2}_{\rm P}}\simeq\frac{|q^{\text{NS-NS}}|}{|h|}\,m_{\phi}\,.$ (7.34)

Therefore, the energetic condition including R-R membranes and leaving out the
NS-NS membranes is

$\frac{{\cal T}_{\rm memb}}{M^{2}_{\rm P}}\lesssim m_{\phi}\,.$ (7.35)

Namely, it is just sufficient to choose the low energy cut-off to be just
slightly above $m_{\phi}$, while still being well below the Kaluza-Klein mass
scale:

$m_{\phi}\lesssim\Lambda_{\rm UV}\lesssim m_{\rm KK}\,,$ (7.36)

as foreseen in (7.20). In this case, the condition (7.35) is satisfied by a
_parametrically large_ number of R-R membranes, but only ‘marginally’ by NS-NS
membranes.

On the other hand, one may read (7.35) by following a reverse reasoning. That
is, (7.35) selects the region of validity of the EFT description within
$\Gamma_{\rm EFT}$, which include R-R membranes but exclude the NS-NS ones,
and restricts

$\frac{|q^{\text{R-R}}|}{|h|{\rm Im\hskip 1.00006pt}\tau}\lesssim 1\,.$ (7.37)

Such region will have a minimal radius set by $|h|\,{\rm Im\hskip
1.00006pt}\tau_{\rm min}$ and for large values of this quantity one may
effectively work with a lattice of fluxes $\Gamma_{\rm EFT}$.

As a final remark, we also stress that the semi-classical requirement (7.15)
is indeed satisfied for both R-R and NS-NS membranes, legitimating us to treat
them semi-classically. In fact, from (7.31), we obtain the estimates

$\frac{{\cal T}_{\rm memb}}{\Lambda_{\rm
UV}^{3}}\simeq\frac{|q|\|\Pi\|}{\rho^{3}({\rm Im\hskip
1.00006pt}\tau)^{2}}\left(\frac{M_{\rm P}}{\Lambda_{\rm UV}}\right)^{3}\,,$
(7.38)

so that, in the above range of ${\rm Im\hskip 1.00006pt}\tau$ and $\rho$,
(7.38) is always satisfied.

#####  7.3.3 Including strings and 3-branes

Membranes are crucial to identify the dynamical fluxes, but they are not the
only extended objects that one can include in four-dimensional Type IIB EFTs.
After having chosen the flux sublattice (6.34), in Section 6.2.3, we dualized
the axio-dilaton to a linear multiplet – see (6.47) and below – introducing a
gauge two-form ${\cal B}_{2}$ in the EFT. We should then include an axionic
string to be coupled to ${\cal B}_{2}$, in a similar manner as in (5.47). The
ten-dimensional picture in fact predicts the existence of such a string, by
considering a D7-brane wrapped over the full internal space $X$. Its tension
is easily obtained from (5.35) as

${\cal T}_{\rm string}=|e|M^{2}_{\rm P}\,\ell\simeq\,\frac{|e|M^{2}_{\rm
P}}{{\rm Im\hskip 1.00006pt}\tau}\,.$ (7.39)

However, we also stated that the gauge two-form is gauged as in (6.50), if the
gauge two-form is supposed to encapsulate the superpotential axio-
dilaton/complex structure moduli coupling $W\sim\tau h_{A}\Pi^{A}(\Phi)$
(6.46). The gauging (6.50) indeed couples to a string which serves as a
boundary of membranes, with the string charge $e$ related to the membrane R-R
charges $q_{A}^{\text{R-R}}$ as

$q_{A}^{\text{R-R}}=eh_{A}\,.$ (7.40)

After crossing a membrane so determined, with a ‘hole’ bounded by a string,
both the axio-dilaton and the R-R fluxes shift as

$\tau\rightarrow\tau+e\,,\qquad m_{A}\to m_{A}+eh_{A}\,,$ (7.41)

identifying the vacua on the two sides, owing to the fact that the potential
is equal.

In order include such a string, this has to be treated semiclassically,
satisfying (7.18). Here, the condition (7.18) reads

$\frac{{\cal T}_{\rm string}}{\Lambda_{\rm UV}^{2}}\simeq\frac{|e|}{{\rm
Im\hskip 1.00006pt}\tau}\left(\frac{M_{\rm P}}{\Lambda_{\rm
UV}}\right)^{2}\lesssim 1\;,$ (7.42)

which is clearly satisfied in the weak string coupling limit ${\rm Im\hskip
1.00006pt}\tau\gg 1$.

The picture just described precisely matches the prediction of the Freed-
Witten anomalies [110]. As briefly summarized in Appendix E, a string of
charge $e$ develops an anomaly which is cured by attaching $q$ membranes of
charge $e$. Furthermore, the condition (7.19), as can be easily seen from
(7.39), exactly matches with the membrane condition (7.37) for
$q=e_{\Lambda}h^{\Lambda}$. This is an important consistency check of the dual
formulation of Type IIB EFTs that we proposed in Section 6.2.3: the two-form
gauging (6.50) (allowed by the choice of the lattice (6.34)) naturally leads
to the introduction of BPS–string/membrane junctions, which are in turn
predicted from the ten-dimensional perspective and the Freed-Witten anomaly
cancellation mechanism; additionally, in the chosen perturbative regime
(7.37), in the EFT both the string and the membranes constituting the junction
can coexist.

So far, we have never discussed about the possible origin of the four-form
term as it appears in the second line of (6.104):

${\cal Q}^{\rm bg}\int_{\Sigma}C_{4}\,,$ (7.43)

where $C_{4}$ is trivially closed. Its ten-dimensional origin may be
understood as follows. Let us consider the GKP Type IIB background [175]:
there, the five-form field strength is given by

$\tilde{F}_{5}=(1+*){\rm{d}}e^{4A(y)}\wedge{\rm{d}}^{4}x$ (7.44)

where, we recall, $e^{2A(y)}$ is the warp factor that appears in the
background metric (6.6). Then, one may think of the four-form potential
$C_{4}$, related to $\tilde{F}_{5}$ as in (6.5), as composed by a closed part
${\rm{d}}_{10}C_{4}^{(0)}=0$ in the full ten-dimensional space and a non-
closed part $C_{4}$. Then, $C_{4}|_{\rm ext}=e^{4A(y)}{\rm{d}}^{4}x$ and the
action of a D3-brane in the Einstein-frame simplifies as

$S_{\rm D3}=-\frac{2\pi}{\ell_{\rm
s}^{4}}\int_{\Sigma_{4}}{\rm{d}}^{4}\sigma\sqrt{-\det\left(P[g_{mn}]\right)}+\frac{2\pi}{\ell_{\rm
s}^{4}}\int_{\Sigma_{4}}(C_{4}+C_{4}^{(0)})=\frac{2\pi}{\ell_{\rm
s}^{4}}\int_{\Sigma_{4}}C_{4}^{(0)}\,,$ (7.45)

where the Nambu-Goto term has been canceled by the Chern-Simons contribution.
Hence, the ‘topological’ term (7.45) may be understood as coming from
spacetime filling D3-branes.

####  7.4 A hierarchy of objects in Type IIA and M-theory EFTs

We now move the case of Type IIA and M-theory EFTs, which we will treat
together. The perturbative regimes that we consider are distinguished by the
string coupling constant $g_{\rm s}$: in weak coupling limit, the Type IIA
effective description is valid and therethe requirement $g_{s}\ll 1$
determines a hierarchy of membranes, as for the Type IIB case; moving towards
strongly coupled regions, the hierachy of membranes changes and, for $g_{\rm
s}\gg 1$ the Type IIA effective description has to be abandoned, in favor of
an M-theory description.

#####  7.4.1 Weak coupling

The first case that we examine is the weak coupling limit, $g_{s}\ll 1$, where
the Type IIA effective description is expected to be valid. We focus our
attention of flux lattice (6.90) of Section 6.3.3, by turning off the
geometric and non-geometric fluxes as in (6.122) and further considering, as a
simplifying assumption, that no open string moduli are present, turning off
also the D6 flux quanta $n_{a}^{\alpha}=0$ in (6.123). As for the Type IIB
case, we would like to have an understanding of the mass scales entering the
effective theory. It is then convenient to follow the simplification
introduced in [176], introducing the moduli

$\rho\equiv V^{\frac{1}{3}}_{\rm s}\,,\quad\quad\sigma\equiv\frac{1}{g_{\rm
s}}V^{\frac{1}{2}}_{\rm s}\,,$ (7.46)

which have diagonal kinetic terms

$-\frac{3}{4}M^{2}_{\rm
P}\frac{\partial_{\mu}\rho\partial^{\mu}\rho}{\rho^{2}}-M^{2}_{\rm
P}\frac{\partial_{\mu}\sigma\partial^{\mu}\sigma}{\sigma^{2}}\,,$ (7.47)

and generically denoting the remaining moduli, associated with the Kähler and
complex structure deformations, with $\chi^{\alpha}$.

The background is characterized by picking up background fluxes into the
lattice (6.90) and let us assume that, for the moment, the only local sources
which enter the effective description are D6-branes and O6-planes. These
contribute to the ten-dimensional effective action (6.52) as

$S_{\rm loc}=-\mu_{6}\int_{\text{D6}}{\rm{d}}^{7}\xi\sqrt{-\det{\bf
h}}\,e^{\frac{3}{4}\phi}+2\mu_{6}\int_{\text{O6}}{\rm{d}}^{7}\xi\sqrt{-\det{\bf
h}}\,e^{\frac{3}{4}\phi}$ (7.48)

where we have just restricted to their Nambu-Goto term, expressed in the
Einstein frame. Recalling the identification (7.2), we can rewrite the
effective potential in the generic form [176]

$V=M^{4}_{\rm
P}\left[\frac{A_{3}(\chi)}{\rho^{3}\sigma^{2}}+\sum_{n=0,2,4,6}\frac{A_{n}(\chi)}{\rho^{n-3}\sigma^{4}}+\frac{A_{\rm
D6}(\chi)-A_{\rm O6}(\chi)}{\rho^{3}\sigma^{3}}\right]\,.$ (7.49)

Here $A_{3}(\chi)$ comes from the NS-NS internal flux and then scales as
$|h|^{2}$, while the $A_{n}$s quadratically depend on the R-R fluxes $e_{a}$,
$m^{a}$. We also notice that in the weak-coupling limit $g_{\rm s}\ll 1$, it
is the first term of (7.49) to dominate and thus we can get the following
estimation for the mass of the moduli

$m_{\chi}\simeq\sqrt{\frac{A_{3}(\chi)}{\rho^{3}\sigma^{2}}}M_{\rm
P}=\frac{g_{\rm s}}{\rho^{3}}\sqrt{A_{3}(\chi)}M_{\rm P}\,,$ (7.50)

which agrees with its IIB counterpart given in (7.23). The Kaluza-Klein mass
scale $m_{\rm KK}$ can again be estimated by (7.21). Hence, the arguments
leading to (7.25) (with ${\rm Im\hskip 1.00006pt}\tau=\frac{1}{g_{\rm s}}$)
can be applied to the Type IIA case as well.

Let us now pass to investigate the membrane content that the four-dimensional
theory may be endowed with. First, there can be included membranes that
originate from compactifying D$p$-branes, with $p=2,4,6,8$, over an internal
$(p-2)$-cycle. These wrap, respectively, zero-, two-, four- and six-cycles in
the internal manifold $X$. The volume of such cycles are

${\rm vol}(\Sigma_{2})=\int_{\Sigma_{2}}J_{\rm c}\;,\qquad{\rm
vol}(\Sigma_{4})=\frac{1}{2}\int_{\Sigma_{4}}J_{\rm c}^{2}\;,\qquad{\rm
vol}(X)=\frac{1}{3!}\int_{X}J_{\rm c}^{3}\;.$ (7.51)

By then recalling that $J_{c}=\varphi^{i}\omega_{i}$, the tensions of such
membranes have the following schematic behaviors in terms of the Kähler moduli

${\cal T}^{(2)}_{\rm memb}\sim q_{0}\,,\qquad{\cal T}^{(4)}_{\rm memb}\sim
q_{i}\varphi^{i}\,,\qquad{\cal T}^{(6)}_{\rm
memb}\sim\kappa_{ijk}p^{i}\varphi^{j}\varphi^{k}\,,\qquad{\cal T}^{(8)}_{\rm
memb}\sim\kappa_{ijk}p^{0}\varphi^{i}\varphi^{j}\varphi^{k}\,,$ (7.52)

with $(q_{a},p^{a})$ a set of quantized constants, ultimately identified with
the charges of the membranes. Therefore, given a generic four-dimensional
membrane originating from a set of $p$-cycles chosen as above, this gives rise
to a tension of the form

${\cal T}^{\text{R-R}}_{\rm memb}=2M_{\rm P}^{3}|q_{A}{\cal V}^{A}(\varphi)|$
(7.53)

where $q_{A}=(q_{a},p^{a})$ are the quantized charges and we introduced the
periods as in (6.85) with the prepotential chosen as in (6.58), with
$a_{ab}=0$. The relation (7.53) matches with the pure four-dimensional
counterpart (5.16).

A D$p$-brane generates a _jump_ for the the $(8-p)$-form background fluxes
$N_{A}=(e_{a},m^{a})$ in (6.76) – see also Table 7.2.333This can be easily
inferred by the Bianchi identities associated to the $F_{p+1}={\rm{d}}C_{p}$,
which read  ${\rm{d}}F_{p+1}=q\delta_{p+2}(\Sigma_{8-p})$ (7.54) where
$\delta_{p+1}$ is a $(p+2)$-current form representing the $(8-p)$ cycle
$\Sigma_{8-p}$. One may think of $\delta_{p+1}$ as the Poincaré dual
representative of the cycle $\Sigma_{8-p}$. Integrating the previous relation
along the directions transverse to $\Sigma_{8-p}$, we get the cohomological
relation  $[F_{p+1}]|_{\rm right}-[F_{p+1}]|_{\rm left}=q{\rm
PD}[\Gamma_{7-p}\subset\Sigma_{8-p}]$ (7.55) It is then clear that a
D$p$-brane makes a $(8-p)$–flux jump. Then, following the general expression
for the tension (7.8), we get the following estimation for the membrane
tensions originating from D$p$-branes

${\cal T}^{(p)}_{\rm memb}\simeq\frac{g^{2}_{\rm s}\,{\cal
A}^{(8-p)}(\chi)}{\rho^{\frac{1}{2}(11-p)}}\,M^{3}_{\rm P}\,.$ (7.56)

where we have used the relation

$e^{K_{\rm k}}=\frac{1}{8V_{6}}\ \,,\qquad e^{K_{\rm
cs}}=\frac{e^{4\phi}}{V_{6}^{2}}\,,$ (7.57)

where $V_{6}\equiv\frac{1}{3!}\int_{X}J^{3}$ is the internal volume measured
in string units. The moduli-dependent quantity ${\cal A}^{(8-p)}(\chi)$
linearly depends on the charges making the $(8-p)$–fluxes jump.

However, also ten-dimensional NS5-branes can make four-dimensional membranes
be born, once wrapped over internal three-cycles $\Sigma_{\Lambda}$. Their
tensions, again, can be inferred from the general expression (7.8). Given that
the volume of an internal three-cycle, proportional to
$|\int_{\Sigma_{\Lambda}}e^{\tilde{\phi}}{\cal C}\Omega|$, carry a linear
dependence on the complex structure moduli

${\rm vol}(\Sigma_{\Lambda})\sim T_{\Lambda}\,,$ (7.58)

the tension of a NS-NS membrane is

${\cal T}^{\text{NS-NS}}_{\rm memb}=2M_{\rm
P}^{3}g_{s}^{-1}e^{\frac{K}{2}}|r^{\Lambda}T_{\Lambda}|\,,$ (7.59)

as a particular case of (5.16).

They induce jumps for the NS-NS background fluxes $h^{\Lambda}=(h_{K},h^{Q})$
in (6.80) and we can estimate their tensions as

${\cal T}^{\text{NS-NS}}_{\rm memb}\simeq\frac{g_{\rm s}\,{\cal A}^{\text{NS-
NS}}(\chi)}{\rho^{3}}\,M^{3}_{\rm P}\,.$ (7.60)

An immediate comparison of (7.56) with (7.60) shows off that, in the small
string coupling limit $g_{\rm s}\to 0$, while keeping the volume $V_{\rm s}$
fixed

${\cal T}^{\text{NS-NS}}_{\rm memb}\gg{\cal T}^{(p)}_{\rm memb}$ (7.61)

Brane | wrapped over… | Charge | Flux | Behavior of ${\cal T}_{\rm memb}/M^{3}_{\rm P}$ | Dynamical
---|---|---|---|---|---
D2 | trivial cycle | $q_{0}$ | $e_{0}=\int_{X}\overline{G}_{6}$ | $g_{\rm s}^{2}\rho^{-\frac{9}{2}}$ | ✓
D4 | two-cycles $\tilde{\Sigma}_{i}$ | $q_{i}$ | $e_{i}=\int_{\tilde{\Sigma}^{i}}\overline{G}_{4}$ | $g_{\rm s}^{2}\rho^{-\frac{7}{2}}$ | ✓
D6 | four-cycles $\Sigma^{i}$ | $p^{i}$ | $m^{i}=-\int_{\Sigma_{i}}\overline{G}_{2}$ | $g_{\rm s}^{2}\rho^{-\frac{5}{2}}$ | ✓
D8 | six-cycle $X$ | $p^{0}$ | $m^{0}=\overline{G}_{0}$ | $g_{\rm s}^{2}\rho^{-\frac{3}{2}}$ | ✓
NS5 | three-cycles $\Sigma_{\Lambda}$ | $r^{\Lambda}$ | $h^{\Lambda}=\int_{\Sigma_{\Lambda}}\overline{H}_{3}$ | $g_{\rm s}\rho^{-3}$ | ✗
Table 7.2: The branes originating the membranes which give the jumps over the
R-R fluxes $e_{a}$, $m^{a}$ and the NS-NS fluxes $h_{p}$.

We also notice that

$\frac{{\cal T}_{\rm mem}^{(p)}}{M^{2}_{\rm
P}}\simeq\frac{g_{s}}{\rho^{\frac{1}{2}(11-p)}}\frac{|q_{(p)}|}{|h|}\,m_{\chi}\,,\quad\quad\frac{{\cal
T}_{\rm mem}^{\text{NS-NS}}}{M^{2}_{\rm P}}\simeq\frac{|q^{\text{NS-
NS}}|}{|h|}\,m_{\chi}\,,$ (7.62)

and so the condition (7.61) is satisfied for a parametrically large fraction
of R-R membranes at sufficiently weak coupling, while it essentialy excludes
the NS-NS membranes. Therefore, once again the criterion (7.61) matches the
choice of EFT flux lattice (6.91).

#####  7.4.2 Moderately strong coupling

The hierarchy of membrane tensions illustrated above changes as we change the
perturbative regime that we are scanning. As an example , let us examine the
case of moderately strong coupling $g_{\rm s}\simeq 1$. The estimate (7.56)
for D8-branes, which make the Romans mass $m^{0}$ jump, reads

$\frac{{\cal T}_{\rm memb}^{(8)}}{M^{2}_{\rm P}}\simeq\frac{{\cal
A}^{(0)}(\chi)}{\rho^{\frac{3}{2}}}\,M_{\rm P}\simeq{\cal
A}^{(0)}(\chi)\rho^{\frac{1}{2}}\,m_{\rm KK}>m_{\rm KK}\,,$ (7.63)

which violates the EFT condition (7.15), with $\Lambda_{\rm UV}\simeq m_{\rm
KK}$. This fact is just a manifestation of the usual strong coupling
obstruction for massive IIA, as explained, for instance, in the recent [157].
In our framework, we can interpret this result as stating that those membranes
for which $\frac{{\cal T}_{\rm memb}}{M^{2}_{\rm P}}>m_{\rm KK}$ not only do
not correspond to elements of $\Gamma_{\rm EFT}$, but in fact must be excluded
from the larger flux lattice $\Gamma$. This is to be expected, in the sense
that if one works in the ten-dimensional supergravity approximation such flux
lattice is defined at the compactification scale, being different for each
compact manifold.

Notably, with the choice (6.122), turning off the open string fluxes and
excluding the Romans mass $m^{0}$ from the full lattice $\Gamma$ (6.90), the
flux contributions to the tadpole conditions (6.88) disappear, and so does the
obstruction to dualize all the remaining fluxes to three-form potentials. We
are then led to an ‘unconstrained’ dual three-form theory, with the fluxes
$(e_{i},m^{j})$ regarded as generated by three-form potentials
$(A^{i}_{3},\tilde{A}_{3j})$ which are accommodated into $b_{2}^{-}$ double
three-form multiplets of the kind (4.76a), and the remaining fluxes $e_{0}$
and $h^{\Lambda}$ generated by the three-forms $A^{0}_{3}$ and
$\hat{A}_{3\Lambda}$ which are part of single three-form multiplets as (4.70).

We also notice that, in general, the strong coupling regime does not allow for
identifying the Kähler potential of the effective theory, unless the theory is
interpreted as stemming from M-theory (see below or the previous Section
(6.4.4)). Nevertheless, the structure of the three-form multiplets is dictated
just by the superpotential, thus by holomorphic periods, and so it is expected
to enjoy some protection mechanism against perturbative corrections.

#####  7.4.3 M-theory regime

To conclude, we consider the very strong coupling regime $g_{\rm s}\gg 1$, in
which the Type IIA description is no longer suitable and one must rather
formulate the setup in terms of M-theory compactifications. The eleven-
dimensional M-theory metric ${\rm{d}}s^{2}_{11}$ is related to the IIA string
frame metric ${\rm{d}}s^{2}_{10}$ by

${\rm{d}}s^{2}_{11}=e^{-\frac{2}{3}\phi}{\rm{d}}s^{2}_{\rm 10}+\ell^{2}_{\rm
M}e^{\frac{4}{3}\phi}({\rm{d}}y+C_{1})^{2}\,,$ (7.64)

where $y$ is the $S^{1}$ coordinate $y\simeq y+1$ nontrivially fibered over
the ten-dimensional base manifold, with connection $C_{1}$. For simplicity, we
have chosen a parametrization such that the M-theory Planck length $\ell_{\rm
M}$ coincides with the string length, $\ell_{\rm M}\equiv\ell_{\rm s}$. We can
consider a limit in which the internal seven-dimensional space $\hat{X}$ is
large in natural $\ell_{\rm M}$-units, $V_{\rm M}\gg 1$. Furthermore, as will
be particularly useful to get energy estimates, we assume that $\hat{X}$ is
approximately isotropic and homogeneous. Then, from (7.64) we get the
relations $\rho=V_{\rm M}^{\frac{3}{7}}$ and $g_{\rm s}=\langle
e^{\phi}\rangle=V_{\rm M}^{\frac{3}{14}}=\rho^{\frac{1}{2}}$. With such
assumption, the estimates (7.61)-(7.60) can be rephrased exclusively in terms
of $\rho$:

${\cal T}^{(p)}_{\rm memb}\simeq\frac{{\cal
A}^{(8-p)}}{\rho^{\frac{1}{2}(9-p)}}\,M^{3}_{\rm P}\,,\quad\quad{\cal
T}^{\text{NS-NS}}_{\rm memb}\simeq\frac{{\cal A}^{\text{NS-
NS}}}{\rho^{\frac{5}{2}}}M^{3}_{\rm P}\,.$ (7.65)

In the M-theory regime, with metric (7.64), the KK-scale becomes

$m^{\rm M}_{\rm KK}=\frac{M_{\rm P}}{\rho^{\frac{3}{2}}}\,.$ (7.66)

and therefore we obtain that

$\frac{{\cal T}^{(p)}_{\rm memb}}{M^{2}_{\rm P}}\simeq{\cal
A}^{(n)}\rho^{\frac{1}{2}(p-6)}\,m_{\rm KK}^{\rm M}\,,\quad\quad\frac{{\cal
T}^{\text{NS-NS}}_{\rm memb}}{M^{2}_{\rm P}}\simeq{\cal
A}^{(p)}\rho^{-1}\,m_{\rm KK}^{\rm M}\,.$ (7.67)

Hence, in the geometric regime $\rho\gg 1$, both ${\cal T}^{(8)}_{\rm memb}$
and ${\cal T}^{(6)}_{\rm memb}$ violate the KK scale condition $\frac{{\cal
T}_{\rm memb}}{M^{2}_{\rm P}}\lesssim m_{\rm KK}$. According to our criterion
above the corresponding fluxes, namely the Romans mass and the IIA R-R two-
form fluxes, must not be included. From the M-theory perspective, to geometric
fluxes that vanish on $G_{2}$-holonomy spaces.

The flux lattice $\Gamma$ in this regime is parametrized by the former type
IIA fluxes $e_{0},e_{i}$ and $h^{\Lambda}$. In M-theory language, $e_{0}$ is
identified with the internal $G_{7}$-flux $n_{0}$ over the entire $\hat{X}$,
while $e_{a},h^{\Lambda}$ recombine into the flux quanta $n_{\check{\imath}}$
of $G_{4}\in H^{4}(\hat{X};\mathbb{Z})$ as in (6.137) . The associated
membranes correspond to M2-branes, which are just points in the internal
manifold, and M5-branes, wrapped over internal three-cycles, respectively.

Interestingly, with this restricted choice of $\Gamma$ there are no tadpole
conditions, and so in principle one may take $\Gamma_{\rm EFT}=\Gamma$. More
precisely, one can see that $(n_{0},n_{\check{\imath}})$ can be dualized to
three-form potentials $A^{0}_{3},A^{\check{\imath}}_{3}$ which can be
incorporated, together with $\check{s}^{0}$ and $\check{s}^{\check{\imath}}$,
into single three-form multiplets as in (6.140). One may then see if this
choice is compatible with the hierarchy of corresponding membrane tensions. By
using the Kähler potential of [158] in the one-modulus case, one can obtain a
simple estimate $m_{\phi}\sim M_{\rm P}\rho^{-\frac{5}{2}}$ of the scaling
behaviour of the moduli masses. We then find that

$\frac{{\cal T}^{(2)}_{\rm memb}}{M^{2}_{\rm
P}}\simeq\rho^{-1}\,m_{\phi}\,,\quad\quad\frac{{\cal T}^{(4)}_{\rm
memb}}{M^{2}_{\rm P}}\simeq\frac{{\cal T}^{\text{NS-NS}}_{\rm
memb}}{M^{2}_{\rm P}}\simeq\,m_{\phi}\,.$ (7.68)

Therefore, to set $\Gamma_{\rm EFT}=\Gamma$ one must take a cut-off scale such
that $m_{\phi}\ll\Lambda_{\rm UV}\lesssim m^{\rm M}_{\rm KK}$. As a result,
the corresponding EFT has the attractive feature of including flux transitions
that change significantly the masses of the would-be moduli, unlike in
previous examples. The full picture that has emerged from Type IIA and
M-theory EFTs is depicted in Fig. 7.3.

Figure 7.3: How the membrane content changes across Type IIA and M-theory EFTs
according to the chosen value of $g_{\rm s}$. Darker colors correspond to
‘heavier’ membranes.

While this is a perfectly consistent low-energy effective action, it does not
incorporate anomalous axionic strings in its spectrum of fundamental extended
objects, as well as the associated gauging of the two-forms dual to the
corresponding axions. Indeed, such strings are given by M5-branes wrapping
four-cycles of the compactification manifold. If the integral of the internal
four-form flux does not vanish over the M5-brane, a Freed-Witten anomaly will
be generated on its worldvolume [177], which will then be cured by M2-branes
ending on the corresponding 4d axionic string. In terms of the EFT Lagrangian,
we will have a series of non-trivial gaugings of the 4d two-forms dual to the
$C_{3}$-axions by the three-form potential $A_{3}^{0}$ coupling to the
M2-branes. On the other hand, the gauging coefficients are nothing but the
four-form fluxes $n_{I}$ and, by the reasoning of section 4.4.2, we could
adopt the dual description in terms of two-forms, in which the four-form
fluxes are considered as part of the background fluxes ${\cal N}^{\rm
bg}_{\cal A}$ and specify the two-form gaugings. In this case, we identify
$\Gamma_{\rm EFT}$ with the one-dimensional lattice parametrized by $e_{0}$.

It seems that this class of compactifications allow for two different,
complementary descriptions in terms of the three-form Lagrangians. Either we
describe a 4d EFT containing $b_{3}+1$ classes of membranes, or we have an EFT
with one class of membranes and one anomalous strings. It would be interesting
to consider further examples of compactifications of this sort, and to
understand whether the obstruction we find in incorporating all of these
ingredients simultaneously is fundamental or can be overcome by adopting some
so far unknown alternative scheme.

### Conclusions and future outlook

In this work we have presented a full reformulation of generic ${\cal N}=1$
four-dimensional supersymmetric theories including elements predicted from
higher-dimensional string theory. In both global and local supersymmetric
theories we have introduced a hierarchy of gauge forms, summarized in Table
8.1. In order to construct manifestly ${\cal N}=1$ supersymmetric Lagrangians,
such gauge forms have been properly included among the components of
supersymmetric multiplets: gauge two-forms find their place in linear
multiplets, while gauge three- and four-forms are accommodated into special
three- and four-form multiplets, the construction of some of which was only
recently performed in [6, 7, 10].

${\bm{p}}$-form |  | Data | Multiplet
---|---|---|---
Two-form | ${\cal B}_{2}^{\Lambda}$ | axion | Linear multiplet $L^{\Lambda}$
Three-form | $A_{3}^{A}$ | dynamical background fluxes | Master three-form multiplet $S^{A}$
Four-form | $C_{4}^{I}$ | tadpole cancellation conditions | Four-form multiplet $\Gamma^{I}$
Table 8.1: The role of the $p$-forms in four-dimensional effective theories of
string and M-theory

The introduction of such forms within a four-dimensional theory is related to
the spectrum of the theory. Gauge two-forms may be regarded as ‘dual’ of
axionic fields, for which they provide an alternative representation. The
rather peculiar gauge three-forms, although not carrying any propagating
degrees of freedom, dynamically generate sets of constants which enter
quadratically and linearly the potential. Such constants are arbitrary, but
they may be constrained by means of gauge four-forms. In short, the newly
introduced hierarchy delivers an _alternative_ , possibly more complete
formulation of known ${\cal N}=1$ supersymmetric theories.

The proposed reformulation is necessary when it comes to coupling extended
objects, summarized in Table 8.2. By introducing a set of _super $p$-forms_,
built out of the aforementioned supersymmetric multiplets, we were able to
construct manifestly supersymmetric actions for BPS-strings, membranes and
3-branes. The action for strings and membranes are composed by a Nambu-Goto
part, which includes the field-dependent tension of the object, and a Wess-
Zumino part, which expresses the coupling of the object to the gauge $p$-forms
and depends on its charges. The two terms are tightened together by the
requirement that the action is $\kappa$-symmetric, allowing strings and
membranes to preserve a ${\cal N}=1$ worldvolume supersymmetry once a ground
state is chosen. In turn, this implies that the tensions of the BPS-objects
are completely determined by their charges. Instead, 3-branes appear in our
framework only with a Wess-Zumino term, which depends on the charge, and
preserve the full four-dimensional ${\cal N}=1$ supersymmetry.

BPS–object | coupled to | Data
---|---|---
String | ${\cal B}_{2}^{\Lambda}$ | axion monodromy
Membrane | $A_{3}^{A}$ | shifts in background fluxes
Topological 3-brane | $C_{4}^{I}$ | tadpole cancellation conditions
Table 8.2: The extended BPS–objects included in the 4D EFTs.

The inclusion of such objects has important physical consequences. The
presence of a string signals a monodromy transformation which makes the axions
transverse a certain number of periods. Membranes make the constants which are
dual to the gauge three-forms jump. As a result, _within the same theory_ ,
one can describe a multi-branched potential. Membranes then open the
possibility to get new vacua for a potential and new nonperturbative
transitions among them can be triggered by the very existence of the membrane.
Particular attention in this work has been devoted to the study of stable,
BPS-domain wall solutions, which spontaneously break half of the bulk
supersymmetry.

More complex BPS-objects were also built and are here summarized in Table 8.3.
Membranes can be regarded as boundaries of 3-branes, identifying a
‘BPS–region’, whose naturally defined gauge-invariant object is the gauged
three-form ${\rm{d}}A_{3}^{A}+Q_{I}^{A}C_{4}^{I}$. Such a gauging serves to
take into account how the tadpole cancellation conditions change across the
various spacetime regions separated by the membranes. At the same time,
strings may serve as boundaries for membranes and identify the gauged two-form
${\rm{d}}{\cal B}_{2}^{\Lambda}+c_{A}^{\Lambda}A_{3}^{A}$. The discrete data
which is encoded in this configuration is the cancellation of the Freed-Witten
anomalies, which make the membrane ‘unstable’, forming the hole bounded by the
string.

Gauged ${\bm{p}}$-form | BPS–object | Data
---|---|---
${\rm{d}}{\cal B}_{2}^{\Lambda}+c_{A}^{\Lambda}A_{3}^{A}$ | Membranes ending on a string | Freed-Witten anomaly cancellation
${\rm{d}}A_{3}^{A}+Q_{I}^{A}C_{4}^{I}$ | 3-branes ending on a membrane | changing tadpole cancellation
Table 8.3: The mutual gaugings of the $p$-forms of Table 8.2 and the composite
objects to which they couple.

The results mentioned so far hold for _any_ ${\cal N}=1$ global and local
supersymmetric theory, namely with arbitrary Kähler potential and
superpotential. They find fertile ground in the contexts of effective field
theories stemming from compactifications of string and M-theory. In this work
we focused on Type IIA and Type IIB string theory compactified over an
orientifolded Calabi-Yau three-fold and on M-theory compactified over a seven-
manifold with either $G_{2}$-holonomy or $G_{2}$-structure and briefly
mentioned a possible F-theory uplift. In compactification scenarios, a
potential for the otherwise _moduli_ of the compactification is introduced by
threading the internal manifold with background fluxes and filling it with
sources. Allowing them to vary, the background fluxes determine the maximal
flux lattice

$\Gamma=\\{{\cal N}^{A}\;|\;{\cal N}^{A}\in\mathbb{Z}\\}\,,$ (8.1)

which enter linearly into the superpotential. In a three-form formulation,
where membranes are also coupled, one may then think the lattice $\Gamma$ to
be _dynamically_ spanned via membrane transitions.

However, we found some obstructions in interpreting the _full_ lattice
$\Gamma$ dynamically: only a portion thereof $\Gamma_{\rm EFT}\subset\Gamma$
can be truly generated by gauge three-forms and undertake membrane
transitions. The obstructions to identify $\Gamma_{\rm EFT}$ with $\Gamma$ are
a priori all different, and arise independently from the EFT ingredients
mentioned above:

Supersymmetry

The number of dynamical fluxes must be such that $n+1\leq{\rm
dim}\,\Gamma_{\rm EFT}\leq 2n+2$, where $n$ is the number of chiral fields
entering the superpotential terms generated by the dynamical fluxes. This
comes from the upper and lower bound on the number of three-forms per scalar
in master multiplets.

Tadpoles

Fluxes typically contribute to tadpole conditions quadratically, by means of
symmetric bilinear forms ${\cal I}_{I}$. To implement tadpoles as three-form
gaugings, $\Gamma_{\rm EFT}$ must be an isotropic sublattice of $\Gamma$ with
respect to each of these parings. As a result, at the level of the EFT tadople
cancellation appears as a set of linear conditions on the dynamical fluxes.

Axion-monodromy

In certain regimes of the compactification anomalous axionic strings appear at
low energies, which means that membranes can nucleate holes in their
worldvolume [109, 110]. At the EFT level it makes sense to include the two-
forms coupled electrically to the strings, together with a gauging encoding
the anomaly. The gauging parameters will be fluxes that cannot belong to
$\Gamma_{\rm EFT}$.

As a result, the lattice is somehow ‘perceived differently’ from the four-
dimensional EFT perspective with respect to the higher-dimensional viewpoint.
The most vivid argument is in the correct implementation of the tadpole
cancellation condition. In higher dimensions, the tadpole cancellation
involves quadratic combination of the fluxes in $\Gamma$; in four dimensions,
a proper dynamical implementation of the tadpole require it to be reduced to a
_linear_ constraint for $\Gamma_{\rm EFT}$ – see Table 8.4.

| String Theory | EFT
---|---|---
Flux lattice | $\Gamma$ | $\Gamma_{\rm EFT}$
Tadpole condition | Quadratic | Linear
Table 8.4: Main difference between the string theory and the 4d EFT
perspectives. For each region of field space in the string theory construction
a different sublattice $\Gamma_{\rm EFT}\subset\Gamma$ may be selected, but
the linear dependence of the tadpole condition on the EFT fluxes applies to
all of them.

Then, while $\Gamma_{\rm EFT}\subset\Gamma$ is what can be rendered dynamical,
the quotient $\Gamma/\Gamma_{\rm EFT}$, in one-to-one correspondence with the
background, ‘frozen’ fluxes, is what determines the gauging parameters in
Table 8.3.

The choice of the dynamical sublattice is crucial, for it tells what is the
explorable Landscape within a single effective theory and which
nonperturbative effects may connect the various vacua of the Ladscape.
However, such a choice is not arbitrary and we gave some physical arguments in
Chapter 7 to discriminate the well–motivated choices. For a given perturbative
regime, certain kinds of membranes may be lighter, signaling that the
associated flux transitions require less energy to take place. Choosing
$\Gamma_{\rm EFT}$ then amounts in singling out which objects can be included
within the EFT and which others may not, being too ‘heavy’. The picture that
emerged draws effective field theories with a mutable spectrum of objects,
according to the perturbative regime that is scanned, and with different
amounts of fields which may render a potential multi-branched.

This is not the end of the story and the results presented in this work can be
generalized and extended along diverse directions. A first, interesting
technical development would be to include worldvolume matter supported on the
effective strings, membranes and 3-branes here considered. Such a matter
content is indeed expected from higher-dimensional constructions, predicting,
for example, gauge bosons and adjoint chiral fields living on 3-branes.
Furthermore, it would be interesting to study the gauge theories living on top
of membranes, better exploring the coupling of the Goldstone sector with the
worldvolume fields and how the EFT treats this sector upon membrane-mediated
transitions that change the number of 3-branes. Moreover, the presence of
world-volume matter would allow for the study of possible novel swampland
criteria, along the lines of [178].

The theories here presented could be a starting point to extract models useful
for phenomenology and cosmology. It would be interesting to question which
Physics one should expect from the sub-ensemble of vacua dynamically connected
through membrane nucleations within $\Gamma_{\rm EFT}$ and how physical
observables like the cosmological constant, Yukawa couplings, soft terms vary
significantly sailing through the EFT landscape. Then, for those couplings
that are effectively scanned over by the EFT, one may attempt to see if any
physically relevant information can be extracted from a statistical analysis
[179, 180].

The models here presented could well provide a compelling framework to test
the recent Swampland conjectures [181, 182]. The simultaneous presence of
gravity, gauge fields and extended objects in a controlled setup could indeed
be crucial to inquire how the conjectures intertwine among each other.

## Appendices

###  Appendix A Conventions on differential forms

In generic $D$ dimensions, we define the totally antisymmetric Levi-Civita
symbol such that

$\varepsilon^{01\ldots D}=-\varepsilon_{01\ldots D}=1\;.$ (A.1)

In the differential basis $\\{{\rm{d}}x^{\mu}\\}$, with $\mu=0,\ldots,D-1$, a
generic bosonic $p$–form $\omega_{p}$ has the following expansion

$\omega_{p}=\frac{1}{p!}\omega_{\mu_{1}\ldots\mu_{p}}{\rm{d}}x^{\mu_{1}}\wedge\ldots\wedge{\rm{d}}x^{\mu_{p}}$
(A.2)

The differential ${\rm{d}}$ acts on $\omega_{p}$ as

${\rm{d}}\omega_{p}=\frac{1}{p!}\partial_{[\sigma}\omega_{\mu_{1}\ldots\mu_{p}]}{\rm{d}}x^{\sigma}\wedge{\rm{d}}x^{\mu_{1}}\wedge\ldots\wedge{\rm{d}}x^{\mu_{p}}$
(A.3)

The Hodge-dual of a $p$–form $\omega_{p}$ is the $(D-p)$–form defined as

$*\omega=-\frac{e}{(D-p)!p!}\varepsilon_{\mu_{1}\ldots\mu_{D}}g^{\mu_{1}\nu_{1}}\ldots
g^{\mu_{p}\nu_{p}}\omega_{\nu_{1}\ldots\nu_{p}}{\rm{d}}x^{\mu_{p+1}}\wedge\ldots\wedge{\rm{d}}x^{\mu_{D}}$
(A.4)

where the components given by

$(*\omega)_{\mu_{p+1}\ldots\mu_{D}}=-\frac{e}{p!}\varepsilon_{\mu_{1}\ldots\mu_{D}}g^{\mu_{1}\nu_{1}}\ldots
g^{\mu_{p}\nu_{p}}\omega_{\nu_{1}\ldots\nu_{p}}\,.$ (A.5)

The components $\omega^{\mu_{1}\ldots\mu_{p}}$ are defined by raising the
indices of the components of $\omega_{p}$ with the inverse $g^{\mu\nu}$ of the
metric as usual.

In any dimension $D$, we get the following useful identity

$\begin{split}\omega\wedge*\omega&=-\frac{e}{p!(D-p)!}\omega_{\nu_{1}\ldots\nu_{p}}\varepsilon_{\mu_{1}\ldots\mu_{D}}\omega^{\mu_{1}\ldots\mu_{p}}{\rm{d}}x^{\nu_{1}}\wedge\ldots\wedge{\rm{d}}x^{\nu_{p}}\wedge{\rm{d}}x^{\mu_{p+1}}\wedge\ldots\wedge{\rm{d}}x^{\mu_{D}}\\\
&=\frac{e}{p!}\,\omega^{\mu_{1}\ldots\mu_{p}}\omega_{\mu_{1}\ldots\mu_{p}}{\rm{d}}x^{0}\wedge\ldots\wedge{\rm{d}}x^{D}\equiv
e\,\omega\lrcorner\,\omega\,,\end{split}$ (A.6)

and we have

$*(*\omega_{p})=-(-)^{p(D-p)}\omega_{p}\;.$ (A.7)

Finally, given the embedding of a $(p+1)$-dimensional bosonic manifold
$\Sigma_{p+1}$, we use orientation conventions such that Stokes’ theorem is

$\int_{\Sigma_{p+1}}{\rm{d}}\omega_{p}=\int_{\partial\Sigma_{p+1}}\omega_{p}\;.$
(A.8)

###### Four-dimensional conventions

In four dimensions, we define the Levi-Civita totally anti-symmetric symbol
$\varepsilon^{mnpq}$ such that

$\varepsilon^{0123}=-\varepsilon_{0123}=1\;.$ (A.9)

Then, the four-dimensional volume element is

$*1=e\,{\rm{d}}x^{0}\wedge{\rm{d}}x^{1}\wedge{\rm{d}}x^{2}\wedge{\rm{d}}x^{3}\equiv
e\,{\rm{d}}^{4}x\,.$ (A.10)

Other useful identities that we used throughout this work include

$\displaystyle*F_{4}$
$\displaystyle=-\frac{1}{e}\partial^{m}\left(e(*A_{3})_{m}\right)\;,$ (A.11)
$\displaystyle F_{4}$ $\displaystyle=-e\,(*F_{4})\,{\rm{d}}^{4}x\;.$

###### World-volume conventions

The 3-brane worldvolume differentials enjoy the same conventions as the four-
dimensional ones enlisted above.

For the membrane worldvolume we define the Levi-Civita antisymmetric symbol
$\varepsilon^{ijk}$ such that

$\varepsilon^{012}=-\varepsilon_{012}=1\;,$ (A.12)

and, given the local coordinates $\xi^{i}$, $i=0,1,2$, the worldvolume
integration is performed by

${\rm{d}}^{3}\xi\equiv{\rm{d}}\xi^{0}\wedge{\rm{d}}\xi^{1}\wedge{\rm{d}}\xi^{2}=-\frac{1}{3!}\varepsilon_{ijk}{\rm{d}}\xi^{i}\wedge{\rm{d}}\xi^{j}\wedge{\rm{d}}\xi^{k}\;.$
(A.13)

Equivalently, for the string worldvolume, the two-dimensional Levi-Civita
symbol $\varepsilon^{ij}$ is defined such that

$\varepsilon^{01}=-\varepsilon_{01}=1\;,$ (A.14)

and, denoting with $\zeta^{i}$, $i=0,1$ the local coordinates, the worlvolume
integration is performed by

${\rm{d}}^{2}\zeta\equiv{\rm{d}}\zeta^{0}\wedge{\rm{d}}\zeta^{1}=-\frac{1}{2!}\varepsilon_{ij}{\rm{d}}\zeta^{i}\wedge{\rm{d}}\zeta^{j}\;.$
(A.15)

###  Appendix B Conventions on ${\cal N}=1$ superspace

The superspace conventions that we have used throughout this work coincide
with those of [66]. In this appendix we collects definitions and computations
of the superfield components used in the main text. Furthermore, we also
briefly recall how differential forms are defined in superspace.

####  B.1 Component structure of ${\cal N}=1$ superfields

In this section we provide all the components of the $\mathcal{N}=1$
multiplets considered throughout this work. We will be very schematic.

Chiral multiplets

The chiral superfield $\Phi$ satisfies the constraint

$\bar{D}_{\dot{\alpha}}\Phi=0$ (B.1)

and its expansion in components is

$\Phi=\varphi+\sqrt{2}\theta\psi+\theta^{2}f+{\rm
i}\theta\sigma^{m}\bar{\theta}\partial_{m}\varphi-\frac{{\rm
i}}{\sqrt{2}}\theta^{2}\partial_{m}\psi\sigma^{m}\bar{\theta}+\frac{1}{4}\theta^{2}\bar{\theta}^{2}\Box\varphi\;.$
(B.2)

Here $\varphi$ and $f$ are complex scalar fields, while $\psi$ is a Weyl
spinor. The independent components of $\Phi$ can be defined by the projections

$\displaystyle\Phi|$ $\displaystyle=\varphi\,,$ (B.3) $\displaystyle
D_{\alpha}\Phi$ $\displaystyle|=\sqrt{2}\psi_{\alpha}\,,$
$\displaystyle-\frac{1}{4}D^{2}\Phi|$ $\displaystyle=f\,,$

where the vertical line means that the quantity is evaluated at
$\theta=\bar{\theta}=0$.

Real multiplets

A real scalar superfield $V$, beside being real $V=\bar{V}$, is unconstrained.
It has the following component expansion

$\begin{split}V=&\,u+{\rm i}\theta\chi-{\rm i}\bar{\theta}\bar{\chi}+{{\rm
i}}\theta^{2}\bar{\varphi}-{i}\bar{\theta}^{2}{\varphi}-\theta\sigma^{m}\bar{\theta}B_{m}\\\
&+{\rm i}\theta^{2}\bar{\theta}\left(\bar{\lambda}+\frac{{\rm
i}}{2}\bar{\sigma}^{m}\partial_{m}\chi\right)-i\bar{\theta}^{2}\theta\left(\lambda+\frac{{\rm
i}}{2}{\sigma}^{m}\partial_{m}\bar{\chi}\right)+\frac{1}{2}\theta^{2}\bar{\theta}^{2}\left({\rm
D}-\frac{1}{2}\Box u\right)\,,\end{split}$ (B.4)

where $u$ and $D$ are real scalar fields, $\varphi$ is a complex scalar field,
$v_{m}$ is a real vector field and $\chi$ and $\lambda$ are Weyl spinors. The
independent components of $V$ can be defined by projections

$\displaystyle V|$ $\displaystyle=u\,,$ (B.5) $\displaystyle D_{\alpha}V|$
$\displaystyle={\rm i}\chi_{\alpha}\,,$
$\displaystyle\frac{1}{4}\bar{\sigma}^{\dot{\alpha}\alpha}_{m}[D_{\alpha},\bar{D}_{\dot{\alpha}}]V|$
$\displaystyle=B_{m}\,,$ $\displaystyle\frac{{\rm i}}{4}D^{2}V|$
$\displaystyle=\bar{\varphi}\,,$
$\displaystyle-\frac{1}{4}\bar{D}^{2}D_{\alpha}V|$ $\displaystyle=-{\rm
i}\lambda_{\alpha}\,,$ $\displaystyle\frac{1}{16}D^{2}\bar{D}^{2}V|$
$\displaystyle=\frac{1}{2}\left({\rm D}-{\rm i}\partial^{m}v_{m}\right)\,.$

Real linear multiplets

A real linear superfield $L$ is a real superfield that obeys the constraint

$D^{2}L=0\,,\qquad\bar{D}^{2}L=0\,.$ (B.6)

Its component expansion is

$\begin{split}L&=l+{\rm i}\theta\eta-{\rm
i}\bar{\theta}\bar{\eta}-\frac{1}{2}\theta\sigma_{m}\bar{\theta}\varepsilon^{mnpq}\partial_{n}\Lambda_{pq}\\\
&\quad\,+\frac{1}{2}\theta^{2}\bar{\theta}\bar{\sigma}^{m}\partial_{m}\eta+\frac{1}{2}\bar{\theta}^{2}\theta\sigma^{m}\partial_{m}\bar{\eta}-\frac{1}{4}\theta^{2}\bar{\theta}^{2}\Box
l\,,\end{split}$ (B.7)

where $l$ is a real scalar, $\Lambda_{mn}$ is a rank 2 antisimmetric tensor
and $\eta$ is a Weyl spinor. The independent components of $L$ can be defined
by projections

$\displaystyle L|$ $\displaystyle=l\,,$ (B.8) $\displaystyle D_{\alpha}L|$
$\displaystyle={\rm i}\eta_{\alpha}\,,$
$\displaystyle-\frac{1}{2}\bar{\sigma}^{m\,\dot{\alpha}\alpha}\left[D_{\alpha},\bar{D}_{\dot{\alpha}}\right]L|$
$\displaystyle=\varepsilon^{mnpq}\partial_{n}\Lambda_{pq}\,.$

Complex linear multiplets

A complex linear multiplet $\Sigma$ satisfies the condition

$\bar{D}^{2}\Sigma=0\,.$ (B.9)

Its component expansion is

$\begin{split}\Sigma&=\sigma+\sqrt{2}\theta\psi+\sqrt{2}\bar{\theta}\bar{\rho}-\theta\sigma_{m}\bar{\theta}\mathcal{B}^{m}+\theta^{2}\bar{s}+\sqrt{2}\theta^{2}\bar{\theta}\bar{\zeta}\\\
&\quad\,-\frac{{\rm
i}}{\sqrt{2}}\bar{\theta}^{2}\theta\sigma^{m}\partial_{m}\bar{\rho}+\theta^{2}\bar{\theta}^{2}\left(\frac{{\rm
i}}{2}\partial_{m}\mathcal{B}^{m}-\frac{1}{4}\Box\sigma\right)\,,\end{split}$
(B.10)

where $\sigma$ and $\bar{s}$ are complex scalars, $\rho$, $\psi$ and $\xi$ are
Weyl spinors and $\mathcal{B}^{m}$ is a complex vector. The components of
$\Sigma$ can be defined by the projections

$\displaystyle\Sigma|$ $\displaystyle=\sigma\,,$ (B.11) $\displaystyle
D_{\alpha}\Sigma|$ $\displaystyle=\sqrt{2}\psi_{\alpha}\,,$
$\displaystyle\bar{D}_{\dot{\alpha}}\Sigma|$
$\displaystyle=\sqrt{2}\bar{\rho}_{\dot{\alpha}}\,,$
$\displaystyle\frac{1}{4}\bar{\sigma}^{m\,\dot{\alpha}\alpha}\left[D_{\alpha},\bar{D}_{\dot{\alpha}}\right]\Sigma|$
$\displaystyle=\mathcal{B}^{m}\,,$ $\displaystyle-\frac{1}{4}D^{2}\Sigma|$
$\displaystyle=\bar{s}\,,$ $\displaystyle\bar{D}_{\dot{\alpha}}D^{2}\Sigma|$
$\displaystyle=-4\sqrt{2}\bar{\zeta}_{\dot{\alpha}}+2\sqrt{2}{\rm
i}\,\partial_{m}\psi^{\alpha}\sigma^{m}_{\alpha\dot{\alpha}}\,,$
$\displaystyle\frac{1}{16}\bar{D}^{2}D^{2}\Sigma|$ $\displaystyle={\rm
i}\partial_{m}\mathcal{B}^{m}\,.$

####  B.2 Differential forms in superspace

Also in the whole superspace $\mathbb{R}^{1,3|4}$ differential forms can be
introduced. The most natural basis of differentials
${\rm{d}}{\mathfrak{z}}^{M}=\\{{\rm{d}}x^{m},{\rm{d}}\theta^{\alpha},{\rm{d}}\bar{\theta}_{\dot{\alpha}}\\}$
is however not so useful in order to build supersymmetric objects. It is more
convenient to introduce the differential basis

$\displaystyle E^{a}$ $\displaystyle\equiv{\rm{d}}x^{a}-{\rm
i}{\rm{d}}\theta\sigma^{a}\bar{\theta}+{\rm
i}\theta\sigma^{a}{\rm{d}}\bar{\theta}\,,$ (B.12) $\displaystyle E^{\alpha}$
$\displaystyle\equiv{\rm{d}}\theta^{\alpha}\,,\qquad\quad\;E_{\dot{\alpha}}\equiv{\rm{d}}\bar{\theta}_{\dot{\alpha}}\,.$

which we collectively denote as
$E^{A}=\\{E^{m},E^{\alpha},E_{\dot{\alpha}}\\}$. As can be easily checked,
this basis in invariant under the supersymmetry transformations (3.21).

The vierbein matrices which allow to pass from the basis
${\rm{d}}{\mathfrak{z}}^{M}$ to $E^{A}$ are then defined as

$\displaystyle E^{a}$
$\displaystyle={\rm{d}}x^{m}E_{m}^{a}+{\rm{d}}\theta^{\mu}E_{\mu}^{a}+{\rm{d}}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}a}\,,$
(B.13) $\displaystyle E^{\alpha}$
$\displaystyle={\rm{d}}x^{m}E_{m}^{\alpha}+{\rm{d}}\theta^{\mu}E_{\mu}^{\alpha}+{\rm{d}}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}\alpha}\,,$
$\displaystyle E_{\dot{\alpha}}$
$\displaystyle={\rm{d}}x^{m}E_{m\dot{\alpha}}+{\rm{d}}\theta^{\mu}E_{\mu\dot{\alpha}}+{\rm{d}}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}}_{\dot{\alpha}}\,,$

with

$\displaystyle E_{m}^{a}$ $\displaystyle=\delta^{a}_{m}\,,$ $\displaystyle
E_{\mu}^{a}=-{\rm i}\sigma_{\mu\dot{\mu}}^{a}\bar{\theta}^{\dot{\mu}}\,,$
$\displaystyle E^{\dot{\mu}a}=-{\rm
i}\bar{\sigma}^{a\dot{\mu}\mu}\theta_{\mu}\,,$ (B.14) $\displaystyle
E_{m}^{\alpha}$ $\displaystyle=0\,,$ $\displaystyle
E_{\mu}^{\alpha}=\delta_{\mu}^{\alpha}\,,$ $\displaystyle
E^{\dot{\mu}\alpha}=0\,,$ (B.15) $\displaystyle E_{m\dot{\alpha}}$
$\displaystyle=0\,,$ $\displaystyle E_{\mu\dot{\alpha}}=0\,,$ $\displaystyle
E^{\dot{\mu}}_{\dot{\alpha}}=\delta^{\dot{\mu}}_{\dot{\alpha}}\,.$ (B.16)

We define the _superspace differential_ ${\bf{d}}$

${\bf{d}}=E^{A}D_{A}=E^{a}D_{a}+E^{\alpha}D_{\alpha}+E_{\dot{\alpha}}\bar{D}^{\bar{\alpha}}\;.$
(B.17)

Given generic super $p$-form ${\bm{\Omega}}_{p}$, with the superspace
differential expansion

${\bm{\Omega}}_{p}=E^{A_{1}}\wedge\ldots E^{A_{p}}\Omega_{A_{p}\ldots
A_{1}}\;,$ (B.18)

the action of the superspace differential (B.17) is defined as

${\bf{d}}{\bm{\Omega}}_{p}=E^{A_{1}}\wedge\ldots E^{A_{p}}\wedge
E^{A}D_{A}\Omega_{A_{p}\ldots A_{1}}\ $ (B.19)

We have also introduced the more common differential operator ${\rm{d}}$,
acting on the left on a form as in (A.3). If we focus on the bosonic
components, the relation between the action of ${\rm{d}}$ and ${\bf{d}}$ on a
generic super $p$-form ${\bm{\Omega}}_{p}$ is given by

${\rm{d}}{\bm{\Omega}}_{p}|_{\rm
bos}={\rm{d}}\Omega_{p}=(-)^{p}{\bf{d}}{\bm{\Omega}}_{p}|_{\rm bos}\;.$ (B.20)

Furthermore, in superspace, given the embedding of a $(p+1)$-dimensional
bosonic manifold $\Sigma_{p+1}$ into superspace, we use orientation
conventions such that Stokes’ theorem reads $\int_{\Sigma_{p+1}}{\rm{d}}{\bf
A}_{p}=\int_{\partial\Sigma_{p+1}}{\bf A}_{p}$, compatibly with its purely
bosonic counterpart (A.8).

Whenever the superspace coordinates ${\mathfrak{z}}^{M}$ define a super-
embedding of the form

$\xi^{i}\quad\mapsto\quad{\mathfrak{z}}^{M}=(x^{m}(\xi),\theta^{\alpha}(\xi),\bar{\theta}_{\dot{\alpha}}(\xi))\;,$
(B.21)

with $\xi^{i}$ generic worldvolume local coordinates, the basis $E^{A}$ may be
re-expressed in terms of a basis of worldvolume differentials
$\\{{\rm{d}}\xi^{i}\\}$ as

$\displaystyle E^{a}$
$\displaystyle={\rm{d}}\xi^{i}\left(\partial_{i}x^{m}E_{m}^{a}+\partial_{i}\theta^{\mu}E_{\mu}^{a}+\partial_{i}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}a}\right)\equiv{\rm{d}}\xi^{i}E^{a}_{i}\;,$
(B.22) $\displaystyle E^{\alpha}$
$\displaystyle={\rm{d}}\xi^{i}\left(\partial_{i}x^{m}E_{m}^{\alpha}+\partial_{i}\theta^{\mu}E_{\mu}^{\alpha}+\partial_{i}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}\alpha}\right)\equiv{\rm{d}}\xi^{i}E^{\alpha}_{i}\;,$
$\displaystyle E_{\dot{\alpha}}$
$\displaystyle={\rm{d}}\xi^{i}\left(\partial_{i}x^{m}E_{m\dot{\alpha}}+\partial_{i}\theta^{\mu}E_{\mu\dot{\alpha}}+\partial_{i}\bar{\theta}_{\dot{\mu}}E^{\dot{\mu}}_{\dot{\alpha}}\right)\equiv{\rm{d}}\xi^{i}E_{\dot{\alpha}i}\;.$

with the vierbein $E_{i}^{A}$ defined by the pullback of $E^{A}_{M}$ to the
worldvolume as

$\displaystyle E^{a}_{i}$ $\displaystyle=\partial_{i}x^{m}-{\rm
i}\partial_{i}\theta\sigma^{m}\bar{\theta}-{\rm
i}\partial_{i}\bar{\theta}\bar{\sigma}^{m}\theta\;,$ (B.23) $\displaystyle
E^{\alpha}_{i}$
$\displaystyle=\partial_{i}\theta^{\alpha}\;,\qquad\qquad\;E_{\dot{\alpha}i}=\partial_{i}\bar{\theta}_{\dot{\alpha}}\;.$

###  Appendix C Projections of ${\cal N}=1$ Supergravity superfields

We enlist the component structure of the multiplets in Supergravity that we
have used throughout this work. We focus on the bosonic components only. The
notation is that of [66], except for the scalar curvature, here redefined as
$R\to-R$.

Chiral multiplets

In supergravity, a chiral superfield $\Phi$ is a constrained superfield
satisfying the covariantized chirality condition

$\overline{\cal D}_{\dot{\alpha}}\Phi=0\,.$ (C.1)

Its component fields are defined as

$\displaystyle\Phi|$ $\displaystyle=\varphi\,,$ (C.2a)
$\displaystyle-\frac{1}{4}{\cal D}^{2}\Phi|$ $\displaystyle=F_{(\Phi)}\,.$
(C.2b)

with $\varphi$ and $F$ complex scalar fields. The vertical line means that the
quantity is evaluated at $\theta=\bar{\theta}=0$.

Supergravity multiplet

The bosonic components of the off-shell Supergravity multiplet ${\cal R}$ are
the graviton $g_{mn}$, an auxiliary real vector $G^{a}$ and a complex scalar
auxiliary field $M$. These components may be defined as projections of a
chiral superfield ${\cal R}$ as

$\displaystyle{\cal R}|$ $\displaystyle=-\frac{1}{6}M\,,$ (C.3a)
$\displaystyle-\frac{1}{4}{\cal D}^{2}{\cal R}|$
$\displaystyle=-\frac{1}{12}R+\frac{{\rm i}}{6}{\cal
D}_{a}G^{a}-\frac{1}{18}G_{a}G^{a}-\frac{1}{9}M\bar{M}\,.$ (C.3b)

Real multiplets

An unconstrained, real multiplet $P$, in a Wess-Zumino gauge [66], are

$\displaystyle P|$ $\displaystyle=0\;,$ (C.4a) $\displaystyle-\frac{{\rm
i}}{4}\bar{\cal D}^{2}P|$ $\displaystyle=z$ (C.4b)
$\displaystyle-\frac{1}{4}\bar{\sigma}_{a}^{\dot{\alpha}\alpha}[{\cal
D}_{\alpha},\bar{\cal D}_{\dot{\alpha}}]P|$ $\displaystyle=A_{a}\,,$ (C.4c)
$\displaystyle\frac{{\rm i}}{16}{\cal D}^{2}(\bar{\cal D}^{2}-8{\cal R})P|$
$\displaystyle=\frac{{\rm i}}{2}d-\frac{1}{2}{\cal D}^{a}A_{a}+{\rm Re\hskip
1.00006pt}(\bar{M}z)$ (C.4d)

where $z$ is a complex, propagating scalar field, $d$ a real, auxiliary scalar
field and $A_{a}$ a real vector field, which can although be related, via
Hodge–duality, to a three-form as in (2.7).

Real linear multiplets

A real linear superfield $L$ satisfies the constraint

$(\bar{\cal D}^{2}-8{\cal R})L=0\,,\qquad({\cal D}^{2}-8\bar{\cal R})L=0$
(C.5)

which can be solved in terms of an unconstrained spinorial superfield
$\Psi^{\alpha}$ as [97]

$L={\cal D}^{\alpha}(\bar{\cal D}^{2}-8{\cal
R})\Psi_{\alpha}^{\Lambda}+\bar{\cal D}_{\dot{\alpha}}({\cal D}^{2}-8\bar{\cal
R})\bar{\Psi}^{\Lambda\dot{\alpha}}\,.$ (C.6)

The bosonic components of $L$ are a real scalar $l$ and a gauge two-form
${\cal B}_{2}$, which appear through its field strength ${\cal
H}_{3}={\rm{d}}{\cal B}_{2}$. These are defined via:

$\displaystyle L|$ $\displaystyle=l\;,$ (C.7a)
$\displaystyle-\frac{1}{4}\bar{\cal D}^{2}L|$ $\displaystyle=\frac{1}{3}Ml\;,$
(C.7b) $\displaystyle-\frac{1}{4}\bar{\sigma}_{a}^{\dot{\alpha}\alpha}[{\cal
D}_{\alpha},\bar{\cal D}_{\dot{\alpha}}]L|$ $\displaystyle={\cal
H}_{a}-\frac{2}{3}b_{a}l\,,$ (C.7c)

where ${\cal H}_{a}=\frac{1}{3!}\varepsilon_{abcd}{\cal
H}^{bcd}=\frac{1}{2}\varepsilon_{abcd}\partial^{[b}{\cal B}_{2}^{cd]}$.

Complex linear multiplets

A complex linear superfield $\Sigma$ satisfies the constraint [97]

$(\bar{\cal D}^{2}-8{\cal R})\Sigma=0$ (C.8)

which can be solved in terms of an unconstrained spinorial superfield
$\Psi^{\alpha}$ as

$\Sigma=\bar{\cal D}_{\dot{\alpha}}\bar{\Psi}^{\dot{\alpha}}$ (C.9)

In the Wess-Zumino gauge, the bosonic components of $\Sigma$ read

$\displaystyle\Sigma|$ $\displaystyle=0\,$ (C.10a)
$\displaystyle-\frac{1}{4}{\cal D}^{2}\Sigma|$ $\displaystyle=\bar{s}\,,$
(C.10b) $\displaystyle-\frac{1}{2}\bar{\sigma}_{m}^{\dot{\alpha}\alpha}[{\cal
D}_{\alpha},\bar{\cal D}_{\dot{\alpha}}]\Sigma|$ $\displaystyle=C_{m}\,,$
(C.10c) $\displaystyle\frac{1}{16}{\cal D}^{2}\bar{\cal D}^{2}\bar{\Sigma}|$
$\displaystyle=\frac{{\rm i}}{2}\,{\cal D}_{m}C^{m}+\bar{M}s\,,$ (C.10d)

where $s$ is a complex scalar field and $C_{a}$ are component of a complex
vector which can be related to those of a three-form as in (2.7).

###  Appendix D Super-Weyl invariant Supergravity

In Chapter 4 we have extensively used the super-Weyl invariant formalism. This
approach was firstly formulated in [96] and we refer to [97] for a more
detailed treatment. Here we provide a review on how to construct generic
super-Weyl invariant Lagrangians and how to compute their gauge–fixed
components. For the sake of clarity, in this appendix we set $M_{\rm P}=1$,
which can be re-installed with simple field redefinitions.

####  D.1 Kähler and super-Weyl invariance in Supergravity

Let us consider a set of $n$ chiral multiplets $\Phi^{i}$, whose bosonic
components are

$\Phi^{i}=\\{\varphi^{i},F^{i}_{\Phi}\\}\,,\quad\text{with}\quad
i=1,\ldots,n\,,$ (D.1)

where, as in (C.2), $\varphi^{i}$ are the lowest component, complex scalar
fields and $F^{i}_{\Phi}$ are the highest component, auxiliary complex scalar
fields. These are ‘physical’ fields, containing all the propagating degrees of
freedom associated to the matter fields.

The locally supersymmetric coupling of the matter multiplets $\Phi^{i}$ to
gravity requires the introduction of another chiral multiplet ${\cal R}$, the
supergravity multiplet:

${\cal R}=\\{e_{a}^{m},\psi^{m}_{\alpha},b^{a},M\\}$ (D.2)

which accommodates the veilbein $e_{a}^{m}$, the gravitino $\psi^{m}$ and the
nonpropgating fields, $b^{a}$ and $M$ which are a real vector and a complex
scalar field. Up to two derivatives, the most general ${\cal N}=1$ locally
supersymmetric Lagrangian that we can build in superspace with only these
ingredients is

${\cal
L}=\int{\rm{d}}^{4}\theta\,E\,\Omega(\Phi,\bar{\Phi})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,W(\Phi)+{\rm c.c}\right)\,,$ (D.3)

where $\Omega(\Phi,\bar{\Phi})=-3e^{-\frac{1}{3}K(\Phi,\bar{\Phi})}$, with
$K(\Phi,\bar{\Phi})$ being the ordinary Kähler potential and $W(\Phi)$,
holomorphic in $\Phi^{i}$, is the superpotential. A direct computation of the
bosonic components of the Lagrangian (D.3) leads to

$\begin{split}e^{-1}{\cal L}_{\rm
bos}&=-\frac{1}{6}\Omega\,R-\Omega_{i\bar{\jmath}}\partial_{\mu}\varphi^{i}\partial^{\mu}\bar{\varphi}^{\bar{\jmath}}-\frac{1}{4\Omega}\left(\Omega_{i}\partial_{\mu}\varphi^{i}-\bar{\Omega}_{\bar{\jmath}}\partial_{\mu}\bar{\varphi}^{\bar{\jmath}}\right)^{2}\\\
&\quad\,+\frac{\Omega}{9}\left|M-\frac{3\Omega_{\bar{\imath}}\bar{F}^{\bar{\jmath}}_{\Phi}}{\Omega}\right|^{2}+\left(\Omega_{i\bar{\jmath}}-\frac{\Omega_{i}\Omega_{\bar{\jmath}}}{\Omega}\right)F^{i}_{\Phi}\bar{F}^{\bar{\jmath}}_{\Phi}+\left(W_{i}F^{i}_{\Phi}+\bar{W}_{\bar{\jmath}}\bar{F}^{\bar{\jmath}}_{\Phi}\right)\;.\end{split}$
(D.4)

The Lagrangian (D.3) exhibits a neat geometrical interpretation. The fields
$\varphi^{i}$ parametrize a Hodge–Kähler manifold. This is a Kähler manifold
${\mathscr{M}}_{\rm scalar}$ of restricted type, namely it is defined by the
existence of a line bundle ${\mathscr{L}}\rightarrow{\mathscr{M}}_{\rm
scalar}$, whose first Chern class equals the cohomology class of the Kähler
form $J$: $c_{1}({\mathscr{L}})=[J]$. Practically, this implies the following.
Given a holomorphic section $\xi(z)$ over the line bundle ${\mathscr{L}}$, the
first Chern class is determined by

$c_{1}({\mathscr{L}})=-\frac{{\rm
i}}{2\pi}\partial\bar{\partial}\log|\xi(z)|^{2}$ (D.5)

where $|\xi(z)|^{2}\equiv h(z,\bar{z})\xi(z)\bar{\xi}(\bar{z})$ is with
$h(z,\bar{z})$an hermitian fiber metric over ${\mathscr{L}}$. Then, recalling
that $J=\frac{{\rm
i}}{2\pi}K_{i\bar{\jmath}}{\rm{d}}z^{i}\wedge{\rm{d}}\bar{z}^{\bar{\jmath}}$
is closed, locally, by the Poincaré-Lelong theorem, we may rewrite

$J=-\frac{{\rm i}}{2\pi}\partial\bar{\partial}\rho$ (D.6)

in terms of a _potential_ $\rho(z,\bar{z})$. The comparison of (D.5) with
(D.6) shows that, when ${\mathscr{M}}_{\rm scalar}$ is Hodge–Kähler, then we
may regard $\rho=e^{K}$ as the metric over the line bundle ${\mathscr{L}}$.

A Kähler transformations

$K(\Phi,\bar{\Phi})\rightarrow K(\Phi,\bar{\Phi})+h(\Phi)+\bar{h}(\bar{\Phi})$
(D.7)

with $h(\Phi)$ an arbitrary holomorphic function of $\Phi^{i}$, leaves (D.6)
invariant. However, the Lagrangian (D.4) is not invariant under the sole
(D.7). The reason is that the superpotential and the auxiliary fields are
sections of the ${\mathscr{L}}$-line bundle and, whenever (D.7) acts over the
Kähler potential, the following transformations have also to be performed:

$\displaystyle W(\Phi)\rightarrow e^{-h(\Phi)}W(\Phi)\,$ (D.8a) $\displaystyle
M\rightarrow e^{-\frac{2}{3}(h(\Phi)+\bar{h}(\Phi))}M\,,\qquad
F^{i}_{\Phi}\rightarrow
F^{i}_{\Phi}e^{-\frac{2}{3}(h(\Phi)+\bar{h}(\Phi))}\,.$ (D.8b)

A proper Kähler transformation then combines the three relations (D.7), (D.8)
and (D.8b).

Still, the combined action of (D.7), (D.8) and (D.8b) does not leave the
Lagrangian (D.4) invariant. In fact, they have to be further combined with the
_super-Weyl transformation_ , which acts on the super-vielbeins as [183]

$E^{a}_{M}\rightarrow e^{\Upsilon+\bar{\Upsilon}}E^{a}_{M}\,,\quad
E^{\alpha}_{M}\rightarrow
e^{2\bar{\Upsilon}-\Upsilon}\left(E^{\alpha}_{M}-\frac{{\rm
i}}{4}E^{a}_{M}\sigma^{\alpha\dot{\alpha}}_{a}\bar{\cal
D}_{\dot{\alpha}}\bar{\Upsilon}\right).$ (D.9)

where $(a,\alpha)$ are flat superspace indices, $M=(m,\mu)$ are curved indices
and $\Upsilon$ is an arbitrary chiral superfield parameterizing the super-Weyl
transformation. In particular, (D.9) implies that

$E\rightarrow e^{2\Upsilon+2\bar{\Upsilon}}E\,,\quad{\rm{d}}^{2}\Theta\,2{\cal
E}\rightarrow e^{6\Upsilon}{\rm{d}}^{2}\Theta\,2{\cal E}$ (D.10)

as well as, from the lower bosonic components of (D.9), we recover the usual
Weyl-rescaling $e_{m}^{a}\to e_{m}^{a}e^{\Upsilon+\bar{\Upsilon}}$. It can be
immediately show that the superspace Lagrangian (D.3) is invariant if (D.7)
and (D.8) are supplemented by (D.9) with the choice $6\Upsilon=h(\Phi)$.

However, the Lagrangian (D.4) is not expressed in the Einstein frame. It is
then the lack of invariance of (D.4) under the sole super-Weyl rescaling (D.9)
that can be used to perform just a super-Weyl rescaling (D.9), choosing
$\Upsilon+\bar{\Upsilon}=\frac{1}{6}K(\Phi,\bar{\Phi})$, which recasts the
Lagrangian in the form

$\begin{split}e^{-1}{\cal L}_{\rm
bos}&=\frac{1}{2}R-K_{i\bar{\jmath}}\partial_{\mu}\varphi^{i}\bar{\partial}_{\mu}\bar{\varphi}^{\bar{\jmath}}\\\
&\quad\,+e^{-K}K_{i\bar{\jmath}}F^{i}_{\Phi}\bar{F}^{\bar{\jmath}}_{\Phi}-\frac{e^{-K}}{3}\left|\bar{M}+K_{i}F^{i}_{\Phi}\right|^{2}+\left(W_{i}F^{i}_{\Phi}-W\bar{M}+{\rm
c.c.}\right)\;.\end{split}$ (D.11)

with a canonically normalized Einstein-Hilbert term. Further integrating out
the auxiliary fields $F^{i}_{\Phi}$ and $M$, we arrive at the Lagrangian

$\begin{split}e^{-1}{\cal L}_{\rm
bos}&=\frac{1}{2}R-K_{i\bar{\jmath}}\partial_{\mu}\varphi^{i}\partial_{\mu}\bar{\varphi}^{\bar{\jmath}}-e^{K}\left(K^{\bar{\jmath}i}D_{i}W\bar{D}_{\bar{\jmath}}\bar{W}-3|W|^{2}\right)\;,\end{split}$
(D.12)

where the last term is nothing but the well-known Cremmer et al. potential
[98]. Indeed, this Lagrangian is invariant under (D.7) and (D.8a).

####  D.2 Super-Weyl invariant Lagrangians with chiral multiplets

The super-Weyl invariant formalism takes somewhat another perspective. The
Lagrangian (D.3) is rendered invariant under Kähler transformations
(D.7)-(D.8) and super-Weyl transformations (D.9) _separately_ , while still
arriving at the same component Lagrangian (D.11) or (D.12) when the the super-
Weyl transformations are properly gauge-fixed. At the core of the formalism is
the introduction of an unphysical, chiral compensator $U$, which transforms as

$U\rightarrow e^{-6\Upsilon}U$ (D.13)

under super-Weyl transformations. In turn, we introduce new chiral superfields
$Z^{a}$

$Z^{a}=\\{z^{a},\,F^{a}_{Z}\\}\,\quad\text{with}\quad a=1,\ldots,n+1\,,$
(D.14)

where $z^{a}$ and $f^{a}$ are understood to be functions of the components of
$\Phi^{i}$ and those of $U$, and we assume that, from $Z^{a}$, we can single
out the compensator $U$ as

$Z^{a}=Ug^{a}(\Phi)$ (D.15)

where $g^{a}$ are functions of the physical fields only and are inert under
super-Weyl transformations.

The most general supergravity Lagrangian that we can build out the $Z^{a}$
multiplets has still the form (D.3)

$\begin{split}{\cal L}=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z,\bar{Z})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal W}(Z)+{\rm
c.c.}\right)\end{split}$ (D.16)

where ${\cal K}({Z},\bar{Z})$ is the kinetic potential and ${\cal W}(Z)$ the
superpotential. Additionally, however, they satisfy the following homogeneity
conditions

${\cal K}(\lambda Z,\bar{\lambda}\bar{Z})=|\lambda|^{\frac{2}{3}}{\cal
K}({Z},\bar{Z})\,,\qquad{\cal W}(\lambda Z)=\lambda{\cal W}(Z).$ (D.17)

with $\lambda$ an arbitrary chiral superfield.

The ordinary Kähler potential $K(\Phi,\bar{\Phi})$ and superpotential
$W(\Phi)$ are then recovered by isolating the compensator $U$ as

${\cal
K}({Z},\bar{Z})=-3|U|^{\frac{2}{3}}e^{-\frac{1}{3}K(\Phi,\bar{\Phi})}\,,\qquad{\cal
W}(Z)=U\,W(\Phi)$ (D.18)

where we have set $K(\Phi,\bar{\Phi})\equiv K(g(\Phi),\bar{g}(\bar{\Phi}))$
and $W(\Phi)\equiv W(g(\Phi))$. Such homogeneity properties, along with
(D.13), make the Lagrangian (D.16) manifestly invariant under (D.9). Indeed,
the Lagrangian (D.17) is also invariant under Kähler transformations. In fact,
the split (D.15) is clearly not unique, since we may redefine