However, they are somewhat similar in nature: they cover three spacetime
directions and they are both BPS, spontaneously breaking half of the bulk
generators.

This section is devoted to explore how new domain wall solution can be
generated by membranes. After qualitatively describing how membranes may
induce new domain walls, we will pass to examine the mathematical properties
that configurations which merge membranes and domain walls enjoy.

#####  3.4.1 Membranes creating new domain walls

The membranes that we introduced in Sections 3.1 and 3.2 are _fundamental_
objects, governed by actions of the form (3.6) or (3.39), using which they are
directly included into a supersymmetric theory. Furthermore, they are
_structureless_. That is, owing to their very definition via the embedding
(3.1) as hypersurfaces in the ambient superspace, in the ordinary spacetime,
when they are frozen in their ground state, they are strictly hypersurfaces of
codimension one. In the static gauge, for example, a membrane fully covers the
$x^{i}$–directions, but it is just a point, $z=z^{0}$, along the fourth.

The presence of charged membranes within a theory does not just serve to add
more energy to the system, but also to divide the spacetime into two regions
with two _different_ potentials, as already explained in Section 1.3.
Eventually, the sets of vacua on the two sides of the membrane are different.
Membranes, in other words, provide a new theory, with new vacua and, within
this theory, one might look for the admissible domain wall solutions which
interpolate between the vacua on one side to the others on the other side. In
this sense membranes and domain walls are extended objects that can coexist
and interestingly create new nonperturbative effects within a theory.
Remarkably, an ordinary chiral multiplet theory without membranes, may not
admit any domain wall solution (for example, if the theory has a trivial
$\pi_{0}({\mathscr{M}}_{\rm vac})$, because the vacuum is just a point);
however, once we couple a membrane to the theory, new vacua are born and
domain wall solutions can exist. In Section 3.4.5 we will indeed see a simple
example when this happens.

It is also important to remark that it is not obvious that domain walls
‘enclosing’ a membrane may exist. A domain wall is in fact a _full_ solitonic
solution which includes the backreaction of the scalars of the theory, as
dictated by the flow equations and these, generically, may not be possible to
be solved, preventing the formation of a domain wall. The study of the domain
walls with a membrane at its core is the object of the next sections, where we
compute the energy and the central charges for such a configuration, whenever
a domain wall solution exists.

#####  3.4.2 BPS-condition and flow equations

The analysis of BPS–domain wall solutions performed in (3.3) may be easily
extended to the case where membranes are present. Our aim is to explicitly
show how the presence of membranes can _create_ new domain wall solutions. In
order to ease the exposition, we focus on the case where there is just a
single, flat membrane which divides the spacetime into two distinct regions.
Moreover, we will work in the static gauge (3.12): the membrane stretches
along three spacetime directions, say $x^{i}$ with $i=0,1,2$, and the
transverse direction is most readily identified with $z\equiv x^{3}$. We also
assume the membrane to be fixed at $z=0$, frozen in its ground state.

In the general case, some chiral, _bulk_ superfields $S^{A}$ are coupled to
the membrane, as in Section 3.2. We assume them to be the three-form
superfields (2.109), with chiral expansion

$S^{A}=s^{A}+\sqrt{2}\theta^{\alpha}\psi_{S\alpha}^{A}+\frac{1}{2}\left(*F_{4}^{A}+{\rm
i}d^{A}\right)\theta^{2}\,,$ (3.97)

where the gauge three-forms $A^{A}_{3}$ appear in their highest components
through the field strengths $F_{4}^{A}$. As in the previous section, we will
work with the simplifying assumption that the profile of the fields depends
only on the coordinate $z$ transverse to the membrane

$s^{A}\equiv
s^{A}(z)\,,\qquad\psi^{A}_{S\alpha}\equiv\psi^{A}_{S\alpha}(z)\,,\qquad
A_{3}^{A}\equiv A_{3}^{A}(z)\,,\qquad d^{A}\equiv d^{A}(z)\;.$ (3.98)

However we stress that the scalar fields $s^{A}(\varphi(z))$ do actually
depend on the sets of the physical fields $\varphi^{a}$ through which they can
be expressed (see (2.110a)). We refer to Section 2.4 for a detailed discussion
regarding the correspondence between ordinary chiral fields and three-form
multiplets $S^{A}$. Moreover, the physical fields $s^{A}$ (and $\varphi^{a}$)
and $\psi^{A}_{(S)}$ are continuous in the whole spacetime, although their
derivatives may be discontinuous at $z=0$, where the membrane is encountered.

The full action describing the bulk and the membrane dynamics is

$S=S_{\rm bulk}+S_{\rm memb}\,,$ (3.99)

where the bulk action, defined by the Lagrangian (2.112), is

$\begin{split}S_{\rm bulk}&=\int{\rm{d}}^{4}x\,{\cal L}_{\rm bulk}\\\
&\quad\,=\int{\rm{d}}^{4}x\,\left(\int{\rm{d}}^{4}\theta\,K(\Phi(P),\bar{\Phi}(P))+\left[\int{\rm{d}}^{2}\theta\,\hat{W}(\Phi(P))+\text{c.c.}\right]+\tilde{\cal
L}_{\rm bd}\right)\end{split}$ (3.100)

and $S_{\rm memb}$ is given in (3.39). For simplicity we shall focus just on
bosons and set the fermions to zero. Then, the bulk action is the very same as
that given in (2.121).

In order to get an ‘ordinary’ chiral theory and connect the present discussion
with that of the previous section, we have to get rid of the gauge three-
forms. We then integrate them out as

$T_{AB}(*\hat{F}^{B}_{4}+\Upsilon^{B})=-N_{A}-q_{A}\Theta(z)\;,$ (3.101)

where the last term is due to the Wess-Zumino term of (3.39), which makes the
constants $N_{A}$ jump with the membrane charge.

As explained in Section 2.4, we then arrive at an on-shell Lagrangian of the
form

$S_{\rm bulk}=\int{\rm{d}}^{4}x\,{\cal L}_{\rm
bulk}=\int{\rm{d}}^{4}x\,\left[-K_{a\bar{b}}\partial_{z}\varphi^{a}\partial_{z}\bar{\varphi}^{\bar{b}}-V(\varphi,\bar{\varphi})\right]\;.$
(3.102)

Importantly, due to the presence of the membrane, the potential on the two
sides of the membrane is _different_ , being it specified by different
constants, that is

$\displaystyle
V(\varphi,\bar{\varphi})=\Theta(-z)V_{-}(\varphi,\bar{\varphi})+\Theta(z)V_{+}(\varphi,\bar{\varphi})\;,$
(3.103)

with

$\displaystyle V_{-}(\varphi,\bar{\varphi})$
$\displaystyle=K^{\bar{b}a}(N_{A}{\cal
V}^{A}_{a}(\varphi)+\hat{W}_{a}(\varphi))(N_{B}\bar{\cal
V}^{B}_{\bar{b}}(\bar{\varphi})+\bar{\hat{W}}_{\bar{b}}(\bar{\varphi}))\,,$
(3.104) $\displaystyle V_{+}(\varphi,\bar{\varphi})$
$\displaystyle=K^{\bar{b}a}[(N_{A}+q_{A}){\cal
V}^{A}_{a}(\varphi)+\hat{W}_{a}(\varphi)][(N_{B}+q_{B})\bar{\cal
V}^{B}_{\bar{b}}(\bar{\varphi})+\bar{\hat{W}}_{\bar{b}}(\bar{\varphi})]\,.$

The potential (3.103) can be seen as originating from the superpotential

$W(\varphi)=\Theta(-z)W_{-}(\varphi)+\Theta(z)W_{+}(\varphi)$ (3.105)

with the usual relation $V=K^{\bar{b}a}W_{a}\bar{W}_{\bar{b}}$, where $W_{-}$
and $W_{+}$ are, respectively, the _on–shell superpotential_ on the left and
on the right of the membrane, which are partly dynamically generated by the
gauge three-forms and given by

$\displaystyle W_{-}(\varphi)\equiv N_{A}{\cal
V}^{A}(\varphi)+\hat{W}(\varphi)\qquad{\rm and}\qquad
W_{+}(\varphi)\equiv(N_{A}+q_{A}){\cal V}^{A}(\varphi)+\hat{W}(\varphi)\,.$
(3.106)

It is also convenient to define

$\Delta W(\Phi)\equiv W_{+}(\Phi)-W_{-}(\Phi)=q_{A}{\cal V}^{A}(\Phi)$ (3.107)

so that, when the scalar fields are evaluated at $z=0$, ${\cal
T}_{\text{memb}}=2|\Delta W|$.

Generically, the classical vacua of the potentials
$V_{-}(\varphi,\bar{\varphi})$ and $V_{+}(\varphi,\bar{\varphi})$ are
different and we may study the domain wall solutions which interpolate between
the set of vacua on the left of the membrane to those on the right. Let us
then consider a vacuum $\varphi_{-}^{a}$ on the left of the membrane and
$\varphi_{+}^{a}$ on the right. A domain wall is a solitonic solution of the
field equations which interpolates between $\varphi_{-}^{a}$, which we may
assume to be the asymptotic field configuration at $z\to-\infty$, and
$\varphi_{+}^{a}$, which is reached at $z\to+\infty$; in between, the fields
may nontrivially vary.

We are still interested in BPS–domain walls and the preservation of (part of)
the bulk supersymmetry requires the supersymmetry variations of the fields to
vanish over the domain wall solution. Setting to zero the variations of the
chiralini as in (3.71) leads to the flow equation

$\partial_{z}\varphi^{a}=\mp e^{{\rm
i}\vartheta}K^{a\bar{b}}\bar{W}_{\bar{b}}\;,$ (3.108)

with the superpotential defined in (3.105), provided that the supersymmetry
parameters satisfy the projection condition

$\epsilon_{\alpha}=\mp{\rm i}e^{{\rm
i}\alpha}\sigma^{3}_{\alpha\dot{\alpha}}\bar{\epsilon}^{\dot{\alpha}}\;.$
(3.109)

Consistency with the $\kappa$-symmetry of the membrane requires that, at
$z=0$, the supersymmetry parameter is _identified_ with the
$\kappa$-parameter:

$\epsilon_{\alpha}|_{z=0}\stackrel{{\scriptstyle!}}{{=}}\kappa_{\alpha}=-{\rm
i}\frac{q_{A}{\cal V}^{A}(\varphi)}{|q_{A}{\cal
V}^{A}(\varphi)|}\Big{|}_{z=0}\sigma^{3}_{\alpha\dot{\alpha}}\bar{\kappa}^{\dot{\alpha}}\;.$
(3.110)

Setting $\vartheta\equiv\arg[q_{A}{\cal V}^{A}(\varphi)]|_{z=0}$, the previous
relation implies that

$e^{{\rm i}\alpha}\stackrel{{\scriptstyle!}}{{=}}\pm e^{{\rm i}\vartheta}$
(3.111)

needs to hold over the membrane worldvolume.

In the following sections we first compute, as we did for sourceless domain
walls, the tension of the domain wall-plus-membrane system and later the
central charge. Finally, we present a very simple example where we explicitly
compute the domain wall solution where a membrane is present.

#####  3.4.3 Tension of a BPS-domain wall sourced by a single membrane

Even though a membrane is present, we may still define the tension of a domain
wall enclosing the membrane as in (3.77), where the action to be set on-shell
is the full (3.99). Once we set the three-forms on-shell on the two sides of
the membrane, in order to compute the tension, we can proceed as in Section
3.3.1. In fact, we may still rewrite the action in the BPS–form

$\begin{split}S_{\rm
bos}&=\int{\rm{d}}^{3}x\,{\rm{d}}z\,\Bigg{[}-K_{a\bar{b}}\left(\partial_{z}\varphi^{a}\pm
e^{{\rm
i}\beta}K^{\bar{c}a}\bar{W}_{\bar{c}}\right)\left(\partial_{z}\bar{\varphi}^{\bar{b}}\pm
e^{-{\rm i}\beta}K^{\bar{b}d}W_{d}\right)\Bigg{]}\\\ &\quad\,\pm
2\int{\rm{d}}^{3}x\,{\rm{d}}z\,\frac{{\rm{d}}}{{\rm{d}}z}{\rm Re\hskip
1.00006pt}\left(We^{-{\rm
i}\beta}\right)-\int{\rm{d}}^{3}x\,{\rm{d}}z\,\delta(z){\cal
T}_{\text{memb}}\,.\end{split}$ (3.112)

where $\beta$ is an arbitrary real constant. and the superpotential is given
in (3.105). In order to perform the integration in the last line, let us
notice that, from (3.105), we easily get

$\frac{{\rm{d}}}{{\rm{d}}z}W=W_{a}\partial_{z}\varphi^{a}+\Delta W\delta(z)$
(3.113)

and then (3.112) gives

$\begin{split}S_{\rm
bos}&=\int{\rm{d}}^{3}x\,{\rm{d}}z\,\Bigg{[}-K_{a\bar{b}}\left(\partial_{z}\varphi^{a}\pm
e^{{\rm
i}\beta}K^{\bar{c}a}\bar{W}_{\bar{c}}\right)\left(\partial_{z}\bar{\varphi}^{\bar{b}}\pm
e^{-{\rm i}\beta}K^{\bar{b}d}W_{d}\right)\Bigg{]}\\\ &\quad\,\pm
2\int{\rm{d}}^{3}x\,{\rm Re\hskip
1.00006pt}\left[(W_{+\infty}-W_{-\infty})e^{-{\rm i}\beta}\right]\\\
&\quad\,-\int{\rm{d}}^{3}x\,{\rm{d}}z\,\delta(z)\left[{\cal
T}_{\text{memb}}\mp 2{\rm Re\hskip 1.00006pt}(\Delta We^{-{\rm
i}\beta})\right]\,.\end{split}$ (3.114)

Furthermore, although the superpotential is discontinuous, the integral of
motion (3.83) is present also here

$\frac{{\rm{d}}}{{\rm{d}}z}{\rm Im\hskip 1.00006pt}(We^{-{\rm
i}\alpha})=0\qquad\Rightarrow\qquad{\rm Im\hskip 1.00006pt}\left[e^{-{\rm
i}\alpha}(W_{+\infty}-W_{-\infty})\right]=0\,.$ (3.115)

The third line of (3.114) includes the contributions localized over the
membrane worldvolume. However, upon using the consistency condition (3.111)
between bulk supersymmetry and $\kappa$–symmetry, they cancel out

${\cal T}_{\text{memb}}\mp 2{\rm Re\hskip 1.00006pt}(\Delta We^{-{\rm
i}\beta})|_{z=0}={\cal T}_{\text{memb}}-2{\rm Re\hskip 1.00006pt}(\Delta
We^{-{\rm i}\vartheta})|_{z=0}=0$ (3.116)

Hence, the on-shell action simply reads

${\cal T}_{\rm
DW}=2\,|W_{+\infty}-W_{-\infty}|\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\small\text{\sf{Tension
of a BPS domain wall sourced by a membrane}}}$ (3.117)

which is formally equivalent to (3.117) with, however, the superpotential
asymptotic values computed from (3.105), in turn determined by the presence of
the membrane.

#####  3.4.4 Central charge of a BPS-domain wall sourced by a membrane

In the case where membranes are present, the topological charges which modify
the supersymmetry algebra have a twofold origin: first, they come from the
very existence of the membrane, in analogy with [84] and (3.66); moreover,
another contribution come from the full domain-wall configuration
interpolating among vacua, in analogy with [83, 93] and our (3.91).

By combining (3.66) and (3.91), we immediately get the central charges for the
whole configuration of a domain wall supported by a static membrane

$\displaystyle M_{\alpha\beta}$
$\displaystyle=-8\int{\rm{d}}x^{1}{\rm{d}}x^{2}\,\varepsilon_{\beta\delta}\sigma^{03}{}_{\alpha}{}^{\delta}\,(\bar{W}_{+\infty}-\bar{W}_{-\infty})\,,$
(3.118) $\displaystyle\bar{M}^{\dot{\alpha}\dot{\beta}}$
$\displaystyle=-8\int{\rm{d}}x^{1}{\rm{d}}x^{2}\,\bar{\sigma}^{03\dot{\alpha}}{}_{\dot{\gamma}}\varepsilon^{\dot{\gamma}\dot{\beta}}(W_{+\infty}-W_{-\infty})\,,$

with the superpotential defined as in (3.105).

#####  3.4.5 A simple example

Consider a supersymmetric theory with just a single chiral superfield $\Phi$
and characterized by

$K(\Phi,\bar{\Phi})=\Phi\bar{\Phi}\,,\qquad\qquad
W(\Phi)=n\Phi-\frac{1}{2}\Phi^{2}\;,$ (3.119)

with $n$ a real constant. This model exhibits just a _single_ supersymmetric
vacuum, located at $\varphi=n$. Therefore, according to the discussion of
Section 3.3, it is not possible to construct any nontrivial domain wall within
this theory.

However, in Section 2.4 we saw that the linear part of the superpotential in
(3.119) can be seen as generated by a single three-form multiplet. Namely, we
consider a dual three-form theore, described by the Lagrangian (2.112), with
just one single three-form multiplet $S$ (as (2.78)), including a gauge three-
form $A_{3}$ in its components:

$K(S,\bar{S})=S\bar{S}\,,\qquad\qquad\hat{W}(S)=-\frac{1}{2}S^{2}\;.$ (3.120)

Once the gauge three-form is integrated out, we reget the same theory as the
one specified by (3.119), where $n$ is regarded as an arbitrary integration
constant.

The true advantage of dealing with the theory in the form (3.120) is that now
we can couple a membrane, charged with charge $q$ under the gauge three-form
$A_{3}$. Let us assume, for simplicity, that the membrane is located at $z=0$
and frozen in its static ground state. Then, the membrane influences the
equation of motion for the gauge three-forms as in (3.101), making the
constant $n$ shift to $n+q$ once pass across $z=0$. In turn, this implies that
the superpotential of the dual chiral multiplet theory is

$W(\Phi)=W_{\rm gen}(\Phi)-\frac{1}{2}\Phi^{2}\;,$ (3.121)

where the dynamically generated part _differs_ on the two sides:

$W_{\rm gen}(\Phi)=(n+q\Theta(z))\Phi\,.$ (3.122)

As a result, two different potentials on the two sides of the membrane
originate, each one with its own suspersymmetric vacuum, determined by

$\displaystyle\partial_{\Phi}W_{-}\stackrel{{\scriptstyle!}}{{=}}0\qquad\Rightarrow\qquad\Phi_{-}|=n\;,$
(3.123)
$\displaystyle\partial_{\Phi}W_{+}\stackrel{{\scriptstyle!}}{{=}}0\qquad\Rightarrow\qquad\Phi_{+}|=n+q\;,$

with $W_{+}$ and $W_{-}$ defined from (3.121) as

$W=:W_{-}\Theta(-z)+W_{+}\Theta(z)\;.$ (3.124)

At the vacua, the superpotential is

$W_{-\infty}=\frac{c^{2}}{2}\;,\qquad W_{+\infty}=\frac{(c+q)^{2}}{2}\;.$
(3.125)

Now, domain walls can exist which interpolate between $\Phi_{-}$ and
$\Phi_{+}$ from the left to the right of the membrane.

Let us now determine $\alpha$. The integral of motion (3.115) implies that

${\rm Im\hskip 1.00006pt}(e^{-{\rm i}\alpha}W_{+\infty})\equiv C\equiv{\rm
Im\hskip 1.00006pt}(e^{-{\rm i}\alpha}W_{-\infty})$ (3.126)

with $C$ a fixed constant. Consistency with (3.125) and positivity of the
domain wall tension (3.117) requires $C=0$ and $\alpha=2\pi k$,
$k\in\mathbb{Z}$, for the upper sign choice or $\alpha=(2k+1)\pi$ for the
lower sign in (3.114). Then, the flow equations (3.108) become

$\begin{cases}\partial_{z}\varphi=-\bar{\varphi}+c&{\text{for}}\quad z<0\\\
\partial_{z}\varphi=-\bar{\varphi}+c+k&{\text{for}}\quad z>0\end{cases}$
(3.127)

Further assuming $\varphi$ to be real for simplicity, these are simply solved
by

$\begin{cases}\varphi(z)=n&{\text{for}}\quad z<0\\\
\varphi(z)=n+q(1-e^{-z})&{\text{for}}\quad z>0\end{cases}$ (3.128)

where the integration constants are determined requiring $\varphi$ to be
continuous at $z=0$. The wall profile is plotted in Fig. 3.3.

Figure 3.3: The domain wall profile described by (3.128). The membrane is
located at $z=0$: there, although $\varphi$ is continuous, is derivative is
not. In yellow is highlighted the region where the scalar field varies.

This example should have made concrete an important, remarkable physical fact
that was drawn in the general discussion at the beginning of this section: the
very existence of membranes open the possibilities of getting new vacua,
eventually connected by domain walls as examined here.

####  3.5 Strings in Supersymmetry

In Section 2.5 we have seen how to construct generic supersymmetric
Lagrangians containing some linear multiplets $L^{\Lambda}$, which play the
role of super-field strengths for some gauge two-forms ${\cal
B}_{2}^{\Lambda}$. Within such a framework, we may then consider fundamental,
structureless objects which electrically couple to ${\cal B}_{2}^{\Lambda}$.
These objects are _fundamental strings_ , of the kind which we already
introduced in Section 1.5: they are _structureless_ objects, namely of
strictly codimension two. Here, however, we would like to take a step forward:
we want to write down an action for strings, generically coupled to the ${\cal
B}_{2}^{\Lambda}$ gauge fields, in a manifestly supersymmetric way. The
strategy which we follow is the same as the one we adopted for membranes in
the Sections 3.1 and 3.2.

A fundamental string, whose worldsheet is parametrized by the coordinates
$\zeta^{i}$ ($i=1,2$), spans a superspace hypersurface ${\cal S}$ which is
described by the superspace embedding

$\zeta^{i}\quad\mapsto\quad{\cal
S}:\;{\mathfrak{z}}^{M}(\xi)=\left(x^{m}(\zeta),\theta^{\alpha}(\zeta),\bar{\theta}_{\dot{\alpha}}(\zeta)\right).$
(3.129)

Consequently, we define the induced metric on the string as

${\bm{\gamma}}_{ij}\equiv E_{i}^{a}\eta_{ab}E_{j}^{b}\,,$ (3.130)

where we have introduced the pull–backs of the target superspace supervielbein

$E^{a}_{i}(\zeta)\equiv\partial_{i}{\mathfrak{z}}^{M}(\zeta)E^{a}_{M}({\mathfrak{z}}(\zeta))\,.$
(3.131)

Furthermore, it is necessary to promote the gauge two-forms ${\cal
B}_{2}^{\Lambda}$ to a super–gauge two-form ${\bf B}_{2}^{\Lambda}$ whose
lowest bosonic component just coincides with ${\cal B}_{2}^{\Lambda}$. As we
will see shortly, the super–gauge two-form ${\bf B}_{2}^{\Lambda}$ are defined
in terms of the linear multiplets $L^{\Lambda}$, exactly as ${\bf A}_{3}^{A}$
in (3.35) are defined by the potentials $P^{A}$.

Equipped with these ingredients, we can introduce the supersymmetric action
for a fundamental string:

$S_{\rm string}=-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}L^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}+e_{\Lambda}\int_{\cal
S}{\bf B}^{\Lambda}_{2}\,.$ (3.132)

The first part of the action (3.132) encodes the dynamics of the string and
defines the string tension

${\cal T}_{\rm string}=|e_{\Lambda}l^{\Lambda}|\,.$ (3.133)

The second part of the action (3.132), that is the Wess-Zumino term, expresses
the coupling of the gauge two-forms ${\cal B}_{2}^{\Lambda}$ to the string
with charges $e_{\Lambda}$.

As stressed for membranes, the super-two-form ${\bf B}^{\Lambda}_{2}$ is
defined in terms of closed super-field strength three-form ${\bf
H}^{\Lambda}_{3}={\bf{d}}{\bf B}_{2}^{\Lambda}$,

$\displaystyle{\bf H}^{\Lambda}_{3}={\bf{d}}{\bf B}_{2}^{\Lambda}$
$\displaystyle=-2{{\rm i}}E^{a}\wedge
E^{\alpha}\wedge\bar{E}^{\dot{\alpha}}\sigma_{a\alpha\dot{\alpha}}{L}^{\Lambda}$
(3.134) $\displaystyle\quad\,+E^{b}\wedge E^{a}\wedge
E^{\alpha}\sigma_{ab\;\alpha}{}^{\beta}D_{\beta}{L}^{\Lambda}+E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\bar{\sigma}_{ab}{}^{\dot{\beta}}{}_{\dot{\alpha}}\bar{D}_{\dot{\beta}}{L}^{\Lambda}$
$\displaystyle\quad\,+\frac{1}{24}E^{c}\wedge E^{b}\wedge
E^{a}\epsilon_{abcd}\,\bar{\sigma}{}^{d\dot{\alpha}\alpha}[D_{\alpha},\bar{D}_{\dot{\alpha}}]{L}^{\Lambda}\,,$

and it is the unique closed super-three-form which can be constructed from the
linear multiplets [95]. It can be immediately seen that its lowest bosonic
component is

${\bf H}_{3}^{\Lambda}|={\cal H}_{3}^{\Lambda}={\rm{d}}{\cal B}_{2}^{\Lambda}$
(3.135)

so that (3.132) reduces to (1.71) once we restrict to the bosonic components.

As for the membrane action (3.6) and (3.39), the string action (3.132) is
invariant under worldsheet reparametrizations by construction. Moreover, it
enjoys a local, fermionic $\kappa$-symmetry specified by the parameters
$\kappa_{\alpha}(\zeta)$ and $\bar{\kappa}^{\dot{\alpha}}(\zeta)$. As is
demonstrated in Section 3.5.1, requiring

$\delta_{\kappa}S_{\rm string}\stackrel{{\scriptstyle!}}{{=}}0$ (3.136)

implies that the fermionic parameters $\kappa_{\alpha}(\zeta)$ and
$\bar{\kappa}^{\dot{\alpha}}(\zeta)$ obey the projection conditions

$\kappa_{\alpha}=-\frac{e_{\Lambda}L^{\Lambda}}{|e_{\Lambda}L^{\Lambda}|}\Gamma_{\alpha}{}^{\beta}\kappa_{\beta}\,,\qquad\bar{\kappa}^{\dot{\alpha}}=-\frac{e_{\Lambda}L^{\Lambda}}{|e_{\Lambda}L^{\Lambda}|}\bar{\Gamma}^{\dot{\alpha}}{}_{\dot{\beta}}\bar{\kappa}^{\dot{\beta}}\,,$
(3.137)

where we have defined

$\Gamma_{\alpha}{}^{\beta}\equiv\frac{1}{\sqrt{-\det{\bm{\gamma}}}}\epsilon^{ij}E^{a}_{i}E^{b}_{j}(\sigma_{ab})_{\alpha}{}^{\beta}\,,\qquad\bar{\Gamma}^{\dot{\alpha}}{}_{\dot{\beta}}\equiv\frac{1}{\sqrt{-\det{\bm{\gamma}}}}\epsilon^{ij}E^{a}_{i}E^{b}_{j}(\bar{\sigma}_{ab})^{\dot{\alpha}}{}_{\dot{\beta}}\,.$
(3.138)

satisfying

$\Gamma_{\alpha}{}^{\gamma}\Gamma_{\gamma}{}^{\beta}=\delta^{\beta}_{\alpha}\,,\qquad\bar{\Gamma}^{\dot{\alpha}}{}_{\dot{\gamma}}\bar{\Gamma}^{\dot{\gamma}}{}_{\dot{\beta}}=\delta_{\dot{\beta}}^{\dot{\alpha}}\,.$
(3.139)

#####  3.5.1 Proof of $\kappa$–symmetry

In this section we provide a detailed demonstration of the $\kappa$-symmetry
of the string action (3.132). We want to show that the variation of the string
action (3.132) is zero under local fermionic transformations, specified by the
parameters $\kappa_{\alpha}(\zeta)$ and $\bar{\kappa}^{\dot{\alpha}}(\zeta)$,
provided that the local parameters satisfy (3.137). The action of such
transformations on the super-vielbeins is given by (3.40) and (3.41).

We split the variation of (3.132) into two pieces, the variation of the Nambu-
Goto part and that of the Wess-Zumino part

$\delta_{\kappa}S_{\rm string}=\delta_{\kappa}S_{\rm
string,NG}+\delta_{\kappa}S_{\rm string,WZ}\,.$ (3.140)

The variation of the Nambu-Goto part $S_{\rm string,NG}$ can be easily
computed by first rewriting the world-sheet measure as

${\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}=-\frac{1}{2}{}^{*_{2}}\\!E_{a}\wedge
E^{a}\,,$ (3.141)

where we have defined the action of worldvolume Hodge duality operation
${}^{*_{2}}$ as

${}^{*_{2}}\\!E^{M}\equiv{\rm{d}}\xi^{i}\sqrt{-{\bm{\gamma}}}\epsilon_{ij}\gamma^{jl}E_{l}^{M}\;.\qquad$
(3.142)

Then, using (3.41), we immediately get

$\begin{split}\delta S_{\text{string,NG}}&=\int_{{\cal
S}}{}^{*_{2}}\\!E_{a}\wedge\delta E^{a}\,|e_{\Lambda}L^{\Lambda}|-\int_{{\cal
S}}{\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}\,\frac{e_{\Sigma}L^{\Sigma}}{|e_{\Sigma}L^{\Sigma}|}e_{\Lambda}\left(\kappa^{\alpha}D_{\alpha}L^{\Lambda}+\bar{\kappa}_{\dot{\alpha}}\bar{D}^{\dot{\alpha}}L^{\Lambda}\right)\\\
&=-2{\rm i}\int_{{\cal S}}{}^{*_{2}}\\!E_{a}\wedge
E^{\alpha}\sigma^{a}_{\alpha\dot{\alpha}}\bar{\kappa}^{\dot{\alpha}}|e_{\Lambda}L^{\Lambda}|+2{\rm
i}\int_{{\cal
S}}{}^{*_{2}}\\!E_{a}\wedge\kappa^{\alpha}\sigma^{a}_{\alpha\dot{\alpha}}\bar{E}^{\dot{\alpha}}|e_{\Lambda}L^{\Lambda}|\\\
&\quad\,-\int_{{\cal
S}}{\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}\,\frac{e_{\Sigma}L^{\Sigma}}{|e_{\Sigma}L^{\Sigma}|}e_{\Lambda}\kappa^{\alpha}D_{\alpha}L^{\Lambda}-\int_{{\cal
S}}{\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}\,\frac{e_{\Sigma}L^{\Sigma}}{|e_{\Sigma}L^{\Sigma}|}e_{\Lambda}\bar{\kappa}_{\dot{\alpha}}\bar{D}^{\dot{\alpha}}L^{\Lambda}\;,\end{split}$
(3.143)

The variation of the Wess-Zumino term is

$\delta S_{\text{string,WZ}}=e_{\Lambda}\int_{{\cal S}}\delta_{\kappa}{\bf
B}_{2}^{\Lambda}=e_{\Lambda}\int_{{\cal S}}(i_{\kappa}{\bf{d}}{\bf
B}_{2}^{\Lambda}+{\bf{d}}\,i_{\kappa}{\bf
B}_{2}^{\Lambda})=e_{\Lambda}\int_{{\cal S}}i_{\kappa}{\bf
H}_{3}^{\Lambda}\,.$ (3.144)

where, in order to get the second equality, we have assumed that the string
has no boundary. Using (3.40), from (3.134), we get

$\begin{split}i_{\kappa}{\bf H}_{3}^{\Lambda}=\;&-2{\rm i}E^{a}\wedge
E^{\alpha}(\sigma_{a}\bar{\kappa})_{\alpha}L^{\Lambda}+2{\rm
i}E^{a}\wedge(\kappa\sigma_{a})_{\dot{\alpha}}\bar{E}^{\dot{\alpha}}L^{\Lambda}\\\
&-E^{b}\wedge E^{a}(\kappa\sigma_{ab})^{\beta}D_{\beta}L^{\Lambda}+E^{b}\wedge
E^{a}\bar{D}_{\dot{\beta}}L^{\Lambda}(\bar{\sigma}_{ab}\bar{\kappa})^{\dot{\beta}}\;.\end{split}$
(3.145)

By using

$\displaystyle E_{a}\wedge
E^{\alpha}\,\sigma^{a}_{\alpha\dot{\alpha}}=-{}^{*_{2}}\\!E_{a}\wedge
E^{\beta}\sigma_{\beta\dot{\alpha}}^{a}\bar{\Gamma}^{\dot{\alpha}}{}_{\dot{\alpha}}\;,$
(3.146) $\displaystyle
E^{a}\wedge\bar{E}_{\dot{\alpha}}\,\bar{\sigma}_{a}^{\dot{\alpha}\alpha}={}^{*_{2}}\\!E^{a}\wedge\bar{E}_{\dot{\alpha}}\bar{\sigma}^{\dot{\alpha}\beta}_{a}\Gamma_{\beta}{}^{\alpha}\;,$

the first line may be written as

$\begin{split}&2{\rm i}{}^{*_{2}}\\!E^{a}\wedge
E^{\alpha}(\sigma_{a}\,{\bar{\Gamma}}\bar{\kappa})_{\alpha}{}L^{\Lambda}-2{\rm
i}{}^{*_{2}}\\!E^{a}\wedge\bar{E}_{\dot{\alpha}}(\bar{\sigma}_{a}\,{\Gamma}\kappa)^{\dot{\alpha}}L^{\Lambda}\;;\end{split}$
(3.147)

indeed, by means of the identities

$\displaystyle{\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}\,{\Gamma}=E^{a}\wedge
E^{b}\sigma_{ab}\,,$ (3.148)
$\displaystyle{\rm{d}}^{2}\zeta\sqrt{-{\bm{\gamma}}}\,{\bar{\Gamma}}=-E^{a}\wedge
E^{b}\bar{\sigma}_{ab}\,.$

the second line becomes

$\begin{split}&-{\rm{d}}^{2}\zeta\,\sqrt{-{\bm{\gamma}}}\,(\kappa\Gamma)^{\beta}D_{\beta}-{\rm{d}}^{2}\zeta\,\sqrt{-{\bm{\gamma}}}\,(\bar{\kappa}\Gamma)_{\dot{\beta}}\bar{D}^{\dot{\beta}}L^{\Lambda}\;.\end{split}$
(3.149)

Hence

$\begin{split}\delta S_{\text{string,WZ}}&=2{\rm i}e_{\Lambda}\int_{\cal
S}{}^{*_{2}}\\!E^{a}\wedge
E^{\alpha}(\sigma_{a}\,{\bar{\Gamma}}\bar{\kappa})_{\alpha}{}L^{\Lambda}-2{\rm
i}e_{\Lambda}\int_{\cal S}{}^{*_{2}}\\!E^{a}\wedge
E_{\dot{\alpha}}(\bar{\sigma}_{a}\,{\Gamma}\kappa)^{\dot{\beta}}L^{\Lambda}\\\
&\quad\,-e_{\Lambda}\int_{\cal
S}{\rm{d}}^{2}\zeta\,\sqrt{-{\bm{\gamma}}}\,(\kappa\Gamma)^{\beta}D_{\beta}L^{\Lambda}-e_{\Lambda}\int_{\cal
S}{\rm{d}}^{2}\zeta\,\sqrt{-{\bm{\gamma}}}\,(\bar{\kappa}\bar{\Gamma})_{\dot{\beta}}\bar{D}^{\dot{\beta}}L^{\Lambda}\end{split}$
(3.150)

Then, it is immediate to see that (3.150) cancels (3.143) provided that

$\kappa_{\alpha}=-\frac{e_{\Lambda}L}{|e_{\Lambda}L|}\Gamma_{\alpha}{}^{\beta}\kappa_{\beta}\,,\qquad\bar{\kappa}^{\dot{\alpha}}=-\frac{e_{\Lambda}L}{|e_{\Lambda}L|}\bar{\Gamma}^{\dot{\alpha}}{}_{\dot{\beta}}\bar{\kappa}^{\dot{\beta}}\,.$
(3.151)

#####  3.5.2 Membranes ending on Strings

In Section 3.2 we just considered membranes without any boundary: these could
be, for instance, membranes which are infinitely stretched along three
spacetime direction, as hyperplanes, or closed membranes. This assumption was
crucial, in Section 3.2.1, to demonstrate the $\kappa$-symmetry of the
membrane action (3.39) (see, in particular, (3.49) and (3.48), where this
hypothesis enters). Let us now re-examine the issue, assuming that the
membrane ends on a string, that is

Figure 3.4: A membrane, coupled to the gauge three-forms ${\bf A}_{3}^{A}$,
whose boundary is a BPS–string, coupled to the gauge two-forms ${\bf
B}_{2}^{\Lambda}$.

$\partial{\cal M}={\cal S}$ (3.152)

as depicted in Fig. 3.4. Then, even though we assume the $\kappa$-symmetry
parameter to satisfy (3.38), the membrane action (3.39) is _not_
$\kappa$-symmetric, due to the boundary term

$\delta S_{\text{memb}}|_{\kappa}=q_{A}\int_{\partial{\cal M}}i_{\kappa}{\bf
A}_{3}^{A}\,.$ (3.153)

This lack of variation clearly prevents us to consider such membranes as BPS-
objects. However, this is not the end of the story. Assume that the boundary
is a BPS–string described by an action of the same form as (3.132). The full
variation under $\kappa$–transformations at the junction is then

$(\delta S_{\text{memb}}+\delta S_{\text{string}})|_{\kappa}=\delta
S_{\text{string,NG}}|_{\kappa}+e_{\Lambda}\int_{{\cal S}}i_{\kappa}{\bf
H}_{3}^{\Lambda}+q_{A}\int_{{\cal S}}i_{\kappa}{\bf A}_{3}^{A}$ (3.154)

with $\kappa$ obeying (3.38). It is therefore reasonable to try to cancel the
last membrane contribution by appropriately modifying the string Nambu-Goto
term. In order to see what such a modification has to be, rewrite
$q_{A}=e_{\Lambda}c_{A}^{\Lambda}$, so that (3.154) becomes

$(\delta S_{\text{memb}}+\delta S_{\text{string}})|_{\kappa}=\delta
S_{\text{string,NG}}|_{\kappa}+e_{\Lambda}\int_{{\cal S}}i_{\kappa}\left({\bf
H}_{3}^{\Lambda}+c_{A}^{\Lambda}{\bf A}_{3}^{\Lambda}\right)\,.$ (3.155)

With respect to (3.144) that we encountered in proving the $\kappa$-symmetry
of BPS–strings, here the field strengths ${\bf H}_{3}^{\Lambda}$ are _gauged_
by some combinations of the three-form potentials coupling to the membrane. A
direct inspection of (3.155), recalling the definitions of super-forms (3.35)
and (3.134), shows that the variation (3.155) is zero provided that we modify
the Nambu-Goto term of the BPS-string as

$S_{\text{string,NG}}=-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}\hat{L}^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}\,,$
(3.156)

where the old linear multiplets $L^{\Lambda}$ are replaced by their gauged
versions

$\hat{L}^{\Lambda}\equiv L^{\Lambda}+c_{A}^{\Lambda}P^{A}\,,$ (3.157)

and if the parameters $\kappa$, beside satisfying (3.38), _also_ obey the
projection conditions (3.137). The projections (3.38) and (3.137) are, in
general, two independent conditions that the spinor $\kappa$ has to satisfy.
These both halves the number of the independent parameters, making the
junction a $\frac{1}{4}$–BPS object.

To summarize, the full supersymmetric action of a membrane, coupled to the
bulk scalars $s^{A}$ and charged under the gauge three-forms $A_{3}^{A}$,
whose boundary is a string, charged under the gauge two forms ${\cal
B}_{2}^{\Lambda}$, is

$\begin{split}S_{\text{BPS-junction}}&=-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|e_{\Lambda}c_{A}^{\Lambda}S^{A}|\sqrt{-\det{\bf
h}}+e_{\Lambda}c_{A}^{\Lambda}\int_{\cal M}{\bf A}^{A}_{3}+\\\
&\quad\,-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}\hat{L}^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}+e_{\Lambda}\int_{\cal
S}{\bf B}^{\Lambda}_{2}\;,\end{split}$ (3.158)

with the superfields $S^{A}$ and $\hat{L}^{\Lambda}$ defined in (3.32) and
(3.157).

####  3.6 Spacetime filling 3-branes

Finally, to complete the hierarchy of four-dimensional BPS-objects, let us
consider 3-branes. For the moment, let us assume that these objects fill the
whole spacetime with their worldvolume ${\cal W}$ and do not have boundaries.
We should expect that 3-branes couple to some gauge four-forms $C_{4}^{I}$. As
discussed in Section 2.6, the variant chiral superfields (3.33) are those
which properly contain (real) gauge four-forms in their highest components and
we may use them to construct the super-four-form potentials

$\begin{split}{\bf C}^{I}_{4}&=E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\wedge\bar{E}^{\dot{\beta}}\bar{\sigma}_{ab\;\dot{\alpha}\dot{\beta}}\Gamma^{I}+E^{b}\wedge
E^{a}\wedge E^{\alpha}\wedge
E^{\beta}\sigma_{ab\;\alpha\beta}\bar{\Gamma}^{I}\ \\\
&\quad\,-\frac{1}{6}E^{c}\wedge E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\epsilon_{abcd}\sigma^{d}_{\alpha\dot{\alpha}}D^{\alpha}{\Gamma^{I}}+\frac{1}{6}E^{c}\wedge
E^{b}\wedge E^{a}\wedge
E^{\alpha}\epsilon_{abcd}\sigma^{d}_{\alpha\dot{\alpha}}\bar{D}^{\dot{\alpha}}\bar{\Gamma}^{I}\\\
&\quad\,+\frac{{\rm i}}{96}E^{d}\wedge E^{c}\wedge E^{b}\wedge
E^{a}\epsilon_{abcd}\left(D^{2}\Gamma^{I}-\bar{D}^{2}\bar{\Gamma}^{I}\right)\,.\qquad\end{split}$
(3.159)

Its lowest bosonic component can be immediately computed from (2.173) and is

$\begin{split}{\bf C}^{I}_{4}|&=\frac{1}{4!}(2\,{\rm Im\hskip
1.00006pt}f_{\Gamma}^{I})\varepsilon_{mnpq}{\rm{d}}x^{m}\wedge{\rm{d}}x^{n}\wedge{\rm{d}}x^{p}\wedge{\rm{d}}x^{q}\\\
&=\frac{1}{4!}C^{I}_{mnpq}{\rm{d}}x^{m}\wedge{\rm{d}}x^{n}\wedge{\rm{d}}x^{p}\wedge{\rm{d}}x^{q}=C^{I}_{4}\end{split}$
(3.160)

This is formally similar to the super-field strength (3.33) of three-form
potentials. As such, also ${\bf C}^{I}_{4}$ is closed

${\bf{d}}{\bf C}^{I}_{4}=0\,.$ (3.161)

We can immediately write down a BPS-action for 3-branes as

$S_{\text{3-brane}}=\mu_{I}\int_{\cal W}{\bf C}^{I}_{4}$ (3.162)

where we have not included _any_ Nambu-Goto term. In fact, owing to the
closure of ${\bf C}_{4}^{I}$ and assuming that the 3-brane has no boundary,
the action (3.162) is already $\kappa$–symmetric, the proof of
$\kappa$–symmetry being trivial

$\delta_{\kappa}S_{\text{3-brane}}=\mu_{I}\int_{\cal
W}\left(i_{\kappa}{\bf{d}}{\bf C}^{I}_{4}+{\bf{d}}i_{\kappa}{\bf
C}^{I}_{4}\right)=0\,,$ (3.163)

with no projection condition over the parameters $\kappa$. In other words, we
may directly choose $\kappa$ to be the spacetime supersymmetry parameter and
the 3-brane worldvolume still enjoys the full ${\cal N}=1$ supersymmetry.

#####  3.6.1 3-branes ending on membranes

The (trivial) proof of $\kappa$-symmetry for 3-branes in (3.163) relies on the
fact that 3-branes do not have any boundary. However, let us now assume that
this is not the case and, rather, that the 3-brane ends on a BPS–membrane,
that is $\partial{\cal W}={\cal M}$ (see Fig. 3.5). Then, the variation under
$\kappa$-transformations of the the 3-brane action (3.163) is no more zero,
due to the boundary contribution

$\delta_{\kappa}S_{\text{3-brane}}=-\mu_{I}\int_{\partial{\cal
W}}i_{\kappa}{\bf C}^{I}_{4}=-\mu_{I}\int_{{\cal M}}i_{\kappa}{\bf
C}^{I}_{4}\,.$ (3.164)

Figure 3.5: A 3-brane, coupled to the gauge four-forms ${\bf C}_{4}^{I}$,
whose boundary is a BPS–membrane, coupled to the gauge three-forms ${\bf
A}_{3}^{A}$.

Therefore, over the membrane worldvolume, the full variation of the action is

$\delta S_{\text{memb}}+\delta S_{\text{3-brane}}=\delta
S_{\text{memb,NG}}|_{\kappa}+q_{A}\int_{{\cal M}}i_{\kappa}{\bf
F}_{4}^{A}-\mu_{I}\int_{{\cal M}}i_{\kappa}{\bf C}^{I}_{4}\,.$ (3.165)

We may hope to cancel the last, 3-brane boundary contribution by conveniently
modifying the Nambu-Goto part of the membrane action. In order to achieve
this, let us decompose $\mu_{I}=q_{A}Q_{I}^{A}$, so that the variation reads
(3.165)

$\delta S_{\text{memb}}+\delta S_{\text{3-brane}}=\delta
S_{\text{memb,NG}}|_{\kappa}-q_{A}\int_{{\cal M}}i_{\kappa}\left(-{\bf
F}_{4}^{A}+Q^{A}_{I}{\bf C}_{4}^{I}\right)$ (3.166)

Remarkably, in the last _modified_ Wess-Zumino term appear the gauged four-
forms

$-{\bf F}_{4}^{A}|+Q^{A}_{I}{\bf
C}_{4}^{I}|={\rm{d}}A_{3}^{A}+Q_{A}^{I}C_{4}^{I}$ (3.167)

Recalling the definitions (3.33) and (3.159), from the proof of membrane
$\kappa$-symmetry that we gave in Section 3.2.1, it is immediate to see that
(3.166) vanishes provided that the $\kappa$–parameter does obey (3.38) _and_
the Nambu-Goto part of the membrane action is modified as

$S_{\text{memb,NG}}=-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|q_{A}\hat{S}^{A}|\sqrt{-\det{\bf h}}$ (3.168)

where the chiral three-form multiplets $S^{A}$ are substituted by their gauged
versions

$\hat{S}^{A}\equiv S^{A}-{\rm i}Q^{A}_{I}\Gamma^{I}\,.$ (3.169)

To summarize, the full supersymmetric action of a 3-brane whose boundary is a
membrane, coupled to the bulk scalars $s^{A}$ and charged under the gauge
three-forms $A_{3}^{A}$, is

$\begin{split}S_{\text{BPS-region}}&=-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|q_{A}\hat{S}^{A}|\sqrt{-\det{\bf h}}+q_{A}\int_{\cal
M}{\bf A}^{A}_{3}+q_{A}Q^{A}_{I}\int_{\cal W}{\bf C}^{I}_{4}\end{split}$
(3.170)

with the superfields $\hat{S}^{A}$ defined in (3.169).

Moreover, the action (3.170) is invariant under the combined gauge
transformation:

$\displaystyle{\bf C_{4}}^{I}$ $\displaystyle\quad\rightarrow\quad{\bf
C_{4}}^{I}+{\rm{d}}{\bf\Lambda}^{I}_{3}\,,$ (3.171a) $\displaystyle{\bf
A}^{A}_{3}$ $\displaystyle\quad\rightarrow\quad{\bf
A}^{A}_{3}-q_{A}Q^{A}_{I}{\bf\Lambda}^{I}_{3}\,,$ (3.171b)

where, in analogy with (3.134), ${\bf\Lambda}^{I}_{3}$ is defined as

$\displaystyle{\bf\Lambda}^{I}_{3}=$ $\displaystyle\,{-}2{{\rm i}}E^{a}\wedge
E^{\alpha}\wedge\bar{E}^{\dot{\alpha}}\sigma_{a\alpha\dot{\alpha}}\Xi^{I}$
(3.172) $\displaystyle+E^{b}\wedge E^{a}\wedge
E^{\alpha}\sigma_{ab\;\alpha}{}^{\beta}{D}_{\beta}\Xi^{I}+E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\bar{\sigma}_{ab}{}^{\dot{\beta}}{}_{\dot{\alpha}}\bar{D}_{\dot{\beta}}\Xi^{I}$
$\displaystyle+\frac{1}{24}E^{c}\wedge E^{b}\wedge
E^{a}\epsilon_{abcd}\,\bar{\sigma}{}^{d\dot{\alpha}\alpha}[D_{\alpha},\bar{D}_{\dot{\alpha}}]\Xi^{I}\,.\qquad$

Indeed, (3.171) provides the supersymmetric promotion of the bosonic gauge
transformation foreseen in (1.55).

####  3.7 A hierarchy of objects, a hierarchy of gaugings

We conclude this chapter by providing the complete actions coupling the bulk
theories examined in Chapter 2 with the extended objects just introduced. We
will outline, for each of the cases examined, the physical consequences, for
the bulk theory, of the introduction of the extended objects.

Three-forms coupled to membranes

The complete action describing the coupling of a single membrane (which we
assume to be localized at $z=0$) to gauge three-form is obtained combining
(2.121) with (3.39). We get

$\displaystyle S|_{\rm bos}$
$\displaystyle=-\int_{\Sigma}K_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}$
(3.173)
$\displaystyle\quad\,-\int_{\Sigma}\Big{[}\frac{1}{2}T_{AB}F^{A}_{4}*\\!F^{B}_{4}+T_{AB}\Upsilon^{A}F^{B}_{4}+\Big{(}\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\Big{)}*\\!1\Big{]}$
$\displaystyle\quad\,+\int_{\partial\Sigma}T_{AB}(*F^{A}_{4}+\Upsilon^{A})A^{B}_{3}$
$\displaystyle\quad\,-2\int_{\cal M}{\rm{d}}^{3}\xi\,|q_{A}{\cal
V}^{A}(\varphi)|\sqrt{-\det{\bf h}}+q_{A}\int_{\cal M}A_{3}^{A}\;,$

where $\Sigma$ denotes the four-dimensional spacetime. Integrating out the
gauge three-forms leads to the equations of motion

$T_{AB}(*\\!{F}^{B}_{4}+\Upsilon^{B})=-N_{A}-q_{A}\Theta(z)\;,$ (3.174)

with $N_{A}$ real constants. The resulting on-shell theory is the ordinary
theory

$S|_{\rm
bos}=\int\left[-K_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}-V(\varphi,\bar{\varphi})*1\right]\;.$
(3.175)

where, however, the potential is different on the two sides of the membrane

$\displaystyle
V(\varphi,\bar{\varphi})=\Theta(-z)V_{-}(\varphi,\bar{\varphi})+\Theta(z)V_{+}(\varphi,\bar{\varphi})$
(3.176)

with

$\displaystyle V_{-}(\varphi,\bar{\varphi})$
$\displaystyle=K^{\bar{b}a}(N_{A}{\cal
V}^{A}_{a}(\varphi)+\hat{W}_{a}(\varphi))(N_{B}\bar{\cal
V}^{B}_{\bar{b}}(\bar{\varphi})+\bar{\hat{W}}_{\bar{b}}(\bar{\varphi}))\,,$
(3.177) $\displaystyle V_{+}(\varphi,\bar{\varphi})$
$\displaystyle=K^{\bar{b}a}[(N_{A}+q_{A}){\cal
V}^{A}_{a}(\varphi)+\hat{W}_{a}(\varphi)][(N_{B}+q_{B})\bar{\cal
V}^{B}_{\bar{b}}(\bar{\varphi})+\bar{\hat{W}}_{\bar{b}}(\bar{\varphi})]\,.$

Therefore, the role of a membrane is to make the constants $N_{A}$, which
appear in the superpotential (2.99), jump of an amount given by the charge of
the membrane, as depicted in Fig. 3.6.

Figure 3.6: The presence of a membrane of charges $q_{A}$ under the gauge
three-forms $A_{3}^{A}$ influences the potential, for the constant $N_{A}$
shift making the potential change its shape on the two sides of the membrane.

Three- and four-forms coupled to membranes and 3-branes

Combining (2.199) with (3.170) we can build up the action which couples, at
once, gauge three- and four-forms to membranes and 3-branes, with the latter
having the former as boundaries:

$\displaystyle S|_{\rm bos}$
$\displaystyle=-\int_{\Sigma}K_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}$
(3.178)
$\displaystyle\quad\,-\int_{\Sigma}\Big{[}\frac{1}{2}T_{AB}\hat{F}^{A}_{4}*\\!\hat{F}^{B}_{4}+T_{AB}\Upsilon^{A}\hat{F}^{B}_{4}+\Big{(}\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\Big{)}*\\!1\Big{]}$
$\displaystyle\quad\,+\int_{\partial\Sigma}T_{AB}(*\hat{F}^{A}_{4}+\Upsilon^{A})A^{B}_{3}+\int_{\Sigma}{\cal
Q}_{I}^{\rm bg}C^{I}_{4}$ $\displaystyle\quad\,-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|q_{A}{\cal V}^{A}(\varphi)|\sqrt{-\det{\bf
h}}+q_{A}\int_{\cal M}A^{A}_{3}+q_{A}Q^{A}_{I}\int_{\cal W}C^{I}_{4}\;.$

Again the dual, on-shell chiral theory has the very same form as (3.175) and,
owing to the presence of membranes, the potential is generically different on
the two sides of the membrane. The equations of motion for the gauge three-
forms $A_{3}^{A}$ are as in (3.174); additionally, here, in order to get the
ordinary chiral theory, we also need to integrate out the gauge four-forms
$C_{4}^{I}$, obtaining

$-T_{AB}(*\\!\hat{F}_{4}^{B}+\Upsilon^{B})Q_{I}^{A}+q_{A}Q_{I}^{A}-{\cal
Q}_{I}^{\rm bg}=0\,,$ (3.179)

in all the spacetime. Indeed, the action (3.178) provides the globally
supersymmetric version of the model introduced in Section 1.4.1. The role of
3-branes having membranes as boundaries is to ‘accompany’ the changing of the
tadpole cancellation condition due to the change of the constants $N_{A}$.

Two-forms coupled to strings

In the absence of a superpotential for the chiral fields $\Phi^{a}$, merging
(2.150) with (3.132) we get the full action describing a BPS–string coupled to
a bulk theory where both linear and chiral multiplets are present

$\displaystyle S|_{\rm bos}$
$\displaystyle=\int_{\Sigma}\left(-F_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}+\frac{1}{4}F_{\Lambda\Sigma}{\rm{d}}l^{\Lambda}\wedge*{\rm{d}}l^{\Sigma}+\frac{1}{4}F_{\Lambda\Sigma}{\cal
H}^{\Lambda}_{3}\wedge*{\cal H}^{\Sigma}_{3}\right)$ (3.180)
$\displaystyle\quad\,+\int_{\Sigma}\left(\frac{{\rm
i}}{2}F_{\bar{a}\Sigma}\,{\rm{d}}\bar{\varphi}^{\bar{a}}\wedge{\cal
H}^{\Sigma}_{3}+{\rm c.c.}\right)$ $\displaystyle\quad\,-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}l^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}+e_{\Lambda}\int_{\cal
S}{\cal B}^{\Lambda}_{2}\;.$

In the electro-magnetic dual theory, after transversing a string, the axions
shift with the charge of the string

$a_{\Lambda}\quad\rightarrow\quad a_{\Lambda}+e_{\Lambda}\;.$ (3.181)

Two- and three-forms coupled to a BPS-junction

The full action that governs the coupling of a BPS–junction to a bulk theory
is obtained from (2.170) and (3.158):

$\displaystyle S|_{\rm bos}$
$\displaystyle=\int_{\Sigma}\left(-F_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}+\frac{1}{4}F_{\Lambda\Sigma}{\rm{d}}l^{\Lambda}\wedge*{\rm{d}}l^{\Sigma}+\frac{1}{4}F_{\Lambda\Sigma}\hat{\cal
H}^{\Lambda}_{3}\wedge*\hat{\cal H}^{\Sigma}_{3}\right)$ (3.182)
$\displaystyle\quad\,+\int_{\Sigma}\left(\frac{{\rm
i}}{2}F_{\bar{a}\Sigma}\,{\rm{d}}\bar{\varphi}^{\bar{a}}\wedge\hat{\cal
H}^{\Sigma}_{3}+{\rm c.c.}\right)$
$\displaystyle\quad\,-\int_{\Sigma}\Big{[}\frac{1}{2}T_{AB}F^{A}_{4}*\\!F^{B}_{4}+T_{AB}\Upsilon^{A}F^{B}_{4}+\Big{(}\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\Big{)}*\\!1\Big{]}$
$\displaystyle\quad\,+\int_{\partial\Sigma}T_{AB}(*F^{A}_{4}+\Upsilon^{A})A^{B}_{3}$
$\displaystyle\quad\,-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}l^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}+e_{\Lambda}\int_{\cal
S}{\cal B}^{\Lambda}_{2}$ $\displaystyle\quad\,-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|e_{\Lambda}c^{\Lambda}_{A}{\cal
V}^{A}(\varphi)|\sqrt{-\det{\bf h}}+e_{\Lambda}c^{\Lambda}_{A}\int_{\cal
M}A^{A}_{3}\;.$

Transversing _both_ the string and the membrane, combining the results (3.174)
and (3.181), we get the combined shift

$N_{A}\quad\rightarrow\quad N_{A}+e_{A}c_{A}^{\Lambda}\,,\qquad
a_{\Lambda}\quad\rightarrow\quad a_{\Lambda}+e_{\Lambda}\;.$ (3.183)

In dual, ordinary chiral theory described by the superpotential (2.162)

$W(\Phi;T)=N_{A}{\cal V}^{A}(\Phi)-c^{A}_{\Lambda}T^{\Lambda}{\cal
V}^{A}(\Phi)+\hat{W}(\Phi)$ (3.184)

the shift (3.183) _has no net effect!_ In other words, a BPS–junction so built
connects two equivalent theories sharing the same potential. This is depicted
in Fig. 3.7.

Figure 3.7: Crossing simultaneously a string and a membrane ending on the
string makes the constants $N_{A}$ and the axions $a_{\Lambda}$ to shift so as
the potential is left invariant on the two sides.

Two-, three- and four-forms coupled to a BPS-junction and a BPS-region

As an example of more complete networks that we can build with the objcets
presented in this chapter, we consider the simultaneous presence of a
BPS–junction and a BPS–region within the same theory. Combining (2.199) and
(2.170) with (3.158) and (3.170), we get the full action

$\displaystyle S|_{\rm bos}$
$\displaystyle=\int_{\Sigma}\left(-F_{a\bar{b}}{\rm{d}}\varphi^{a}\wedge*{\rm{d}}\bar{\varphi}^{\bar{b}}+\frac{1}{4}F_{\Lambda\Sigma}{\rm{d}}l^{\Lambda}\wedge*{\rm{d}}l^{\Sigma}+\frac{1}{4}F_{\Lambda\Sigma}\hat{\cal
H}^{\Lambda}_{3}\wedge*\hat{\cal H}^{\Sigma}_{3}\right)$ (3.185)
$\displaystyle\quad\,+\int_{\Sigma}\left(\frac{{\rm
i}}{2}F_{\bar{a}\Sigma}\,{\rm{d}}\bar{\varphi}^{\bar{a}}\wedge\hat{\cal
H}^{\Sigma}_{3}+{\rm c.c.}\right)$
$\displaystyle\quad\,-\int_{\Sigma}\Big{[}\frac{1}{2}T_{AB}\hat{F}^{A}_{4}*\\!\hat{F}^{B}_{4}+T_{AB}\Upsilon^{A}\hat{F}^{B}_{4}+\Big{(}\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\Big{)}*\\!1\Big{]}$
$\displaystyle\quad\,+\int_{\partial\Sigma}T_{AB}(*\hat{F}^{A}_{4}+\Upsilon^{A})A^{B}_{3}+\int_{\Sigma}{\cal
Q}_{I}^{\rm bg}C^{I}_{4}$ $\displaystyle\quad\,-\int_{\cal
S}{\rm{d}}^{2}\zeta\,|e_{\Lambda}l^{\Lambda}|\sqrt{-\det{\bm{\gamma}}}+e_{\Lambda}\int_{\cal
S}{\cal B}^{\Lambda}_{2}$ $\displaystyle\quad\,-2\int_{\cal
M}{\rm{d}}^{3}\xi\,|e_{\Lambda}c^{\Lambda}_{A}{\cal
V}^{A}(\varphi)|\sqrt{-\det{\bf h}}+e_{\Lambda}c^{\Lambda}_{A}\int_{\cal
M}A^{A}_{3}$ $\displaystyle\quad\,-2\int_{{\cal
M}^{\prime}}{\rm{d}}^{3}\xi\,|q_{A}{\cal V}^{A}(\varphi)|\sqrt{-\det{\bf
h}}+q_{A}\int_{{\cal M}^{\prime}}A^{A}_{3}$
$\displaystyle\quad\,+q_{A}Q^{A}_{I}\int_{\cal W}C^{I}_{4}\;.$

The worldvolumes are chosen as

$\partial{\cal M}={\cal S}\,,\qquad\partial{\cal W}={\cal M}^{\prime}\;,$
(3.186)

and the membranes described by the worldvolumes ${\cal M}$ and ${\cal
M}^{\prime}$ span different hypersurfaces (for example, one lozalized at
$z=0$, the other at $z=z_{0}\neq 0$). In fact, a membrane which serves as the
boundary of a 3-brane cannot simultaneously end on a string, otherwise the
latter would be ‘the boundary of a boundary’. Consistency then requires the
compatibility condition

$Q_{I}^{A}c_{A}^{\Lambda}=0$ (3.187)

to hold. In turn, this implies that the gauge three-forms that are gauged by
four-forms as in (2.194) do not simultaneously participate to the gauging of
three-form field strength as in (2.165).

## Hierarchy of forms and extended objects in Supergravity

### Hierarchies of forms and gaugings in supergravity

As of now, we have extensively explored the implementation of $p$–forms into
globally supersymmetric multiplets and how to construct their super-field
strengths, whenever a gauge invariance is present. With such multiplets, we
were also able to couple extended objects to generic ${\cal N}=1$
supersymmetric theories by preserving totally or partially the supersymmetry
of the bulk theory.

This chapter is devoted to the extension of the results of Chapters 2 and 3 to
locally supersymmetric theories. Preliminarily, however, we will briefly
introduce the super-Weyl invariant approach to supergravity [96, 97] which is
further examined in greater details in Appendix D. For our purposes, the
super-Weyl invariant approach has some advantages: still relying on the
superfield formalism, it allows us to write manifestly supersymmetric
invariant action, and the components of super-Weyl invariant Lagrangians, up
to some proper covariantizations, have a striking resemblance to those of
globally supersymmetric Lagrangians. The apparent price to be paid is the
introduction of an unphysical compensator which, as we will see, will indeed
be a useful tool to achieve a canonical normalization of the Einstein-Hilbert
term with a little effort.

Such a formalism will allow us to easily generalize the globally
supersymmetric results to locally supersymmetric theories: we will be able to
introduce gauge three-forms in any given ${\cal N}=1$ supergravity theory to
dynamically generate $F$-term potentials, introduce gauge four-forms to
constrain the constants that we may so generate and couple gauge two-forms,
included in linear multiplets eventually gauged by three-form multiplets. As
in the Chapter 2, we will end up with a now locally supersymmetric theory
where a hierarchy of forms is present. What is developed in this chapter will
serve as the basic framework to study and reformulate effective theories
originating from compactification of string/M-theory.

####  4.1 The super-Weyl invariant approach to Supergravity

Although discussed in details in Appendix D, in order to keep this chapter
self-contained, we briefly summarize the super-Weyl invariant approach to
supergravity. Consider a theory with $n$ chiral multiplets $\Phi^{i}$, whose
components

$\Phi^{i}=\\{\varphi^{i},\chi^{i}_{\alpha},F^{i}_{(\Phi)}\\}$ (4.1)

contain propagating complex scalars $\varphi^{i}$ and Weyl-fermions
$\chi^{i}_{\alpha}$ and the nonpropagating complex scalar fields
$F^{i}_{(\Phi)}$. The promotion of supersymmetry to a local symmetry of the
theory further introduces new degrees of freedom, collected into the
_supergravity multiplet_

${\cal R}=\\{e_{a}^{m},\psi^{m}_{\alpha},G^{a},M\\}$ (4.2)

where $e_{a}^{m}$ is the veilbein encoding the propagating bosonic degrees of
the graviton, $\psi^{m}_{\alpha}$ is the gravitino, the fermionic superpartner
of the graviton, and the nonpropagating fields $G^{a}$ and $M$ are a real
vector and a complex scalar, required for a proper definition of ${\cal R}$ as
an off-shell multiplet. Up to two derivatives, the most general, locally
supersymmetric Lagrangian which describes the coupling of the chiral
multiplets $\Phi^{i}$ to the supergravity multiplet ${\cal R}$ is

${\cal
L}=\int{\rm{d}}^{4}\theta\,E\,\Omega(\Phi,\bar{\Phi})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,W(\Phi)+{\rm c.c.}\right)\,.$ (4.3)

Here we have denoted with $E$ the Berezinian super-determinant of the super-
vielbein $E^{A}_{M}$ and ${\rm{d}}^{2}\Theta\,2{\cal E}$ the chiral superspace
measure, which is the covariantization of $\int{\rm{d}}^{2}\theta$ of global
supersymmetry. The Lagrangian is then fully specified by the _kinetic section_
$\Omega(\Phi,\bar{\Phi})$, which defines the Kähler potential
$K(\Phi,\bar{\Phi})$ as

$\Omega(\Phi,\bar{\Phi})\equiv-3\,e^{-\frac{1}{3}K(\Phi,\bar{\Phi})}\,,$ (4.4)

and the holomorphic _superpotential_ $W(\Phi)$.

In ${\cal N}=1$ supergravity, the chiral fields $\Phi^{i}$ describe a
Hodge–Kähler manifold, namely a Kähler manifold ${\mathscr{M}}_{\rm scalar}$
such that there exists a line bundle
${\mathscr{L}}\rightarrow{\mathscr{M}}_{\rm scalar}$, whose first Chern class
equals the cohomology class of the Kähler form $J$:
$c_{1}({\mathscr{L}})=[J]$. The superpotential $W(\Phi)$, as well as the
chiralini $\chi^{i}_{\alpha}$, the gravitino $\psi^{m}_{\alpha}$ and the
auxiliary fields $F^{i}_{\Phi}$, $M$ are then interpreted as sections of the
line bundle ${\mathscr{L}}$. Then, once a Kähler transformation is performed

$K(\Phi,\bar{\Phi})\rightarrow
K(\Phi,\bar{\Phi})+h(\Phi)+\bar{h}(\bar{\Phi})\,,$ (4.5)

also the superpotential nontrivially transforms

$W(\Phi)\rightarrow e^{-h(\Phi)}W(\Phi)\,,$ (4.6)

along with the fermions and the auxiliary fields. This makes the Kähler
invariance of the Lagrangian (4.3) not manifest. In fact, if we want to
recover its Kähler invariance, we additionally need to exploit _super-Weyl
transformations_. These, which are a superspace generalization of the more
common Weyl transformations, act on the supervielbeins so that the superspace
measures get transformed to

${\rm{d}}^{4}\theta\,E\rightarrow{\rm{d}}^{4}\theta\,E\,e^{-\frac{\Upsilon+\bar{\Upsilon}}{3}}\;,\quad{\rm{d}}^{2}\Theta\,2{\cal
E}\rightarrow{\rm{d}}^{2}\Theta\,2{\cal E}\,e^{\Upsilon}\,,$ (4.7)

with $\Upsilon$ a generic chiral field, compensating (4.5) and (4.6) by
choosing $\Upsilon=h(\Phi)$.

One might wonder whether there exists a smarter way to implement the Kähler
invariance in supergravity Lagrangians of the kind of (4.3), which does not
need the pass through super-Weyl transformations. The answer resides in the
so-called super-Weyl invariant approach, first introduced in [96]. At the core
of the approach is the introduction of an unphysical chiral field $U$, serving
the role of a _compensator_ , which transforms, under super-Weyl
transformations, as

$U\rightarrow e^{\Upsilon}U\;.$ (4.8)

We then introduce a new set of $n+1$ chiral superfields$Z^{a}$,
$a=1,\ldots,n$. These carry a dependence on both the compensator $U$ and the
physical superfields $\Phi^{i}$, which can however be split as

$Z^{a}(U,\Phi)=Ug^{a}(\Phi)$ (4.9)

where $g^{a}(\Phi)$ are holomorphic sections, which depend on the physical
fields only and are inert under super-Weyl transformations. We rewrite (4.3)
as

${\cal L}=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z,\bar{Z})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal W}(Z)+{\rm
c.c.}\right)\,,$ (4.10)

Here, ${\cal K}(Z,\bar{Z})$ and ${\cal W}(Z)$ satisfy the homogeneity
properties

${\cal K}(\lambda Z,\bar{\lambda}\bar{Z})=|\lambda|^{\frac{2}{3}}{\cal
K}({Z},\bar{Z})\;,\qquad{\cal W}(\lambda Z)=\lambda{\cal W}(Z)\;.$ (4.11)

and, extracting the compensator $U$ using (4.9), are then related to the usual
Kähler potential $K(\Phi,\bar{\Phi})$ and superpotential $W(\Phi)$ via

${\cal
K}(Z,\bar{Z})\equiv-3|U|^{\frac{2}{3}}e^{-\frac{K(\Phi,\bar{\Phi})}{3}}\;,\qquad{\cal
W}(Z)\equiv\frac{1}{M_{\rm P}^{3}}UW(\Phi)\;,$ (4.12)

where we have set $K(\Phi,\bar{\Phi})\equiv K(g(\Phi),\bar{g}(\bar{\Phi}))$
and $W(\Phi)\equiv W(g(\Phi))$. It is immediate to check that, using (4.9) and
(4.11), the Lagrangian (4.10) is invariant under the sole super-Weyl
transformations. Moreover, the Kähler invariance is here manifest, in fact
emerging as the residual freedom in the parametrization (4.9). That is, (4.9)
is unaffected by the transformations

$U\rightarrow e^{h(\Phi)}U\,,\qquad g(\Phi)\rightarrow e^{-h(\Phi)}g(\Phi)\,,$
(4.13)

which, applied to (4.12), clearly reproduce the transformations (4.5)–(4.6).

As anticipated above, the super-Weyl invariant approach has also virtues at
the level of the bosonic components. From (4.10) we get the bosonic Lagrangian

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}{\cal K}\,R-{\cal
K}_{a\bar{b}}D_{\mu}z^{a}\bar{D}^{\mu}\bar{z}^{b}+{\cal
K}_{a\bar{b}}f^{a}\bar{f}^{b}+\left(\hat{\cal W}_{a}f^{a}+{\rm
c.c.}\right)\,.\end{split}$ (4.14)

where we have defined $f^{a}\equiv\bar{M}z^{a}-F^{a}_{\cal Z}$ and introduced
the derivatives

$D_{\mu}z^{a}=\partial_{\mu}z^{a}+{\rm i}A_{\mu}z^{a}\,\qquad{\rm with}\quad
A_{\mu}=\frac{3{\rm i}}{2{\cal K}}({\cal K}_{a}\partial_{\mu}z^{b}-{\cal
K}_{\bar{b}}\partial_{\mu}\bar{z}^{b})\,.$ (4.15)

which are covariant with respect to the $U(1)$–bundle associated to
${\mathscr{L}}$.

The bosonic components (4.14) have an astonishingly simple form: beside the
introduction of the covariant derivatives and the curvature term, they are
formally equal to those of rigid superymmetry (see, for example, (2.41))!
However, such a neat appearance is just the result of a hidden redundancy in
the Lagrangian (4.14), that is the dependence on the super-Weyl compensator
$U$. The presence of $U$ can indeed be used in our favor in order to
immediately pass to the Einstein-frame. Concretely, we gauge fix $U$ in such a
way that its lowest bosonic component $u$ is

$u=M_{\rm P}^{3}\,e^{\frac{1}{2}K(\varphi,\bar{\varphi})}\,,$ (4.16)

which sets, for instance, ${\cal K}=-3M^{2}_{\rm P}$ and, as explained in
Section D.2, leads to the usual supergravity Lagrangian

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=\frac{1}{2}M_{\rm P}^{2}R-M_{\rm
P}^{2}K_{i\bar{\jmath}}\partial_{m}\varphi^{i}\partial^{m}\bar{\varphi}^{\bar{\jmath}}-V(\varphi,\bar{\varphi})\;,\end{split}$
(4.17)

where the potential $V(\varphi,\bar{\varphi})$ is given by the usual Cremmer
et al. formula [98]

$V(\varphi,\bar{\varphi})=\frac{e^{K}}{M_{\rm
P}^{2}}\left(K^{\bar{\jmath}i}D_{i}W\bar{D}_{\bar{\jmath}}\bar{W}-3|W|^{2}\right)\;.$
(4.18)

####  4.2 Gauge three-forms and F-term potential

In Chapter 2, across the Sections 2.1 to 2.4, we have seen how to implement
gauge three-forms into globally supersymmetric theories with arbitrary Kähler
potential and superpotential. Presently, we shall see how to generalize those
constructions to supergravity theories by using the super-Weyl invariant
formalism introduced in the previous section. Here, rather than reproducing
each of the cases examined in Chapter 2 separately one by one, we will
directly use the more general recipe given in Section 2.4. Namely, we shall
start with a master Lagrangian with a generic number of chiral multiplets and
three-forms and relate them via master three-form multiplets. We shall however
see how, from this generic viewpoint, the local versions of the models of
Sections 2.3 and 2.2 are retrieved.

Consider a superpotential of the form

${\cal W}(Z)=N_{A}{\cal V}^{A}(Z)+\hat{\cal W}(Z)\;.$ (4.19)

Here $N_{A}$, $A=0,\ldots,N$, are _real_ constants and consistency with the
homogeneity properties (4.11) require the _periods_ ${\cal V}^{A}(Z)$ and the
additional superpotential contribution $\hat{\cal W}(Z)$ to be homogeneous of
degree one. Our goal, as in Section 2.4, is to generate, via gauge three-
forms, a contribution to the potential which generates the same effect as the
superpotential

${\cal W}_{\rm gen}(Z)=N_{A}{\cal V}^{A}(Z)\,.$ (4.20)

By extracting the compensator $U$ as

${\cal V}^{A}(Z)=U{\cal V}^{A}(g(\Phi))\equiv U\Pi^{A}(\Phi)\,,$ (4.21)

with $\Pi^{A}$ holomorphic in $\Phi^{i}$, and gauge fixing the super-Weyl
invariance as in (4.16), the physical superpotential that we are going to
generate via gauge three-forms is

$W_{\rm gen}(\Phi)=M_{\rm P}^{3}N_{A}\Pi^{A}(\Phi)\,.$ (4.22)

The recipe of Section 2.4 can now be applied to the local case by a simple
covariantization. We start with a chiral multiplet theory (4.10), with the
superpotential of the class (4.19). In order to build a master Lagrangian, the
term (4.20) of the superpotential that we are going to generate is promoted to

$N_{A}{\cal V}^{A}(Z)\quad\rightarrow\quad X_{A}{\cal V}^{A}(Z)$ (4.23)

where $X_{A}$ are chiral Lagrange multipliers with components
$\\{x_{A},\psi^{\alpha(X)}_{A},F_{A}^{(X)}\\}$. We then introduce the master
Lagrangian

$\displaystyle{\cal L}_{\rm master}$
$\displaystyle=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z,\bar{Z})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal W}(Z)+{\rm
c.c.}\right)\,$ (4.24)
$\displaystyle\quad\,+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,X_{A}{\cal
V}^{A}(Z)+\frac{{\rm i}}{8}\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(\bar{\cal
D}^{2}-8{\cal R})\left[(X_{A}-\bar{X}_{A})P^{A}\right]+\text{c.c.}\right)\,.$

The last term contains a set real superfields $P^{A}$. As we will see shortly,
as in Section 2.4, the role of the last term is either to set $X_{A}$ to
constants or to provide the correct duality between ordinary and three-form
chiral multiplets. In the following, we will also need the bosonic components
of the master Lagrangian (4.24), which read

$\begin{split}e^{-1}{\cal L}_{\rm master}|_{\text{bos}}&=-\frac{1}{6}{\cal
K}R-{\cal K}_{a\bar{b}}D_{m}z^{a}D^{m}z^{\bar{b}}+{\cal
K}_{a\bar{b}}f^{a}\bar{f}^{\bar{b}}+\left(\hat{\cal W}_{a}f^{a}+{\rm
c.c.}\right)\\\ &\quad\,+\Bigg{[}x_{A}{\cal V}^{A}_{a}f^{a}-\frac{1}{2\cdot
3!\,e}\varepsilon^{mnpq}A^{A}_{npq}\partial_{m}x_{A}-\frac{{\rm
i}}{2}d^{A}x_{A}\\\ &\quad\quad\quad\,+f_{XA}({\cal
V}^{A}-s^{A})+x_{A}\left(\bar{M}{\cal V}^{A}-{\rm Re\hskip
1.00006pt}(\bar{M}s^{A})\right)+{\rm c.c.}\Bigg{]}\,,\end{split}$ (4.25)

which are formally similar to those of (2.102), if we turn off the fields of
the supergravity multiplet. Two paths can be followed which can lead either to
an ordinary formulation (4.10) or to a three-form formulation.

Ordinary formulation

Integrating the real superfields $P^{A}$ from the Lagrangian (4.24)
immediately imposes

$\delta P^{A}:\qquad{\rm Im\hskip 1.00006pt}X_{A}=0\;.$ (4.26)

The chirality of $X_{A}$ further constraints them to be real constants $N_{A}$

$X_{A}=N_{A}\;.$ (4.27)

Once (4.27) is plugged into (4.24), we immediately obtain the Lagrangian
(4.10) with the superpotential (4.19)

${\cal L}=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z,\bar{Z})+\left[\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(N_{A}{\cal
V}^{A}(Z)+{\cal W}(Z))+{\rm c.c.}\right]\,.$ (4.28)

After gauge-fixing the super-Weyl invariance, the superpotential is

$W(\Phi)=M_{\rm P}^{3}N_{A}\Pi^{A}(\Phi)+\hat{W}(\Phi)\,,$ (4.29)

which defines the potential as in (4.17).

The possibility of recovering the ordinary formulation from the master
Lagrangian (4.24) can also be seen at the level of bosonic components.
Integrating out the auxiliary fields $d^{A}$, $f_{XA}$ and the three-forms
$C_{3}^{A}$ from (4.25), we immediately obtain that $x_{A}$ are real
constants.

This proves the consistency of the master Lagrangian with the ordinary
formulation (4.10).

Three-form formulation

In order to get a three-form Lagrangian, we need to integrate out the Lagrange
multipliers $X_{A}$ and (part of) the ordinary chiral fields $Z^{a}$.
Integrating out the Lagrange multipliers $X_{A}$ results in the following
relation between the periods and the real superfields $P^{A}$

${\cal V}^{A}(Z)=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P^{A}\equiv
S^{A}\qquad{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\text{\sf{Master
three-form multiplet}}}$ (4.30)

whose components are

$\displaystyle{\cal V}^{A}|$ $\displaystyle={\cal V}^{A}(z)$ $\displaystyle=$
$\displaystyle\left(-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal
R})P^{A}\right)\Big{|}$ $\displaystyle\equiv$ $\displaystyle\,s^{A}\,,$
(4.31a) $\displaystyle-\frac{1}{4}{\cal D}^{2}{\cal V}^{A}|$
$\displaystyle={\cal V}^{A}_{a}(z)F^{a}_{(Z)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{\cal D}^{2}\left(-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R})P^{A}\right)\Big{|}$ $\displaystyle\equiv$
$\displaystyle\,F^{A}_{(S)}\,,$ (4.31b)

with

$\displaystyle F^{A}_{(S)}=\frac{1}{2}\left(*\\!F_{4}^{A}+{\rm
i}d^{A}\right)+{\rm Re\hskip 1.00006pt}(\bar{M}s^{A})\,.$ (4.32)

Loosely speaking, (4.30), via the component relations (4.31), (4.32) provides
the translation between ordinary chiral fields $Z^{a}$ and three-form
multiplets. As we shall see in the example of the next section, the number of
$Z^{a}$ multiplets which can be promoted to three-form multiplets is
determined by the periods ${\cal V}^{A}(Z)$. For this reason, we shall call
$S^{A}$ _master three-form multiplets_.

Integrating out the chiral superfields $\Phi^{a}$ gives

${\cal V}^{A}_{a}X_{A}=\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R}){\cal
K}_{a}-\hat{\cal W}_{a}\,,$ (4.33)

which, plugged back into the master Lagrangian (4.24), gives

$\tilde{\mathcal{L}}=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z(P),\bar{Z}(P))+\left[\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal
W}(Z(P))+{\rm c.c.}\right]+\tilde{\mathcal{L}}_{\rm bd},$ (4.34)

with the boundary terms

$\tilde{\mathcal{L}}_{\rm bd}=-\frac{{\rm
i}}{8}\left[\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(\bar{\cal D}^{2}-8{\cal
R})-\int{\rm{d}}^{2}\bar{\Theta}\,2\bar{\cal E}\,({\cal D}^{2}-8\bar{\cal
R})\right]\left(X_{A}P^{A}\right)+\text{c.c.}\,,$ (4.35)

with $X_{A}$ defined by (4.33). The superspace Lagrangian (4.34) is then
specified in terms of the superfields $Z^{a}$, some of which do depend on
$P^{A}$ through (4.30) and their highest components contain gauge three-forms.
At first sight, it is clear that the Lagrangian (4.34), due to the nontrivial
dependence on the real potentials $P^{A}$, does not fit well to compute the
bosonic components of generic models containing an arbitrary number of gauge
three-forms. In order to more easily extract the bosonic components of the
three-form Lagrangian, it is better to start from the master Lagrangian
(4.24), with components (4.25). Rather then integrating whole superfields at
once, as done above, we integrate out the components one by one. First,
integrating out the auxiliary fields of the Lagrange multipliers $X_{A}$
immediately gives

$\delta\,f_{XA}:\qquad{\cal V}^{A}(\varphi)=s^{A}\,,$ (4.36)

compatibly with (4.31a); further integrating $d^{A}$ results in setting
$x_{A}$ to be real Lagrange multipliers

$\delta\,d^{A}:\qquad{\rm Im\hskip 1.00006pt}x_{A}=0\,.$ (4.37)

Up to this point, the master Lagrangian becomes

$\begin{split}e^{-1}{\cal L}_{\rm master}|_{\text{bos}}&=-\frac{1}{6}{\cal
K}R-{\cal K}_{a\bar{b}}D_{m}\varphi^{a}D^{m}\varphi^{\bar{b}}+{\cal
K}_{a\bar{b}}f^{a}\bar{f}^{\bar{b}}+\left(\hat{\cal W}_{a}f^{a}+{\rm
c.c.}\right)\\\ &\quad\,+\Bigg{[}x_{A}{\cal V}^{A}_{a}f^{a}-\frac{1}{2\cdot
3!\,e}\varepsilon^{mnpq}A^{A}_{npq}\partial_{m}x_{A}+{\rm
c.c.}\Bigg{]}\,.\end{split}$ (4.38)

Let us now proceed by integrating out the fields $f^{a}$

$\delta\,f^{a}:\qquad\bar{f}^{\bar{b}}=-{\cal K}^{\bar{b}a}(x_{A}{\cal
V}^{A}_{a}+\hat{\cal W}_{a})\,,$ (4.39)

which recast the Lagrangian (4.38) in the simple form

$\begin{split}e^{-1}{\cal L}_{\rm master}|_{\text{bos}}&=-\frac{1}{6}{\cal
K}R-{\cal K}_{a\bar{b}}D_{m}\varphi^{a}D^{m}\bar{\varphi}^{\bar{b}}-{\cal
K}^{\bar{b}a}\left(\hat{\cal W}_{a}+x_{A}{\cal
V}^{A}_{a}\right)\left(\bar{\hat{\cal W}}_{\bar{b}}+x_{B}\bar{\cal
V}^{B}_{\bar{b}}\right)\\\ &\quad\,+\left[-\frac{1}{2\cdot
3!e}\varepsilon^{mnpq}A^{A}_{npq}\partial_{m}x_{A}+{\rm
c.c.}\right]\;.\end{split}$ (4.40)

From this Lagrangian, it is then easy to get a three-form Lagrangian. It is
just sufficient to integrate out the now–real Lagrange multipliers $x_{A}$

$\delta\,x_{A}:\qquad x_{A}=-T_{AB}(*F_{4}^{B}+\Upsilon^{B})\,,$ (4.41)

where we have defined

$\displaystyle T^{AB}$ $\displaystyle=2\,{\rm Re\hskip
1.00006pt}\left(K^{\bar{b}a}{\cal V}^{A}_{a}\bar{\cal
V}^{\bar{B}}_{\bar{b}}\right)\,,$ (4.42a) $\displaystyle\Upsilon^{A}$
$\displaystyle\equiv 2\,{\rm Re\hskip 1.00006pt}\left(K^{\bar{b}a}{\cal
V}^{A}_{a}\bar{\hat{\cal W}}_{\bar{b}}\right)\,,$ (4.42b)

and $T_{AB}$ is the inverse of $T^{AB}$. Finally, we arrive to the Lagrangian

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=-\frac{1}{6}{\cal K}R-{\cal
K}_{a\bar{b}}D_{m}z^{a}D^{m}\bar{z}^{\bar{b}}+e^{-1}{\cal
L}_{\text{3-form}}|_{\rm bos}\,,\end{split}$ (4.43)

Here we have collected the three-form and potential contributions in
$e^{-1}{\cal L}_{\text{3-form}}|_{\rm bos}$, which reads

$\begin{split}e^{-1}{\cal L}_{\text{3-form}}|_{\rm
bos}&=\frac{1}{2}T_{AB}*\\!F_{4}^{A}*\\!F_{4}^{B}+T_{AB}\Upsilon^{A}*\\!F_{4}^{B}\\\
&\quad\,-\left(\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\right)+e^{-1}{\cal
L}_{\rm bd}\end{split}$ (4.44)

with the boundary terms

$\begin{split}{\cal L}_{\rm
bd}=\frac{1}{3!}\partial_{m}\left[\varepsilon^{mnpq}{A}^{A}_{npq}T_{AB}(*\\!F_{4}^{B}+\Upsilon^{B})\right]\,,\end{split}$
(4.45)

and

$\displaystyle\hat{V}$ $\displaystyle\equiv{\cal K}^{\bar{b}a}\hat{\cal
W}_{a}\bar{\hat{\cal W}}_{\bar{b}}\,.$ (4.46a)

Up to now, the steps that we have followed to obtain a generic three-form
Lagrangian are the same – up to covariantization and some field redefinitions
– to the ones of Section 2.4. But this was expected, since we have worked
within super-Weyl invariant Lagrangians.

In supergravity, however, if we want to get Lagrangians expressed solely in
terms of physical fields, an additional step is required: we need to gauge-fix
the super-Weyl invariance so as to eliminate the dependence on the unphysical
compensator $U$. As explained in the previous section, the gauge-fixing (4.16)
allows us to get a Lagrangian in the Einstein-frame. Employing (4.16), as
explained in more details in Appendix D, the Lagrangian (4.44) becomes

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}M_{\rm
P}^{2}R-M_{\rm
P}^{2}K_{i\bar{\jmath}}\partial_{m}\varphi^{i}\partial^{m}\bar{\varphi}^{\bar{\jmath}}+\frac{1}{2}T_{AB}*\\!F_{4}^{A}*\\!F_{4}^{B}\\\
&\quad\,+T_{AB}\Upsilon^{A}*\\!F_{4}^{B}-\left(\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\right)+e^{-1}\tilde{\cal
L}_{\rm bd}\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=\frac{1}{3!}\partial_{m}\left[\varepsilon^{mnpq}{A}^{A}_{npq}T_{AB}(*\\!F_{4}^{B}+\Upsilon^{B})\right]\,,\end{split}$
(4.47)

where now the three-form part of the Lagrangian is also re-expressed in terms
of the physical quantities $K(\Phi,\bar{\Phi})$, $\Pi^{A}(\Phi)$ and
$\hat{W}(\Phi)$ as

$\displaystyle T^{AB}$ $\displaystyle=2M^{4}_{\rm P}\,e^{K}{\rm Re\hskip
1.00006pt}\left(K^{\bar{\jmath}i}D_{i}\Pi^{A}\bar{D}_{\bar{\jmath}}\bar{\Pi}^{B}-3\Pi^{A}\bar{\Pi}^{B}\right)\,,$
(4.48a) $\displaystyle\Upsilon^{A}$ $\displaystyle=2M_{\rm P}\,e^{K}{\rm
Re\hskip
1.00006pt}\left(K^{\bar{\jmath}i}D_{i}\hat{W}\bar{D}_{\bar{\jmath}}\bar{\Pi}^{A}-3\hat{W}\bar{\Pi}^{A}\right)\,,$
(4.48b) $\displaystyle\hat{V}$ $\displaystyle=\frac{e^{K}}{M^{2}_{\rm
P}}\left(K^{\bar{\jmath}i}D_{i}\hat{W}\bar{D}_{\bar{\jmath}}\overline{\hat{W}}-3|\hat{W}|^{2}\right)\,.$
(4.48c)

It is immediate to see, as a consistency check, that integrating out the gauge
three-forms as

$\delta A_{3}^{A}:\qquad T_{AB}(*\\!\hat{F}_{4}^{B}+\Upsilon^{B})=-N_{A}\,,$
(4.49)

reduces (4.47) to (4.17) with the superpotential (4.29).

#####  4.2.1 Relevant examples

Before moving on and see how gauge four- and two-forms may be implemented into
supergravity theories, let us pause for a moment to examine some relevant
examples where the previous discussion finds concrete applications. Some of
the models here presented will naturally emerge in EFT originating from
string/M-theory compactifications. As a summary, the examples here examined
are collected in Table 4.1.

Scalars | Three-forms | $\bm{W_{\rm gen}}$ | Superfields |  $\bm{\tilde{{\cal L}}}$
---|---|---|---|---
$1$ | $1$ | $\lambda$ | $S=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P$ | (4.55)
$1$ | $2$ | $\omega$ | $S=-\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R})\bar{\Sigma}$ | (4.65)
$n+1$ | $n+1$ | $r_{0}+r_{i}\Phi^{i}$ | $S^{a}=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P^{a}$ | (4.71)
$n+1$ | $2(n+1)$ | $e_{0}+e_{i}\Phi^{i}+m^{a}{\cal G}_{a}(\Phi)$ | $S^{a}=\frac{{\rm i}}{2}(\bar{\cal D}^{2}-8{\cal R})({\cal M}^{ab}{\rm Im\hskip 0.90005pt}\Sigma_{b})$ | (4.81)
Table 4.1: The relevant examples of supergravity models where (part of) the
potential is dynamically generated by gauge three-forms. The number of scalars
include the super-Weyl compensator.

###### Minimal single three-form Supergravity

As stressed in the introductory Chapter 1, one of the main reasons why gauge
three-forms were first considered into any effective theory was to generate a
cosmological constant, eventually dynamically neutralized by subsequent
membrane nucleation. One of the first appearances of gauge three-forms in
supegravity was to find out similar models in contexts where supersymmetry is
preserved. For instance, in [99, 29, 30, 28, 31, 32, 47, 100, 101, 102], the
cosmological constant is understood as originating from a real gauge three-
form; in the later [61, 103] is inserted into the supergravity multiplet by
preserving supersymmetry. Here, we rephrase that simple model in our super-
Weyl invariant approach.

Our aim is to dynamically generate a field–independent cosmological constant,
namely we wish to generate a potential $W_{\rm gen}=M_{\rm P}^{3}\lambda$,
with $\lambda$ a real, dimensionless constant. We then assume that no physical
fields $\Phi^{i}$ are present and we set $K(\Phi,\bar{\Phi})=0$ and
$\hat{W}(\Phi)=0$. In the super-Weyl invariant approach, such requests
translate into setting

${\cal K}(U,\bar{U})=-3|U|^{\frac{2}{3}}\,,\qquad\hat{\cal W}(U)=0\,,$ (4.50)

and, since we are going to generate just a single constant, there is only one
period, which, by homogeneity, can be chosen to be just

${\cal V}(U)=U\,.$ (4.51)

Generating a single real constant $\lambda$ requires only one real three-form
and we need to consider only a single real prepotential $P$. Then, the super-
Weyl invariant Lagrangian (4.34) simply becomes

$\tilde{\cal L}=\int{\rm{d}}^{4}\theta\,E\,|S|^{\frac{2}{3}}+\tilde{\cal
L}_{\rm bd}$ (4.52)

with $S$ given by

$S=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P$ (4.53)

and the boundary terms are given by (4.35) with $X=-\frac{1}{4}(\bar{\cal
D}^{2}-8{\cal R})U^{-\frac{2}{3}}\bar{U}^{\frac{1}{3}}$. In components, from
(4.44), we immediately get

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm
bos}&=\frac{1}{2}{|u|^{\frac{2}{3}}}R+\frac{1}{3|u|^{\frac{4}{3}}}\partial_{m}u\partial^{m}\bar{u}-\frac{1}{12|u|^{\frac{4}{3}}}(*\\!F_{4})^{2}+e^{-1}\tilde{\cal
L}_{\rm bd}\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=-\partial_{m}\left(\frac{1}{6\cdot
3!|u|^{\frac{4}{3}}}\varepsilon^{mnpq}{A}_{npq}*\\!F_{4}\right)\,.\end{split}$
(4.54)

Gauge fixing $u=M_{\rm P}^{3}$ as in (4.16), we arrive at

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}{M_{\rm
P}^{2}}R-\frac{1}{12M_{\rm P}^{4}}(*\\!F_{4})^{2}+e^{-1}\tilde{\cal L}_{\rm
bd}\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=-\partial_{m}\left(\frac{1}{6\cdot 3!M_{\rm
P}^{4}}\varepsilon^{mnpq}{A}_{npq}*\\!F_{4}\right)\,.\end{split}$ (4.55)

as can be also directly computed from (4.47), considering just a single period
$\Pi=1$.

At the superfield level, the action (4.55) is described by just one
superfield, namely the supergravity one (4.2), where, however, the imaginary
part of the complex auxiliary field $M$ has been replaced with the real gauge
three-form $A_{3}$:

${\cal R}_{\rm single}=\\{e_{a}^{m},\psi^{m}_{\alpha},G^{a},{\rm Re\hskip
1.00006pt}M,*F_{4}\\}\,,$ (4.56)

Its action is

${\cal L}_{\rm single}=\left(-3\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
R}_{\rm single}+{\rm c.c.}\right)+\tilde{\cal L}_{\rm bd}$ (4.57)

In this sense, (4.55) is a _variant_ minimal supergravity, which we call
_single three-form supergravity_ [61, 6].

By integrating out the gauge three-form $A_{3}$, we immediately get

$*F_{4}=6M_{\rm P}^{4}\lambda\,,$ (4.58)

whence, on-shell, (4.55) reduces to

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}{M_{\rm
P}^{2}}R+3M_{\rm P}^{4}\lambda^{2}\,,\qquad V=-3M_{\rm
P}^{4}\lambda^{2}\end{split}$ (4.59)

with a constant, negative potential.

###### Minimal double three-form Supergravity

A slight generalization of the previous example consists in generating a
cosmological constant term which, although still negative definite, depends on
a complex parameter $\omega\equiv\lambda_{1}+{\rm i}\lambda_{2}$ (with
$\lambda_{1}$ and $\lambda_{2}$ real). Namely, our goal is to generate, with
the help of two real gauge three-forms, a superpotential $W_{\rm gen}=M_{\rm
P}^{3}\omega$. In super-Weyl invariant language, this means that in (4.20) we
need two periods

${\cal W}(U)=N_{A}{\cal V}^{A}(U)\qquad{\rm with}\qquad
N_{A}\equiv\begin{pmatrix}\lambda_{1}\\\
\lambda_{2}\end{pmatrix}\,,\qquad{\cal V}^{A}(U)\equiv U\begin{pmatrix}1\\\
{\rm i}\end{pmatrix}\,.$ (4.60)

We still work with the assumptions that no matter fields are present, so that,
in the super-Weyl invariant approach, (4.50) still holds.

In order to regard ${\cal W}(U)$ as generated by gauge three-forms, we need to
introduce two real potentials, $P^{A}=({\cal P},\tilde{\cal P})^{t}$, to
generate the real constants $\lambda_{1}$ and $\lambda_{2}$, building the
complex $\omega$. The trading prescribed by (4.30) then reads

$U=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R}){\cal P}\,,\qquad{\rm
i}U=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})\tilde{\cal P}\,.$ (4.61)

These two conditions may be rearranged as

$(\bar{\cal D}^{2}-8{\cal R})({\cal P}-{\rm i}\tilde{\cal P})=0\,,$ (4.62)

which is solved by setting

${\rm i}{\cal P}+\tilde{\cal P}=-2\Sigma\,,$ (4.63)

with $\Sigma$ a complex linear multiplet. Then, $U$ is fully determined by the
complex linear multiplet as

$U=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})(-2{\rm Im\hskip
1.00006pt}\Sigma)=-\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R})\Sigma\equiv S\,,$
(4.64)

which provides the Supergravity generalization of the double three-form
multiplet (2.24).

In the super-Weyl invariant approach, the superspace Lagrangian has exactly
the same form as (4.52), where now $S$ is given by (4.64). After gauge-fixing
the super-Weyl invariance as in (4.16), we arrive at the Lagrangian

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}{M_{\rm
P}^{2}}R-\frac{1}{12M_{\rm P}^{4}}|*\\!G_{4}|^{2}+e^{-1}\tilde{\cal L}_{\rm
bd}\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=-\partial_{m}\left(\frac{1}{6\cdot 3!M_{\rm
P}^{4}}\varepsilon^{mnpq}{C}_{npq}*\\!G_{4}+{\rm c.c.}\right)\,,\end{split}$
(4.65)

where $G_{4}={\rm{d}}C_{3}$ is the field strength of the complex three-form
entering the components of $\Sigma$ as in (C.10). At the superfield level, the
action (4.65) is described by just the supergravity multiplet (4.2), where now
the full complex auxiliary field $M$ has been traded with the complex gauge
three-form $C_{3}$:

${\cal R}_{\rm double}=\\{e_{a}^{m},\psi^{m}_{\alpha},G^{a},*G_{4}\\}\,,$
(4.66)

Its action is

${\cal L}_{\rm double}=\left(-3\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
R}_{\rm double}+{\rm c.c.}\right)+\tilde{\cal L}_{\rm bd}$ (4.67)

In this sense, (4.55) is another _variant_ version of the minimal
supergravity, which we call _double three-form supergravity_ [6].

Integrating out the complex gauge three-form $C_{3}$ we immediately get

$*G_{4}=6M_{\rm P}^{4}\lambda\,,$ (4.68)

with $\lambda$ a complex constant, whence, on-shell, (4.65) becomes

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}{M_{\rm
P}^{2}}R+3M_{\rm P}^{4}|\omega|^{2}\,,\qquad V=-3M_{\rm
P}^{4}(\lambda_{1}^{2}+\lambda_{2}^{2})\,.\end{split}$ (4.69)

with again a constant, negative potential.

###### Linear superpotential

Consider now the more involved case where a generic number $n+1$ of complex
scalars $z^{a}$, $a=0,\ldots,n$, including the super-Weyl compensator, enter
the superpotential ${\cal W}(z)=N_{A}{\cal V}^{A}(z)$, $A=1,\ldots,N\leq n+1$.
We may assume the matrix ${\cal V}^{A}_{a}(z)$ to have maximal rank $N$. It is
however convenient choose ‘adapted’ coordinates
$z^{a}=(z^{A},\tilde{z}^{\tilde{a}})$ such that ${\cal V}^{A}(z)\equiv z^{A}$
and the homogeneous superpotential generated by the gauge three-forms takes
the form ${\cal W}_{\rm gen}=N_{A}z^{A}$. In this picture, clearly,
$\tilde{z}^{\tilde{a}}$ constitute a spectator sector, with zero super-Weyl
weight.

In the $z^{A}$-sector, we may isolate the compensator $u$ by setting $z^{0}=u$
and $z^{i}=u\varphi^{i}$, with $i=1,\ldots,n$, so that $\Pi^{0}=1$ and
$\Pi^{i}=\varphi^{i}$. In the spectator sector, the fields
$\tilde{z}^{\tilde{a}}$ can be immediately identified with the physical fields
$\varphi^{\tilde{a}}$. We will then collect
$\varphi^{\alpha}=(\varphi^{i},\varphi^{\tilde{a}})$. Then, before
dualization, the superpotential takes the form $W=M^{3}_{\rm
P}(N_{0}+N_{i}\varphi^{i})$. For instance, we will encounter this kind of
superpotential when we will discuss M-theory compactifications on
$G_{2}$-holonomy spaces in Section 6.4.

In the dual three-forms picture, where $N_{A}$ are promoted to dynamical
variables, we then introduce $N$ gauge three-forms $A^{A}_{3}$ included in the
same number of real multiplets $P^{A}$. Then, the constraint (4.30) reads

${\cal V}^{A}(Z)=Z^{A}=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P^{A}$
(4.70)

The dual Lagrangian is

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}M_{\rm
P}^{2}R-M_{\rm
P}^{2}K_{\alpha\bar{\beta}}\partial_{m}\varphi^{\alpha}\partial^{m}\bar{\varphi}^{\bar{\beta}}+\frac{1}{2}T_{AB}*\\!F_{4}^{A}*\\!F_{4}^{B}\\\
&\quad\,+T_{AB}\Upsilon^{A}*\\!F_{4}^{B}-\left(\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\right)+e^{-1}\tilde{\cal
L}_{\rm bd}\,,\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=\frac{1}{3!}\partial_{m}\left[\varepsilon^{mnpq}{A}^{A}_{npq}T_{AB}(*\\!F_{4}^{B}+\Upsilon^{B})\right]\,.\end{split}$
(4.71)

The three-form kinetic matrix $T_{AB}$ is the inverse of $T^{AB}$ as defined
in (4.48a), This can be most readily computed by observing that
$D_{i}\Pi^{0}=K_{i}$ and $D_{i}\Pi^{j}=\delta_{i}^{j}+\varphi^{i}K_{j}$, which
gives

$\begin{split}T^{AB}_{\rm single}&=\left(\begin{array}[]{cc}T^{00}&T^{0j}\\\
T^{i0}&T^{ij}\end{array}\right)\\\ &=2M^{4}_{\rm P}e^{K}{\rm Re\hskip
1.00006pt}\left(\begin{array}[]{cc}\alpha&K^{\alpha\bar{\jmath}}K_{\alpha}+\alpha\bar{\phi}^{\bar{\jmath}}\\\
K^{i\bar{\beta}}K_{\bar{\beta}}+\alpha\phi^{i}&K^{i{\bar{\jmath}}}+K^{i\bar{\beta}}K_{\bar{\beta}}\bar{\phi}^{\bar{\jmath}}+K^{\alpha\bar{\jmath}}K_{\alpha}\phi^{i}+\alpha\phi^{i}\bar{\phi}^{\bar{\jmath}}\end{array}\right)\end{split}$
(4.72)

where we have defined the quantity

$\alpha=K^{\bar{\beta}\alpha}K_{\bar{\beta}}K_{\alpha}-3\,.$ (4.73)

###### Maximally nonlinear case

We now consider the maximal case, where the number of gauge three-forms $N$
equals the number of _real_ scalars $2n+2$ entering the superpotential ${\cal
W}=N_{A}{\cal V}^{A}(z)$. and we are still assuming ${\cal V}^{A}_{a}$ be of
maximal rank. For the sake of generality, we keep the spectator sector
$\tilde{z}^{\tilde{a}}$, with zero super-Weyl weights. Locally in field space,
we can make a field-redefinition such that

${\cal V}^{A}(z)\equiv\left(\begin{array}[]{c}z^{a}\\\ {\cal
G}_{a}(z)\end{array}\right)\,.$ (4.74)

and conveniently introduce a set of $2(n+1)$ real prepotential

$P^{A}\equiv\left(\begin{array}[]{r}{\cal P}^{a}\\\ \tilde{\cal
P}_{b}\end{array}\right)\,,$ (4.75)

whose $[{\cal D},\bar{\cal D}]$–components are the gauge three-forms
$A_{3}^{A}$ and $\tilde{A}_{3A}$. The constraint (4.30) then splits as
follows:

$\displaystyle Z^{a}$ $\displaystyle=S^{a}\equiv-\frac{{\rm i}}{4}(\bar{\cal
D}^{2}-8{\cal R}){\cal P}^{a}\,,$ (4.76a) $\displaystyle{\cal G}_{a}(Z)$
$\displaystyle=\tilde{S}_{a}=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal
R})\tilde{\cal P}_{a}\,.$ (4.76b)

By substituting the first condition into the second, we get

$(\bar{\cal D}^{2}-8{\cal R})\big{[}{\cal G}_{ab}({\cal Z}){\cal
P}^{b}-\tilde{\cal P}_{a}\big{]}=0\,,$ (4.77)

which can be explicitly solved by setting

${\cal G}_{ab}(Z){\cal P}^{b}-\tilde{\cal P}_{a}\equiv-2\Sigma_{a}\,,$ (4.78)

where $\Sigma_{a}$ are arbitrary complex linear multiplets, obeying the
constraint (C.8). We can then invert (4.78) into

${\cal P}^{a}=-2{\cal M}^{ab}{\rm Im\hskip
1.00006pt}\Sigma_{b}\;,\quad\quad\tilde{\cal P}_{a}=-2{\rm Im\hskip
1.00006pt}(\bar{\cal G}_{ab}{\cal M}^{bc}\Sigma_{c})\;.$ (4.79)

Hence, at the superspace level, the full Lagrangian is given by

$\tilde{\mathcal{L}}=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(S,\bar{S};\tilde{Z},\bar{\tilde{Z}})+\left[\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,\hat{\cal W}(Z;T)+{\rm c.c.}\right]+\tilde{\mathcal{L}}_{\rm bd},$ (4.80)

with the boundary terms properly given by (4.35).

Gauge-fixing the super-Weyl invariance by first setting $z^{0}=u$,
$z^{i}=u\varphi^{i}$ and imposing (4.16), we arrive at the dual Lagrangian

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}M_{\rm
P}^{2}R-M_{\rm
P}^{2}K_{\alpha\bar{\beta}}\partial_{m}\varphi^{\alpha}\partial^{m}\bar{\varphi}^{\bar{\beta}}+\frac{1}{2}T_{AB}*\\!F_{4}^{A}*\\!F_{4}^{B}\\\
&\quad\,+T_{AB}\Upsilon^{A}*\\!F_{4}^{B}-\left(\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\right)+e^{-1}{\cal
L}_{\rm bd}\\\ &{\rm with}\qquad\tilde{\cal L}_{\rm
bd}=\frac{1}{3!}\partial_{m}\left[\varepsilon^{mnpq}{A}^{A}_{npq}T_{AB}(*\\!F_{4}^{B}+\Upsilon^{B})\right]\,,\end{split}$
(4.81)

where now the three-form kinetic matrix $T_{AB}$ is the inverse of the matrix

$\begin{split}T^{AB}&=\left(\begin{array}[]{c c c}&&T^{0}_{b}\\\
<EMAIL_ADDRESS>single}^{ab}$}}\hfil\lx@intercol&T^{i}_{b}\\\
T^{0}_{a}&T^{j}_{a}&T_{ab}\end{array}\right)\,,\end{split}$ (4.82)

with $T_{\rm single}^{ab}$ as given in (4.72) and the other matrix elements

$\displaystyle T^{0}_{a}$ $\displaystyle=2M^{4}_{\rm P}e^{K}{\rm Re\hskip
1.00006pt}\left(K^{\bar{\jmath}\beta}K_{\beta}\bar{\cal
G}_{\bar{a}\bar{\jmath}}+\alpha\bar{\cal G}_{\bar{a}}\right)\;,$ (4.83)
$\displaystyle T^{i}_{a}$ $\displaystyle=2M^{4}_{\rm P}e^{K}{\rm Re\hskip
1.00006pt}\left(K^{\bar{\jmath}i}\bar{\cal
G}_{\bar{a}\bar{\jmath}}+K^{\bar{\beta}i}K_{\bar{\beta}}\bar{\cal
G}_{\bar{a}}\bar{\varphi}^{\bar{\jmath}}+K^{\bar{\jmath}\beta}K_{\beta}\varphi^{i}\bar{\cal
G}_{\bar{a}\bar{\jmath}}+\alpha\varphi^{i}\bar{\cal G}_{\bar{a}}\right)\;,$
$\displaystyle T_{ab}$ $\displaystyle=2M^{4}_{\rm P}e^{K}{\rm Re\hskip
1.00006pt}\left(K^{\bar{\jmath}i}{\cal G}_{ia}\bar{\cal
G}_{\bar{\jmath}\bar{b}}+2K^{\bar{\jmath}\beta}K_{\beta}{\cal G}_{a}\bar{\cal
G}_{\bar{\jmath}\bar{b}}+\alpha{\cal G}_{a}\bar{\cal G}_{\bar{b}}\right)\;.$

####  4.3 Gauge four-forms and constraints

The four-form multiplets introduced in Section 2.6 in global supersymmetry may
be straightforwardly introduced in supergravity as well. Since their
introduction is a simple generalization of the global case, here we will be
briefer, recalling the basic features of such multiplets.

Let us consider a chiral multiplet $\Gamma$, which satisfies the constraint
$\bar{\cal D}_{\dot{\alpha}}\Gamma=0$. In order to implement a gauge four-form
among its components, it is just sufficient to trade – say – the imaginary
part of the auxiliary field $F_{(\Gamma)}$ of $\Gamma$ with the Hodge-dual of
a real four-form $C_{4}$:

${\rm Im\hskip 1.00006pt}F_{(\Gamma)}=\frac{1}{2}*\\!C_{4}\;.$ (4.84)

Hence, the components of a four-form multiplet are

$\displaystyle\Gamma|$ $\displaystyle=\gamma\,,$ (4.85) $\displaystyle{\cal
D}^{\alpha}\Gamma|$ $\displaystyle=\chi^{\alpha}_{(\Gamma)}\,,$
$\displaystyle-\frac{1}{4}{\cal D}^{2}\Gamma|$
$\displaystyle=F_{(\Gamma)}\equiv{\rm Re\hskip
1.00006pt}F_{(\Gamma)}+\frac{{\rm i}}{2}*\\!C_{4}\,.$

Interpreting $\Gamma$ as a super-gauge potential, it has to transform via a
super-gauge transformation. Such a transformation is of the kind

$\Gamma\rightarrow\Gamma+\Xi$ (4.86)

where $\Xi$ is a single three-form multiplet as (4.53), that is

$\Xi=\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R})P$ (4.87)

whose components are

$\displaystyle\Xi|$ $\displaystyle=\left[\frac{1}{4}(\bar{\cal D}^{2}-8{\cal
R})U\right]\bigg{|}$ $\displaystyle\equiv$ $\displaystyle\,\xi\,,$ (4.88)
$\displaystyle{\cal D}^{\alpha}\Xi|$ $\displaystyle={\cal
D}^{\alpha}\left[\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R})U\right]\bigg{|}$
$\displaystyle\equiv$ $\displaystyle\,\chi^{\alpha}\,,$
$\displaystyle-\frac{1}{4}{\cal D}^{2}\Xi|$ $\displaystyle=-\frac{1}{4}{\cal
D}^{2}\left[\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R})U\right]\bigg{|}$
$\displaystyle=$ $\displaystyle\,\frac{1}{2}\left({\rm
i}*\\!{\rm{d}}\Lambda_{3}-d\right)+{\rm i}{\rm Im\hskip
1.00006pt}(\bar{M}\xi)\,.$

It is then clear that a transformation of the kind (4.86) may gauge away all
the components of $\Gamma$, but the gauge four-form, which transforms, as
expected, as

$C_{4}\rightarrow C_{4}+{\rm{d}}\Lambda_{3}\,.$ (4.89)

#####  4.3.1 Implementing constraints via three-form gaugings

Consider a three-form theory as (4.47). Once gauge three-forms are integrated
out as in (4.49), they dynamically generate any real arbitrary constants
$N_{A}$. However, as we shall see in details in the next chapters, in typical
compactification scenarios, the constants $N_{A}$ are indeed constrained. In
particular, assume that they satisfy the _linear_ constraint

$Q_{I}^{A}N_{A}+{\cal Q}_{I}^{\rm bg}=0\,.$ (4.90)

with ${\cal Q}_{I}^{\rm bg}$, $I=1,\ldots,L$, fixed real constants and
$Q_{I}^{A}$ real $L\times(N+1)$ matrices. As already pointed out in Section
2.6.1, such constraints can indeed be implement dynamically into the three-
form Lagrangian (4.47) and it is the peculiar structure of the four-form
multiplets which, regarding them as Lagrange multipliers, serve to set the
constraints (4.90).

Preliminarily, we notice that, also in Supergravity, a term of the form

${\cal L}_{\rm constraint}={\rm i}\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,\left(Q_{I}^{A}N_{A}+{\cal Q}_{I}^{\rm bg}\right)\Gamma^{I}+{\rm c.c.}\,,$
(4.91)

locally supersymmetric generalization of (2.184), where $\Gamma^{I}$ are a set
of $L$ four-form multiplets, serves at the scope of setting the constraint
(4.90) once we integrate out the superfields $\Gamma^{I}$. This is also clear
at the component level, where, due to the fact that we can gauge away all the
components of $\Gamma^{I}$ but the four-forms $C_{4}^{I}$ in their highest
components, (4.91) simply reduces to

$e^{-1}{\cal L}_{\rm constraint}|=-\left(Q_{I}^{A}N_{A}+{\cal Q}_{I}^{\rm
bg}\right)*\\!C_{4}^{I}\,.$ (4.92)

It is then immediate to recognize that integrating out the gauge four-forms
$C_{4}^{I}$ sets the constraints (4.90).

However, the terms (4.91) and (4.92), as they stand, are not fit to be
implemented into the three-form Lagrangians (4.47). There, the fluxes $N_{A}$
are dynamically generated, as such they are not present directly in (4.47) and
it is pointless to constraint them as in (4.90) straightly from the begininng.
Rather, we need a mechanism which is able to generate at once the fluxes
$N_{A}$ along with the constraints (4.90). We therefore promote (4.91) to

${\cal L}_{\rm constraint}={\rm i}\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,\left(Q_{I}^{A}X_{A}+{\cal Q}_{I}^{\rm bg}\right)\Gamma^{I}+{\rm c.c.}\,,$
(4.93)

where $X_{A}$ are Lagrange multipliers. The term (4.93) can then be straightly
added to the master Lagrangian (4.24), resulting in

$\displaystyle{\cal L}_{\rm master}$
$\displaystyle=\int{\rm{d}}^{4}\theta\,E\,{\cal
K}(Z,\bar{Z})+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal W}(Z)+{\rm
c.c.}\right)\,$ (4.94)
$\displaystyle\quad\,+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,X_{A}{\cal
V}^{A}(Z)+\frac{{\rm i}}{8}\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(\bar{\cal
D}^{2}-8{\cal R})\left[(X_{A}-\bar{X}_{A})P^{A}\right]+\text{c.c.}\right)$
$\displaystyle\quad\,+\left[{\rm i}\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,\left(Q_{I}^{A}X_{A}+{\cal Q}_{I}^{\rm bg}\right)\Gamma^{I}+{\rm
c.c.}\right]\;,$

whose components are

$\begin{split}e^{-1}{\cal L}_{\rm master}|_{\text{bos}}&=-\frac{1}{6}{\cal
K}R-{\cal K}_{a\bar{b}}D_{m}\varphi^{a}D^{m}\varphi^{\bar{b}}+{\cal
K}_{a\bar{b}}f^{a}\bar{f}^{\bar{b}}+\left(\hat{\cal W}_{a}f^{a}+{\rm
c.c.}\right)\\\ &\quad\,+\Bigg{[}x_{A}{\cal V}^{A}_{a}f^{a}-\frac{1}{2\cdot
3!\,e}\varepsilon^{mnpq}A^{A}_{npq}\partial_{m}x_{A}-\frac{{\rm
i}}{2}d^{A}x_{A}\\\ &\quad\quad\quad\,+f_{XA}({\cal
V}^{A}-s^{A})+x_{A}\left(\bar{M}{\cal V}^{A}-{\rm Re\hskip
1.00006pt}(\bar{M}s^{A})\right)+{\rm c.c.}\Bigg{]}\\\
&\quad\,-(Q_{I}^{A}\,{\rm Re\hskip 1.00006pt}x_{A}+{\cal Q}_{I}^{\rm
bg})*\\!C_{4}^{I}\;.\end{split}$ (4.95)

Given the master Lagrangian, we may decide to either re-get an ordinary
formulation in terms of chiral multiplets or obtain a formulation where
(gauged) three-forms are present. The steps to get both the formulations
closely follow those of global supsersymmetry of Section 2.6.1 and we will
here be very brief.

Ordinary formulation

Integrating out the real superfields $P^{A}$ and the four-form multiplets
$\Gamma^{I}$ from the master Lagrangian (4.94), we get $X_{A}=N_{A}$, with
$N_{A}$ real constants constrained by (4.90). We therefore go back at (4.28).

Gauged three-form formulation

More interestingly, we may keep the potentials $P^{A}$ and $\Gamma^{I}$ and
rather integrate out the Lagrange multipliers $X_{A}$ and the ordinary chiral
superfields $Z^{a}$. The latter integration reproduces (4.33), while the
former gives

${\cal V}^{A}(Z)=-\frac{{\rm i}}{4}(\bar{\cal D}^{2}-8{\cal R})P^{A}-{\rm
i}Q_{I}^{A}\Gamma^{I}\equiv\hat{S}^{A}\qquad\parbox{100.00015pt}{\color[rgb]{0.5703125,0.0234375,0.0234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.5703125,0.0234375,0.0234375}\sf{Gauged
master \\\ three-form multiplet}}$ (4.96)

which are the master three-form multiplets introduced in (4.30), augmented
with a gauging of the three-forms. their components differ form (4.31) only
for (4.32), which here becomes

$\displaystyle F^{A}_{(S)}=\frac{1}{2}\left(*\\!\hat{F}_{4}^{A}+{\rm
i}d^{A}\right)+{\rm Re\hskip 1.00006pt}(\bar{M}s^{A})\,,$ (4.97)

where we have defined the gauged four-form field strengths

$\hat{F}_{4}^{A}\equiv{\rm{d}}A_{3}^{A}+Q_{I}^{A}C_{4}^{I}\,.$ (4.98)

In order to recover the full bosonic Lagrangian, it is convenient to work with
the bosonic components of the master Lagrangian (4.95). By closely following
the same steps as in Section 2.6.1, after gauge-fixing the super-Weyl
invariance as in (4.16), we get

$\begin{split}e^{-1}\tilde{\cal L}|_{\rm bos}&=\frac{1}{2}M_{\rm
P}^{2}R-M_{\rm
P}^{2}K_{i\bar{\jmath}}\partial_{m}\varphi^{i}\partial^{m}\bar{\varphi}^{\bar{\jmath}}+\frac{1}{2}T_{AB}*\\!\hat{F}_{4}^{A}*\\!\hat{F}_{4}^{B}\\\
&\quad\,+T_{AB}\Upsilon^{A}*\\!\hat{F}_{4}^{B}-\left(\hat{V}-\frac{1}{2}T_{AB}\Upsilon^{A}\Upsilon^{B}\right)-{\cal
Q}_{I}^{\rm bg}*\\!C_{4}^{I}+e^{-1}{\cal L}_{\rm bd}\\\ &{\rm
with}\qquad\tilde{\cal L}_{\rm
bd}=\frac{1}{3!}\partial_{m}\left[\varepsilon^{mnpq}{A}^{A}_{npq}T_{AB}(*\\!\hat{F}_{4}^{B}+\Upsilon^{B})\right]\,,\end{split}$
(4.99)

where we have employed the same definitions as in (4.48). It can be easily
shown, in complete analogy with the global case (2.200)-(2.201), that
integrating out the gauge three- and four-forms we indeed get the usual
Cremmer et al. potential, specified by the superpotential (4.29), where the
constants $N_{A}$ generated by the gauge three-forms are constrained by
(4.90).

####  4.4 Gauge two-forms and axions

Consider a theory where, among the matter multiplets, we can single out an
axionic sector made up by the chiral multiplets $T_{\Lambda}$. Explicitly, the
kinetic part of the Lagrangian reads

${\cal L}_{\rm chiral}=\int{\rm{d}}^{4}\theta\,E\,{\cal K}(Z,\bar{Z};{\rm
Im\hskip 1.00006pt}T)\;,$ (4.100)

which depends on the sector $T_{\Lambda}$ only through their imaginary parts,
while keeping a generic dependence on the other chiral multiplets $Z^{a}$,
including the super-Weyl compensator. Consistency of super-Weyl transformation
with the chirality of $T_{\Lambda}$ requires the axionic multiplets
$T_{\Lambda}$ to carry zero super-Weyl weight and the homogeneity properties
of the kinetic section are

${\cal K}(\lambda Z,\bar{\lambda}\bar{Z};{\rm Im\hskip
1.00006pt}T)=|\lambda|^{\frac{2}{3}}{\cal K}(Z,\bar{Z};{\rm Im\hskip
1.00006pt}T)\,.$ (4.101)

As seen in global supersymmetry in Section 2.5, there exists a dual
description where the axionic multiplets are traded with linear multiplets
$L^{\Lambda}$. In supergravity, linear multiplets satisfy the covariant
constraints

$({\cal D}^{2}-8\bar{\cal R})L^{\Lambda}=0\,,\qquad(\bar{\cal D}^{2}-8{\cal
R})L^{\Lambda}=0\,,$ (4.102)

and their bosonic components are a real scalar field $l^{\Lambda}$ and a real
field strength ${\cal H}_{3}^{\Lambda}$ of a gauge two form ${\cal
B}_{2}^{\Lambda}$, which can be regarded as the electro-magnetic dual of the
axions ${\rm Re\hskip 1.00006pt}T_{\Lambda}|=a_{\Lambda}$. The kinetic
Lagrangian (4.100) gets replaced by

${\cal L}_{\rm linear}=\int{\rm{d}}^{4}\theta\,E\,{\cal F}(Z,\bar{Z};L)\,,$
(4.103)

where ${\cal F}(Z,\bar{Z};L)$, similarly to $F(\Phi,\bar{\Phi};L)$ in (2.148)
for the global case, is the Legendre transform of the kinetic function ${\cal
K}(Z,\bar{Z};{\rm Im\hskip 1.00006pt}T)$. Unlike their chiral counterparts,
the linear multiplets $L^{\Lambda}$ are required to have super-Weyl weights
$(\frac{1}{3},\frac{1}{3})$, whence the homogeneity property of the kinetic
function are

${\cal F}(\lambda
Z,\bar{\lambda}\bar{Z};|\lambda|^{\frac{2}{3}}L)=|\lambda|^{\frac{2}{3}}{\cal
F}(Z,\bar{Z};L)\,.$ (4.104)

In the following, after introducing a first order formalism which relates the
Lagrangians (4.100) and (4.103), we shall see how linear multiplets can be
gauged by the three-form multiplets introduced in Section 4.2, along the lines
of the global case examined in Section 2.5.1.

#####  4.4.1 Axion/two-form duality in supergravity

In order to establish a connection between the ordinary formulation in terms
of chiral multiplets with a dual formulation where the axionic multiplets are
replaced with linear multiplets, we consider the super-Weyl invariant
Lagrangian

${\cal L}_{\rm dual}=\int{\rm{d}}^{4}\theta\,E\,{\cal
F}(Z,\bar{Z};L)-2\int{\rm{d}}^{4}\theta\,E\,L^{\Lambda}{\rm Im\hskip
1.00006pt}T_{\Lambda}\,.$ (4.105)

where here $L$ are generic real multiplets. The two pictures are related as
follows.

Ordinary chiral formulation

The formulation in terms of the multiplets $T_{\Lambda}$ is obtained by
integrating out the real multiplets $L^{\Lambda}$ from (4.105) as

$\delta L^{\Lambda}\,:\qquad{\rm Im\hskip
1.00006pt}T_{\Lambda}=\frac{1}{2}\frac{\partial{\cal F}}{\partial
L^{\Lambda}}\,.$ (4.106)

Substituting this relation in (4.105) gives

$\displaystyle{\cal L}_{\rm dual}$
$\displaystyle=\int{\rm{d}}^{4}\theta\,E\,\left({\cal
F}(Z,\bar{Z};L)-\frac{\partial{\cal F}}{\partial
L^{\Lambda}}L^{\Lambda}\right)$ (4.107)
$\displaystyle=\int{\rm{d}}^{4}\theta\,E\,{\cal K}(Z,\bar{Z};{\rm Im\hskip
1.00006pt}T)={\cal L}_{\rm chiral}\,.$

The purely chiral multiplet formulation is then obtained requiring that the
kinetic functions ${\cal K}(\Phi,\bar{\Phi};{\rm Im\hskip 1.00006pt}T)$ and
${\cal F}(\Phi,\bar{\Phi};L)$ are related by a Legendre transformation.

Formulation with linear multiplets

In order to integrate out the chiral multiplets $T_{\Lambda}$, we first need
to solve the chirality constraints ${\cal D}_{\alpha}T_{\Lambda}=0$,
$\bar{\cal D}^{\dot{\alpha}}\bar{T}_{\Lambda}=0$ and re-express them as

${\rm Im\hskip 1.00006pt}T_{\Lambda}=\frac{1}{2{\rm i}}\left[(\bar{\cal
D}^{2}-8{\cal R})\bar{\Xi}_{\Lambda}-({\cal D}^{2}-8\bar{\cal
R})\Xi_{\Lambda}\right]$ (4.108)

with $\Xi_{\Lambda}$ unconstrained (complex) superfields. Varying (4.105) with
repect to $\Xi_{\Lambda}$, we get the constraints

$\delta\Xi_{\Lambda}\,,\delta\bar{\Xi}_{\Lambda}\,:\qquad({\cal
D}^{2}-8\bar{\cal R})L^{\Lambda}=0\,,\quad(\bar{\cal D}^{2}-8{\cal
R})L^{\Lambda}=0$ (4.109)

which tell that $L^{\Lambda}$ are linear multiplets, retrieving (4.103).

In the super-Weyl invariant formalism, the bosonic components of the
Lagrangian (4.103) are

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal F}\,R-{\cal
F}_{a\bar{b}}Dz^{a}\bar{D}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}+\frac{1}{4\cdot
3!}{\cal F}_{\Lambda\Sigma}{\cal H}_{\mu\nu\rho}^{\Lambda}{\cal
H}^{\Sigma\mu\nu\rho}\\\ &\quad\,+\left(\frac{{\rm i}}{2\cdot 3!}{\cal
F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}{\cal
H}^{\Sigma}_{\nu\rho\sigma}\bar{D}_{\mu}\bar{z}^{\bar{a}}+{\rm
c.c.}\right)\;.\end{split}$ (4.110)

Before gauge fixing the super-Weyl invariance, we introduce the Legendre
transform of the Kähler potential

$F(\varphi,\bar{\varphi};\ell)=K+2\ell^{\Lambda}{\rm Im\hskip
1.00006pt}t_{\Lambda}\,,$ (4.111)

where we have introduced the variables

$\ell^{\Lambda}\equiv-\frac{1}{2}\frac{\partial K}{\partial{\rm Im\hskip
1.00006pt}t_{\Lambda}}\,,$ (4.112)

related to the lowest components of the linear multiplets $l^{\Lambda}$ as

$l^{\Lambda}=M^{2}_{\rm P}\,\ell^{\Lambda}\,.$ (4.113)

In (4.110), we may then gauge fix the super-Weyl invariance by setting

$u=M_{\rm P}^{3}\,e^{\frac{1}{2}\tilde{F}(\phi,\bar{\phi})}\,,\quad{\rm
with}\quad\tilde{F}=F-\ell^{\Lambda}F_{\Lambda}\,,$ (4.114)

which is indeed analogous to (4.16), because of the identification
$\tilde{F}(\varphi,\bar{\varphi};\ell)=K(\varphi,\bar{\varphi};{\rm Im\hskip
1.00006pt}t(\ell))$. However, we stress that gauge-fixing the super-Weyl
invariance in the presence of linear multiplets is a quite involved procedure
[10], which is discussed in details in Appendix D. The bosonic components of
the gauge-fixed Lagrangian reassemble into a rather simple form

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=\frac{M_{\rm P}^{2}}{2}R-M_{\rm
P}^{2}F_{i\bar{\jmath}}\partial\phi^{i}\bar{\partial}\bar{\phi}^{\bar{\jmath}}+\frac{M_{\rm
P}^{2}}{4}F_{\Lambda\Sigma}\partial_{\mu}\ell^{\Lambda}\partial^{\mu}\ell^{\Sigma}\\\
&\quad\,+\frac{1}{4\cdot 3!M_{\rm P}^{2}}{\cal H}_{\mu\nu\rho}^{\Lambda}{\cal
H}^{\Sigma\mu\nu\rho}+\left\\{\frac{{\rm i}}{2\cdot
3!}F_{\bar{\imath}\Sigma}\varepsilon^{\mu\nu\rho\sigma}{\cal
H}^{\Sigma}_{\nu\rho\sigma}\partial_{\mu}\bar{\phi}^{\bar{\imath}}+{\rm
c.c.}\right\\}\,.\end{split}$ (4.115)

#####  4.4.2 Gauged linear multiplets and F-term couplings

Linear multiplets do not only provide an alternative description for axionic
multiplets, but they can also dynamically generate F-term couplings between
the axionic multiplets and another chiral multiplet sector. To be concrete,
let us consider a theory which, as above, is made up by two sectors: an
axionic one $T_{\Lambda}$ and an ordinary chiral one $Z^{a}$, which also
includes the super-Weyl compensator. Consider a superpotential of the form

${\cal W}(Z;T)=N_{A}{\cal V}^{A}(Z)-c_{A}^{\Lambda}T_{\Lambda}{\cal
V}^{A}(Z)+\hat{\cal W}(Z)\,.$ (4.116)

In Section 4.2 we saw that $N_{A}$ can be interpreted as integration constants
coming from integrating out a set of gauge three-forms. Hence, the first part
of the superpotential is dynamically generated by exchanging, according to the
recipe (4.30), (part of) the multiplets $Z^{a}$ with three-form multiplets
$S^{A}$.

We want to show that also the second piece of the superpotential (4.116) may
be indeed dynamically generated in the following sense: if we move to the dual
picture where the axionic multiplets $T_{\Lambda}$ are exchanged with linear
multiplets $L^{\Lambda}$, the coupling term $c_{A}^{\Lambda}T_{\Lambda}{\cal
V}^{A}(Z)$ is fully re-absorbed into a gauging of the linear multiplets
$L^{\Lambda}$ with the real multiplets $P^{A}$ [36]. Therefore, in a gauge
fixed picture, our aim is to show that the full superpotential in a purely,
ordinary chiral multiplet theory

$W(\Phi;T)=\underbrace{M_{\rm P}^{3}N_{A}\Pi^{A}(\Phi)}_{\text{generated by
$A_{3}^{A}$}}-\underbrace{M_{\rm
P}^{3}c_{A}^{\Lambda}T_{\Lambda}\Pi^{A}(\Phi)}_{\text{generated by gauging
$L^{\Lambda}$}}+\underbrace{\hat{W}(\Phi)}_{\text{spectator}}\,.$ (4.117)

collapses to just the spectator part $\hat{W}(\Phi)$ once we pass to a parent
theory with three-form multiplets and gauged linear multiplets.

The joining link between a chiral theory and the dual one that we are looking
for is provided once again by a master Lagrangian, which here is

$\displaystyle{\cal L}_{\rm master}$
$\displaystyle=\int{\rm{d}}^{4}\theta\,E\,{\cal
F}(Z,\bar{Z};\hat{L})-2\int{\rm{d}}^{4}\theta\,E\,\hat{L}^{\Lambda}{\rm
Im\hskip 0.90005pt}T_{\Lambda}$ (4.118)
$\displaystyle\quad\,+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal
W}(Z)+{\rm c.c.}\right)$
$\displaystyle\quad\,+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,X_{A}{\cal
V}^{A}(Z)+\frac{{\rm i}}{8}\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(\bar{\cal
D}^{2}-8{\cal R})\left[(X_{A}-\bar{X}_{A})P^{A}\right]+\text{c.c.}\right)$
$\displaystyle\quad\,+\left(\frac{{\rm i}}{8}\int{\rm{d}}^{2}\Theta\,2{\cal
E}\,(\bar{\cal D}^{2}-8{\cal
R})\left[c_{A}^{\Lambda}(T_{\Lambda}-\bar{T}_{\Lambda})P^{A}\right]+\text{c.c.}\right)\,.\guilsinglleft\guilsinglright$

The first line is just (4.105) and the second line contains the spectator
super-Weyl superpotential $\hat{\cal W}(Z)$. The third line, directly coming
from (4.24), provides the trading between the multiplets $Z^{a}$ and $S^{A}$.
The last line is new and it will be responsible for the generation of the
F-term coupling. Let us now see how, from (4.118), we may retrieve either an
ordinary theory or a dual one.

Ordinary chiral formulation

A formulation where only chiral multiplets are present is obtained integrating
out the the real superfields $P^{A}$ and $\hat{L}^{\Lambda}$ from the master
Lagrangian (4.118). The former integration sets

$\delta P^{A}\,:\qquad{\rm Im\hskip 1.00006pt}X_{A}+c_{A}^{\Lambda}{\rm
Im\hskip 1.00006pt}T_{\Lambda}=0\,,$ (4.119)

which, employing the chirality of both $X_{A}$ and $T_{\Lambda}$, implies

$X_{A}=N_{A}-c_{A}^{\Lambda}T_{\Lambda}\,,$ (4.120)

with $N_{A}$ arbitrary real constants. The variation of the master Lagrangian
(4.118) with respect to the real superfields $\hat{L}^{\Lambda}$ instead gives

$\delta\hat{L}^{\Lambda}\,:\qquad{\rm Im\hskip
1.00006pt}T_{\Lambda}=\frac{1}{2}\frac{\partial{\cal
F}}{\partial\hat{L}^{\Lambda}}\,.$ (4.121)

Plugging both (4.120) and (4.121) into (4.118) we arrive at the Lagrangian

${\cal L}_{\rm chiral}=\int{\rm{d}}^{4}\theta\,E\,{\cal K}(Z,\bar{Z};{\rm
Im\hskip 1.00006pt}T)+\left(\int{\rm{d}}^{2}\Theta\,2{\cal E}\,{\cal
W}(Z;T)+{\rm c.c.}\right)$ (4.122)

with the super-Weyl invariant superpotential

${\cal W}(Z;T)=N_{A}{\cal V}^{A}(\Phi)-c_{A}^{\Lambda}T_{\Lambda}{\cal
V}^{A}(\Phi)+\hat{\cal W}(Z)\;.$ (4.123)

Formulation with linear multiplets

The steps which allow for obtaining a Lagrangian fully given in terms of
(gauged) linear multiplets and three-form multiplets is definitely much more
involved. At the superspace level, the first step is to integrate out the
axionic chiral superfields $T_{\Lambda}$. Re-expressing them in terms of
unconstrained complex superfields $\Xi_{\Lambda}$ as in (4.108), the
variations with respect to the latter give

$\delta\Xi_{\Lambda}\,,\delta\bar{\Xi}_{\Lambda}\,:\qquad({\cal
D}^{2}-8\bar{\cal
R})(\hat{L}^{\Lambda}-c_{A}^{\Lambda}P^{A})=0\,,\quad(\bar{\cal D}^{2}-8{\cal
R})(\hat{L}^{\Lambda}-c_{A}^{\Lambda}P^{A})=0\,,$ (4.124)

which are solved by

$\hat{L}^{\Lambda}=L^{\Lambda}+c_{A}^{\Lambda}P^{A}\,,$ (4.125)

with $L^{\Lambda}$ generic linear multiplets. The relation (4.125) defines a
_gauged_ counterpart of the linear multiplets $L^{\Lambda}$ and their $[{\cal
D},\bar{\cal D}]$–components contain the three-form field strengths ${\cal
H}_{3}^{\Lambda}$ gauged by the three-forms $A_{3}^{A}$:

$\hat{\cal H}_{3}^{\Lambda}={\cal
H}_{3}^{\Lambda}+c_{A}^{\Lambda}A_{3}^{A}\,.$ (4.126)

The variation with respect to the Lagrange multipliers $X_{A}$ indeed gives
the same trading as in (4.30) between the _old_ chiral superfields $Z^{a}$ and
the three-form multiplets $S^{A}$. A final integration of the chiral
multiplets $Z^{a}$ leads to

${\cal V}^{A}_{a}X_{A}=\frac{1}{4}(\bar{\cal D}^{2}-8{\cal R}){\cal
F}_{a}-\hat{\cal W}_{a}\,.$ (4.127)

The master Lagrangian (4.118) becomes

$\tilde{\mathcal{L}}=\int{\rm{d}}^{4}\theta\,E\,{\cal
F}(Z(P),\bar{Z}(P),\hat{L})+\left[\int{\rm{d}}^{2}\Theta\,2{\cal E}\,\hat{\cal
W}(Z(P))+{\rm c.c.}\right]+\tilde{\mathcal{L}}_{\rm bd},$ (4.128)

with the boundary terms

$\tilde{\mathcal{L}}_{\rm bd}=-\frac{{\rm
i}}{8}\left[\int{\rm{d}}^{2}\Theta\,2{\cal E}\,(\bar{\cal D}^{2}-8{\cal
R})-\int{\rm{d}}^{2}\bar{\Theta}\,2\bar{\cal E}\,({\cal D}^{2}-8\bar{\cal
R})\right]\left(X_{A}P^{A}\right)+\text{c.c.}\,,$ (4.129)

with $X_{A}$ defined by (4.127).

At the level of components, it is however better to work with the master
Lagrangian (4.118). Within the super-Weyl invariant formalism, the procedure
of integrating out the fields proceeds along the very same lines as in the
previous Section 4.4.1 and we refer to the Appendix D for further details
regarding how that procedure is generalized to supergravity. The super-Weyl
invariant Lagrangian which includes both gauge two- and three-forms is

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=-\frac{1}{6}\tilde{\cal F}\,R-{\cal
F}_{a\bar{b}}Dz^{a}\bar{D}\bar{z}^{b}+\frac{1}{4}{\cal
F}_{\Lambda\Sigma}\partial_{\mu}l^{\Lambda}\partial^{\mu}l^{\Sigma}\\\
&\quad\,+\frac{1}{4\cdot 3!}{\cal F}_{\Lambda\Sigma}\hat{\cal
H}_{\mu\nu\rho}^{\Lambda}\hat{\cal H}^{\Sigma\mu\nu\rho}+\left(\frac{{\rm
i}}{2\cdot 3!}{\cal F}_{\bar{a}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}\bar{D}_{\mu}\bar{z}^{\bar{a}}+{\rm c.c.}\right)\\\
&\quad\,+e^{-1}{\cal L}_{\text{3-forms}}\,,\end{split}$ (4.130)

where the three-form Lagrangrian has the same form as (4.44) with, however

$\displaystyle T^{AB}(z,\bar{z})$ $\displaystyle\equiv 2\,{\rm Re\hskip
0.90005pt}\left({\cal F}^{\bar{b}a}\,{\cal V}_{a}^{A}\bar{\cal
V}_{\bar{b}}^{B}\right)\,,$ (4.131a) $\displaystyle\Upsilon^{A}(z,\bar{z})$
$\displaystyle\equiv 2{\rm Re\hskip 0.90005pt}\left[{\cal
F}^{\bar{b}a}\left(\bar{\hat{\cal W}}_{\bar{b}}+\frac{{\rm i}}{2}{\cal
F}_{\Lambda}c_{B}^{\Lambda}\bar{\cal V}^{B}_{\bar{b}}+{\rm i}{\cal
F}_{\Lambda\bar{b}}c_{F}^{\Lambda}\bar{\cal V}^{\bar{F}}\right){\cal
V}_{a}^{A}\right]\,,$ (4.131b)
$\displaystyle\begin{split}\hat{V}(z,\bar{z})&\equiv{\cal
F}^{\bar{b}a}\left(\hat{\cal W}_{a}-\frac{{\rm i}}{2}{\cal
F}_{\Lambda}c_{A}^{\Lambda}{\cal V}^{A}_{a}-{\rm i}{\cal F}_{\Lambda
a}c_{E}^{\Lambda}{\cal V}^{E}\right)\times\\\
&\quad\quad\,\times\left(\bar{\hat{\cal W}}_{\bar{b}}+\frac{{\rm i}}{2}{\cal
F}_{\Lambda}c_{B}^{\Lambda}\bar{\cal V}^{B}_{\bar{b}}+{\rm i}{\cal
F}_{\Lambda\bar{b}}c_{F}^{\Lambda}\bar{\cal V}^{\bar{F}}\right)-{\cal
F}_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}{\cal V}^{A}\bar{\cal
V}^{B}\,.\end{split}$ (4.131c)

We can now proceed to gauge-fixing the super-Weyl invariance by following the
same procedure described above for ungauged linear multiplets. After having
imposed the Einstein frame condition (4.114), redefined the linear multiplets
as in (4.113) and introduced the Legendre transform (4.111) of the Kähler
potential, we arrive at

$\begin{split}e^{-1}{\cal L}_{\rm bos}&=\frac{M_{\rm P}^{2}}{2}R-M_{\rm
P}^{2}F_{i\bar{\jmath}}\partial\phi^{i}\bar{\partial}\bar{\phi}^{\bar{\jmath}}+\frac{M_{\rm
P}^{2}}{4}F_{\Lambda\Sigma}\partial_{\mu}\ell^{\Lambda}\partial^{\mu}\ell^{\Sigma}\\\
&\quad\,+\frac{1}{4\cdot 3!M_{\rm P}^{2}}\hat{\cal
H}_{\mu\nu\rho}^{\Lambda}\hat{\cal H}^{\Sigma\mu\nu\rho}+\left\\{\frac{{\rm
i}}{2\cdot 3!}F_{\bar{\imath}\Sigma}\varepsilon^{\mu\nu\rho\sigma}\hat{\cal
H}^{\Sigma}_{\nu\rho\sigma}\partial_{\mu}\bar{\phi}^{\bar{\imath}}+{\rm
c.c.}\right\\}\\\ &\quad\,+e^{-1}{\cal L}_{\text{three-forms}}\,,\end{split}$
(4.132)

where the three-form Lagrangian has the same form as (4.44) with now

$\displaystyle T^{AB}$ $\displaystyle\equiv 2M^{4}_{\rm
P}\,e^{\tilde{F}}\,{\rm Re\hskip
0.90005pt}\left({F}^{\bar{\jmath}i}\,D_{i}\Pi^{A}\bar{D}_{\bar{\jmath}}\bar{\Pi}^{B}-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})\Pi^{A}\bar{\Pi}^{B}\right)\,,$
(4.133a) $\displaystyle\begin{split}\Upsilon^{A}&\equiv 2M_{\rm
P}\,e^{\tilde{F}}{\rm Re\hskip
0.90005pt}\Bigg{[}{F}^{\bar{\jmath}i}\,\left(D_{\bar{\jmath}}\bar{\hat{W}}+\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{B}^{\Lambda}D_{\bar{\jmath}}\bar{\Pi}^{B}+{\rm
i}M_{\rm{P}}^{3}F_{\Lambda\bar{\jmath}}c_{B}^{\Lambda}\bar{\Pi}^{\bar{B}}\right)D_{i}\Pi^{A}\\\
&\quad\quad\,-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})\left(\bar{\hat{W}}+\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{D}^{\Lambda}\bar{\Pi}^{D}\right)\Pi^{A}-{\rm
i}M_{\rm{P}}^{3}\tilde{F}_{\Lambda}c_{B}^{\Lambda}\bar{\Pi}^{\bar{B}}\Pi^{A}\Bigg{]}\,,\end{split}$
(4.133b) $\displaystyle\begin{split}\hat{V}&\equiv\frac{e^{\tilde{F}}}{M_{\rm
P}^{2}}{F}^{\bar{\jmath}i}\left(D_{i}{\hat{W}}-\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{A}^{\Lambda}D_{i}\Pi^{A}-{\rm
i}M_{\rm{P}}^{3}F_{\Lambda i}c_{A}^{\Lambda}\Pi^{A}\right)\times\\\
&\quad\,\qquad\qquad\times\left(D_{\bar{\jmath}}\bar{\hat{W}}+\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{B}^{\Lambda}D_{\bar{\jmath}}\bar{\Pi}^{B}+{\rm
i}M_{\rm{P}}^{3}F_{\Lambda\bar{\jmath}}c_{B}^{\Lambda}\bar{\Pi}^{\bar{B}}\right)\\\
&\quad\,-(3-\ell^{\Lambda}\tilde{F}_{\Lambda})\frac{e^{\tilde{F}}}{M_{\rm
P}^{2}}\left|\hat{W}-\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{A}^{\Lambda}\Pi^{A}\right|^{2}-M_{\rm{P}}^{4}e^{\tilde{F}}F_{\Lambda\Sigma}c_{A}^{\Lambda}c_{B}^{\Sigma}\Pi^{A}\bar{\Pi}^{B}\\\
&\quad\,-M_{\rm{P}}e^{\tilde{F}}\left[-{\rm
i}c_{B}^{\Sigma}\tilde{F}_{\Sigma}\Pi^{B}\left(\bar{\hat{W}}+\frac{{\rm
i}}{2}M_{\rm{P}}^{3}F_{\Lambda}c_{A}^{\Lambda}\bar{\Pi}^{A}\right)+{\rm
c.c.}\right].\end{split}$ (4.133c)

####  4.5 Quantization conditions and dualities

We conclude this chapter discussing two important properties that the actions
introduced above enjoy: the compactness of the gauge symmetry, which allows us
to infer the quantization conditions for the constants $N_{A}$, and the
symmetries of the actions, which will be related, in the next chapter, to the
BPS–extended objects.

#####  4.5.1 Three-forms and quantization conditions

In the discussion so far, the integration constants $N_{A}$ introduced in
(4.49) in the dual three-form picture, or in the superpotential (4.22) in the
ordinary chiral multiplet theory, have been treated as arbitrary real
parameters. However, this is really so if the gauge three-forms $A_{3}^{A}$
are associated to non-compact gauge symmetries. On the other hand,
constructions from string theory, as well as purely four-dimensional arguments
(for instance in [104]), indicate that in consistent quantum gravitational
theories all gauge symmetries should be compact. In practice, given any
compact three-dimensional submanifold ${\cal E}$, this means that the
integrals

${\mathfrak{q}}^{A}\equiv\int_{\cal E}A^{A}_{3}\;,$ (4.134)

are periodic. It is then natural to normalize the gauge three-form fields so
that (4.134) have $2\pi$–periodicity.

The compactness of the gauge symmetries implies quantization conditions on the
corresponding field strengths. As in [34], a simple way to identify these
conditions is to relate our system to a $1$-dimensional theory by performing
the dimensional reduction of the four-dimensional theory
$\mathbb{R}\times{\cal E}$ along the compact space-like ${\cal E}$. Let us
focus on the four-form part of the bosonic action (4.47), namely

$\int_{\mathbb{R}}{\rm{d}}t\,L_{\rm
flux}\equiv-\frac{1}{2}\int_{\mathbb{R}\times{\cal
E}}T_{AB}F_{4}^{A}*F_{4}^{B}$ (4.135)

where $t$ parametrizes the time direction $\mathbb{R}$. In the one-dimensional
effective theory we can compute the momenta conjugated to the angles
${\mathfrak{q}}^{A}$, namely

$\displaystyle{\mathfrak{p}}_{A}$ $\displaystyle\equiv\frac{\partial L_{\rm
flux}}{\partial\dot{\mathfrak{q}}^{A}}=-T_{AB}*F_{4}^{B}\;,$ (4.136)

where $\dot{\mathfrak{q}}^{A}$ denote the derivatives of the angles with
respect to $t$. Quantum mechanically, the momenta must be integrally
quantized, that is ${\mathfrak{p}}_{A}\in\mathbb{Z}$, since the angles are
$2\pi$-periodic. On the other hand, by comparing (4.136) and (4.49) we see
that ${\mathfrak{p}}_{A}$ coincide with the integration constants $N_{A}$,
leading to the quantization condition

$N_{A}\in\mathbb{Z}\,.$ (4.137)

This shows how the compactness of the gauge symmetries implies the
quantization of the constants appearing in the effective superpotential
(4.22). This is in agreement with what is expected from explicit string theory
models, in which $N_{A}$ usually measure quantized internal fluxes, as we will
see in details in Chapter 6. However, we stress that the three-form
formulation has allowed for a purely four-dimensional derivation of this fact.

#####  4.5.2 Symmetries of the dual action

In general, there may exist a duality group $G_{\rm dual}$ of transformations
which acts on the quantized constants $N_{A}$ as well as on the EFT fields.
Hence, in a traditional EFT depending on fixed non-vanishing fluxes $N_{A}$,
part or all of the duality group is explicitly broken. By using the dual
formulation in terms of three-forms, the constants $N_{A}$ are traded for
dynamical three-forms $A^{A}_{3}$. In such a formulation $G_{\rm dual}$ acts
as an actual symmetry group of the action, which is only spontaneously broken.

More concretely, given a homogeneous kinetic section ${\cal K}(z,\bar{z})$ and
a superpotential (4.20), a class of dualities are given by isometries
$z^{a}\rightarrow z^{\prime a}$ which leave ${\cal K}(z,\bar{z})$ and
$\hat{\cal W}(z)$ invariant and act linearly on the periods

${\cal V}^{A}(z)\rightarrow{\cal V}^{A}(z^{\prime})=R^{A}{}_{B}{\cal
V}^{B}(z)\,.$ (4.138)

In the presence of non-vanishing flux quanta $N_{A}$, the superpotential
(4.20) is clearly not invariant under such a transformation, which may be
regarded as a ‘spurionic’ symmetry if $N_{A}$ transform in an opposite way:
$N_{A}\rightarrow N_{A}^{\prime}=(R^{-1})^{B}{}_{A}N_{B}$.111Notice that, by
imposing the Weyl-fixing and splitting the homogeneous periods ${\cal V}^{A}$
as in (4.21), the periods $\Pi^{A}$ may transform linearly only up to a pre-
factor, which should then be compensated by a Kähler tranformation of $K$.
Recalling the procedure followed for constructing the three-form formulation,
or more directly from explicit formulas like (4.47), with (4.48), or (4.130),
with (4.131), it is clear that the three-form theory is exactly invariant
under the duality transformation provided that the three-form potentials
transform as follows,

$A^{A}_{3}\rightarrow A^{\prime A}_{3}=R^{A}{}_{B}A^{B}_{3}\,.$ (4.139)

Such a symmetry is spontaneously broken once a certain vacuum sector specified
by (4.49) is selected.

Another class of possible duality transformations appear in the models with
superpotentials of the forms (4.116). These are associated with the integral
shifts $t_{\Lambda}\rightarrow t_{\Lambda}+n_{\Lambda}$, with
$n_{\Lambda}\in\mathbb{Z}$, and are explicitly broken, even though the
explicit breaking may be compensated by shifting $N_{A}$ to
$N_{A}^{\prime}=N_{A}+c_{A}^{\Lambda}n_{\Lambda}$. To analyze this case, let
us consider the corresponding Weyl-invariant EFT with three-forms $A_{3}^{A}$
and chiral fields $(z^{a},t_{\Lambda})$. By extending the formulas
(4.42a)–(4.42b) in order to include the chiral fields $t_{\Lambda}$ in an
obvious way and to replace $\hat{\cal W}$ with $\hat{\cal W}^{\prime}$ as
defined in (4.116), it is immediate to check that a shift
$t_{\Lambda}\rightarrow t_{\Lambda}+n_{\Lambda}$ induces the shifts
$\Upsilon^{A}\rightarrow\Upsilon^{A}-n_{\Lambda}c^{\Lambda}_{B}T^{AB}$ and
$\hat{V}\rightarrow\hat{V}-n_{\Lambda}c^{\Lambda}_{B}h^{B}+\frac{1}{2}n_{\Lambda}n_{\Sigma}c^{\Lambda}_{A}c^{\Sigma}_{B}T^{AB}$.
It follows that the three-form action (4.47) is exactly invariant under such
transformations. Hence, also in this case, the duality transformations are
_proper_ symmetries of the three-form action, which are only spontaneously
broken by the choice of a vacuum sector. We also notice that, on the one hand,
the same conclusion would hold also in the presence of corrections depending
on $e^{2\pi{\rm i}k^{\Lambda}t_{\Lambda}}$, with $k^{\Lambda}\in\mathbb{Z}$,
which break the continuous shift symmetries, but preserve the discrete ones.
On the other hand, in absence of such corrections, one may make a further step
and dualize the chiral fields $t_{\Lambda}$ to the linear multiplet bosons
$(l^{\Lambda},{\cal B}^{\Lambda}_{2})$. In the resulting formulation the
original shift symmetries are traded for the compact gauge symmetry of the
two-form potentials ${\cal B}^{\Lambda}_{2}$.

### BPS–extended objects and domain walls in supergravity

Having introduced gauge two-, three- and four-forms into generic supergravity
theories, we may couple them to BPS–strings, membranes and 3-branes. As was
explained in Chapter 3 for globally supersymmetric theories, the key elements
that we need to introduce to properly couple such BPS–objects are super-gauge
forms. These allow us to write down the actions coupling the BPS–objects to
the bulk sector in a manifestly supersymmetric manner. Once the charges of the
objects are specified, the form of their action is strictly determined by
$\kappa$–symmetry. Indeed, as we will see, the construction of their BPS
actions proceeds along the same lines as for the global case of Chapter 3.
Therefore, in this chapter we will be very brief and we will focus on
highlighting the differences of the local case with the respect to the
globally supersymmetric one.

Particular attention will be devoted to membranes. As already explored in the
previous chapters, membranes determine spacetime regions with different
potential for the scalar fields. As shown in Section 3.4 for global
supersymmetry, new domain wall solutions, which are supported by membranes,
may appear within a theory. Presently, we will generalize those constructions
to supergravity, highlighting the modifications induced by gravity.

####  5.1 Extended BPS–objects in supergravity

In supergravity we may introduce the very same hierarchy of BPS–objects that
we included in globally supersymmetric theories in Chapter 3. We shall
introduce the BPS–objects by increasing codimension, starting with 3-branes,
then moving to membranes and finally strings. We will also show how to
construct BPS string/membrane junctions as well as BPS membrane/3-brane
regions.

#####  5.1.1 Supersymmetric 3-branes

We start with supersymmetric spacetime filling 3-branes, with no boundary.
They are specified by the super-embedding

$\xi^{i}\quad\mapsto\quad{\mathfrak{z}}^{M}=(x^{m}(\xi),\theta^{\alpha}(\xi),\bar{\theta}_{\dot{\alpha}}(\xi))\;,$
(5.1)

where $\xi^{i}$, with $i=0,\ldots,3$, are the 3-brane worlvolume coordinates.
The action of a supersymmetric spacetime filling 3-brane is given by

$S_{\text{3-brane}}=\mu_{I}\int_{\cal W}{\bf C}^{I}_{4}\;,$ (5.2)

where ${\cal W}$ is the 3-brane world-volume and $\mu_{I}$ denotes the charges
of the 3-branes. The super-four-forms ${\bf C}^{I}_{4}$ are a covariantization
of (3.159)

$\displaystyle{\bf C}^{I}_{4}$ $\displaystyle=$ $\displaystyle 2E^{b}\wedge
E^{a}\wedge E^{\alpha}\wedge
E^{\beta}\sigma_{ab\;\alpha\beta}\bar{\Gamma}^{I}+2E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\wedge\bar{E}^{\dot{\beta}}\bar{\sigma}_{ab\;\dot{\alpha}\dot{\beta}}\Gamma^{I}\qquad$
(5.3) $\displaystyle{-}\frac{1}{6}E^{c}\wedge E^{b}\wedge
E^{a}\wedge\bar{E}^{\dot{\alpha}}\epsilon_{abcd}\sigma^{d}_{\alpha\dot{\alpha}}{\cal
D}^{\alpha}{\Gamma^{I}}+\frac{1}{6}E^{c}\wedge E^{b}\wedge E^{a}\wedge
E^{\alpha}\epsilon_{abcd}\sigma^{d}_{\alpha\dot{\alpha}}\bar{{\cal
D}}^{\dot{\alpha}}\bar{\Gamma}^{I}$ $\displaystyle+\frac{{\rm
i}}{96}E^{d}\wedge E^{c}\wedge E^{b}\wedge E^{a}\epsilon_{abcd}\left[({\cal
D}^{2}-24\bar{{\cal R}}){\Gamma^{I}}-(\bar{{\cal D}}^{2}-24{\cal
R})\bar{\Gamma}^{I}\right]\,.$

These are constructed out of the four-form multiplets $\Gamma^{I}$ introduced
in Section 4.3, which contain, among their components (4.85), real gauge four-
forms $C_{4}^{I}$. It is immediate to show that, using (4.85), the purely
bosonic components of (5.3) are

${\bf
C}^{I}_{4}|=C_{4}^{I}=\frac{1}{4!}C^{I}_{mnpq}{\rm{d}}x^{m}\wedge{\rm{d}}x^{n}\wedge{\rm{d}}x^{p}\wedge{\rm{d}}x^{q}\;.$
(5.4)

The super-four-forms (5.3) are the _unique_ closed super-forms which can be
constructed with the four-form multiplets $\Gamma^{I}$. As explained in
Section 3.6, the closure of ${\bf C}^{I}_{4}$, combined with the fact that the
3-brane has no boundary, makes the action (5.2) to be trivially invariant
under any superdiffeomorphism and hence, in particular, under any
$\kappa$-transformation. Being the $\kappa$-parameters unconstrained, we can
trivially choose the $\kappa$-parameters to coincide, over the full 3-brane
worldvolume, with the local supersymmetry parameters. Therefore, the 3-brane
specified by the action (5.2) preserves the _complete_ ${\cal N}=1$ bulk
supersymmetry.

It may seem quite awkward that we have not included any Nambu-Goto term to the
action (5.2). In Section 7.3.3, for Type IIB EFTs, we will give a top-down
justification of why (and under which hypotheses) an action of the sort of
(5.2) may be born from a complete 3-brane action, which also include a Nambu-
Goto term. For the moment let us notice that, eventually, we may also add a
supersymmetric Nambu-Goto term in the action (5.2), but it turns out that such
an addition comes with a huge price: supersymmetry is fully spontaneously
broken, being it realized only non-linearly. Such a 3-brane can be identified
with a _Goldstino brane_ of the kind introduced in [105, 106].

As we shall see in Chapter 7, the aforementioned problem of including a
Nambu–Goto term intertwines with the definition of the charges $\mu_{I}$.
Within the four-dimensional description that we are considering throughout
this chapter, nothing seems to strictly fix the sign of the charges $\mu_{I}$:
whatever sign we choose for $\mu_{I}$, the action (5.2) is still invariant
under the full ${\cal N}=1$ local supersymmetry. But we do know that, in
typical (orientifolded) string compactifications, where both 3-branes and
anti-3-branes are present, only one choice of signs for $\mu_{I}$ is
consistent with a linear realization of the ${\cal N}=1$ local supersymmetry,
the opposite choice being associated to the Goldstino branes. It would then be
interesting to see whether such correlation could be understood also at pure
four-dimensional level.

Furthermore, since the 3-branes that we are considering here do fill entire
spacetime regions, the worldvolume fields living on them cannot be generically
neglected. Nevertheless, in this work, we shall assume that this is the case.
We treat the action (5.2) as a topological term once coupled to a bulk action
and then, apart from contributing to the total four-form charge ${\cal Q}_{I}$
by $\mu_{I}$, it does not contain much more physical content. Due to its
peculiar nature of being completely decoupled from the rest, the action (5.2)
can be added to any supergravity action.

#####  5.1.2 Supersymmetric membranes

We now pass to supersymmetric objects of codimension one, the BPS–membranes.
Membranes are described by three worldvolume coordinates $\xi^{i}$, with
$i=0,1,2$ and, as objects living in the ambient superspace, are defined via
the super-embedding

$\xi^{i}\quad\mapsto\quad{\mathfrak{z}}^{M}=(x^{m}(\xi),\theta^{\alpha}(\xi),\bar{\theta}_{\dot{\alpha}}(\xi))\;.$
(5.5)

As in Section 3.2, we would like to couple membranes minimally to some gauge
three-forms $A_{3}^{A}$ with charges $q_{A}$. Therefore, to begin with, we
need a proper generalization to superspace Wess-Zumino term which appears in
(1.31). This promotion of the Wess-Zumino term is obtained, as explained in
Section 3.2, by introducing a super-gauge three-form ${\bf A}^{A}_{3}$:

$S_{\text{memb,WZ}}=q_{A}\int_{\cal M}{\bf A}^{A}_{3}\;,$ (5.6)

where ${\cal M}$ is the membrane worldvolume. We already saw in Section 4.2
that the real multiplets $P^{A}$ are the proper suspersymmetric objects that
contain a real three-form among their components (see, for example, (C.4)).
The super-three-form ${\bf A}^{A}_{3}$ is then built as a manifestly
supersymmetric object with the supervielben $E^{A}$ and the real multiplets
$P^{A}$ as

$\displaystyle{\bf A}^{A}_{3}=$ $\displaystyle\,{-}2{{\rm i}}E^{a}\wedge
E^{\alpha}\wedge\bar{E}^{\dot{\alpha}}\sigma_{a\alpha\dot{\alpha}}P^{A}+E^{b}\wedge