On the Stability of String Theory Vacua

Suvendu Giri,1,2 Luca Martucci,3 Alessandro Tomasiello2,4

1 Dipartimento di Fisica, Università di Milano-Bicocca,
Piazza della Scienza 3, I-20126 Milano, Italy
2 INFN, sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano,
Italy
3 Dipartimento di Fisica e Astronomia, Università degli Studi di Padova
& INFN sezione di Padova, Via F. Marzolo 8, I-35131 Padova, Italy
4 Dipartimento di Matematica, Università di Milano-Bicocca,
Via Cozzi 55, I-20126 Milano, Italy

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

Vacuum compactifications may suffer from instabilities under small
perturbations or tunnel effects; both are difficult to analyze. In this paper
we consider the issue from a higher-dimensional perspective. We first look at
how stability works for supersymmetric vacua, where it is widely expected to
hold. We first show that the nucleation of brane bubbles in type II AdS
compactifications is forbidden in the probe approximation by a simple argument
involving pure spinors and calibrations. We then adapt familiar positive-
energy theorems directly to M-theory and type II supergravity, rather than to
their effective lower-dimensional reductions, also showing how to consistently
include localized sources. We finally initiate an analysis of how these
arguments might be extended to non-supersymmetric vacua. In M-theory, at the
lower-derivative level, we find that the most natural modifications fail to
stabilize the skew-whiffed and Englert vacua.

###### Contents

  1. 1 Introduction and summary
  2. 2 Probe brane bubbles
    1. 2.1 Preliminary general remarks
    2. 2.2 Brane bubbles in type II $\mathrm{AdS}_{4}$ vacua
  3. 3 Stability in four-dimensional (super)gravity
    1. 3.1 $\mathrm{Mink}_{4}$ stability from supergravity
    2. 3.2 $\mathrm{AdS}_{4}$ stability
    3. 3.3 Breaking supersymmetry
    4. 3.4 Some comments
  4. 4 Positivity and stability in M-theory
    1. 4.1 The main identity
    2. 4.2 Positivity in the absence of branes
    3. 4.3 Inclusion of M2 branes
    4. 4.4 Inclusion of M5-branes
    5. 4.5 Positivity and stability of $G_{2}$-compactifications
    6. 4.6 Stability of $\mathrm{AdS}_{4}$ vacua
  5. 5 Positivity and stability in type II
    1. 5.1 BPS energy
    2. 5.2 Main identity
    3. 5.3 Positivity in the absence of branes
    4. 5.4 Inclusion of localized sources
  6. 6 Supersymmetry breaking
    1. 6.1 An operator in M-theory
    2. 6.2 Skew-whiffed and Englert vacua
    3. 6.3 Lack of positivity
    4. 6.4 Type II
  7. A Useful spinorial identities
  8. B Some details on M-theory
    1. B.1 Supercharge from the Noether theorem
    2. B.2 Noether charge in M-theory
    3. B.3 The divergence identity
    4. B.4 The supersymmetry-breaking case
    5. B.5 Equations of motion from supersymmetry
  9. C Inclusion of M2-branes
    1. C.1 M2 BPS-energy
    2. C.2 M2 bound
  10. D Details on type II
    1. D.1 Linear terms
    2. D.2 Quadratic tems
  11. E From M-theory to IIA

##  1 Introduction and summary

In a gravitational theory, spacetime becomes dynamical. This creates the
possibility that it might change drastically, perhaps with a catastrophic
change in vacuum energy. For the Minkowski vacuum of general relativity, such
worries were put to rest long ago by mathematical theorems [1, 2]. If matter
satisfies the dominant energy condition, an appropriately defined total
energy, computed in terms of the boundary behavior of the metric, is non-
negative, and can only be zero for Minkowski space itself; so not even a
quantum tunneling event can trigger spacetime decay.

This result was later generalized in various directions [3, 4, 5, 6]. One of
the lessons was that gravity can help stabilize scalar potentials that in
field theory would lead to vacuum decay, confirming considerations from small
fluctuations [7] and instantons [8]. Another is that supersymmetry often plays
a crucial role, ultimately because the Hamiltonian is the square of the
supercharges. (Indeed even the argument in [2] for pure gravity relies on
introducing auxiliary spinors which are covariantly constant at infinity,
inspired by the vacuum supersymmetric spinors for the simplest supergravity
models.) In particular, it was shown in various $d=4$ examples that
supersymmetric vacua are stable even if the model does not satisfy the
dominant energy condition [4].

In theories with extra dimensions such as string theory, however, new decay
channels open up. This is demonstrated in a spectacular fashion by the
$\mathrm{Mink}_{4}\times S^{1}$ Kaluza–Klein (KK) vacuum of pure five-
dimensional gravity, which can decay via the nucleation of a bubble of
different topology, created by recombining internal and external directions;
from the four-dimensional perspective this appears as a _bubble of nothing_
where spacetime ends [9]. (The supersymmetry-inspired argument in [2] does not
apply, because on the new topology there exist no spinors that are
asymptotically covariantly constant.) In string theory, many other similar
decay channels are provided by brane bubbles, where again spacetime can end or
across which the flux quanta can change (with early examples in [10, 11, 12]).
While an effective four-dimensional approach might go a long way towards
establishing vacuum stability, it cannot capture such processes, which involve
the dynamics of higher KK modes.111Effective theories of compactifications
often not only violate the dominant energy condition, but naively seem to have
a scalar potential that is unbounded from below, coming from the internal
integral of the scalar curvature [13]. This issue was however resolved in
[14]. This issue is made all the more pressing by the absence of a scale
separation between the KK scale and the vacuum energy in many AdS vacua. While
in some theories there was recent progress (beginning with [15]) in computing
the full infinite KK mass spectrum, this still only covers small fluctuations.

In this paper, we attack this problem directly in ten or eleven dimensions.
For supersymmetric $\mathrm{Mink}_{4}\times M_{d}$ or $\mathrm{AdS}_{4}\times
M_{d}$ vacua, $d=6,7$, stability is of course widely expected for the reasons
we recalled earlier; but quite surprisingly, it was never explicitly shown
before. Our original motivation came from vacua that break supersymmetry: de
Sitter vacua are expected to be metastable at best, while anti-de Sitter (AdS)
vacua have been conjectured to be either unstable or metastable by a
generalization of the weak gravity conjecture [16, 17] – see for instance [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28] for a sample of recent papers
discussing non-perturbative (in)stability in string theory.

In the supersymmetric case, as an appetizer we first consider brane bubbles in
the probe approximation. Such processes have been shown to destabilize various
non-supersymmetric vacua, beginning with [11, 12]; we show why these can never
happen in the supersymmetric case, with a simple argument involving the type
II pure spinor equations [29].

We then attack the problem in full generality by adapting the strategy of the
positive energy theorem [2, 30, 3, 4, 5, 6] to compactifications. We consider
a spinor $\epsilon$ that is _asymptotically supersymmetric_ : namely, it is
asymptotic to a supersymmetric spinor of the vacuum. From this we define a
notion of _BPS energy_ $I(\epsilon)$ by integrating the Hodge-dual of a
certain two-form $E_{2}$, bilinear in $\epsilon$, on the $S^{2}\times
M_{d}=\partial S$ asymptotic boundary of a spatial slice $S$. Using Stokes’
theorem, $I(\epsilon)=\int_{S}{\rm d}*\\!E_{2}$. Our main result is a formula
for the divergence $*\,{\rm d}*\\!E_{2}$, which we obtained both for eleven-
dimensional and type II supergravity. In the former case, for example, we show
that on-shell and in absence of branes the divergence satisfies the simple
identity
$\nabla_{{M}}E^{{{MN}}}=\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}_{{P}}\epsilon$,
where $\delta\psi_{{P}}={\mathcal{D}}_{{P}}\epsilon$ is the gravitino
supersymmetry transformation; several flux and curvature terms conspire on the
right-hand side to reconstruct the bulk equations of motion, which we can then
set to zero. (The type II result is similar, but also contains the operator
${\mathcal{O}}$ in the dilatino transformations,
$\delta\lambda={\mathcal{O}}\epsilon$.)

The divergence $\nabla_{{M}}E^{{{MN}}}$ has the same form as in $d=4$, so the
remaining steps are standard. Namely, let $\epsilon$ be asymptotically
supersymmetric and such that the _Witten condition_
$\Gamma^{a}{\mathcal{D}}_{a}\epsilon=0$ holds, where $a$ is a flat index
tangent to $S$. A gamma matrix identity now implies
$I(\epsilon)=\int_{S}({\mathcal{D}}^{a}\epsilon)^{\dagger}({\mathcal{D}}_{a}\epsilon)\geq
0$, which is non-negative and only vanishes for supersymmetric vacua, as
required.

In presence of branes, the bulk equations of motion produce delta-like
contributions to $I(\epsilon)$; using supersymmetry and the theory of
calibrations as generalized in [31], we show that these contributions are
still all positive. (This is similar in spirit to how fields in multiplets not
containing the graviton give extra contributions to $\nabla_{\mu}E^{\mu\nu}$
in $d=4$ supersymmetric models [4].)

This proves stability in the supersymmetric case, up to a couple of technical
details which are expected to hold and were checked in similar situations in
existing literature. First, for AdS vacua one needs to check that
$I(\epsilon)$ is independent of the boundary of the spatial slice $\partial
S$; or in other words, that the BPS energy is conserved. In $d=4$ this can be
done for example by relating it to the energy defined in the covariant phase
space formalism, which is conserved [32]. Second, one needs to prove that an
$\epsilon$ that is asymptotically supersymmetric and satisfies the Witten
condition $\Gamma^{a}{\mathcal{D}}_{a}\epsilon=0$ always exists. Here we check
a standard formal argument; a rigorous analytical proof was given in [33] for
the $d=4$ case (with an AdS4 generalization in [34]), and in [35, 36] for
Calabi–Yau compactifications.

Having worked out the argument for stability of supersymmetric vacua, we
finally proceed to non-supersymmetric ones. The idea is now to look for an
operator ${\mathcal{D}}^{\prime}_{{M}}$ different from the one appearing in
supersymmetry, trying to i) solve ${\mathcal{D}}^{\prime}_{{M}}\epsilon=0$ and
ii) to make sure that the positivity argument still holds. In $d=4$ models [6]
this leads to the idea of _fake superpotential_ : a $W$ which is related to
the scalar potential by the usual supergravity formula
$V=2(\partial_{\phi}W)^{2}-3W^{2}$, but which isn’t the supersymmetric one.
There is no direct analogue of this in our higher-dimensional setting; in
M-theory, the most general gauge-invariant operator
${\mathcal{D}}^{\prime}_{{M}}$ at one-derivative level is obtained by changing
the coefficients of the flux terms in ${\mathcal{D}}_{{M}}$, and adding a new
term proportional to $\Gamma_{{M}}$.

We are able to satisfy condition i) above on some simple AdS4 vacua: the
_skew-whiffed_ ones [37, 38], obtained from supersymmetric ones by reversing
the sign of $G$, and the _Englert_ vacua [39], where internal flux components
are added. In both of these cases we find multiple violations to the stability
argument. One might want to conclude from this absence of protection that
these vacua are in fact unstable. Of course it is possible, however, that a
more sophisticated modification of the supersymmetric argument does work for
them, or that a completely different protection mechanism is at play. We hope
our methods can be refined in the future to settle the issue one way or
another.

In section 2 we warm up, as we mentioned, by giving an argument that forbids
brane bubbles in the probe approximation for AdS supersymmetric vacua. In
section 3 we switch gears and review positive-energy theorems and their
applications to stability in $d=4$ theories. This provides a blueprint for
subsequent developments: in section 4 we show positivity for M-theory, and in
section 5 for type II supergravity. Finally we consider supersymmetry-breaking
in section 6, mainly in M-theory. Many technical aspects of the computations
are provided in detail in the appendices.

##  2 Probe brane bubbles

A vacuum may be unstable under tunneling effects. The probability of such a
decay can be computed in Euclidean signature, by looking for bubbles that
connect the old vacuum at infinity with a new one at their core. In this
section we consider the simple case of decays mediated by branes in AdS vacua,
in the probe approximation, namely disregarding their back-reaction. This is
of course not always appropriate, but we consider it here as a warm-up:
focusing for concreteness on D-brane bubbles in type II AdS4 vacua, we are
able to give a very simple argument that such bubbles cannot be nucleated. The
argument is generalizable to other settings and directly exploits the ten-
dimensional geometrical structure of the background. In other words, it does
not rely on any effective four-dimensional description, which may be
problematic in absence of scale separation.

###  2.1 Preliminary general remarks

We start by reviewing AdSd tunneling mediated by a probe $(d-2)$-brane charged
under a $d$-form field strength $F_{d}={\rm d}A_{d-1}$.222In the simplest
$d$-dimensional setting in which only the metric and $A_{d-1}$ appear as
dynamical bulk fields, it is relatively easy to take into account backreaction
effects [40]. This is much harder for generic string/M-theory AdS vacua, in
particular in absence of scale separation. We first set up the formalism in a
$d$-dimensional theory, and then discuss the modifications in the presence of
extra dimensions.

In global coordinates, the Euclidean AdSd metric is

$\mathrm{d}s^{2}_{\mathrm{EAdS}_{d}}=L^{2}(\mathrm{d}r^{2}+\sinh^{2}r\,\mathrm{d}s^{2}_{S^{d-1}})\,,$
(2.1)

where $L$ is the AdS radius. This metric has an $\mathrm{SO}(1,d)$ isometry
group. As in [8, 41], the dominant instantonic bubbles are expected to be
invariant under the largest possible amount of symmetries. Hence, without loss
of generality we can focus on Euclidean bubbles wrapping the $S^{d-1}$ in
(2.1), which are invariant under the isometry subgroup $\mathrm{SO}(d)$.333The
bubbles considered in [12] would superficially not look spherically symmetric,
but in fact they are; see [21, Sec. 4.1.2].

The semiclassical decay rate is controlled by a $(d-1)$-dimensional Euclidean
action of the form

$S=\tau\int\mathrm{d}^{d-1}\sigma\sqrt{-g}-q\int A_{d-1}\,,$ (2.2)

with some $\tau$ and $q$ representing the tension and charge of the brane. By
spacetime symmetry, the $d$-form flux in a vacuum will be of the form

$\begin{split}F_{d}=\mathrm{d}A_{d-1}=L^{-1}h\,\mathrm{vol}_{\mathrm{EAdS}_{d}}=L^{d-1}h\sinh^{d-1}r\,\mathrm{d}r\wedge\mathrm{vol}_{S^{d-1}}\,;\\\
A_{d-1}=L^{d-1}h\,c(r)\,\mathrm{vol}_{S^{d-1}}\,,\qquad
c^{\prime}(r)=\sinh^{d-1}r\,,\end{split}$ (2.3)

for some constant $h$, which in a quantum gravity theory is expected to
satisfy an appropriate quantization condition. Now (2.2) becomes

$S=L^{d-1}\mathrm{vol}_{S^{d-1}}\left[\tau\sinh^{d-1}r-qhc(r)\right]\,.$ (2.4)

This is extremized at $r=r_{0}$ if an only if

$\tanh r_{0}=\frac{(d-1)\tau}{qh}\,.$ (2.5)

This condition can be satisfied, and then the tunnel effect is allowed, only
if

$qh>(d-1)\tau\,.$ (2.6)

Intuitively, this means that the WZ term, which wants to expand the bubble,
wins over the gravitational term, which wants to crush it.

We can now uplift these simple $d$-dimensional arguments to higher
$D$-dimensional vacua of the (possibly warped) product form AdS${}_{d}\times
M$, where $M$ is a compact $(D-d)$-dimensional space, having in mind the
string/M-theory models ($D=10,11$). We assume that the $(d-1)$-dimensional
bubble corresponds to a microscopic $p$-brane of the form
$S^{d-1}\times\Sigma$, where $\Sigma\subset M$ is an internal closed
$(p-d+2)$-dimensional submanifold, possibly supporting non-trivial world-
volume fields. By invoking again the SO($d$) symmetry, the internal cycle
$\Sigma$ as well as the world-volume fields cannot depend on the position
along the external $S^{d-1}$. Hence, they must satisfy some internal equations
of motion, decoupled from the external ones, which fix them to some particular
configuration. One is then reduced to a $(d-1)$-dimensional action of the form
(2.2), with $\tau$ and $q$ fixed by the on-shell internal $p$-brane
configuration. After having computed $\tau$ and $q$ in this way, one can then
check whether the instability condition (2.6) is satisfied or not. Of course
we expect that (2.6) never holds for supersymmetric vacua. In the following
section we will highlight the microscopic geometric origin of this stability
in the case of D-brane bubbles in AdS4-vacua.

We also remark that (2.6) admits an alternative interpretation. Put the
Lorentzian version of the same $(d-2)$-brane along a $\mathbb{R}^{1,2}$ slice
of the Lorentzian AdSd in Poincaré coordinates. If (2.6) holds, the brane will
experience a run-away potential, which will push it to the AdS boundary. This
brane ejection effect is indeed often adopted as an alternative criterion to
detect the instability of an AdS vacuum; it was argued for holographically in
[42, 22] and by using (2.6) in [21, Sec. 5.1]. In the last two references, the
role of the dependence of $\tau$, $q$ on the internal coordinates was
emphasized.

###  2.2 Brane bubbles in type II $\mathrm{AdS}_{4}$ vacua

We now focus on D-brane bubbles in type II vacua of the form
AdS${}_{4}\times_{\mathrm{{{W}}}}M_{6}$, with the metric given by ${\rm
d}s^{2}=e^{2A}\,{\rm d}s^{2}_{\mathrm{AdS}_{4}}+{\rm d}s^{2}_{M_{6}}$. We will
work in string units $2\pi\sqrt{\alpha^{\prime}}=1$. We begin by recalling
that the conditions for such a vacuum to have ${\cal N}=1$ supersymmetry can
be written as [29]

$\displaystyle\mathrm{d}_{H}(\mathrm{e}^{2A-\phi}\mathrm{Re}\Phi_{\mp})=0\,,$
(2.7a)
$\displaystyle\mathrm{d}_{H}(\mathrm{e}^{3A-\phi}\Phi_{\pm})=\frac{2}{L}\mathrm{e}^{2A-\phi}\mathrm{Re}\Phi_{\mp}\,,\hskip
113.81102pt\overset{\text{\tiny{IIA}}}{\text{\tiny{IIB}}}$ (2.7b)
$\displaystyle\mathrm{d}_{H}(\mathrm{e}^{4A-\phi}\mathrm{Im}\Phi_{\mp})=\frac{3}{L}\mathrm{e}^{3A-\phi}\mathrm{Im}\Phi_{\pm}-\mathrm{e}^{4A}*\lambda
f\,,$ (2.7c)

where $*$ is the Hodge-star along $M_{6}$. The $\Phi_{\pm}$ are polyforms that
obey a certain algebraic purity condition, defined as bilinears of the
internal supersymmetry parameters $\eta^{a}$. We use the normalization
$\|\eta^{a}_{+}\|^{2}=8\mathrm{e}^{A}$, and $\Phi_{\pm}$ such that
$(\bar{\Phi}_{\pm},\Phi_{\pm})=8{\rm i}$; this is convenient but not
necessary. $f=\sum_{k}f_{k}$ is a polyform collecting the internal RR fluxes
$f_{k}$, $H$ is the NSNS flux, $\lambda\alpha_{k}\equiv(-1)^{\lfloor
k/2\rfloor}\alpha_{k}$ and $\mathrm{d}_{H}\equiv\mathrm{d}-H\wedge$. The total
RR flux is related to the internal one by

$F=f+\mathrm{vol}_{4}\wedge\mathrm{e}^{4A}*\lambda f\,.$ (2.8)

Crucially for us, the equations (2.7) can be interpreted as closure of the
(generalized) calibrations for D-branes extended in AdS4 along time plus one,
two, and three space coordinates respectively [43, 44].

Consider now a Euclidean D$p$-brane bubble of the form $S^{3}\times\Sigma$,
where $\Sigma\subset M_{6}$ is a $(p-2)$-cycle. $\Sigma$ can support a world-
volume two-form flux ${\cal F}$, satisfying the Bianchi identity ${\rm d}{\cal
F}=H|_{\Sigma}$, where $|_{\Sigma}$ denotes the pull-back to $\Sigma$. By
reducing the usual D-brane DBI action on $\Sigma$, we easily get

$\tau=2\pi\int_{\Sigma}{\rm d}^{p-2}\xi\,\mathrm{e}^{3A}\sqrt{\det(h+{\cal
F})}\,.$ (2.9)

Here $\xi^{i}$ are world-volume coordinates along $\Sigma$ and $h_{ij}$ is the
pull-back of the string frame metric along $M_{6}$. The factor of
$\mathrm{e}^{3A}$ comes about because the brane is extended along three
directions in EAdS4.

In order to obtain $qh$, we must analogously reduce the D-brane CS-term

$-2\pi\int_{S^{3}\times\Sigma}C\wedge\mathrm{e}^{{\cal F}}$ (2.10)

where $C$ is the polyform of RR potentials, such that $F={\rm d}_{H}C$. We can
take

$C=c_{\mathrm{int}}+L^{4}c(r)\mathrm{vol}_{S^{3}}\wedge\mathrm{e}^{4A}*\lambda
f$ (2.11)

with $\mathrm{d}c_{\mathrm{int}}=f$ (which will not play any role) and $c(r)$
as in (2.3). By reducing (2.10) along $\Sigma$ and matching it against (2.2)
with $A_{d-1}$ as in (2.3), we get

$qh=2\pi L\int_{\Sigma}e^{4A}*\lambda f\wedge\mathrm{e}^{{\cal
F}}=6\pi\int_{\Sigma}e^{3A-\phi}{\rm Im\hskip
1.00006pt}\Phi_{\pm}\wedge\mathrm{e}^{\cal F}\,.$ (2.12)

In the second step, we used (2.7c).

We now recall that $\Phi_{\pm}$ can be regarded as D-brane calibrations [43,
44]. This allows us to write the bound:

${\rm d}^{p-2}\xi\,\sqrt{\det(h+{\cal
F})}\geq\left[\mathrm{Re}(\mathrm{e}^{{\rm
i}\theta}\Phi_{\pm})|_{\mathrm{\Sigma}}\wedge\mathrm{e}^{{\cal F}}\right]_{\rm
top}\quad~{}~{}~{}\text{(for any $\theta$)}$ (2.13)

where the inequality applies to the coefficients of the local top-form ${\rm
d}^{p-2}\xi$ on $\Sigma$. By using (2.13) inside (2.12) and recalling (2.9),
we get the bound

$qh\leq 3\tau$ (2.14)

which shows that (2.6) (with the correct numerical coefficient $d-1=3)$ is
indeed never attained in these vacua.

Note also that the calibration appearing in (2.12) corresponds to the phase
$e^{{\rm i}\theta}=-{\rm i}$ in (2.13). Precisely with such a choice ${\rm
Re\hskip 1.00006pt}(e^{{\rm i}\theta}\Phi_{\pm})$ is the calibration for a
domain-wall-like D-brane filling the Poincaré $\mathbb{R}^{1,2}$ slice of the
Lorentzian AdS4. Such a configuration is supersymmetric precisely when (2.13)
is saturated. In this case $qh=3\tau$ and then we have threshold stability
under the brane ejection mechanism discussed in the previous subsection.

This argument clearly illustrates how the non-perturbative stability of
supersymmetric AdS vacua under nucleation of probe Euclidean brane bubbles is
controlled by calibrations. Hence, it is natural to guess that any more
general stability argument involving backreacting branes, which possibly
contribute to both the classical background and the semiclassical nucleated
bubbles, must crucially depend on the existence of appropriate calibration
structures. The following sections will show how this guess is precisely
realized.

##  3 Stability in four-dimensional (super)gravity

In order to extend the above stability argument beyond the probe regime, we
will adapt methods successfully used in gravity without extra dimensions.

The method stems from the expectation [45, 46] that a supersymmetric state in
supergravity saturates a quantum BPS bound of the schematic form

$I\equiv\langle\\{Q,Q\\}\rangle\geq 0\,,$ (3.1)

where $Q$ is any of the real supercharges preserved by the supersymmetric
state. One may then loosely conclude that the supersymmetric state must be
stable, or at most threshold-decay to other supersymmetric states saturating
the same bound.

A more precise realization of this formal positivity argument was given by
Witten [2] and Nester [30]. The first application in these papers was to the
Minkowski vacuum in Einstein gravity, resulting in a new proof of the result
in [1]. Indeed the argument is sufficiently robust that it may be applied to
theories without supersymmetry, in which case the spinors involved in the
proof can just be taken to be auxiliary objects. The proof was later extended
to non-zero cosmological constant $\Lambda$ in gravity and supergravity [3, 4,
5], and later to various models in different dimensions, which may also arise
as EFTs or consistent truncations of string/M-theory compactifications.
However, somewhat surprisingly, no successful attempt has so far been made to
either apply this approach directly to 10/11-dimensional string/M-theory
models, or to include local backreacting brane sources in the argument.

In the next sections we will explicitly fill these gaps. Since the discussion
will inevitably be quite technical, in this section we will first provide a
self-contained review of some key points of this formalism in simpler four-
dimensional settings.

###  3.1 $\mathrm{Mink}_{4}$ stability from supergravity

Consider first the minimal four-dimensional supergravity action

$\frac{1}{2}\int{\rm d}^{4}x\sqrt{-g}\left(R+{\rm
i}\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}D_{\nu}\psi_{\rho}\right)$ (3.2)

where the gravitino $\psi_{\mu}$ is in the Majorana representation, and we
work in Planck units $8\pi G=M^{-2}_{\rm P}=1$. As reviewed in App. B, by
applying the standard Noether procedure we obtain the following formula for
the supercharge associated with a given supersymmetry generator $\varepsilon$:

$Q(\varepsilon)=-{\rm
i}\int_{\partial\Sigma}\bar{\varepsilon}\,\gamma_{5}\gamma_{\mu}\psi_{\nu}\,{\rm
d}x^{\mu}\wedge{\rm d}x^{\nu}$ (3.3)

where $\gamma_{5}\equiv{\rm i}\gamma^{\underline{0123}}$ is the four-
dimensional chiral operator and $\partial\Sigma$ is the boundary of a three-
dimensional non-timelike hypersurface. Now, the Dirac brackets of two such
generators must give
$\\{Q(\varepsilon^{\prime}),Q(\varepsilon)\\}=\delta_{\varepsilon^{\prime}}Q(\varepsilon)$,
which can be computed from
$\delta_{\varepsilon}\psi_{\mu}=D_{\mu}\varepsilon$. Hence, the usual quantum
argument suggests that the positive definite quantity associated with
$Q(\varepsilon)$ can be identified with
$I(\varepsilon)\equiv\delta_{\varepsilon}Q(\varepsilon)$, with $\varepsilon$
now considered as a commuting spinor. This logic leads to the following
quantity:

$I(\varepsilon)=\int_{\partial\Sigma}*E_{2}\,,$ (3.4)

where

$E_{2}\equiv-\frac{1}{2}\,\bar{\varepsilon}\gamma_{\mu\nu}{}^{\rho}D_{\rho}\varepsilon\,{\rm
d}x^{\mu}\wedge{\rm d}x^{\nu}\,.$ (3.5)

Note that in (3.3) and $\delta_{\varepsilon}\psi_{\mu}=D_{\mu}\varepsilon$ the
spinor $\varepsilon$ is anticommuting, while in (3.5) it must be considered as
a commuting spinor.

On asymptotically Minkowski spacetimes, one can choose $\Sigma$ to reduce
asymptotically to a plane with radial coordinate $r$, and identify
$\partial\Sigma$ with a sphere of constant radius $R\rightarrow\infty$. If one
imposes that $\varepsilon$ asymptotically reduces to a constant spinor
$\varepsilon_{0}$ up to ${\cal O}(1/r)$ terms, then $I(\varepsilon)$ coincides
with the Nester–Witten energy [30], which is still valid in the presence of
additional matter coupled to the minimal gravity multiplet through an energy-
momentum tensor $T^{\text{(mat)}}_{\mu\nu}$. More precisely, in an appropriate
normalization, we can make the identification

$I(\varepsilon)=-k_{0}^{\mu}P_{\mu}\,,$ (3.6)

where $P_{\mu}$ is the four-momentum of the system and

$k^{\mu}\equiv\bar{\varepsilon}\gamma^{\mu}\varepsilon\quad\Rightarrow\quad
k_{0}^{\mu}\equiv\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}\,.$ (3.7)

In particular, $I(\varepsilon)=I(\varepsilon_{0})$: it only depends on the
asymptotic value of the spinor. Note that here we are using a Majorana spinor
$\varepsilon$ (as for instance in [6]), while in [30, 2] a complex
$\varepsilon$ is used. This implies that in our case $k^{\mu}$ is a future-
pointing null vector. However, since $\varepsilon_{0}$ is arbitrary,
$k_{0}^{\mu}$ spans the entire future light-cone of the asymptotic flat space.
This means that, if $I(\varepsilon)\geq 0$ for any $\varepsilon_{0}$, then
$P^{\mu}$ must be necessarily non-spacelike and future-pointing. In
particular, the energy is non-negative in any frame.

In order to prove that $I(\varepsilon)\geq 0$ if $T^{\text{(mat)}}_{\mu\nu}$
satisfies the dominant energy condition, we now assume that $\Sigma$ is a
regular space-like surface as in [2, 30]. (Horizons may be included as in [47]
and the discussion may be generalized to null surfaces $\Sigma$ following
[48].) Then by using Stokes’ theorem and the identity
$\bar{\varepsilon}\gamma_{\mu\nu\rho}D^{\nu}D^{\rho}\varepsilon=G_{\mu\nu}k^{\nu}$
(which we will show in detail in sec. 4.1), we can write (3.4) as follows:

$\displaystyle I(\varepsilon)$ $\displaystyle=\int_{\Sigma}{\rm
d}*E_{(2)}=\int_{\Sigma}\nabla^{\nu}E_{\mu\nu}*{\rm
d}x^{\mu}=\int_{\Sigma}\left(D^{\nu}\bar{\varepsilon}\gamma_{\mu\nu\rho}D^{\rho}\varepsilon+\frac{1}{2}G_{\mu\nu}k^{\nu}\right)*{\rm
d}x^{\mu}$ (3.8) $\displaystyle\overset{\textrm{on-
shell}}{=}\int_{\Sigma}D^{\nu}\bar{\varepsilon}\gamma_{\mu\nu\rho}D^{\rho}\varepsilon~{}n^{\mu}~{}\textrm{vol}_{\Sigma}+\frac{1}{2}\int_{\Sigma}T_{\mu\nu}^{\textrm{(mat)}}k^{\nu}n^{\mu}~{}\textrm{vol}_{\Sigma}\,.$

In the last line we have rewritten $*{\rm
d}x^{\nu}|_{\Sigma}=\textrm{vol}_{\Sigma}n^{\nu}$, where $n^{\nu}$ is a time-
like unit vector orthogonal to the surface $\Sigma$ and
$\textrm{vol}_{\Sigma}$ is the induced volume form on $\Sigma$, and we have
imposed Einstein’s equations $G_{\mu\nu}=T_{\mu\nu}^{\textrm{(mat)}}$. The
last term of (3.8) is manifestly non-negative if
$T_{\mu\nu}^{\textrm{(mat)}}k^{\nu}n^{\mu}\geq 0$ for any null future-directed
vector $k^{\mu}$ and then, by linearity, for any non-space-like vector
$k^{\mu}$. This is precisely the definition of the dominant energy condition,
which we are assuming. In order to prove that $I(\varepsilon)\geq 0$, it then
remains to show that also the first term in the last line of (3.8) is non-
negative.

At this point it is convenient to pick an adapted vielbein
$e^{{A}}=e^{{A}}_{\mu}{\rm d}x^{\mu}=(e^{\underline{0}},e^{a})$ (with spatial
flat indices $a\in\\{\underline{1},\underline{2},\underline{3}\\}$) and the
dual frame $e_{{A}}=e_{{A}}^{\mu}\partial_{\mu}=(e_{\underline{0}},e_{a})$
such that $e^{\underline{0}}|_{\Sigma}=0$ and then $n^{\mu}\equiv
e_{\underline{0}}^{\mu}$. The first term in the last line of (3.8) can then be
decomposed into:

$\int_{\Sigma}D^{a}\bar{\varepsilon}\gamma_{\underline{0}ab}D^{b}\varepsilon\textrm{vol}_{\Sigma}=\int_{\Sigma}\left(D^{a}\varepsilon\right)^{\dagger}\left(D_{a}\varepsilon\right)\textrm{vol}_{\Sigma}-\int_{\Sigma}|\gamma^{a}D_{a}\varepsilon|^{2}\textrm{vol}_{\Sigma}.$
(3.9)

The first contribution is manifestly positive; the second vanishes upon
imposing the _Witten condition_ :

$\gamma^{a}D_{a}\varepsilon=0.$ (3.10)

This has a natural interpretation from the supergravity viewpoint [5]: it can
be regarded as following from a “transverse” gauge choice
$\gamma^{a}\psi_{a}=0$ [49, 50] on the gravitino. In other words, the last
term in (3.9) can be associated with “longitudinal” unphysical degrees of
freedom, which can be gauged away.

Let us for the moment assume that (3.10) can be solved for any choice of the
spatial surface $\Sigma$ and asymptotic $\varepsilon_{0}$. Combining (3.8) and
(3.9), we can then conclude that $I(\varepsilon)\geq 0$. Moreover, if
$I(\varepsilon)=0$, $D_{a}\varepsilon$ needs to vanish; if this is true for
any $\varepsilon_{0}$, by varying $\Sigma$ we find four constant spinors on
our spacetime, which implies that it is Minkowski. Since the only spacetime
where $I(\varepsilon)=0$ for any $\varepsilon_{0}$ is Minkowski, and
$I(\varepsilon)$ is conserved (it does not depend on $\Sigma$), Minkowski
space cannot evolve into anything else: it is stable.

The remaining key step is then to prove that (3.10) always admits a solution
for the given boundary condition
$\varepsilon\to\varepsilon_{0}+\mathcal{O}\left(1/r\right)$. This question can
be addressed at various levels of mathematical rigor [2, 33, 47], but for our
purposes we will just focus on the most crucial requirement: the elliptic
operator $\gamma^{a}D_{a}$ must have no normalizable zero modes. If this
holds, $\gamma^{a}D_{a}$ can be inverted into a Green’s operator. We can then
start with any trial $\varepsilon_{1}$ asymptotic to $\varepsilon_{0}$ and
find a normalizable solution $\varepsilon_{2}$ to
$\gamma^{a}D_{a}\varepsilon_{2}=-\gamma^{a}D_{a}\varepsilon_{1}$; now
$\varepsilon_{1}+\varepsilon_{2}$ solves (3.10) [2, 33]. The absence of a
normalizable zero mode $\varepsilon_{\rm zm}$ of $\gamma^{a}D_{a}$ can be
understood as follows. Such an $\varepsilon_{\rm zm}$ would decrease
sufficiently fast to make $I(\varepsilon_{\rm zm})$ as defined in (3.4)
vanish. But then, by rewriting $I(\varepsilon_{\rm zm})$ as in (3.8) and using
$\gamma^{a}D_{a}\varepsilon_{\rm zm}=0$ inside the identity (3.9), we would
get

$\int_{\Sigma}\left(D^{a}\varepsilon_{\rm
zm}\right)^{\dagger}\left(D_{a}\varepsilon_{\rm
zm}\right)+\frac{1}{2}\int_{\Sigma}T_{\mu\nu}^{\textrm{(mat)}}k_{\rm
zm}^{\nu}n^{\mu}~{}\textrm{vol}_{\Sigma}=0\,.$ (3.11)

This identity is clearly impossible unless $D_{a}\varepsilon_{\rm zm}=0$ for
any $\Sigma$, and then $\varepsilon_{\rm zm}\equiv 0$ (since $\varepsilon_{\rm
zm}$ must vanish asymptotically). We will also apply the same kind of argument
to more complicated settings.

###  3.2 $\mathrm{AdS}_{4}$ stability

The logic outlined in the previous subsection can be adapted to AdS vacua in
theories of gravity and supergravity [3, 4, 5]. The simplest, “minimal” model
is obtained by adding a constant superpotential $W_{0}$, which generates a
negative cosmological constant $\Lambda=-3|W_{0}|^{2}$ (in Planck units).
Without loss of generality, we assume that $W_{0}$ is real and positive, so
that $W_{0}=\sqrt{-\Lambda/3}\equiv 1/L$, where $L$ is the AdS radius. In
global coordinates, the AdS metric reads

${\rm d}s^{2}_{\text{AdS}_{4}}=-\left(1+\frac{\rho^{2}}{L^{2}}\right){\rm
d}t^{2}+\left(1+\frac{\rho^{2}}{L^{2}}\right)^{-1}{\rm d}\rho^{2}+\rho^{2}{\rm
d}\Omega^{2}\,;$ (3.12)

at $\rho\to\infty$, the rescaled $\rho^{-2}{\rm d}s^{2}_{\text{AdS}_{4}}$
induces on the boundary the metric of $\mathbb{R}\times S^{2}$. The formula
(3.3) for the supercharge is unchanged while the gravitino transformation is
modified into $\delta_{\varepsilon}\psi_{\mu}={\cal D}_{\mu}\varepsilon$,
where

${\cal D}_{\mu}\equiv D_{\mu}+\frac{1}{2}W_{0}\gamma_{\mu}\,.$ (3.13)

Recall that an $\varepsilon$ annihilated by ${\cal D}_{\mu}$ is said to be a
Killing spinor. AdS has the maximal number of such spinors: they behave
asymptotically as [7, (3.17)] $\varepsilon\sim\rho^{1/2}\varepsilon_{0}$, with
$\varepsilon_{0}$ projecting on the boundary to a conformal Killing spinor of
$\mathbb{R}\times S^{2}$.

Changing $D\to{\cal D}$ everywhere in the argument (3.8)–(3.10) in the
previous subsection now again establishes in this minimal model that
$I(\varepsilon)\geq 0$, and that $I(\varepsilon)=0$ only in the AdS vacuum.
The same procedure can be repeated for non-minimal gauged supergravities [4]:
the only difference is the appearance of additional contributions to
$I(\varepsilon)$, also manifestly non-negative, containing the supersymmetry
variations of additional fermions in the supersymmetric multiplets. (See also
[51, 52] for applications to the calculation of BPS bounds for non-vacuum
states in ${\cal N}=2$ gauged supergravities.) However, the different global
structure of AdS changes the rest of the argument. The presence of the
$\mathbb{R}\times S^{2}$ timelike boundary makes the spacetime not globally
hyperbolic — one cannot predict the future from the data on a Cauchy slice
alone, but rather one also needs to know what happens at spatial infinity. A
priori this might make it unclear whether $I(\varepsilon)$ is conserved. A
natural boundary condition for the metric is to impose that $\rho^{-2}{\rm
d}s^{2}$ be conformal to $\mathbb{R}\times S^{2}$ at the boundary, as happens
for the AdS vacuum. In [32] it was shown (in a pure gravity model) that with
this boundary condition, $I(\varepsilon)$ is equal to the conserved charge
${\cal E}(k_{0})$ associated by the covariant phase space formalism [53] to
the asymptotic isometry
$k_{0}^{\mu}=\bar{\varepsilon}_{0}\gamma^{\mu}\varepsilon_{0}$. The latter are
conformal Killing vectors of $\mathbb{R}\times S^{2}$, and together they
generate its conformal isometry group ${\rm SO}(2,3)$. Moreover, ${\cal
E}(k_{0})$ does not depend on the space slice $\Sigma$ (i.e., it is
conserved).444A detailed comparison of $I(\varepsilon)$ to alternative notions
of energy in asymptotically AdS spacetimes is also given in [32]; see also
[54] for a review.

That the asymptotically supersymmetric $\varepsilon$ can satisfy the Witten
condition (now $\gamma^{a}{\cal D}_{a}\varepsilon=0$) was shown in a slightly
more general context in [34, Sec. V]. So we have in fact several non-negative
conserved quantities, to which we can apply the argument in the previous
subsection.

The covariant phase space method can also be applied to supersymmetry itself,
in which case it reproduces once again the expression (3.3) for the
supercharge $Q(\varepsilon)$ [55]. The formalism is built so that the
conserved charge is a Hamiltonian generator for the associated symmetry; from
the (super-)Jacobi identity it follows that $Q(\varepsilon)$ and the ${\rm
SO}(2,3)$ generators $J_{{{AB}}}$ together form a superalgebra, as expected.
In particular [3, 4, 5]

$I(\varepsilon)=\frac{1}{2}J_{{{AB}}}\,\bar{\varepsilon}_{0}\sigma^{{{AB}}}\varepsilon_{0}\,,$
(3.14)

where $\sigma^{ab}\equiv\frac{1}{2}\gamma^{ab}$ and
$\sigma^{a4}=\frac{1}{2}\gamma^{a}$. The AdS energy $E_{\rm AdS}\equiv J_{04}$
is obtained by tracing (3.14) over a basis of $\varepsilon_{0}$; it vanishes
only when both $T_{\mu\nu}^{\textrm{(mat)}}\equiv 0$ and
$D_{\mu}\varepsilon\equiv 0$, i.e., maximally supersymmetric AdS is the unique
$E_{\rm AdS}=0$ configuration of these models.

###  3.3 Breaking supersymmetry

As we mentioned at the beginning, the arguments in this section are robust
enough that they don’t need supersymmetry: the spinors $\varepsilon$ can be
auxiliary variables, unrelated to any fermionic symmetry. Indeed, even the
first application [2, 30] was to pure Einstein gravity. One can follow this
idea to give stability arguments both for vacua in non-supersymmetric
theories, and for supersymmetry-breaking vacua in supersymmetric ones.

A systematic analysis was initiated in [6] (and generalized to $d>4$ in [56]).
To illustrate the idea, consider a model with a single scalar $\phi$ and
action

$\frac{1}{2}\int{\rm
d}^{4}x\sqrt{-g}\left(R-\partial_{\mu}\phi\partial^{\mu}\phi-2V(\phi)\right)\,.$
(3.15)

There is no supersymmetry, but we may nevertheless consider spinors
$\varepsilon$ that satisfy ${\cal D}^{\prime}_{\mu}\varepsilon=0$, with

${\cal D}^{\prime}_{\mu}\equiv D_{\mu}+\frac{1}{2}W(\phi)\gamma_{\mu}\,,$
(3.16)

generalizing (3.13). Trying to prove an analogue of (3.8), one finds that it
works if

$V=2(\partial_{\phi}W)^{2}-3W^{2}\,.$ (3.17)

This has the same structure of an ${\cal N}=1$ model; but one may also look
for such a $W$ in absence of supersymmetry, in which case it is known as a
fake superpotential.

The discussion of boundary conditions for $\phi$ is quite intricate; see for
example [34]. As is well known, near the boundary it behaves like
$\alpha_{-}\rho^{-\Delta_{-}}+\alpha_{+}\rho^{-\Delta_{+}}$, where
$\Delta_{-}\leq\Delta_{+}$ are the two roots of $\Delta(\Delta-d+1)=m^{2}$.
For “fast” boundary conditions $\alpha_{-}=0$, the BPS energy again is equal
to the conserved energy from the covariant phase space formalism, but for
choices of the type $\alpha_{+}=f(\alpha_{-})$ the two differ. In fact there
are in general two local solutions $W_{-}\leq W_{+}$ to (3.17) for each given
$V$ around the vacuum; $W_{+}$ always exists globally, but when
$\alpha_{-}\neq 0$ the BPS energy associated to it is infinite. So one has to
use $W_{-}$, but its global existence is guaranteed only for some potentials.

###  3.4 Some comments

We close this section with some remarks.

  * •

Even though the identification of the positive definite $I(\varepsilon)$
exploits the supersymmetric structure, the stability argument regards only
purely bosonic configurations. This is not really an issue at the classical
level, since non-vanishing fermionic profiles do not admit a classical
interpretation. Hence, the above stability arguments are complete as long as
we are interested in classical instabilities or semiclassical ones mediated by
Coleman–de Luccia instantonic bubbles [8, 41]. Indeed, the portion of
spacetime created after nucleation of a bubble can be considered as a
localized classical excitation of the Mink4 or AdS4 vacuum, preserving the
appropriate boundary conditions.

  * •

These four-dimensional arguments can be applied to string/M-theory models as
long as the four-dimensional theory is a consistent EFT and the possible decay
processes are describable within the EFT regime of validity. But many examples
of string/M-theory compactifications to AdS4 do not admit such an EFT
description, since for instance the AdS scale $\sqrt{-\Lambda}$ and the KK
scale are of the same order. In such cases, there often exist some consistent
truncations which lead to extended gauged supergravities of the kind
considered in [4]. However, any conclusion on the stability of the vacuum
based on these consistent truncations is inconclusive.

  * •

To our knowledge, the models considered so far in the literature do not
include possible charged membranes, which for instance catalyse the kinds of
decays discussed in section 2 (in the probe approximation). Such membranes
could be incorporated into a four-dimensional ${\cal N}=1$ EFT as in [57, 58]
and should be taken into account. Moreover, in the absence of scale
separation, such membranes should actually be regarded as higher dimensional
branes as in section 2.2.

In the next section we will address some of the open issues raised in the
second and third items by explicitly showing how the above stability arguments
can be uplifted to string/M-theory. Moreover, we will see how various
(backreacting) branes can be taken into account, hence providing a first
important set of examples on how such extended charged objects can be
consistently incorporated into these kinds of arguments.

##  4 Positivity and stability in M-theory

We now move on to the (expected) stability of supersymmetric string/M-theory
supergravity backgrounds, including possible localized sources. In this
section we will consider the M-theory case, while in the next we will discuss
type II theories. The extension of our results to other string theories should
be straightforward.

Some useful properties of eleven-dimensional supergravity are reviewed in App.
B, in which we also fix our conventions. Here we just recall that the bulk
fields are the metric $g_{{{MN}}}$, a three-form potential $C$ with closed
four-form field-strength $G=\mathrm{d}C$, and a Majorana gravitino
$\psi_{{{M}}}$. Furthermore, we will denote the eleven-dimensional gamma
matrices by $\Gamma_{{{M}}}$, and we will work in units $\ell_{\textrm{\tiny
P}}=1$, in which the supergravity Einstein–Hilbert term is $2\pi\int R*1$ and
the M2-brane tension is $2\pi$.

Let us follow the same steps of section 3. First, one can again apply the
Noether procedure reviewed in App. B.2, to get the following expression for
the supercharge:

$Q(\epsilon)=-\int_{\partial S}\bar{\epsilon}\,\Gamma_{(8)}\wedge\psi\,,$
(4.1)

where (commuting) Majorana spinor $\epsilon$ defines an arbitrary
supersymmetry transformation and $\partial S$ is the boundary of a ten-
dimensional hypersurface $S$. We are using the intrinsic notation where
$\psi\equiv\psi_{{{M}}}\mathrm{d}x^{{{M}}}$ and

$\Gamma_{(p)}\equiv\frac{1}{p!}\Gamma_{{{M_{1}\cdots
M_{p}}}}\mathrm{d}x^{{{M_{1}}}}\wedge\cdots\wedge\mathrm{d}x^{{{M_{p}}}}\,.$
(4.2)

The supersymmetry transformation of the gravitino reads

$\delta\psi_{{{M}}}={\mathcal{D}}_{{{M}}}\epsilon+{\cal O}(\psi^{2})\,,$ (4.3)

where $\epsilon$ is a Majorana spinor and

${\mathcal{D}}_{{{M}}}\equiv
D_{{{M}}}+\frac{1}{12}(G\Gamma_{{{M}}}-G_{{{M}}})=D_{{{M}}}+\frac{1}{24}(-\Gamma_{{{M}}}G+3G\Gamma_{{{M}}})\,,$
(4.4)

with $D_{{{M}}}$ being the usual spinorial covariant derivative. In this
expression, $G$ should actually be written as
$G_{/}\equiv\frac{1}{4!}G_{{{MNPQ}}}\Gamma^{{{MNPQ}}}$ (often also denoted by
$\not{G}$), the bispinor associated to the four-form $G$ under the Clifford
map / in (A.2). To get cleaner spinorial equations, we will often omit the
slash symbols, hoping it will be clear from context whether we are referring
to a form or to the associated bispinor. For instance, in (4.4) we also have
$G_{{{M}}}\equiv(\iota_{{{M}}}G)_{/}=\frac{1}{3!}G_{{{MNPQ}}}\Gamma^{{{NPQ}}}$
– again see App. B for more details on our notation.

The argument outlined in section 3.1 leads to the following natural candidate
for a positive definite quantity associated with $\epsilon$:

$I(\epsilon)\equiv\int_{\partial S}*E_{2}$ (4.5)

where

$E_{2}\equiv-\frac{1}{2}\,\bar{\epsilon}\,\Gamma_{{{MN}}}{}^{{{P}}}{\mathcal{D}}_{{{P}}}\epsilon\,\mathrm{d}x^{{{M}}}\wedge\mathrm{d}x^{{{N}}}\,.$
(4.6)

This two-form is the eleven dimensional counterpart of (3.5).

These formulas have already been exploited together with the usual quantum
supersymmetry argument to formally argue for some BPS bounds as in [59] – see
also [60] for a more recent discussion in terms of calibrations which is more
directly relevant for what follows. For this reason, we will refer to (4.5) as
the BPS energy, having in mind a non-negative linear combination of energy and
possible central charges that vanishes when supersymmetry is preserved.
However, even though it is expected, positivity of the BPS energy (4.5) has
not been proved to date. This is precisely one of our main goals.

In order to prove positivity, as in the four-dimensional case we assume that
$S$ is a regular space-like hypersurface and use Stokes’ theorem

$I(\epsilon)=\int_{S}{\rm
d}*E_{2}=\int_{S}\,\nabla^{{{M}}}E_{{{NM}}}\,*\mathrm{d}x^{{{N}}}\,.$ (4.7)

Horizons and asymptotically null hypersurfaces $S$ may be also considered
following [61] and [48] respectively, but we will not do it here. On the other
hand, we will allow for M2 and M5 branes, showing how to properly take into
account their backreaction.

###  4.1 The main identity

Our next crucial step is to compute the divergence appearing in (4.7). We
start by presenting the final result:

$\framebox{$\nabla_{{M}}E^{{{MN}}}=\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}_{{P}}\epsilon+\frac{1}{2}{\cal
E}^{{{NP}}}K_{{P}}+\frac{1}{4}\bar{\epsilon}[\mathrm{d}x^{{N}}\wedge(-{\rm
d}G+{\rm d}*G+\frac{1}{2}G\wedge G)]_{/}\epsilon\,,$}$ (4.8)

where we have introduced the future-pointing causal vector

$K\equiv
K^{{M}}\partial_{{M}}=\bar{\epsilon}\,\Gamma^{{M}}\epsilon\,\partial_{{M}}\,,$
(4.9)

and ${\mathcal{E}}_{{{MN}}}$ denotes the bulk contribution to the Einstein
equations:

$\displaystyle{\mathcal{E}}_{{{MN}}}$ $\displaystyle\equiv
R_{{{MN}}}-\frac{1}{2}g_{{{MN}}}R-\frac{1}{2}T_{{{MN}}}^{{{(G)}}}\,$ (4.10)
$\displaystyle\quad~{}~{}~{}\text{where}\quad~{}~{}~{}~{}T_{{{MN}}}^{{{(G)}}}\equiv
G_{{M}}\cdot G_{{N}}-\frac{1}{2}|G|^{2}g_{{{MN}}}\,.$

Note that (4.8) holds identically for any off-shell configuration of
$g_{{MN}}$ and $G$, in the sense that one does not need to impose either the
equations of motion or the Bianchi identities.

While until here our logic followed quite closely the steps in section 3, the
proof of (4.8) is quite specific to M-theory. We present here the main steps
of the argument, leaving several details to App. B.3. We begin with

$\begin{split}\nabla_{{M}}E^{{{MN}}}=&-\nabla_{{M}}(\bar{\epsilon}\Gamma^{{{MNP}}}{\mathcal{D}}_{{P}}\epsilon)=-\overline{D_{{M}}\epsilon}\Gamma^{{{MNP}}}{\mathcal{D}}_{{P}}\epsilon+\bar{\epsilon}\,\Gamma^{{{NMP}}}D_{{M}}{\mathcal{D}}_{{P}}\epsilon\\\
=&-\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MNP}}}{\mathcal{D}}_{{P}}\epsilon-\frac{1}{24}\bar{\epsilon}\,A^{{{NP}}}D_{{P}}\epsilon\\\
&-\bar{\epsilon}\,\left[\Gamma^{{{MNP}}}\left(D_{{M}}D_{{P}}+\frac{1}{24}[D_{{M}},-\Gamma_{{P}}G+3G\Gamma_{{P}}]\right)+\frac{1}{24^{2}}Q^{{N}}\right]\epsilon\,,\end{split}$
(4.11)

where

$\begin{split}A^{{{NP}}}&=(-G\Gamma_{{M}}+3\Gamma_{{M}}G)\Gamma^{{{MNP}}}-\Gamma^{{{NPM}}}(-\Gamma_{{M}}G+3G\Gamma_{{M}})\,,\\\
Q^{{N}}&=(-G\Gamma_{{M}}+3\Gamma_{{M}}G)\Gamma^{{{MNP}}}(-\Gamma_{{P}}G+3G\Gamma_{{P}})\,.\end{split}$
(4.12)

We show in App. B.3 that $A^{{{NP}}}=0$. The first term on the last line of
(4.11) can be evaluated recalling that the commutator of two spinorial
covariant derivatives involves the Riemann tensor, and some relatively
standard gamma matrix identities:

$\displaystyle\Gamma^{{{MNP}}}[D_{{N}},D_{{P}}]$
$\displaystyle=\frac{1}{4}R^{{{AB}}}{}_{{{NP}}}\Gamma^{{{MNP}}}\Gamma_{{{AB}}}=\frac{1}{4}R^{{{AB}}}{}_{{{NP}}}(\Gamma^{{{MNP}}}_{{{AB}}}+6\delta^{[{{M}}}_{{A}}\Gamma^{{{NP}}]}_{{B}}-6\delta^{[{{M}}}_{{A}}\delta^{\mathchoice{\raisebox{1.5pt}{$\displaystyle{{N}}$}}{\raisebox{1.5pt}{$\textstyle{{N}}$}}{\raisebox{1.5pt}{$\scriptstyle{{N}}$}}{\raisebox{1.5pt}{$\scriptscriptstyle{{N}}$}}}_{{B}}\Gamma^{\mathchoice{\raisebox{1.5pt}{$\displaystyle{{P}}]$}}{\raisebox{1.5pt}{$\textstyle{{P}}]$}}{\raisebox{1.5pt}{$\scriptstyle{{P}}]$}}{\raisebox{1.5pt}{$\scriptscriptstyle{{P}}]$}}})$
$\displaystyle=\left(R^{{{MN}}}-\frac{1}{2}Rg^{{{MN}}}\right)\Gamma_{{N}}\,.$
(4.13)

The commutator term on the last line in (4.11) is computed in the appendix as

$\Gamma^{{{NMP}}}[D_{{M}},-\Gamma_{{P}}G+3G\Gamma_{{P}}]=6(-\mathrm{d}x^{{N}}\wedge\mathrm{d}G+\iota^{{N}}*\mathrm{d}*G)_{/}\,.$
(4.14)

Finally we also show in App. B.3 that $Q^{{N}}$ evaluates precisely to the
quadratic terms in the equations of motion of the metric and $G$, leading to
(4.8).

###  4.2 Positivity in the absence of branes

We are now in a position to present our positivity argument, following the
four-dimensional case reviewed in section 3. In this subsection we make the
simplifying assumption that no M2 or M5 branes are present or can be
nucleated. The inclusion of branes will be discussed in the following
subsection.

In the absence of branes, the equations of motion and Bianchi identities of
the supergravity fields are

${\cal E}_{{{MN}}}=0\,,\qquad{\rm d}G=0\,,\qquad{\rm d}*G+\frac{1}{2}G\wedge
G=0\,.$ (4.15)

Hence, the identity (4.8) reduces on-shell to

$\nabla_{{M}}E^{{{MN}}}\ \overset{\textrm{on-shell}}{=}\
\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}_{{P}}\epsilon\quad~{}~{}~{}\text{(no
branes)}\,,$ (4.16)

and the BPS energy (4.7) becomes

$I_{0}(\epsilon)=\int_{S}\,\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MN}}}{}_{{P}}{\mathcal{D}}_{{N}}\epsilon\,n^{{P}}\text{vol}_{S}\,,$
(4.17)

where $\text{vol}_{S}$ is the volume form induced on $S$ and
$n=n^{{M}}\partial_{{M}}$ is a future-pointing unit vector orthogonal to $S$.
We can now proceed as in four dimensions, by using an adapted vielbein

$e^{{A}}=(e^{\underline{0}},e^{a})\quad\text{with}\quad
e^{\underline{0}}|_{S}=0\,,$ (4.18)

and the dual frame $e_{{A}}=(e_{\underline{0}},e_{a})$ with
$e_{\underline{0}}=n$. Now (4.17) can be rewritten as

$\displaystyle I_{0}(\epsilon)$
$\displaystyle=\int_{S}\text{vol}_{S}\,\left[({\mathcal{D}}^{a}\epsilon)^{\dagger}({\mathcal{D}}_{a}\epsilon)-|\Gamma^{a}{\mathcal{D}}_{a}\epsilon|^{2}\right]\,.$
(4.19)

By choosing an $\epsilon$ that satisfies the M-theory version of the Witten
condition

$\Gamma^{a}{\mathcal{D}}_{a}\epsilon=0\,,$ (4.20)

we see that the BPS energy is manifestly non-negative: $I_{0}(\epsilon)\geq
0$. Furthermore, $I_{0}(\epsilon)=0$ if and only if ${\cal D}_{a}\epsilon=0$.
Since $S$ can be freely deformed, this condition actually requires that ${\cal
D}_{{M}}\epsilon=0$. Hence, the BPS energy is vanishing if and only if
$\epsilon$ satisfies the supersymmetry equations.

Note that, as in the four-dimensional case, we can again associate (4.20) with
a transverse gauge-fixing condition $\Gamma^{a}\psi_{a}=0$ for the gravitino.
Furthermore, one can adapt the argument around (3.11) to argue that for any
$S$ the operator $\Gamma^{a}{\cal D}_{a}$ does not admit any normalizable zero
mode $\epsilon_{\rm zm}$. Indeed, by combining (4.19) with
$I_{0}(\epsilon_{\rm zm})=0$ and $\Gamma^{a}{\cal D}_{a}\epsilon_{\rm zm}=0$
one gets ${\cal D}_{a}\epsilon_{\rm zm}=0$. Since $\epsilon_{\rm zm}$ must
vanish as one approaches the boundary $\partial S$, we then expect that the
conditions ${\cal D}_{a}\epsilon_{\rm zm}=0$ impose that $\epsilon_{\rm
zm}\equiv 0$, at least for reasonable asymptotic structures. This supports the
expectation that (4.20) indeed admits a solution, although an explicit proof
of this result would require a more detailed case by case investigation which
is beyond the scope of the present paper.

###  4.3 Inclusion of M2 branes

We would now like to allow M2 or M5 branes to be present in the background
configuration, or to be generated by possible fluctuations around it.

We first assume the presence of only M2-branes. This means that the Bianchi
identity ${\rm d}G=0$ is still satisfied, while the bulk equations of motion
are modified. The bosonic action of an M2-brane is

$S_{\text{\tiny(M2)}}=-2\pi\int_{\mathcal{C}}{\rm
d}^{3}\sigma\sqrt{-h}+2\pi\int_{\mathcal{C}}C\,,$ (4.21)

where $h_{\alpha\beta}\equiv
g_{{{MN}}}\partial_{\alpha}X^{{M}}\partial_{\beta}X^{{N}}$ is the metric
induced on the world-volume ${\cal C}$ by the embedding
$\sigma^{\alpha}\mapsto X^{{M}}(\sigma)$. The WZ term in (4.21) provides a
localized source term to the $C$ equations of motion:

${\rm d}*G+\frac{1}{2}G\wedge G=\delta^{(8)}({\mathcal{C}})\,,$ (4.22)

where $\delta^{(8)}({\mathcal{C}})$ is a delta-like eight-form such that
$\int_{\cal C}C\equiv\int C\wedge\delta^{(8)}({\mathcal{C}})$ for any three-
form $C$.555More generically, for any $q$-dimensional submanifold $\Sigma$ of
a $d$-dimensional space, $\delta^{(d-q)}(\Sigma)$ is defined by
$\int_{\Sigma}\omega=\int\omega\wedge\delta^{(d-q)}(\Sigma)$ for any $q$-form
$\omega$. Analogously, the Nambu–Goto term of (4.21) enters the Einstein
equations by a localized source term $T^{{{MN}}}_{\text{\tiny(M2)}}$, such
that:

${\cal E}^{{{MN}}}=\frac{1}{2}T^{{{MN}}}_{\text{\tiny(M2)}}\,,$ (4.23)

where ${\cal E}^{{{MN}}}$ is defined in (4.10). In order to describe
$T^{{{MN}}}_{\text{\tiny(M2)}}$, it is convenient to pick an adapted vielbein

$e^{{A}}=(e^{\underline{\alpha}},e^{\tilde{a}})\quad~{}~{}~{}\text{with
$e^{\tilde{a}}|_{\cal C}=0$}\,,$ (4.24)

and the corresponding dual frame
$e_{{A}}=e_{{{A}}}^{{M}}\partial_{{M}}=(e_{\underline{\alpha}},e_{\tilde{a}})$,
with $A=0,\ldots,10$, $\alpha=0,1,2$ and $\tilde{a}=3,\ldots,8$. Then one can
verify that

$T^{{{MN}}}_{\text{\tiny(M2)}}=\eta^{\underline{\alpha\beta}}\,e_{\underline{\alpha}}^{{M}}e_{\underline{\beta}}^{{N}}*[e^{\underline{012}}\wedge\delta^{(8)}({\mathcal{C}})]\,,$
(4.25)

see App. C. Note that the right-hand side of (4.22) and (4.23) can be
immediately generalized to multiple M2-branes by summing over the
corresponding contributions. Hence, we can focus on a single M2-brane without
loss of generality.

By using (4.22) and (4.23) (together with ${\rm d}G=0$) in the identity (4.8),
we now get

$\displaystyle\nabla_{{M}}E^{{{MN}}}$ $\displaystyle\overset{\textrm{on-
shell}}{=}\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}_{{P}}\epsilon+\frac{1}{4}T^{{{NP}}}_{\text{\tiny(M2)}}K_{{P}}+\frac{1}{4}\bar{\epsilon}[{\rm
d}x^{{N}}\wedge\delta^{(8)}({\cal C})]\epsilon$ (4.26)
$\displaystyle\quad=\quad\overline{{\mathcal{D}}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}_{{P}}\epsilon+\frac{1}{4}T^{{{NP}}}_{\text{\tiny(M2)}}K_{{P}}+\frac{1}{4}\big{(}*\big{[}{\rm
d}x^{{N}}\wedge\delta^{(8)}({\mathcal{C}})\big{]}\big{)}\cdot\Omega^{\text{\tiny(M2)}}\,,$

where we have introduced the two-form

$\Omega^{\text{\tiny(M2)}}\equiv\bar{\epsilon}\,\Gamma_{(2)}\epsilon=\frac{1}{2}\bar{\epsilon}\,\Gamma_{{{MN}}}\epsilon\,{\rm
d}x^{{M}}\wedge{\rm d}x^{{N}}\,.$ (4.27)

By using (4.26) inside (4.7) we find that the total BPS energy can be split
into bulk and brane contributions:

$I(\epsilon)=I_{0}(\epsilon)+I_{\text{\tiny(M2)}}(\epsilon)\,.$ (4.28)

Here $I_{0}(\epsilon)$ is defined in (4.17) and represents the bulk
contribution. On the other hand, the brane contribution

$I_{\text{\tiny(M2)}}(\epsilon)=\frac{1}{4}\int_{{\mathcal{C}}\cap
S}\left(K^{\underline{0}}\,{\rm
vol}_{{\mathcal{C}}\cap\Sigma}-\Omega^{\text{\tiny(M2)}}\right)$ (4.29)

comes from the last two terms in (4.26) – see App. C for more details. In
(4.29) $K^{\underline{0}}=K^{{M}}e^{\underline{0}}_{{M}}$ is the zeroth
component of $K$ in the local frame (4.18), that is, $K^{\underline{0}}$ is
the component of $K$ normal to $S$.

In App. C we prove, following [31], that the integrated forms in (4.29)
satisfy the algebraic bound

$K^{\underline{0}}\,{\rm vol}_{{\mathcal{C}}\cap
S}\geq\Omega^{\text{\tiny(M2)}}|_{{\mathcal{C}}\cap S}\,,$ (4.30)

which can be regarded as a local BPS bound.666The bound (4.30) involves top-
forms on ${\cal C}\cap S$ and should more precisely be read as a bound for the
corresponding coefficients with respect to a given reference top-form, fixing
the overall sign ambiguity by requiring that $\text{vol}_{{\cal C}\cap S}$ is
positive. Hence we can conclude that

$I_{\text{\tiny(M2)}}(\epsilon)\geq 0$ (4.31)

and then the inclusion of fully backreacting M2-branes preserves the
positivity of BPS energy:

$I(\epsilon)\geq 0\,.$ (4.32)

Furthermore, remembering the conclusion of subsection 4.2, $I(\epsilon)=0$ if
and only if $\epsilon$ satisfies the supersymmetry equations and the bound
(4.30) is saturated:

$K^{\underline{0}}\,{\rm vol}_{{\mathcal{C}}\cap
S}=\Omega^{\text{\tiny(M2)}}|_{{\mathcal{C}}\cap S}\,.$ (4.33)

As explained in App. C, this condition is equivalent to requiring that the
M2-brane does not break the supersymmetry generated by $\epsilon$.

As in [31], the local BPS bound (4.30) provides a generalization to non-static
settings of the ordinary calibration bounds [62], which instead apply only to
static backgrounds. These local bounds are usually exploited to study the
energetics of probe branes on fixed supersymmetric vacua. Our result (4.28)
makes it clear how, even after the brane backreaction is taken into account
and more general on-shell bulk deformations are allowed, the combination
entering the local bound (4.30) still determines the M2 contribution
$I_{\text{\tiny(M2)}}(\epsilon)$ to the positive BPS energy $I(\epsilon)$.
Furthermore, we explicitly see how the inclusion of M2-branes cannot be
represented just by their energy-momentum tensor (4.25) (which satisfies the
dominant energy condition), since this would not take into account the second
(possibly negative) contribution in (4.29). We will explicitly show how these
observations also hold for other branes in string/M-theory, and we indeed
expect them to be universal properties of supergravity models including
charged branes.

###  4.4 Inclusion of M5-branes

Conceptually, the inclusion of M5-branes should proceed as for the M2 case. In
particular, we expect an M5 to contribute to the BPS energy with a non-
negative contribution $I_{\text{\tiny(M5)}}(\epsilon)$ analogous to (4.29).
This contribution may be computed along the lines of what we did for
M2-branes, starting from the M5 action of [63]. One technical difficulty comes
from the presence of the self-dual three-form on the M5-brane, which induces
an M2-charge on the M5.

In order to alleviate the presentation, we start by simply ignoring the self-
dual three-form contribution. Then the bosonic action associated with an
M5-brane wrapping a six-dimensional submanifold ${\mathcal{C}}$ reduces to

$S_{\text{\tiny(M5)}}=-2\pi\int_{\mathcal{C}}{\rm
d}^{6}\sigma\sqrt{-h}+2\pi\int_{\mathcal{C}}\tilde{C}+\ldots$ (4.34)

where $\tilde{C}$ is the dual six-form potential, locally defined by ${\rm
d}\tilde{C}=-(*G+\frac{1}{2}C\wedge G)$, and the ellipsis refers to the
omitted terms depending on the world-volume self-dual 3-form. The WZ term
contributes to the Bianchi identity for $G$, which becomes:

${\rm d}G=\delta^{(5)}({\mathcal{C}})\,.$ (4.35)

Also the Einstein equations get modified, by a contribution similar to the
M2-brane one appearing on the right-hand side of (4.23). By repeating the same
steps followed for the M2 case, we arrive at the following M5 contribution to
BPS energy:

$I_{\text{\tiny(M5)}}(\epsilon)=\frac{1}{4}\int_{{\mathcal{C}}\cap\Sigma}\left(K^{\underline{0}}\,{\rm
vol}_{{\mathcal{C}}\cap\Sigma}-\Omega^{\text{\tiny(M5)}}\right)\,,$ (4.36)

where

$\Omega^{\text{\tiny(M5)}}\equiv\bar{\epsilon}\,\Gamma_{(5)}\epsilon\,.$
(4.37)

We have again a calibration bound

$K^{\underline{0}}\,{\rm
vol}_{{\mathcal{C}}\cap\Sigma}\geq\Omega^{\text{\tiny(M5)}}|_{{\mathcal{C}}\cap\Sigma}\,,$
(4.38)

which is saturated by the supersymmetric configuration. This implies the
positivity bound

$I_{\text{\tiny(M5)}}(\epsilon)\geq 0\,,$ (4.39)

for the M5 contribution to the BPS energy.

Coming back to the self-dual 3-form, it induces an M2-charge on the M5-brane.
Hence, it affects the $C$ equations of motion and contributes to the Einstein
equations by further terms localized on the M5-brane. These can be
straightforwardly computed starting from the action of [63], but we will not
do this exercise here, since in section 5.4 we will do it for D-branes
supporting general world-volume fluxes. Since M5-branes and D-branes are
related by dualities, the results of section 5.4 are sufficient to conclude
that (4.39) indeed still holds once the M5 self-dual three-form flux is taken
into account, and is saturated only by supersymmetric configurations.

In summary, the total BPS energy (4.5) can be written as the sum of three non-
negative terms

$I(\epsilon)=I_{0}(\epsilon)+I_{\text{\tiny(M2)}}(\epsilon)+I_{\text{\tiny(M5)}}(\epsilon)\,,$
(4.40)

where $I_{0}(\epsilon)$ becomes manifestly non-negative if we impose the
Witten condition (4.20). Furthermore $I(\epsilon)=0$ if and only if the three
terms separately vanish. This can happen only if the complete configuration is
supersymmetric, that is, if $\epsilon$ is a bulk supercharge and the branes
preserve it.

###  4.5 Positivity and stability of $G_{2}$-compactifications

We can illustrate the above general results by discussing energy positivity
and stability in M-theory compactifications on special holonomy spaces. As we
have mentioned, this stability is widely expected, but checking it is a good
exercise towards attacking vacua without supersymmetry. The energy positivity
associated with these kinds of configurations has been already discussed in
[35, 36], which however focused on the gravitational sector (plus a possible
energy-momentum tensor satisfying the dominant energy condition). Our results
allow us to include the effects of the $G$ flux and of the M2/M5 branes as
well.

For concreteness we focus on vacua of the form $M_{0}=\mathbb{M}_{4}\times Y$,
where $\mathbb{M}_{4}$ is four-dimensional Minkowski space and $Y$ admits a
given $G_{2}$-holonomy metric. (Compactifications to $\mathbb{M}_{d\neq 4}$,
or on $Y$ with special holonomy $\neq G_{2}$, are analogous.) In these purely
geometrical vacua, the eleven-dimensional metric is a direct sum ${\rm
d}s^{2}_{0}={\rm d}s^{2}_{\mathbb{M}_{4}}+{\rm d}s^{2}_{Y}$, and preserves a
supercharge

$\epsilon_{0}=\varepsilon_{0}\otimes\eta_{0}$ (4.41)

where $\varepsilon_{0}$ is a constant four-dimensional Majorana spinor, while
$\eta_{0}$ is the covariantly constant Majorana spinor along the
$G_{2}$-manifold $Y$, which we choose to have unit norm
$\eta_{0}^{\dagger}\eta_{0}=1$. We are interested in general deformations
which asymptotically reduce to the $M_{0}=\mathbb{M}_{4}\times Y$ vacuum. On
$M_{0}$ we can introduce some adapted coordinates $(x^{\mu},y^{m})$, where
$y^{m}$ are coordinates along $Y$ and $x^{\mu}=(t,r,\theta,\phi)$ are standard
spherical coordinates over $\mathbb{M}_{4}$. Asymptotically, we can use these
coordinates also on the deformed backgrounds and choose the nine-dimensional
space-like surface $S$ to take the form $\Sigma\times Y$, where $\Sigma$ is an
asymptotically flat three-dimensional surface of constant time $t$.
Correspondingly, the nine-dimensional boundary $\partial S$ appearing in the
BPS energy (4.5) takes the form $\partial S=\partial\Sigma\times Y$, where
$\partial\Sigma$ is the two-sphere $S^{2}_{r}$ of constant radius $r$, in the
limit $r\rightarrow\infty$.

We impose boundary conditions where the eleven-dimensional metric and spinor
of the deformed background are asymptotically $g=g_{0}+\Delta g$ and
$\epsilon=\epsilon_{0}+\Delta\epsilon$, with $\Delta g$ and $\Delta\epsilon$
of order ${\cal O}(r^{-1})$ and such that ${\cal
D}^{0}_{{M}}\epsilon\equiv\nabla_{{M}}\Delta\epsilon\sim{\cal O}(r^{-2})$ (in
the natural asymptotically flat frame). In order to guarantee that (4.5) is
finite, we also assume that the variation of the spin-connection
$\Delta\omega^{{{AB}}}\equiv\Delta\omega_{{M}}{}^{{{AB}}}{\rm d}x^{{M}}$ and
of the field-strength $\Delta G\equiv G$ are at least of order ${\cal
O}(r^{-2})$. From the identity

$*E_{2}=-\bar{\epsilon}\Gamma_{(8)}\wedge{\mathcal{D}}\epsilon\,,$ (4.42)

with ${\mathcal{D}}\epsilon\equiv{\cal D}_{{M}}\epsilon\,{\rm d}x^{{M}}$, we
see that the integrand appearing in (4.5) reduces asymptotically to

$-\bar{\epsilon}_{0}\Gamma_{0(8)}\wedge({\mathcal{D}}-{\mathcal{D}}_{0})\epsilon_{0}+{\rm
d}(\ldots)+{\cal O}(r^{-3})\,.$ (4.43)

It follows that $I(\epsilon)$ can be written in the following form — see also
[60, Eq. (5.18)]:

$\displaystyle I(\epsilon)=$
$\displaystyle\,\frac{1}{4}\int_{\partial\Sigma\times Y}*(e^{{A}}\wedge
e^{{B}}\wedge K^{\flat})\wedge\Delta\omega_{{{AB}}}$ (4.44)
$\displaystyle+\frac{1}{4}\int_{\partial\Sigma\times
Y}\left(\Omega^{\text{\tiny(M2)}}\wedge*\Delta
G+\Omega^{\text{\tiny(M5)}}\wedge\Delta
G\,-2*\Omega^{\text{\tiny(M5)}}\wedge\Delta\omega\right)\,,$

where we have introduced the one-form $K^{\flat}\equiv K_{{M}}{\rm d}x^{{M}}$,
obtained by lowering the index of (4.9), and the three-form
$\Delta\omega\equiv\frac{1}{2}\Delta\omega_{{{AB}}}\wedge e^{{A}}\wedge
e^{{B}}$, while $\Omega^{\text{\tiny(M2)}}$ and $\Omega^{\text{\tiny(M5)}}$
are defined in (4.27) and (4.37) respectively. These can be expressed in terms
of the associative form $\phi$ and the co-associative form $*_{7}\phi$ on the
$G_{2}$ manifold as:

$K^{\flat}=v,\,\qquad\Omega^{\textrm{\tiny{(M2)}}}=w,\,\qquad\Omega^{\textrm{\tiny{(M5)}}}=v\wedge*_{7}\phi+*_{4}w\wedge\phi,\,\qquad*\Omega^{\textrm{\tiny{(M5)}}}=w\wedge*_{7}\phi+*_{4}v\wedge\phi,$
(4.45)

where $v_{\mu}\equiv\bar{\varepsilon}\gamma_{\mu}\varepsilon$,
$w_{\mu\nu}\equiv\bar{\varepsilon}\gamma_{\mu\nu}\varepsilon$, and
$\varepsilon$ is a four dimensional Majorana spinor.

Note that in (4.44), the forms $K^{\flat}$, $\Omega^{\text{\tiny(M2)}}$ and
$\Omega^{\text{\tiny(M5)}}$, as well as the Hodge-star operator, can in fact
be computed by using the supersymmetric vacuum metric $g_{0}$ and supercharge
$\epsilon_{0}$. We can then identify the first two terms appearing in the
second line of (4.44) as measuring the asymptotic central charges associated
to M2-branes stretching along $\mathbb{M}_{4}$ and M5-branes wrapping internal
three- and four-cycles. Since we want to consider fluctuations around the
vacuum configurations, we can further restrict the boundary conditions in such
a way that these terms vanish.

Similarly, the three-form $\Delta\omega$ may be interpreted as measuring the
flux of a KK6-brane [59], and $*\Omega^{\text{\tiny(M5)}}$ can be considered
as a corresponding calibration — see section 5 of [60]. Then the last term of
(4.44) may be interpreted as measuring the central charge of a KK6-branes
wrapping internal four-cycles and does not contribute if we assume vacuum
boundary conditions.

On the other hand, the first term on the right-hand side of (4.44) can be
interpreted as defining the conserved component $-K^{{M}}P_{{M}}$ of the total
momentum of the system [30, 59]. By using the fact that
$K^{{M}}=(k^{\mu}_{0},0)$, we see that

$I(\epsilon)=I(\varepsilon_{0})=-k^{\mu}_{0}P_{\mu}$ (4.46)

as in (3.6). Our previous discussion on the positivity of $I(\epsilon)$ then
translates into a fully eleven-dimensional proof of the positivity of the
energy of this class of compactifications.

Moreover, if $I(\epsilon)=0$ for any $\varepsilon_{0}$, spacetime coincides
with the vacuum. To see this, first observe that, just as in the pure four-
dimensional case, the combination of (4.19), (4.20) and $I(\epsilon)=0$ imply
(when applied to all possible $\Sigma$) that there are four independent
spinors $\epsilon$ such that ${\cal D}_{{M}}\epsilon=0$. This in turn implies
that there are four independent Killing vectors
$\bar{\epsilon}\Gamma^{{M}}\epsilon$. These coincide at infinity with the four
translations of Minkowski space; so their orbits need to be four-dimensional.
Since the finite transformation generated by a Killing vector is an isometry,
the metric is invariant along these four-dimensional orbits; so on the union
of these orbits the metric is in fact equal to its value at infinity
$\mathbb{M}_{4}\times Y$.

We now use the usual logic: since $I(\epsilon)$ does not depend on $\Sigma$,
it is conserved; but since it vanishes for any $\varepsilon_{0}$ only on the
vacuum, the latter is fully stable. As usual, this conclusion assumes the
existence of a solution of the Witten condition (4.20). In the absence of
fluxes and branes, this has been rigorously proven in [35, 36]. As at the end
of subsections 3.1 and 4.2, one could argue that $\Gamma^{a}{\cal D}_{a}$ does
not have any normalizable zero-mode $\epsilon_{\rm zm}$.

Finally, note that we may in fact relax the above restrictions on the
asymptotic behaviour of $G$ and $*G$ and allow for possible non-trivial
asymptotic fluxes thereof. In this case the positivity of (4.44) would
correspond to an extremality bound as in [64, 61], which would include the
contribution of four-dimensional string and membrane central charges.

###  4.6 Stability of $\mathrm{AdS}_{4}$ vacua

Again in the spirit of gaining experience that might be later useful for
supersymmetry-breaking vacua, we can adapt the discussion of the previous
subsection to M-theory compactifications down to AdSd. For $d=4$, the vast
majority of such supersymmetric vacua are of the famous Freund–Rubin (FR)
type:

${\rm
d}s^{2}_{0}=\mathrm{d}s^{2}_{\mathrm{AdS}_{4}}+4L^{2}\mathrm{d}s^{2}_{Y}\,,\qquad
G_{4}=\frac{3}{L}\mathrm{vol}_{\mathrm{AdS}_{4}}\,.$ (4.47)

The AdS radius is $L^{6}=\frac{N}{3\cdot 2^{7}\mathrm{Vol}(Y)}$, where $N$ is
a positive integer; the internal space $Y$ admits an internal Killing spinor
$\eta$, satisfying $(D_{m}-\frac{{\rm i}}{2}m\gamma_{m})\eta=0$; these include
weak $G_{2}$ ($\mathcal{N}=1$), Sasaki–Einstein manifolds ($\mathcal{N}=2$),
3-Sasaki ($\mathcal{N}=3$). The total supercharge is then
$\epsilon_{0}=\hat{\varepsilon}\otimes\eta$, where
$\hat{\varepsilon}(x)=S(x)\varepsilon_{0}$ is one of AdS Killing spinors
mentioned after (3.13). We will modify these solutions in section 6.2 to break
supersymmetry.

In order to address the stability issue, one again needs to specify
appropriate boundary conditions. The discussion in sections 3.2 and 3.3
suggests that we should require the metric at infinity to factorize as ${\rm
d}s^{2}_{4}+{\rm d}s^{2}_{7}$, with $\rho^{-2}{\rm d}s^{2}_{4}$ inducing on
the boundary a metric conformal to $\mathbb{R}\times S^{2}$, and ${\rm
d}s^{2}_{7}=4L^{2}{\rm d}s^{2}_{Y}+{\cal O}(\rho^{-2})$. Moreover, the spinors
$\epsilon$ are such that $K^{{M}}=\bar{\epsilon}\Gamma^{{M}}\epsilon$ is
asymptotically $(k_{0}^{\mu}=\bar{\varepsilon}\gamma^{\mu}\varepsilon,0)$.
These boundary conditions should be confirmed by uplifting the analysis in
[32, 34, 55] to the case with extra dimensions. In particular one should check
that the BPS energy is still conserved with this choice, and that the Witten
condition can be imposed; we hope to return to this in the future.

The logic then proceeds along lines that are by now familiar. We choose a
space-like surface $S$ that asymptotically takes the form $\Sigma\times Y$,
with $\Sigma$ being a space-like three-dimensional slice with boundary
$\partial\Sigma\simeq S^{2}$. The BPS energy (4.5) is defined by an integral
over $\partial S=\partial\Sigma\times Y\simeq S^{2}\times Y$. We may also
apply (4.44) and provide a more explicit geometrical formula for
$I(\epsilon)$, but the supersymmetry argument which led from (4.1) to (4.5)
anyway guarantees that $I(\epsilon)=I(\varepsilon_{0})$ must take the form
(3.14) in terms of the ${\rm SO}(2,3)$ charges $J_{{{AB}}}$. As in subsection
3.2, this implies the positivity of $E\equiv J_{04}\geq 0$. A version of the
argument in the previous subsection shows that the only zero-energy
configuration is the vacuum AdS${}_{4}\times Y$. As usual, if the energy is
conserved it now follows that such a vacuum is stable.777It was found in [65]
that small perturbations in AdS can refocus until they collapse into a black
hole; this is a different notion of stability from the one of the vacuum we
consider in this paper.

##  5 Positivity and stability in type II

In this section we will show that the main points of the above M-theory
discussion hold also for ten-dimensional type II string theories, up to some
technical but interesting details.

The type II fields are: the string frame metric $g_{{{MN}}}$; a two-form
potential $B$ with closed three-form field-strength $H=\mathrm{d}B$; RR
potentials $C_{k}$ with field-strengths $F_{k}=\mathrm{d}C_{k}-H\wedge
C_{k-2}$; two Majorana–Weyl (MW) gravitinos $\psi_{{{M}}}^{a}$; two MW
dilatinos $\lambda^{a}$. It will be convenient to use the democratic
formalism, where all RR field-strengths $F_{k}$ are included for all $0\leq
k\leq 10$, even in IIA and odd in IIB, and the formal total sum
$F\equiv\sum_{k}F_{k}$ satisfies

$F=*\lambda F\,.$ (5.1)

Here $\lambda$ acts on a form by reversing the order its indices, i.e.,
$\lambda F_{k}=(-)^{\frac{k(k-1)}{2}}F_{k}$, and the Hodge-star is computed in
the string frame, which is more natural for the democratic formulation. The
theory is defined by a pseudo-action [66], whose equations of motion must be
supplemented by the self-duality conditions (5.1).

The supersymmetry parameters are two MW spinors $\epsilon_{a}$, $a=1,2$. It is
convenient to collect all the spinors into doublets:
$\psi_{{M}}\equiv\binom{\psi_{1{{M}}}}{\psi_{2{{M}}}}$, and so on. The
supersymmetry transformations of the fermions, with fermions set to zero, now
read

$\delta_{\epsilon}\psi_{{M}}={\mathcal{D}}_{{M}}\epsilon\,,\qquad\delta_{\epsilon}\lambda={\mathcal{O}}\epsilon\,,$
(5.2)

where888Our conventions are as in [60]. They differ from those in [67] by a
sign change $H\rightarrow-H$.

$\begin{split}{\mathcal{D}}_{{M}}&\equiv
D_{{M}}\otimes\mathbf{1}_{2}-\frac{1}{4}H_{{M}}\otimes\sigma_{3}+{\mathcal{F}}\,\Gamma_{{M}}\,,\\\
{\mathcal{O}}&\equiv\mathrm{d}\phi\otimes\mathbf{1}_{2}-\frac{1}{2}H\otimes\sigma_{3}+\Gamma^{{M}}{\mathcal{F}}\,\Gamma_{{M}}\,,\\\
&\quad\text{with}\quad{\mathcal{F}}\equiv\frac{\mathrm{e}^{\phi}}{16}\left(\begin{smallmatrix}0&F\\\
\pm\lambda(F)&0\end{smallmatrix}\right)\
\text{in}\,\quad\overset{\text{\tiny{IIA}}}{\text{\tiny{IIB}}}\,.\end{split}$
(5.3)

Here and in what follows, the upper/lower sign will refer to IIA/IIB. We work
in string units $2\pi\sqrt{\alpha^{\prime}}=1$.

###  5.1 BPS energy

In order to identify the BPS energy, we will start from the supercharge
generator, following the same strategy used in M-theory. The explicit form of
the supercharge can be obtained by various means. For instance, it can be
identified in IIA by dimensional reduction of the M-theory supercharge (4.1) –
see App. E – and then extrapolated to IIB. The result is

$\begin{split}Q(\epsilon)=\int_{\partial
S}\mathrm{e}^{-2\phi}\,\left(\mathrm{d}x^{{M}}\wedge\overline{\epsilon}\,\Gamma\Gamma_{(7)}\psi_{{M}}+\overline{\epsilon}\,\Gamma\Gamma_{(8)}\lambda\right)\,,\end{split}$
(5.4)

again in the notation of (4.2). In (5.4), $S$ is a nine-dimensional spacelike
surface and we have introduced the chiral operator
$\Gamma\equiv\Gamma^{\underline{01\ldots 9}}$. 999 The supercharge (5.4)
admits a simpler form in terms of the Einstein frame gravitino. Denoting the
Einstein frame quantities with a hat symbol $\hat{\ }$, they are related to
those in the string frame by $g_{{{MN}}}=\mathrm{e}^{\phi/2}\hat{g}_{{{MN}}}$,
$\epsilon=\mathrm{e}^{\phi/8}\hat{\epsilon}$,
$\lambda=\mathrm{e}^{-\phi/8}\hat{\lambda}$,
$\psi_{{M}}=\mathrm{e}^{\phi/8}\left(\hat{\psi}_{{M}}+\frac{1}{8}\hat{\Gamma}_{{M}}\hat{\lambda}\right)$.
Then (5.4) takes the form $Q(\hat{\epsilon})=\int_{\partial
S}\mathrm{d}x^{{M}}\wedge\overline{\hat{\epsilon}}\,\Gamma\hat{\Gamma}_{(7)}\hat{\psi}_{{M}}$.

We can then compute
$\delta_{\epsilon}Q(\epsilon)=\\{Q(\epsilon),Q(\epsilon)\\}$ by using (5.2),
and obtain in this way the BPS energy:

$\displaystyle I(\epsilon)$ $\displaystyle=\int_{\partial S}*E_{2}$ (5.5)

where we have introduced the Nester-like two form

$E_{2}\equiv-\frac{1}{2}\mathrm{e}^{-2\phi}\bar{\epsilon}(\Gamma_{{{MN}}}{}^{{P}}{\mathcal{D}}_{{P}}-\Gamma_{{{MN}}}{\mathcal{O}})\epsilon\,{\rm
d}x^{{{M}}}\wedge{\rm d}x^{{{N}}}\,.$ (5.6)

More explicitly, we can write the components of
$E_{2}=\frac{1}{2}E_{{{MN}}}\mathrm{d}x^{{M}}\wedge\mathrm{d}x^{{N}}$ as
follows:

$E_{{{MN}}}=-\mathrm{e}^{-2\phi}\bar{\epsilon}\,\Gamma_{{{MN}}}{}^{{P}}(D_{{P}}+{\mathcal{A}}_{{P}})\epsilon\,,$
(5.7)

where $D_{{M}}$ is the ordinary spinor covariant derivative and
${\mathcal{A}}_{{M}}$ is defined by the relation

${\cal D}_{{M}}-\frac{1}{8}\Gamma_{{M}}{\cal O}=D_{{M}}+{\cal A}_{{M}}\,.$
(5.8)

In the following we will also need the identity

$\Gamma^{{P}}{}_{{{MN}}}{\mathcal{A}}_{{P}}=\frac{1}{4}\Gamma_{[{{M}}}H\Gamma_{{{N}}]}\otimes\sigma_{3}-\Gamma_{{{MN}}}\mathrm{d}\phi-2\Gamma_{[{{M}}}{\mathcal{F}}\,\Gamma_{{{N}}]}\,.$
(5.9)

This identification can be obtained from (5.7) by using (A.1) with $k=2$ and
$H_{{M}}=\frac{1}{2}\\{\Gamma_{{M}},H\\}$.

###  5.2 Main identity

Once again we need to compute the divergence of $E_{2}$. We begin with

$\displaystyle\mathrm{e}^{2\phi}$
$\displaystyle\nabla_{{M}}E^{{{MN}}}=-2\partial_{{M}}\phi
E^{{{MN}}}-\overline{D_{{M}}\epsilon}\,\Gamma^{{{MNP}}}{\mathcal{A}}_{{P}}\epsilon-\bar{\epsilon}\,D_{{M}}(\Gamma^{{{MNP}}}{\mathcal{A}}_{{P}}\epsilon)$
(5.10)
$\displaystyle=-\overline{D_{{M}}\epsilon}\Gamma^{{{MNP}}}(D_{{P}}\epsilon+2{\mathcal{A}_{{P}}}\epsilon)-\bar{\epsilon}\,\Gamma^{{{MNP}}}(D_{{M}}D_{{P}}-2\partial_{{M}}\phi{\mathcal{A}}_{{P}}+[D_{{M}},\Gamma^{{{MNP}}}{\mathcal{A}}_{{P}}])\epsilon\,.$

We have skipped a few steps similar to (4.11), and we have used (A.14) for
$k=2$ to
derive$\bar{\epsilon}\,\Gamma^{{{MNP}}}{\mathcal{A}}_{{P}}D_{{M}}\epsilon=\overline{D_{{M}}\epsilon}\Gamma^{{{MNP}}}({\mathcal{A}}_{{P}}+2\partial_{{P}}\phi)\epsilon$.

The term $\bar{\epsilon}\,(\cdots)D_{{M}}\epsilon$ now does not vanish
automatically (unlike $A^{{{NP}}}$ in (4.12)). But by (5.8) we expect the
appearance of the square of this operator. The remainder looks at first quite
complicated, but a little experimentation shows that it simplifies if we
subtract a further
$\overline{{\mathcal{O}}\epsilon}\Gamma^{{N}}{\mathcal{O}}\epsilon$. In order
to write the final result in the most transparent way, we first use the spinor
$\epsilon$ to construct the future-pointing time-like or null vector

$K\equiv\frac{1}{2}\,\bar{\epsilon}\,\Gamma^{{M}}\epsilon\,\partial_{{M}}$
(5.11)

and the (poly)forms:

$\displaystyle\Omega^{\text{\tiny(F1)}}\equiv\frac{1}{2}\,\bar{\epsilon}\,\Gamma_{(1)}\otimes\sigma_{3}\epsilon\quad,\quad\Omega^{\text{\tiny(NS5)}}\equiv\frac{1}{2}\mathrm{e}^{-2\phi}\,\bar{\epsilon}\,\Gamma_{(5)}\otimes\sigma_{3}\epsilon\,,$
(5.12a) $\displaystyle\Omega^{\text{\tiny(D)}}=\sum_{k\
\text{even/odd}}\Omega^{\text{\tiny(D)}}_{k}\equiv\sum_{k}\mathrm{e}^{-\phi}\,\bar{\epsilon}_{1}\Gamma_{(k)}\epsilon_{2}\,.$
(5.12b)

Then, with some further computations presented in App. D, we are able to show
that

$\framebox{$\begin{aligned}
\nabla_{{M}}E^{{{MN}}}=&\mathrm{e}^{-2\phi}\overline{\left({\mathcal{D}}_{{M}}-\frac{1}{8}\Gamma_{{M}}{\mathcal{O}}\right)\epsilon}\,\Gamma^{{{MPN}}}\left({\mathcal{D}}_{{P}}-\frac{1}{8}\Gamma_{{P}}{\mathcal{O}}\right)\epsilon-\frac{1}{8}\mathrm{e}^{-2\phi}\overline{{\mathcal{O}}\epsilon}\,\Gamma^{{N}}{\mathcal{O}}\epsilon\\\
+&{\cal E}^{{{NP}}}K_{{P}}+\frac{1}{2}{\cal
H}^{{{NP}}}\Omega^{\text{\tiny(F1)}}_{{P}}-\frac{1}{2}(\mathrm{d}H\wedge{\rm
d}x^{{N}})\cdot\Omega^{\text{\tiny(NS5)}}+\frac{1}{2}(\mathrm{d}_{H}F\wedge{\rm
d}x^{{N}})\cdot\Omega^{\text{\tiny(D)}}\,.\end{aligned}$}$ (5.13)

Here ${\rm d}_{H}\equiv{\rm d}-H\wedge$ and the tensors ${\cal E}_{{{MN}}}$
and ${\cal H}_{{{MN}}}$ are defined by

$\displaystyle\mathrm{e}^{2\phi}{\cal E}_{{{MN}}}$ $\displaystyle\equiv
R_{{{MN}}}-\frac{1}{2}g_{{{MN}}}R+2[(\nabla_{{M}}\nabla_{{N}}-g_{{{MN}}}\nabla^{2})\phi+g_{{{MN}}}|\mathrm{d}\phi|^{2}]-\frac{1}{2}T^{{{(HF)}}}_{{{MN}}}\,,$
(5.14a) $\displaystyle{\cal H}_{2}$ $\displaystyle=\frac{1}{2}{\cal
H}_{{{MN}}}{\rm d}x^{{M}}\wedge{\rm
d}x^{{N}}\equiv*\left[\mathrm{d}(\mathrm{e}^{-2\phi}*H)-\frac{1}{2}(F\wedge\lambda
F)_{8}\right]\,,$ (5.14b)

where $T^{{{(HF)}}}_{{{MN}}}\equiv H_{{M}}\cdot
H_{{N}}-\frac{1}{2}g_{{{MN}}}|H|^{2}+\frac{1}{2}\mathrm{e}^{2\phi}F_{{M}}\cdot
F_{{N}}$. Here we are using a notation similar to that in (4.10), e.g.
$H_{{M}}\equiv\iota_{{M}}H$ and $F_{{M}}\equiv\iota_{{M}}F$. Furthermore
$(\cdots)_{8}$ denotes taking the eight-form part of the polyform.

We emphasize that (5.13) is valid for both type II theories and holds
identically for any (possibly off-shell) configuration.

###  5.3 Positivity in the absence of branes

As in section 4.2, we first consider the positivity of the BPS energy (5.5) in
the absence of branes (and orientifolds). In this case the Einstein and
$B$-field equations of motion require that ${\cal E}_{{{MN}}}=0$ and ${\cal
H}_{{{MN}}}=0$, respectively. Furthermore the $B$-field Bianchi identity is
${\rm d}H=0$ and the democratic RR equations of motion/Bianchi identities read
${\rm d}_{H}F=0$.101010In order to get an on-shell configuration, these
equations must be supplemented by the dilaton equation of motion
$R-\frac{1}{2}|H|^{2}-4\mathrm{e}^{\phi}\nabla^{2}\mathrm{e}^{\phi}=0$. Then
only the first line in (5.13) survives; the second vanishes by the theory’s
equations of motion and Bianchi identities.

If we again introduce an adapted vielbein $e^{{A}}=(e^{\underline{0}},e^{a})$
such that $e^{\underline{0}}|_{S}=0$ and use Stokes’ theorem, we can write
(5.5) as $I_{0}(\epsilon)=\int_{S}{\rm
d}*E_{2}=\int_{S}\text{vol}_{S}\nabla_{{M}}E^{\underline{0}M}$. By using
(5.13) we then get the following on-shell BPS energy:

$\displaystyle I_{0}(\epsilon)$
$\displaystyle=\int_{S}\text{vol}_{S}\,\left[\left({\mathcal{D}}^{a}\epsilon-\frac{1}{8}\Gamma^{a}{\mathcal{O}}\epsilon\right)^{\dagger}\left({\mathcal{D}}_{a}\epsilon-\frac{1}{8}\Gamma_{a}{\mathcal{O}}\epsilon\right)+\frac{1}{8}({\mathcal{O}}\epsilon)^{\dagger}{\mathcal{O}}\epsilon-\left|\left(\Gamma^{a}{\mathcal{D}}_{a}-\frac{9}{8}{\mathcal{O}}\right)\epsilon\right|^{2}\right]\,.$
(5.15)

So, the BPS energy $I_{0}(\epsilon)$ can be made manifestly non-negative if we
can choose an $\epsilon$ satisfying the modified Witten identity:

$\left(\Gamma^{a}{\mathcal{D}}_{a}-\frac{9}{8}{\mathcal{O}}\right)\epsilon=0\,.$
(5.16)

Hence, under this assumption we get $I_{0}(\epsilon)\geq 0$. Furthermore this
bound is saturated if and only if the background is supersymmetric.

Note that, as in the previous cases, the condition (5.16) can be related to a
gauge-fixing condition for the gravitino [5], which now reads
$\Gamma^{a}\psi_{a}=\frac{9}{8}\lambda$. This does not look like a standard
transverse gauge, but only because we are working in the string frame, in
which the gravitino and dilatino kinetic terms are not diagonal. Indeed the
condition $\Gamma^{a}\psi_{a}=\frac{9}{8}\lambda$ corresponds to the
transverse gauge $\Gamma^{a}\hat{\psi}_{a}=0$ in the Einstein frame – see
footnote 9. Furthermore, one can argue for the absence of normalizable zero
modes of the operator appearing in (5.16), and then for the existence of a
solution of (5.16), by adapting the arguments proposed at the end of sections
3.1 and 4.2.

###  5.4 Inclusion of localized sources

As in the M-theory case, one can include branes, which can be part of the
background configuration or can contribute to the fluctuations around it. As
we will see, the presence of O-planes can be similarly taken into account.

The simplest example is provided by F1-strings, which works very similarly to
the M2-brane case of section 4.3. In particular, the presence of an F1-string
along a two-dimensional world-sheet ${\cal C}$ modifies the $B$-field
equations of motion to

${\cal H}_{2}=*\delta^{(8)}({\cal C})\,,$ (5.17)

as well as the Einstein equations. The bottom line is that the first two terms
in the second line of (5.13) do not vanish anymore, but provide the following
localized contribution to $I(\epsilon)=\int_{S}{\rm
d}*E_{2}=\int_{S}\text{vol}_{S}\nabla_{{M}}E^{{{\underline{0}M}}}$:

$I_{\text{\tiny(F1)}}(\epsilon)=\frac{1}{2}\int_{{\cal C}\cap
S}\left(K^{\underline{0}}\,\text{vol}_{{\cal
C}\cap\Sigma}-\Omega^{\text{\tiny(F1)}}\right)\,.$ (5.18)

One can adapt the calculation of App. C.2 for M2-branes to rewrite the local
bound found in [31, Eq. (3.25)] in the form

$K^{\underline{0}}\text{vol}_{{\cal C}\cap
S}\geq\Omega^{\text{\tiny(F1)}}|_{{\cal C}\cap S}\,.$ (5.19)

This shows that (5.18) is non-negative. Furthermore, (5.19) is saturated if
and only if the F1-string does not break the supersymmetry generated by
$\epsilon$ [31]. Hence, we get the bound
$I(\epsilon)=I_{0}(\epsilon)+I_{\text{\tiny(F1)}}(\epsilon)\geq 0$ for the
total BPS energy, which is saturated if and only if the complete configuration
is supersymmetric. For comparison with the D-brane discussion below, we
observe that we can rewrite (5.18) in the form

$I_{\text{\tiny(F1)}}(\epsilon)=\frac{1}{2}\int_{{\cal C}\cap
S}\Big{[}{V}(K)\text{vol}_{{\cal C}\cap
S}-\Omega^{\text{\tiny(F1)}}\Big{]}\,,$ (5.20)

where we have introduced the following one-form defined along ${\cal C}$:

${V}\equiv
h^{\alpha\beta}g_{{{MN}}}e_{\alpha}^{\underline{0}}\partial_{\beta}X^{{M}}{\rm
d}x^{{N}}\,,\text{ section of }T^{*}M|_{\cal C}\,,$ (5.21)

with $M$ denoting the ten-dimensional spacetime. (Recall that, according to
our general notation, $h_{\alpha\beta}$ denotes the induced metric on the
brane.) This equivalence can be understood by picking a ${\cal C}$ adapted
vielbein $e^{{A}}=(e^{\underline{\alpha}},e^{\hat{a}})$, with
$e^{\hat{a}}|_{\cal C}=0$ (and still imposing $e^{\underline{0}}|_{S}=0$).

As with the discussion for M5-branes in section 4.4, the incorporation of
D-branes and NS5-branes is complicated by the presence of additional world-
volume fluxes, but these branes are still expected to provide non-negative
localized contributions $I_{\text{\tiny(D)}}\geq 0$ and
$I_{\text{\tiny(NS5)}}\geq 0$ respectively. As a preliminary easier check, let
us consider an NS5-brane along a submanifold ${\cal C}$ and neglect again its
world-volume flux, as we did for M5-branes in section 4.4. The Einstein
equations and the $B$-field Bianchi identity get modified – for instance, the
latter becomes ${\rm d}H=\delta^{(4)}({\cal C})$ – and the first and third
term of the second line of (5.13) produce a non-negative contribution to the
BPS energy of the form

$I_{\text{\tiny(NS5)}}(\epsilon)=\frac{1}{2}\int_{{\cal C}\cap
S}\left(e^{-2\phi}K^{{{\underline{0}}}}\,\text{vol}_{{\cal C}\cap
S}-\Omega^{\text{\tiny(NS5)}}\right)\geq 0\,,$ (5.22)

which vanishes if and only if the (fluxless) NS5-branes preserve the bulk
supersymmetry generated by $\epsilon$.

In order to better understand the possible role of world-volume fluxes, we now
discuss in more detail the presence of D-branes. Since they are related to
NS5-branes and M5-branes by dualities, our conclusions will provide a
convincing support to some of our previous claims.

Consider the inclusion of a D$p$-brane. In the democratic formulation, it
modifies the RR Bianchi identities/equations of motion as follows:

${\rm d}_{H}F=-\lambda\left[\delta^{(9-p)}({\cal
C})\right]\wedge\mathrm{e}^{-{\cal F}}$ (5.23)

where ${\cal C}$ now denotes the D$p$-brane world-volume and ${\cal F}$ is the
gauge invariant world-volume field strength, such that ${\rm d}{\cal
F}=-H|_{\cal C}$. Furthermore, by using the D-brane effective action

$S_{\rm{{D}}}=-2\pi\int_{\cal C}\mathrm{e}^{-\phi}\sqrt{-\det{\cal
M}}+2\pi\int_{\cal C}C\wedge\mathrm{e}^{{\cal F}}$ (5.24)

where ${\cal M}_{\alpha\beta}\equiv h_{\alpha\beta}+{\cal F}_{\alpha\beta}$
and $C\equiv\sum_{k}C_{k}$, one can check that the Einstein equations and the
$B$-field equations of motion get modified into

$\displaystyle{\cal E}^{{{MN}}}$
$\displaystyle=\frac{1}{4}e^{-\phi}\sqrt{-\det{\cal
M}_{\underline{\alpha\beta}}}\,{\cal
M}^{(\underline{\alpha\beta})}e^{{{M}}}_{\underline{\alpha}}e^{{{N}}}_{\underline{\beta}}*[e^{\underline{0\ldots
p}}\wedge\delta^{(9-p)}({\cal C})]\,,$ (5.25a) $\displaystyle{\cal
H}^{{{MN}}}$ $\displaystyle=-\frac{1}{2}e^{-\phi}\sqrt{-\det{\cal
M}_{\underline{\alpha\beta}}}\,{\cal
M}^{[\underline{\alpha\beta}]}e^{{{M}}}_{\underline{\alpha}}e^{{{N}}}_{\underline{\beta}}*[e^{\underline{0\ldots
p}}\wedge\delta^{(9-p)}({\cal C})]\,,$ (5.25b)

where we are using the adapted vielbein
$e^{{A}}=(e^{\underline{\alpha}},e^{\hat{a}})$ such that $e^{\hat{a}}|_{{\cal
C}}=0$.

As in [31], it is useful to combine $K$ and $\Omega^{\text{\tiny(F1)}}$ into
the ‘generalized vector’

${\cal K}\equiv K+\Omega^{\text{\tiny(F1)}}\,,\text{ section of }TM\oplus
T^{*}M\,.$ (5.26)

Along ${\cal C}$, we can also introduce another generalized vector field:

${\cal V}\equiv\mathrm{e}^{-\phi}\frac{\sqrt{-\det{\cal M}}}{\sqrt{-\det
h}}\left({\cal M}^{(\alpha\beta)}\partial_{\alpha}X^{{M}}\,g_{{{MN}}}{\rm
d}x^{{N}}+{\cal
M}^{[\alpha\beta]}\partial_{\alpha}X^{{M}}\,\partial_{{M}}\right)e^{\underline{0}}_{\beta}\,,$
(5.27)

which is a section of $(TM\oplus T^{*}M)|_{\cal C}$ and can be regarded as the
D-brane counterpart of (5.21). By using (5.23) and (5.25), we see that the
first, second and fourth terms in the second line of (5.13) are non-vanishing
in presence of the D$p$-brane, which then contributes to
$I(\epsilon)=\int_{S}\text{vol}_{S}\nabla_{{M}}E^{{{\underline{0}M}}}$ by

$\displaystyle I_{\text{\tiny(D)}}(\epsilon)=\frac{1}{2}\int_{{\cal C}\cap
S}\Big{[}{\cal V}({\cal K})\,{\rm d}\text{vol}_{{\cal C}\cap
S}-\Omega^{\text{\tiny(D)}}\wedge\mathrm{e}^{{\cal F}}\Big{]}\,,$ (5.28)

where ${\cal V}({\cal K})$ denotes the natural pairing between the generalized
vectors ${\cal V}$ and ${\cal K}$:

${\cal V}({\cal K})=\mathrm{e}^{-\phi}\frac{\sqrt{-\det{\cal M}}}{\sqrt{-\det
h}}\left({\cal M}^{(\alpha\beta)}K_{{M}}\partial_{\alpha}X^{{M}}+{\cal
M}^{[\alpha\beta]}\Omega^{\text{\tiny(F1)}}_{{M}}\partial_{\alpha}X^{{M}}\right)e^{{{\underline{0}}}}_{\beta}\,.$
(5.29)

Note the formal analogy between (5.28) and (5.20). Now, adapting again the
steps of App. C.2, one can rewrite the local bound (4.33) of [31] in the form

${\cal V}({\cal K})\,{\rm d}\text{vol}_{{\cal C}\cap
S}\geq\left(\Omega^{\text{\tiny(D)}}\wedge\mathrm{e}^{{\cal F}}\big{|}_{{\cal
C}\cap S}\right)_{\rm top}\,,$ (5.30)

which is saturated precisely if the D-brane preserves $\epsilon$. Hence, as
expected, the complete D-brane contribution (5.28) is always non-negative and
vanishes only if the D-brane preserves $\epsilon$. By duality, this strongly
supports our claim that these properties hold also for the complete
$I_{\text{\tiny(NS5)}}$ (and $I_{\text{\tiny(M5)}}$).

We then conclude that, if we choose an $\epsilon$ satisfying (5.16), the total
BPS energy

$I(\epsilon)=I_{0}(\epsilon)+I_{\text{\tiny(F1)}}(\epsilon)+I_{\text{\tiny(D)}}(\epsilon)+I_{\text{\tiny(NS5)}}(\epsilon)$
(5.31)

satisfies the bound

$I(\epsilon)\geq 0\,,$ (5.32)

which is saturated if and only if $\epsilon$ satisfies the bulk supersymmetry
equations and all the branes preserve $\epsilon$, that is, if and only if the
complete configuration is supersymmetric.

We would now like to consider the possible presence of orientifolds. Suppose
that $\iota:M\rightarrow M$ denotes the orientifold involution associated with
an O$p$-plane, located at the corresponding fixed locus ${\cal C}$. The
O$p$-plane contributes to the effective action as a D$p$-brane with a
tension/charge rescaled by a factor $-2^{p-5}$ and a vanishing ${\cal F}$,
which is consistent with the fact that the $B$-field is odd under the
orientifold involution and then has vanishing pull-back on the O$p$-plane. So,
the O$p$-plane contribution $I_{\text{\tiny(O)}}$ to the BPS energy takes the
form (5.28) with ${\cal F}=0$, up to an overall $-2^{p-5}$ factor.
Furthermore, the orientifold projection on the supersymmetry generator
$\epsilon\equiv\binom{\epsilon_{1}}{\epsilon_{2}}$ is

$\iota^{*}\epsilon_{2}=\epsilon_{1}\,,\qquad\iota^{*}\epsilon_{1}=(-)^{\lfloor\frac{p-1}{2}\rfloor}\epsilon_{2}\,,$
(5.33)

see for instance [68]. Hence, from (5.12b) we see that
$\Omega^{\text{\tiny(D)}}$ is invariant under the orientifold involution:

$\iota^{*}\Omega^{\text{\tiny(D)}}=\Omega^{\text{\tiny(D)}}\,.$ (5.34)

Furthermore, since the O$p$-plane world-volume ${\cal C}$ is fixed under the
orientifold involution, compatibility with (5.33) requires that along the
O$p$-plane one must impose the boundary condition

$(\epsilon_{1}-\Gamma_{\underline{0\ldots p}}\epsilon_{2})|_{\cal C}=0\,,$
(5.35)

where we are again using an adapted frame
$e^{{A}}=(e^{\underline{\alpha}},e^{\hat{a}})$ with $e^{\hat{a}}|_{\cal C}=0$.
This condition is precisely equivalent to requiring that the local bound
(5.30) (with ${\cal F}=0$) is saturated along the O$p$-plane. Hence
$I_{\text{\tiny(O)}}(\epsilon)\equiv 0$ and the BPS energy $I(\epsilon)$ is
still given by the non-negative combination (5.31).

We close this section by emphasising that so far we have neglected higher
order curvature corrections to the brane effective actions, which would modify
the form of the corresponding localized BPS energy.111111World-volume higher
curvature corrections play for instance and key role in the D-brane mediated
instabilities recently discussed in [27]. For instance, the curvature
corrections of [69, 70, 71] would modify the second term appearing in the
r.h.s. of (5.28) into something of the form $-\frac{1}{2}\int_{{\cal C}\cap
S}\Omega^{\text{\tiny(D)}}\wedge\mathrm{e}^{{\cal
F}}\wedge(\text{curvature})$. The expected local brane supersymmetry suggests
that ${\cal V}$ is also modified, so that $I_{\text{\tiny(D)}}(\epsilon)\geq
0$ still holds and is saturated precisely if the D-brane preserves $\epsilon$.
It would be interesting to better investigate this point, also in connection
with bulk higher-derivative corrections.

##  6 Supersymmetry breaking

In the previous sections, we clarified the mechanism that makes supersymmetric
vacua stable. We will now try to use this knowledge to provide stability
arguments for vacua that break supersymmetry. Our attempts will fail to
exhibit stable non-supersymmetric vacua, but fail in interesting ways that
might point the way to better future strategies.

###  6.1 An operator in M-theory

The idea is a bit similar to that in section 3.3. A supersymmetric vacuum is
defined by having a solution $\epsilon$ to ${\mathcal{D}}_{{M}}\epsilon=0$ in
M-theory and ${\mathcal{D}}_{{M}}\epsilon={\mathcal{O}}\epsilon=0$ in type II
(recalling (4.4), (5.3)). We would like to find a modification
${\mathcal{D}}^{\prime}_{{M}}$, ${\mathcal{O}}^{\prime}$ of these
supersymmetry operators such that

  * i)

an $\epsilon$ annihilated by them exists on some non-supersymmetric vacuum,
and

  * ii)

the BPS energy $I^{\prime}(\epsilon)$ defined via the analogues of (4.6),
(5.5) is still positive and related to the physical energy.

This is a hard set of requirements to satisfy, but let us try anyway. There
are infinitely many operators ${\mathcal{D}}^{\prime}_{{M}}$ that one might
consider, but the problem becomes more tractable if we demand only objects
with a single derivative to appear. As reviewed in App. B.5, the consistency
conditions $[{\mathcal{D}}_{{M}},{\mathcal{D}}_{{N}}]\epsilon=0$ for
supersymmetry imply most of the equations of motion. Having more than one
derivative in the modified ${\mathcal{D}}^{\prime}_{M}$ would imply more than
two derivatives in the consistency conditions
$[{\mathcal{D}}_{{M}}^{\prime},{\mathcal{D}}_{{N}}^{\prime}]\epsilon=0$, and
it might be hard to make these in turn compatible with the equations of
motion. Another restriction that seems reasonable is that, in order to avoid
gauge dependence, only the field-strengths appear and not the potentials.

In M-theory, the most general allowed operator obtained by deforming $D_{{M}}$
would now be121212One additional term we could add here is
$a_{4}\Gamma_{{M}}D$, with $D=\Gamma^{{N}}D_{{N}}$ the Dirac operator.
Consider $\Gamma^{{M}}{\mathcal{D}}^{\prime}_{{M}}\epsilon=0$ with this
modification; for $a_{4}\neq-1/11$, this equation determines $D\epsilon$,
which can be fed back into ${\cal D}^{\prime}_{{M}}\epsilon=0$ to obtain a new
equation without the new term. If $a_{4}=-1/11$, ${\cal D}^{\prime}_{{M}}$
contains the _conformal Killing operator_ $D_{{M}}-\frac{1}{d}\Gamma_{{M}}D$.
We will ignore this particular case in what follows.

${\mathcal{D}}^{\prime}_{{M}}=D_{{M}}+\frac{1}{24}(a_{1}\Gamma_{{M}}G+a_{2}G\Gamma_{{M}})+a_{3}\Gamma_{{M}}\,,$
(6.1)

where the $a_{i}$ are constants. Comparing with (the second expression in)
(4.4), we see that the supersymmetric case is recovered for $a_{1}=-1$,
$a_{2}=3$, $a_{3}=0$.

Notice that in particular we are not considering a modification of the type

${\mathcal{D}}^{\prime}_{{M}}={\mathcal{D}}_{{M}}+V_{{M}}\,.$ (6.2)

This would have the interesting feature that the two-form
$(E^{\prime}_{2})_{{MN}}=-\bar{\epsilon}\Gamma_{{{MN}}}{}^{{P}}{\mathcal{D}}^{\prime}_{{P}}\epsilon$
defined with this operator is in fact equal to the $E_{2}$ in (4.6); so the
positivity argument would still apply. Moreover solving
${\mathcal{D}}^{\prime}_{{M}}\epsilon=0$ would be superficially reminiscent of
$(D_{m}+{\rm i}A_{m})\eta_{+}=0$, which does have a solution on a Kähler
manifold (although with a crucial difference of an ${\rm i}$). However, such a
vector $V_{{M}}$ cannot be an external piece of data: for the positivity
argument to apply, it has to be defined for all possible configurations with a
certain boundary condition. So it has to be somehow defined in terms of the
fields. Given the requirements we listed at the beginning of this section,
there does not seem to be any such vector field, and hence we will ignore the
possibility (6.2).

###  6.2 Skew-whiffed and Englert vacua

We will first look at requirement i) above: namely, whether we can solve
${\mathcal{D}}^{\prime}_{{M}}\epsilon=0$ on any non-supersymmetric vacua.

Skew-whiffed vacua are obtained from a supersymmetric FR vacuum (section 4.6)
by reversing the orientation of the internal space, or in other words by
mapping $G\to-G$ [37, 38]. As reviewed in section 4.6, in a FR solution
AdS${}_{4}\times Y$ supersymmetry requires solving the internal Killing spinor
equation $(D_{m}-\frac{{\rm i}}{2}m\gamma_{m})\eta=0$ on $Y$. Reversing the
sign of $G$ flips the sign of $m$, and the new spinor equation now has no
solution (except when $Y=S^{7}$); so supersymmetry is broken. But the
equations of motion are still satisfied because they are quadratic in $G$. As
noted already in [37, 38], these solutions are automatically stable under
small perturbations.131313Further analysis of stability of these solutions
under squashing was carried out in [72, 73]. On these non-supersymmetric
solutions, we can clearly solve ${\mathcal{D}}_{{M}}^{\prime}\epsilon=0$ with

$a_{1}=1\,,\qquad a_{2}=-3\,,\qquad a_{3}=0\,;$ (6.3)

in other words, with the operator obtained from the supersymmetric
${\mathcal{D}}_{{M}}$ by reversing the sign of $G$.

We next consider the Englert vacua [39]. Here, $Y$ is a weak $G_{2}$ manifold,
namely one with a $G_{2}$-structure $\phi$ such that141414Another class of
solutions uses instead the ${\rm SU}(3)$-structure on a Sasaki–Einstein
manifold [74, 75, 76]. It would be interesting to apply our methods to this
case as well; we thank N. Bobev for the suggestion.

${\rm d}\phi=-4*_{7}\phi\,.$ (6.4)

This implies that the cone over $Y$ has $\mathrm{Spin}(7)$ holonomy, so $Y$
admits a Killing spinor $\eta_{0}$ with $m=1$. As a consequence, $Y$ is also
Einstein: more specifically $R_{mn}=6g_{mn}$. Taking the fields as

${\rm d}s^{2}_{11}=L^{2}{\rm d}s^{2}_{\mathrm{AdS}_{4}}+r_{0}^{2}{\rm
d}s^{2}_{Y}\,,\qquad G=g_{0}\mathrm{vol}_{\mathrm{AdS}_{4}}+g_{1}*_{7}\phi\,,$
(6.5)

the equations of motion have two branches of solutions: one with $g_{1}=0$,
leading back to FR, and one with

$g_{0}=\frac{9}{25}r_{0}^{3}\,,\qquad g_{1}^{2}=4r_{0}^{6}\,,\qquad
L=\sqrt{\frac{3}{10}}r_{0}\,.$ (6.6)

A perturbative instability was found for these vacua in [77], but only when
$Y$ has more than one Killing spinor. One can check from (6.5) that the brane
nucleation condition (2.6) is not satisfied for the easiest case of M2-branes
(as noted in [28] for some particular cases).

A solution of ${\mathcal{D}}_{{M}}^{\prime}\epsilon=0$ is found as
$\epsilon=\zeta_{+}\otimes\eta+\zeta_{-}\otimes\eta^{\mathrm{c}}$.
$\zeta_{-}=(\zeta_{+})^{\mathrm{c}}$, with c denoting Majorana conjugation; as
usual we assume the Killing spinor equation
$D_{\mu}\zeta_{\pm}=\frac{1}{2}\gamma_{\mu}\zeta_{\mp}$ also along AdS4; and
$\eta=\mathrm{e}^{{\rm i}\alpha}\eta_{0}$, with $\alpha$ a constant and
$\eta_{0}$ Killing. With these assumptions, both the internal and external
components of our equation become algebraic, and can be solved for

$a_{2}=\frac{3\pm\sqrt{114}}{10}\,,\qquad a_{1}=3-a_{2}\,,\qquad
a_{3}=\pm\frac{21-2a_{2}}{3}\,,\qquad\tan 2\alpha=\frac{3-2a_{2}}{4a_{2}}\,;$
(6.7)

the two signs are independent.

###  6.3 Lack of positivity

Having found some examples of solutions to
${\mathcal{D}}_{{M}}^{\prime}\epsilon=0$, we now look at requirement ii)
above, namely that one can still prove positivity and stability with the
modified operator ${\mathcal{D}}_{M}^{\prime}$.

In principle there are various terms that can ruin positivity. In (4.11), all
the terms beyond the first should now be re-examined. Already the second term
$\bar{\epsilon}\,A^{{{NP}}}D_{{P}}\epsilon$ is worrisome: it is unlikely to
have a fixed sign because of the derivative of $\epsilon$, so it had better
vanish.151515One might want to take care of this term by “completing the
square” to reabsorb it in the term $\overline{{\cal
D}^{\prime}_{M}\epsilon}\Gamma^{MPN}{\cal D}_{P}\epsilon$; but this requires
looking for a $X_{M}$ such that $X_{M}\Gamma^{MNP}=[G,\Gamma^{NP}]$; there
appears to be no natural choice of such an object. Repeating the steps in
(B.3) we see that in our present more general setting
$2A^{{{NP}}}=-3(3a_{1}+a_{2})[\Gamma^{{{NP}}},G]$. In particular this imposes

$a_{2}=-3a_{1}\,.$ (6.8)

We see from (6.3), (6.7) that the skew-whiffed vacua do satisfy this, while
the Englert vacua do not. While it is conceivable that a positivity property
might be proven by more ingenious methods without assuming (6.8), in the rest
of this subsection we will assume that it must hold.

With this assumption, defining
$E^{\prime}_{{{MN}}}=-\bar{\epsilon}\Gamma_{{{MN}}}{}^{{P}}{\mathcal{D}}^{\prime}_{{P}}\epsilon$
similar to (4.6), (4.8) is modified to

$\begin{split}\nabla_{{M}}E^{\prime{{MN}}}&=\overline{{\mathcal{D}}^{\prime}_{{M}}\epsilon}\Gamma^{{{MPN}}}{\mathcal{D}}^{\prime}_{{P}}\epsilon+\frac{1}{4}(2{\cal
G}^{{{NP}}}-a_{1}^{2}T^{{{NP}}}_{{{(G)}}})K_{{P}}\\\
&+\frac{1}{4}\bar{\epsilon}\left[\mathrm{d}x^{{N}}\wedge\left(a_{1}{\rm
d}G-a_{1}{\rm d}*G+\frac{1}{2}a_{1}^{2}G\wedge
G-12a_{1}a_{3}G-360a_{3}^{2}\right)\right]\epsilon\,,\end{split}$ (6.9)

where ${\cal G}_{{{MN}}}\equiv R_{{{MN}}}-\frac{1}{2}g_{{{MN}}}R$ is the usual
Einstein tensor; see App. B.4 for the details of this computation. In the
absence of branes, the terms multiplying $K_{{P}}$ can be reassembled using
the equations of motion as $(1-a_{1}^{2})T^{{{NP}}}_{{{(G)}}}$. By using (4.7)
we see that, if we assume $|a_{1}|\leq 1$, the contribution of this term to
$I^{\prime}(\epsilon)$ is positive because the flux stress-energy tensor obeys
the dominant energy condition.

Of more concern are the terms on the second line of (6.9). We can use the
equations of motion (4.15) to get rid of ${\rm d}G$ and ${\rm d}*G$, turning
the quadratic flux term into $\frac{1}{2}a_{1}(a_{1}+1)G\wedge G$. However,
there is no reason to believe this term to have a definite sign for all
solutions.161616Consider for example a solution where $K$ is timelike; here
$\epsilon$ will define an $\mathrm{SU}(5)$-structure $(J_{2},\Omega_{5})$ in
the remaining directions (studied in [78]), in terms of which the term of
interest is proportional to $J_{2}\wedge G\wedge G$, a top-form in $d=10$.
Choosing a holomorphic vielbein $h^{a}$, in terms of which $J_{2}=\frac{{\rm
i}}{2}\sum_{a}h^{a}\wedge\overline{h^{a}}$, it is easy to see that the forms
$G=2{\rm Re}(h^{1}\wedge h^{2}\wedge h^{3}\wedge\overline{h^{4}})$ and
$G=J_{2}\wedge J_{2}$ result in opposite signs for $J_{2}\wedge G\wedge G$.
Furthermore, if $a_{3}\neq 0$ we also have a linear $G$ term, which is even
more clearly of indefinite sign.

It is also interesting to consider the effect of localized sources. The same
steps leading to (4.29) now give a localized contribution to the BPS-energy

$I^{\prime}_{\text{\tiny(M2)}}(\epsilon)=\frac{1}{4}\int_{{\mathcal{C}}\cap
S}\left(K^{\underline{0}}\,{\rm
vol}_{{\mathcal{C}}\cap\Sigma}-|a_{1}|\Omega^{\text{\tiny(M2)}}\right)$ (6.10)

where the overall sign of the term including $\Omega^{\text{\tiny(M2)}}$ has
been fixed without loss of generality, since it can be changed by inverting
the M2 orientation, that is, by swapping the M2 for an anti-M2 brane, or
viceversa.

The sign of $\int_{{\mathcal{C}}\cap S}\Omega^{\text{\tiny(M2)}}$ depends on
the M2 orientation. Clearly $I^{\prime}_{\text{\tiny(M2)}}(\epsilon)$ is
manifestly positive if $\int_{{\mathcal{C}}\cap
S}\Omega^{\text{\tiny(M2)}}\leq 0$. If instead $\int_{{\mathcal{C}}\cap
S}\Omega^{\text{\tiny(M2)}}>0$, the second term on the r.h.s. of (6.10) is
negative. However, we can still use the algebraic local bound (4.30), which
implies that

$I^{\prime}_{\text{\tiny(M2)}}(\epsilon)\geq\frac{1}{4}(1-|a_{1}|)\int_{{\mathcal{C}}\cap
S}\Omega^{\text{\tiny(M2)}}\,.$ (6.11)

We then see that $I^{\prime}_{\text{\tiny(M2)}}(\epsilon)$ has a definite sign
if $|a_{1}|\leq 1$, which is the same condition that we found above. This
suggests that vacua can be unstable under M2 nucleation precisely if
$|a_{1}|>1$. Note that this conclusion does not apply to the skew-whiffed case
(in which $a_{1}=1$). A similar argument could be repeated for the non-
supersymmetric generalization of the M5 contribution (4.36). However, in this
case the neglected world-volume fluxes could affect the final conclusion.

We saw earlier that the Englert vacua don’t satisfy (6.8). Even if we ignore
this issue and repeat (6.9) with $a_{2}\neq-3a_{1}$, it turns out that the
sign of the $T^{{{NP}}}_{{{(G)}}}$ is negative; so this class of solutions is
unlikely to enjoy a positivity theorem by the spinorial strategy we are
considering.

The skew-whiffed vacua seem more promising in this respect, but they still
have a non-vanishing $G\wedge G$ term, which as we saw earlier does not have a
definite sign. So we again conclude that no positivity theorem exists with the
present spinorial method. This does not prove that these vacua are unstable,
but certainly seems to give evidence in that direction.

###  6.4 Type II

The supersymmetry transformations in M-theory already contain all the possible
terms that one can write with at most one derivative, except for one term;
this led us to (6.1).

In contrast, in the type II operators (5.3) it would be possible to add
several new terms. For example one could change the $2\times 2$ matrices
acting on the index $a$ of the spinor doublet $\epsilon_{a}$. Or one could add
entirely new terms, such as $H\Gamma_{{M}}$, $\Gamma_{{M}}{\mathcal{F}}$,
$\partial_{{M}}\phi$ to ${\mathcal{D}}_{{M}}$ or ${\mathcal{F}}$ to
${\mathcal{O}}$, tensored by any $2\times 2$ matrix compatible with the
chirality of the spinors. Even more drastically one could violate the
democratic structure of the operator and add a term only involving a single
RR-form degree. This gives rise to a bewildering array of possibilities, that
we will not explore here. For example it would be interesting to try to
reproduce for example the GKP supersymmetry breaking [79], or the
supersymmetry-breaking mechanisms discussed in [67, 80], from this point of
view.

Instead of attempting a general analysis, it would be reasonable to add one
such term at a time. The simplest possibilities would be

${\mathcal{D}}^{\prime}_{{M}}={\mathcal{D}}_{{M}}+\Gamma_{{M}}\otimes
a\,,\qquad{\mathcal{O}}^{\prime}={\mathcal{O}}+1\otimes b\,,$ (6.12)

where $a$, $b$ are some $2\times 2$ matrices. Notice however that this only
makes sense in IIA provided $a$ and $b$ are off-diagonal, since in IIB
${\mathcal{D}}_{{M}}\epsilon$ and $\Gamma_{{M}}\otimes a\epsilon$ have
opposite chiralities, and so do ${\mathcal{O}}\epsilon$ and $1\otimes
b\epsilon$. It would be interesting to apply to (6.12) in IIA the same program
outlined earlier in this section for M-theory.

## Acknowledgements

We would like to thank N. Bobev, D. Cassani, B. De Luca, G. Dibitetto, C.
Hull, G. Lo Monaco and B. Nilsson for useful discussions and correspondence.
We are supported in part by INFN and by MIUR-PRIN contract 2017CC72MK003.

##  Appendix A Useful spinorial identities

Here we recall some standard definitions and techniques useful in any number
$d$ of dimensions; we assume Lorentz signature.

The easiest gamma matrix identities are

$\begin{split}\Gamma_{{M}}\Gamma^{{{N_{1}\cdots
N_{k}}}}&=\Gamma_{{M}}{}^{{{N_{1}\cdots
N_{k}}}}+k\delta_{{{M}}}^{{{[N_{1}}}}\Gamma^{{{\cdots N_{k}]}}}\,,\\\
\Gamma^{{{N_{1}\cdots
N_{k}}}}\Gamma_{{M}}&=(-1)^{k}(\Gamma_{{M}}{}^{{{N_{1}\cdots
N_{k}}}}-k\delta_{{{M}}}^{{{[N_{1}}}}\Gamma^{{{\cdots
N_{k}]}}})\,.\end{split}$ (A.1)

Viewing $\Gamma^{{{N_{1}\cdots N_{k}}}}$ as the image of the Clifford map /

$[\mathrm{d}x^{{{M_{1}}}}\wedge\cdots\wedge\mathrm{d}x^{{{M_{k}}}}]_{/}=\Gamma^{{{M_{1}\cdots
M_{k}}}}\,,$ (A.2)

equations (A.1) can be thought of as operator identities:

$\stackrel{{\scriptstyle\to}}{{\Gamma^{{M}}}}=g^{{{MN}}}\iota_{{N}}+\mathrm{d}x^{{M}}\wedge\,,\qquad\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{M}}}}=(-g^{{{MN}}}\iota_{{N}}+\mathrm{d}x^{{M}}\wedge)(-1)^{\mathrm{deg}}\,.$
(A.3)

The arrows denote action from the left and right:
${\Gamma^{{M}}}\alpha_{/}=(\stackrel{{\scriptstyle\to}}{{\Gamma^{{M}}}}\\!\\!\alpha)_{/}$,
$\alpha_{/}\Gamma^{{M}}=(\stackrel{{\scriptstyle\leftarrow}}{{\Gamma^{{M}}}}\\!\\!\alpha)_{/}$.
Moreover $\deg\alpha_{k}\equiv k\alpha_{k}$, with the subscript denoting form
degree;
$\iota_{{M}}(\mathrm{d}x^{{{N_{1}}}}\wedge\cdots\wedge\mathrm{d}x^{{{N_{k}}}})\equiv
k\delta_{{{M}}}^{{{[N_{1}}}}\mathrm{d}x^{{{N_{2}}}}\wedge\cdots\wedge\mathrm{d}x^{{{N_{k}]}}}$
is the contraction operator.

A useful identity following from (A.3) is

$\Gamma_{{M}}\alpha_{k}\Gamma^{{M}}=(-1)^{k}(d-2k)\alpha_{k}\,.$ (A.4)

One particular consequence of this and (A.1) for $k=2$ is

$\Gamma_{{M}}\Gamma^{{{MNP}}}=\Gamma^{{{MNP}}}\Gamma_{{M}}=(d-2)\Gamma^{{{NP}}}\,.$
(A.5)

Sometimes, in order to avoid ambiguities, we underline flat indices. For
instance, the chiral matrix is $\Gamma=c\Gamma^{\underline{01\ldots d-1}}$. In
general $c$ is chosen such that $\Gamma^{2}=1$, and in both $d=10$ and 11 we
can just take $c=1$. Under the Clifford map (A.2), left multiplication by
$\Gamma$ is related to the Hodge star:

$\stackrel{{\scriptstyle\to}}{{\Gamma}}=c*\lambda\,,$ (A.6)

where $\lambda\alpha_{k}\equiv(-1)^{\lfloor k/2\rfloor}\alpha_{k}$. Our Hodge-
star operator $*$ is defined by

$*(e^{{{A_{1}}}}\wedge\ldots\wedge
e^{{{A_{k}}}})=\frac{1}{(d-k)!}\epsilon_{{{B_{1}\cdots
B_{d-k}}}}{}^{{{A_{1}\cdots A_{k}}}}e^{{{B_{1}}}}\wedge\cdots\wedge
e^{{{B_{d-k}}}}$ (A.7)

where $e^{{{A}}}=e^{{{A}}}_{{{M}}}{\rm d}x^{{M}}$ is a vielbein and the
totally antisymmetric $\epsilon_{{{A_{1}\cdots A_{d}}}}$ is such that
$\epsilon_{\underline{012\cdots}}=1$. Notice that
$(*\lambda)^{2}=-(-)^{\lfloor d/2\rfloor}$ in Lorentzian signature. Using this
and (A.3) we also obtain that the Hodge operator exchanges wedges with
contractions:

$*\lambda\,\mathrm{d}x^{{M}}\wedge=(-1)^{d-1}\iota^{{M}}*\lambda\,,\qquad*\lambda\,\iota^{{{M}}}=(-1)^{d-1}\mathrm{d}x^{{M}}\wedge*\lambda\,.$
(A.8)

In the slightly different notation introduced in (4.2), we can also show

$\Gamma\Gamma_{(k)}=c(-1)^{\lfloor d/2\rfloor}*\lambda(\Gamma_{(d-k)})\,.$
(A.9)

The natural inner product and norm-squared of forms are:

$\alpha_{k}\cdot\beta_{k}\equiv\frac{1}{k!}\alpha_{{{M_{1}\cdots
M_{k}}}}\beta^{{{M_{1}\cdots
M_{k}}}}\,,\qquad|\alpha_{k}|^{2}\equiv\alpha_{k}\cdot\alpha_{k}\,.$ (A.10)

There is also a natural inner product among bispinors, related under (A.2) to
the one among forms:

$\mathrm{Tr}((\lambda\alpha_{k})_{/}(\beta_{k})_{/})=2^{\lfloor
d/2\rfloor}\alpha_{k}\cdot\beta_{k}\,;\qquad(\lambda\alpha_{k})_{/}=(-1)^{k}\Gamma_{\underline{0}}(\alpha_{k}{}_{/})^{\dagger}\Gamma^{\underline{0}}\,.$
(A.11)

Repeated application of (A.3) gives the other useful identities

$\displaystyle(\iota^{{M}}\alpha_{k})\cdot(\iota^{{N}}\beta_{k})-\frac{1}{2}\alpha_{k}\cdot\beta_{k}\,g^{{{MN}}}$
$\displaystyle=\frac{(-1)^{k+1}}{2\cdot 2^{\lfloor
d/2\rfloor}}\mathrm{Tr}\left((\lambda\alpha_{k})_{/}\Gamma^{{{(M}}}\not{\beta}_{k}\Gamma^{{{N)}}}\right)\,,$
(A.12a)
$\displaystyle(\iota^{{M}}\alpha_{k})\cdot(\mathrm{d}x^{{N}}\wedge\alpha_{k})$
$\displaystyle=\frac{(-1)^{k}}{2\cdot 2^{\lfloor
d/2\rfloor}}\mathrm{Tr}\left((\lambda\alpha_{k})_{/}\Gamma^{{{[M}}}\not{\alpha}_{k}\Gamma^{{{N]}}}\right)\,.$
(A.12b)

Notice that the left-hand side of (A.12a) has the form of a stress-energy
tensor associated to a $k$-form field-strength.

We will sometimes use Fierz identities. These come about by expanding a
bispinor along the $\Gamma^{{{N_{1}\cdots N_{k}}}}$. When $d=$even, the set of
such objects $k=0,\,\cdots,\,d$ is a basis; when $d=\mathrm{odd}$, there are
redundancies relating $k$ to $d-k$, so we can expand along only
$k=0,\,\cdots,\,\left(d-1\right)/2$.

The coefficients of such expansions often involve inner products
$\bar{\epsilon}\epsilon^{\prime}$, where

$\bar{\epsilon}\equiv\epsilon^{\dagger}\Gamma_{\underline{0}}\,.$ (A.13)

We also recall the intertwiner property
$\Gamma_{{{M}}}^{\dagger}\Gamma^{\underline{0}}=-\Gamma^{\underline{0}}\Gamma_{{M}}$.
In both $d=10$ and $11$, one can work in a basis where all gamma matrices are
real; a Majorana spinor is then such that $\epsilon^{*}=\epsilon$, and for two
Majorana spinors one then easily sees

$\bar{\epsilon}^{\prime}\,\Gamma_{{{M_{1}\cdots
M_{k}}}}\epsilon=(-1)^{\lfloor(k-1)/2\rfloor}\bar{\epsilon}\,\Gamma_{{{M_{1}\cdots
M_{k}}}}\epsilon^{\prime}\,.$ (A.14)

It is also useful to note that

$\overline{\Gamma_{{{M_{1}\cdots
M_{k}}}}\epsilon}=(-1)^{k}\bar{\epsilon}\,\Gamma_{{{M_{k}\cdots
M_{1}}}}=(-1)^{\lfloor(k-1)/2\rfloor}\bar{\epsilon}\,\Gamma_{{{M_{1}\cdots
M_{k}}}}\,.$ (A.15)

The Clifford map (A.2) also works well at the differential level. One can
extend the spinorial covariant derivative to bispinors $C$ as
$D_{{M}}C\equiv\partial_{{M}}C+\frac{1}{4}\omega_{{{M}}}^{{{AB}}}[\Gamma_{{{AB}}},C]$;
this is then related to the ordinary bosonic covariant derivative
$\nabla_{{M}}$ by

$D_{{M}}(\alpha_{/})=(\nabla_{{M}}\alpha)_{/}\,.$ (A.16)

##  Appendix B Some details on M-theory

The two derivative action of 11 dimensional supergravity reads:

$\begin{split}S=\frac{2\pi}{\ell_{\textrm{\tiny
P}}^{9}}\bigintssss\Big{(}R*1&-2{\rm i}\overline{\psi}\wedge\Gamma_{(8)}\wedge
D\psi-\frac{1}{2}G\wedge*G-\frac{1}{6}G\wedge G\wedge C\\\
&-\frac{i}{6}\overline{\psi}\wedge\Gamma_{(8)}\wedge\left(\Gamma_{(1)}\not{G}-3\left(\iota_{(1)}G\right)_{/}\right)\psi+\mathcal{O}\left(\psi^{4}\right)\Big{)},\end{split}$
(B.1)

where $D$ is the spin covariant derivative, and $G={\rm d}C$, and $\not{G}$
and $\left(\iota_{{N}}G\right)_{/}$ are defined below (4.4) in agreement with
the general (A.2). The supersymmetry variation of the $3$-form gauge field and
the gravitini are given by:

$\delta_{\epsilon}C_{{{MNP}}}=-3\bar{\epsilon}\,\Gamma_{{{\left[MN\right.}}}\psi_{{{\left.P\right]}}},\qquad\delta_{\epsilon}\psi_{{M}}=\mathcal{D}_{{M}}\epsilon\,,$
(B.2)

where $\mathcal{D}_{{M}}$ is defined in (4.4).

###  B.1 Supercharge from the Noether theorem

Let us now briefly summarize a derivation of the supercurrent following an
infinitesimal variation under local supersymmetry, along the lines of [81].

Given a Lagrangian $L\left(X,\partial X\right)$ depending on fields $X$ and
their derivatives $\partial_{{M}}X$, its variation is given by

$\displaystyle\delta L$ $\displaystyle=\sum_{X}\left(\frac{\delta L}{\delta
X}\delta X+\frac{\delta L}{\delta\partial_{{M}}X}\partial_{{M}}\delta
X\right)$ (B.3) $\displaystyle=\sum_{X}\left(\frac{\delta L}{\delta X}\delta
X-\left(\partial_{{M}}\frac{\delta L}{\delta\partial_{{M}}X}\right)\delta
X\right)+\sum_{X}\partial_{{M}}\left(\frac{\delta
L}{\delta\partial_{{M}}X}\delta X\right)$
$\displaystyle\equiv\sum_{X}\left(\mathcal{E}_{X}\,\delta
X+\partial_{{M}}N^{{M}}\right),$

where $\mathcal{E}_{X}$ represents the equations of motion for $X$. Now, the
variation of the Lagrangian under an infinitesimal supersymmetry
transformation gives $\delta_{\epsilon}L\equiv\partial_{{M}}V^{{M}}$.
Identifying this with the above gives

$\partial_{M}\underbrace{\left(V^{{M}}-N^{{M}}\right)}_{\equiv
J^{{M}}}=\sum_{X}\mathcal{E}_{X}\,\delta X\overset{\textrm{on-shell}}{=}0,$
(B.4)

which gives the supercurrent $J^{{M}}=V^{{M}}-N^{{M}}$ that is conserved on
shell. The corresponding supercharge is $Q=\int{\rm d}^{d-1}xJ^{0}$, which
also generates the supersymmetry transformations of the fields:
$\delta_{\epsilon}X=\left\\{Q,X\right\\}$.

###  B.2 Noether charge in M-theory

Applying to M-theory the Noether procedure outlined in B.1 we get,

$N^{{M}}=\frac{\delta
L}{\delta\left(\partial_{{M}}\psi_{{N}}\right)}\delta_{\epsilon}\psi_{{N}}+\frac{1}{3!}\frac{\delta
L}{\delta\left(\partial_{{M}}C_{{{NPQ}}}\right)}\delta_{\epsilon}C_{{{NPQ}}}+\cdots,$
(B.5)

where the ellipsis represents variations with respect to the vielbien and the
spin connection. These eventually cancel out and so we will not write them
here. Up to an overall rescaling by $\ell_{\textrm{\tiny P}}^{9}/(8\pi)$, this
gives

$\displaystyle N^{{M}}$ $\displaystyle=-\frac{{\rm
i}e}{2}\overline{\psi}_{{P}}\Gamma^{{{PMN}}}\mathcal{D}_{{N}}\epsilon+\frac{3e}{4\cdot
4!}G^{{{MNPQ}}}\bar{\epsilon}\,\Gamma_{[{{NP}}}\psi_{{{Q}}]}$ (B.6)
$\displaystyle+\frac{1}{4\cdot 12^{3}}\epsilon^{{{MNPQM_{5}\ldots
M_{11}}}}G_{{{M_{5}\ldots
M_{8}}}}C_{{{M_{9}M_{10}M_{11}}}}\bar{\epsilon}\,\Gamma_{[{{NP}}}\psi_{{{Q}}]}+\cdots,$

where again the ellipsis represents, in addition to the variation with respect
to $e_{{M}}^{{A}}$ and $\omega$ above, quadratic terms in $\psi$. The surface
terms arising from an infinitesimal variation of the action under the local
supersymmetry transformation give
$\delta_{\epsilon}L=\frac{8\pi}{\ell_{\textrm{\tiny
P}}^{9}}\partial_{{M}}V^{{M}}$, where

$\displaystyle V^{{M}}$ $\displaystyle=\frac{{\rm
i}e}{2}\overline{\psi}_{{P}}\Gamma^{{{PMN}}}\mathcal{D}_{{N}}\epsilon+\frac{3e}{4\cdot
4!}G^{{{MNPQ}}}\bar{\epsilon}\,\Gamma_{[{{NP}}}\psi_{{{Q}}]}$ (B.7)
$\displaystyle+\frac{1}{4\cdot 12^{3}}\epsilon^{{{MNPQM_{5}\ldots
M_{11}}}}G_{{{M_{5}\ldots
M_{8}}}}C_{{{M_{9}M_{10}M_{11}}}}\bar{\epsilon}\,\Gamma_{[{{NP}}}\psi_{{{Q}}]}+\cdots,$

and the ellipsis has the same meaning as above. Together with (B.6), this
gives the supercurrent

$J^{{M}}=V^{{M}}-N^{{M}}={\rm
i}e\overline{\psi}_{{P}}\Gamma^{{{PMN}}}\mathcal{D}_{{N}}\epsilon.$ (B.8)

For our purposes, it is convenient to use the fermionic real supercurrent
obtained from (B.8) by considering $\epsilon$ to be commuting and dropping the
overall ${\rm i}$. The corresponding fermionic supercharge $Q(\epsilon)$ is
then given by the integral:

$Q=\int_{S}\overline{\psi}_{{P}}\Gamma^{{{PMN}}}\mathcal{D}_{{N}}\epsilon*{\rm
d}x_{{M}}=\int_{\partial
S}\overline{\psi}_{{P}}\Gamma^{{{P}}}{}_{{{MN}}}\epsilon*{\rm
d}x^{{{MN}}}=-\int_{\partial S}\overline{\epsilon}\,\Gamma_{(8)}\wedge\psi\,.$
(B.9)

In the second step, we have used the gravitino equations of motion
$\Gamma^{{{MNP}}}{\cal D}_{{N}}{\psi}_{{P}}=0+{\cal O}(\psi^{3})$.

###  B.3 The divergence identity

We will now give some more details about the computations leading to (4.8).
For simplicity in the rest of this appendix, as in most of the paper, we
mostly drop the Clifford map symbol /, whose implicit presence should be clear
from the context.

As a warm-up we first note that, from (A.4):

$\Gamma^{{M}}G\Gamma_{{M}}=3G\,,\qquad\Gamma^{{M}}(-\Gamma_{{M}}G+3G\Gamma_{{M}})=(-11+3\cdot
3)G=-2G\,.$ (B.10)

We then evaluate the commutator term in (4.11):

$\begin{split}\Gamma^{{{NMP}}}&[D_{{M}},-\Gamma_{{P}}G+3G\Gamma_{{P}}]\buildrel(\ref{eq:Dnabla})\over{=}\Gamma^{{{NMP}}}(-\Gamma_{{P}}\nabla_{{M}}G+3\nabla_{{M}}G\Gamma_{{P}})\\\
&\buildrel(\ref{eq:g1g3}),(\ref{eq:lrG0}),(\ref{eq:GammaG})\over{=}6\Gamma^{{{[M}}}\nabla_{{M}}G\Gamma^{{{N]}}}\buildrel(\ref{eq:lrG})\over{=}6[(\mathrm{d}x\wedge+\iota)(\mathrm{d}x\wedge-\iota)]^{{{[MN]}}}\nabla_{{M}}G\\\
&=6(\mathrm{d}x^{{M}}\wedge\mathrm{d}x^{{N}}\wedge-\iota^{{M}}\iota^{{N}})\nabla_{{M}}G\buildrel(\ref{eq:*dx})\over{=}6(-\mathrm{d}x^{{N}}\wedge\mathrm{d}G+\iota^{{N}}*\mathrm{d}*G)\,.\end{split}$
(B.11)

We now look at the terms in (4.11) that are quadratic in $G$. Using (A.1) for
$k=2$, it is now easy to evaluate $A^{NP}$ in (4.12):

$\displaystyle A^{{{NP}}}=$
$\displaystyle(-G\Gamma_{{M}}+3\Gamma_{{M}}G)(\Gamma^{{M}}\Gamma^{{{NP}}}-2g^{{{M[N}}}\Gamma^{{{P]}}})$
$\displaystyle-(\Gamma^{{{NP}}}\Gamma^{{M}}-2\Gamma^{{{[N}}}g^{{{P]M}}})(-\Gamma_{{M}}G+3G\Gamma_{{M}})$
(B.12) $\displaystyle=$
$\displaystyle-2G\Gamma^{{{NP}}}-2(-G\Gamma^{{{[N}}}+3\Gamma^{{{[N}}}G)\Gamma^{{{P]}}}+2\Gamma^{{{NP}}}G+2\Gamma^{{{[N}}}(-\Gamma^{{{P]}}}G+3G\Gamma^{{{P]}}})=0\,.$

Another useful consequence of (A.1) is:

$\Gamma^{{{MNP}}}=\Gamma^{{M}}\Gamma^{{N}}\Gamma^{{P}}-g^{{{MN}}}\Gamma^{{P}}-\Gamma^{{M}}g^{{{NP}}}+\Gamma^{{N}}g^{{{MP}}}\,.$
(B.13)

Using this and (B.10), we also get

$Q^{{N}}=9(G\Gamma^{{N}}G+\Gamma_{{M}}G\Gamma^{{N}}G\Gamma^{{M}})\,.$ (B.14)

For our purposes it is now useful to apply the Fierz identity, i.e., to expand
(B.14) in the basis $\Gamma^{{{N_{1}\cdots N_{k}}}}$,
$k=0,\,\cdots,\,\left(d-1\right)/2=5$. This works out to

$Q^{{N}}=\sum_{k=0}^{5}\frac{1}{32k!}\mathrm{Tr}(Q^{{N}}\Gamma_{{{N_{k}\cdots
N_{1}}}})\Gamma^{{{N_{1}\cdots N_{k}}}}\,.$ (B.15)

Actually, it is easy to see that
$\Gamma_{\underline{0}}(Q^{{N}})^{\dagger}\Gamma^{\underline{0}}=-Q^{{N}}$,
and $\Gamma_{\underline{0}}(\Gamma^{{{N_{1}\cdots
N_{k}}}})^{\dagger}\Gamma^{\underline{0}}=(-1)^{\lfloor
k+1/2\rfloor}\Gamma^{{{N_{1}\cdots N_{k}}}}$; so in fact only $k=1,2,5$ appear
in (B.15). The traces can now be simplified using (A.4) again. The $k=1$ and
$k=2$ terms give the quadratic $G$ contributions to the equations of motion
(4.10), (4.15), while in $k=5$ the two terms cancel each other:

$\begin{split}Q^{{N}}&=\frac{1}{32}\left(\mathrm{Tr}(Q^{{N}}\Gamma^{{P}})\Gamma_{{P}}+\frac{1}{2}\mathrm{Tr}(Q^{{N}}\Gamma^{{{QP}}})\Gamma_{{{PQ}}}\right)\\\
&=36\left(T^{{{NP}}}_{{{(G)}}}\Gamma_{{P}}-\frac{1}{4(4!)^{2}}\epsilon^{{{NPQM_{1}\cdots
M_{8}}}}G_{{{M_{1}\cdot M_{4}}}}G_{{{M_{5}\cdot
M_{8}}}}\Gamma^{{{PQ}}}\right)\\\
&=36T^{{{NP}}}_{{{(G)}}}\Gamma_{{P}}-18\iota^{{N}}*(G\wedge G)\,.\end{split}$
(B.16)

Thus as promised, this gives the quadratic terms in the equations of motion in
(4.8).

###  B.4 The supersymmetry-breaking case

Here we will show briefly how the computation is modified for the operator
(6.1) with $a_{2}=-3a_{1}$, namely

${\mathcal{D}}^{\prime}_{{M}}=D_{{M}}+\frac{a_{1}}{24}(\Gamma_{{M}}G-3G\Gamma_{{M}})+a_{3}\Gamma_{{M}}\,.$
(B.17)

Following the same steps as in (4.11), the explicit $G$ terms there are
rescaled by $-a_{1}$; (4.12) are replaced by

$A^{\prime}{}^{{{NP}}}=-a_{1}A^{{{NP}}}+24a_{3}(\Gamma_{{M}}\Gamma^{{{MNP}}}-\Gamma^{{{MNP}}}\Gamma_{{M}})\buildrel(\ref{eq:ANP=0}),(\ref{eq:g1g3})\over{=}0$
(B.18)

and $Q^{\prime}{}^{{{N}}}$ given by

$\displaystyle\frac{Q^{\prime}{}^{{N}}+a_{1}Q^{{N}}}{24a_{3}}$
$\displaystyle=-a_{1}\Gamma_{{M}}\Gamma^{{{MNP}}}(-\Gamma_{{P}}G+3G\Gamma_{{P}})-a_{1}(-G\Gamma_{{M}}+3\Gamma_{{M}}G)\Gamma^{{{MNP}}}\Gamma_{{P}}+24a_{3}\Gamma_{{M}}\Gamma^{{{MNP}}}\Gamma_{{P}}$
$\displaystyle\buildrel(\ref{eq:g1g3})\over{=}-9a_{1}(\Gamma^{{N}}\Gamma^{{P}}-g^{{{NP}}})(-\Gamma_{{P}}G+3G\Gamma_{{P}})-9a_{1}(-G\Gamma_{{M}}+3\Gamma_{{M}}G)(\Gamma^{{M}}\Gamma^{{N}}-g^{{{MN}}})$
$\displaystyle\qquad+90\cdot
24a_{3}\Gamma^{{N}}\buildrel(\ref{eq:GammaG})\over{=}36a_{1}\\{\Gamma^{{N}},G\\}+90\cdot
24a_{3}\Gamma^{{N}}$ (B.19)
$\displaystyle\buildrel(\ref{eq:lrG})\over{=}12(3a_{1}{\rm d}x^{{N}}\wedge
G+180a_{3}{\rm d}x^{N})_{/}\,.$

Putting it all together, one arrives at (6.9).

###  B.5 Equations of motion from supersymmetry

Finally we show here how the Einstein equations are (almost completely)
implied by supersymmetry and the $G$ equations. This was shown in [78, App.
B], but some results we saw in this appendix allow us to derive it more
quickly. This is useful for us in the context of modifying
${\mathcal{D}}_{{M}}$ to break supersymmetry, in section 6.

We consider the operator

$\Gamma^{{M}}[{\mathcal{D}}_{{M}},{\mathcal{D}}_{{N}}]\,.$ (B.20)

The purely gravitational term $\Gamma^{{M}}[D_{{M}},D_{{N}}]$ is well-known to
be $\frac{1}{2}R_{{{NP}}}\Gamma^{{P}}$; this is how a manifold with a
covariantly constant spinor in Euclidean signature is shown to be Ricci-flat.
The remaining terms can again be separated into those involving $G$ linearly,
and those involving it quadratically.

The linear term proceeds similar to (B.11):

$\begin{split}&\Gamma^{{M}}[D_{{{[M}}},-\Gamma_{{{N]}}}G+3G\Gamma_{{{N]}}}]=-\Gamma^{{M}}\Gamma_{{{[N}}}\nabla_{{{M]}}}G+3\Gamma^{{M}}\nabla_{{{[M}}}G\Gamma_{{{N]}}}\\\
&\buildrel(\ref{eq:gag})\over{=}\frac{1}{2}\Gamma_{{N}}\Gamma^{{M}}\nabla_{{M}}G+\frac{3}{2}\Gamma^{{M}}\nabla_{{M}}G\Gamma_{{N}}\buildrel(\ref{eq:lrG})\over{=}2(-{\rm
d}x_{{N}}\wedge+2\iota_{{N}})({\rm d}G+*{\rm d}\\!*G)\,.\end{split}$ (B.21)

For the quadratic term, we first use (B.10) repeatedly to obtain:

$\Gamma^{{M}}[-\Gamma_{{M}}G+3G\Gamma_{{M}},-\Gamma_{{N}}G+3G\Gamma_{{N}}]=-9(G\Gamma_{{N}}G+\Gamma^{{M}}G\Gamma_{{N}}G\Gamma_{{M}})+9\Gamma_{{N}}G^{2}-3\Gamma_{{N}}\Gamma^{{M}}G^{2}\Gamma_{{M}}\,.$
(B.22)

The parenthesis on the right-hand side is just $Q^{{N}}$ from (B.14), so we
can evaluate it as in (B.16). For the other two terms, we first notice that
$G^{2}=\frac{1}{2}\\{G,G\\}=(G^{2})_{0}+(G^{2})_{4}+(G^{2})_{8}$: it only has
zero-, four- and eight-form parts. From (A.4) we then have

$3G^{2}-\Gamma^{{M}}G^{2}\Gamma_{{M}}=-8(G^{2})_{0}-8(G^{2})_{8}=-8(|G|^{2}+G\wedge
G)\,.$ (B.23)

Recalling now the first of (A.1) and putting everything together, we arrive at

$\Gamma^{{M}}[{\mathcal{D}}_{{M}},{\mathcal{D}}_{{N}}]=\frac{1}{2}{\mathcal{E}}_{{{NP}}}\Gamma^{{P}}+\frac{1}{12}({\rm
d}x_{{N}}\wedge-2\iota_{{N}})\left({\rm d}G+*\left({\rm
d}*G+\frac{1}{2}G\wedge G\right)\right)\,,$ (B.24)

where recall that ${\mathcal{E}}_{NP}=0$ is the Einstein equation of motion
from (4.10).

The argument is now standard, and we only repeat it here for completeness. If
the $G$ Bianchi identity and equations of motion hold (away from M2 and M5