###### Abstract

Starting from a peculiar orientifold projection proposed long ago by
Angelantonj and Cardella, we elaborate on a novel perturbative scenario that
involves only D-branes, together with the two types of orientifold planes
${\rm O}_{\pm}$ and anti-orientifold planes $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}\mkern-1.5mu}\mkern 1.5mu_{\pm}$. We
elucidate the microscopic ingredients of such models, connecting them to a
novel realization of brane supersymmetry breaking. Depending on the position
of the D-branes in the internal space, supersymmetry can be broken at the
string scale on branes, or alternatively only at the massive level. The main
novelty of this construction is that it features no NS-NS disk tadpoles, while
avoiding open-string instabilities. The one-loop potential, which depends on
the positions of the D-branes, is minimized for maximally broken, nonlinearly
realized supersymmetry. The orientifold projection and the effective field
theory description reveal a soft breaking of supersymmetry in the closed-
string sector. In such models it is possible to decouple the gravitino mass
from the value of the scalar potential, while avoiding brane instabilities.

CPHT-RR026.032021

Geometry of orientifold vacua and
supersymmetry breaking

Thibaut<EMAIL_ADDRESS>Emilian
<EMAIL_ADDRESS>and Hervé
<EMAIL_ADDRESS>

CPHT, CNRS, Ecole Polytechnique, IP Paris,
F-91128 Palaiseau, France

##  1 Introduction

Supersymmetry breaking in string theory is notoriously difficult to achieve in
a controllable manner. There are several challenging and well-known problems
to overcome at the string level and at the effective field theory one.

A generic issue, both at string perturbative and effective supergravity
levels, is that supersymmetry breaking generates potentials for some moduli
fields that are of runaway type, which typically drive the dynamics towards
zero or strong string coupling, and also lead to decompactification or
compactification of the internal space [1]. The state of the art is to
generate a local minimum somewhere far from the runaway regime, that is
computationally reliable and such that the corresponding lifetime is beyond
the age of the universe. Such a minimum is very hard to obtain in string
perturbation theory, and easier in practice to obtain at the effective field-
theory level, adding extra ingredients like fluxes or nonperturbative effects.

At the string perturbative level, supersymmetry breaking generates a vacuum
energy (more precisely, a scalar potential) at some order in perturbation
theory. In the first models of supersymmetry breaking, so-called
Scherk–Schwarz or breaking by compactification [2, 3, 4, 5, 6, 7], this arises
at one loop.111Scherk–Schwarz compactifications also have often additional,
tachyonic-like instabilities in some range of parameters. Tachyon-free
examples however exist, see e.g. [8, 9, 10]. The generated scalar potential is
typically of runaway type222The cosmological evolution of the moduli fields
can be studied in a thermal [11, 12, 13, 14, 15] or cold [16, 17] universe.
and the classical vacuum used in perturbation theory is therefore not valid
anymore. It is however possible, in a more refined construction, to stabilize
the corresponding modulus, yielding a negative scalar potential [9]. In a
subclass of models, which satisfy a classical Bose/Fermi degeneracy at the
massless level, this one-loop potential turns out to be exponentially
suppressed at low supersymmetry breaking scale [18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30], but not vanishing [31].

Later on, tachyon-free orientifold string models where supersymmetry is broken
at the string scale in the open-string (gauge) sector, whereas the closed-
string (gravity) sector is supersymmetric at lowest perturbative order were
constructed [32, 33, 34, 35, 36, 37]. Since in such frameworks a massless
gravitino is present, supersymmetry has to be nonlinearly realized in the open
sector and this was indeed shown explicitly in [38, 39]. Such models contain
non-BPS tachyon-free configurations. An important application of such setups
is the KKLT scenario of moduli stabilization [40], see e.g. [41, 42, 43, 44,
45, 46, 47]. There was also a recent simplification in constructing
supergravity models with nonlinear supersymmetry [48, 49, 50, 51, 52],
stimulated in part by “Brane Supersymmetry Breaking” (BSB) type models. In
such settings, there is a runaway scalar potential generated at the disk
level. Ignoring the true vacuum state and working naively at fixed values of
moduli fields leads to so-called NS-NS tadpoles for the corresponding moduli,
which ruin perturbation theory since they generate unphysical UV divergences.
It is widely believed that this does not signal any inconsistency of the
theory, but just the fact that naive perturbation theory is performed around a
point in field space that is not an extremum. Indeed, all models of this type
constructed in the literature satisfy all known consistency conditions.
Mechanisms of shifting the vacuum, in analogy with field-theory examples, were
proposed in the literature [53, 54, 55, 56]. However their practical
implementation is limited to toy examples or to special models with small
tadpoles. Hence, whereas the BSB models are tachyon-free, the presence of NS-
NS tadpoles raises the question of the validity of perturbation theory and the
fate of such constructions [57, 58, 59, 60]. Let us also mention that the
coexistence of massless gravitinos and broken supersymmetry in the open sector
in BSB models is shared by compactifications with internal magnetic fields
that break supersymmetry [61, 62, 63, 64].

In another class of non-supersymmetric models based on type II asymmetric
orbifolds or their orientifold descendants [65, 66, 67, 68, 69, 70, 71], a
classical Bose/Fermi degeneracy valid at any mass level implies that the
potential arises only at two loops [72, 73]. In such frameworks, there are no
tadpoles at one loop and no need to shift scalar expectation values for
describing vacua at this order of perturbation theory. However, stability at
one loop of the moduli fields has not been analyzed.

The goal of the present paper is the construction of BSB string vacua without
NS-NS disk tadpoles. Recently, it was conjectured that massless gravitinos in
string theory with broken supersymmetry implies a breakdown of the effective
field theory [74, 75]. It is clearly of interest to check this conjecture in
explicit string models, by trying to avoid NS-NS tadpoles. In this paper we
identify constructions in which the supersymmetry breaking scale in the gauge
(open) sector is much higher than in the closed-string sector. This was
actually achieved previously in [76]. However, our construction avoids the
open-string tachyonic instabilities typically present in such constructions. A
runaway behaviour for internal radii is however still present. We show that
the limit of vanishing gravitino mass is inconsistent. The existence of such
constructions was already anticipated in the pioneering paper [77] in an
algebraic construction using the tools of the Tor-Vergata school [78, 79, 80,
81, 82, 83, 84, 85, 86]. We provide here the correct geometric interpretation
of the eight-dimensional class of models proposed in [77], which turns out to
contain several types of perturbative orientifold and anti-orientifold planes.
We also point out that the simultaneous presence of orientifold and
antiorientifold planes suggests that the closed-string sector is not exactly
supersymmetric at tree-level, but has softly broken supersymmetry. The basic
mechanism goes as follows:

One starts with a supersymmetric orientifold model containing both O-
(negative tension, negative RR charge) and O+-planes (positive tension,
positive RR charge). A consistent supersymmetry-breaking deformation of the
model turns one O- \- O+ pair into an ${\mkern 1.5mu\overline{\mkern-1.5mu\rm
O\mkern-1.5mu}\mkern 1.5mu_{-}}$ \- ${\mkern 1.5mu\overline{\mkern-1.5mu\rm
O\mkern-1.5mu}\mkern 1.5mu_{+}}$ pair, which is mutually BPS but preserves the
other half of the supersymmetries compared to the O±-planes and D-branes.
Since both the initial O- \- O+ pair and its SUSY breaking avatar ${\mkern
1.5mu\overline{\mkern-1.5mu\rm O\mkern-1.5mu}\mkern 1.5mu_{-}}$ \- ${\mkern
1.5mu\overline{\mkern-1.5mu\rm O\mkern-1.5mu}\mkern 1.5mu_{+}}$ have zero
total tension and charge, there will be no RR or NS-NS tadpoles generated in
the non-supersymmetric case. Depending on where the background D-branes sit in
the internal space, their massless spectrum can be supersymmetric (if they sit
on top of O- or O+-planes or in the bulk) or non-supersymmetric (if they sit
on top of ${\mkern 1.5mu\overline{\mkern-1.5mu\rm O\mkern-1.5mu}\mkern
1.5mu_{-}}$ or ${\mkern 1.5mu\overline{\mkern-1.5mu\rm O\mkern-1.5mu}\mkern
1.5mu_{+}}$-planes, in which case supersymmetry is nonlinearly realized in
their worldvolume). Such models also have a supersymmetric limit, when a
certain radius is taken to zero.333A similar option is available in IIB flux
compactifications[87]. We thank J. Mourad and A. Sagnotti for sharing their
results with us. For small values of this radius, the breaking can be
interpreted as spontaneous, whereas for large values, supersymmetry breaking
can be considered as nonlinearly realized if branes sit on top of anti-
orientifold planes. Interestingly, naive energetic considerations on brane-
orientifold plane interactions suggest that the branes move towards stable
configurations with maximal (string scale) breaking of supersymmetry. Whereas
at first sight the closed-string spectrum could be supersymmetric, we show
that a detailed look at the orientifold projections leading to the geometry of
O-planes and, independently, considerations from low-energy effective field
theory suggest that the correct option is a specific soft supersymmetry
breaking deformation in the closed-string sector. A more detailed analysis of
the effective field theory of this class of models deserves however a
dedicated study.

The structure of the paper is the following: In Sect. 2, we review the 8d
$\text{USp}(16)$ supersymmetric orientifold theory and introduce the novel
Brane Supersymmetry Breaking (BSB) mechanism. In particular, we discuss the
consistency between the soft breaking of supersymmetry in the closed-string
spectrum and the supersymmetry breaking deformation in the open-string sector.
The generalization of the construction to dimensions lower than 8 turns out to
be rich but straightforward. We give various examples in Sect. 3. Sect. 4
discusses consistency conditions coming from probe branes as well as
nonperturbative constraints to be satisfied by these models. In Sect. 5, we
study the supersymmetry breaking mass scales in the closed- and open-string
sectors for different positions of stacks of D7-branes in 8d. We also comment
on the limit of vanishing gravitino mass and the connexion with the gravitino
mass conjecture put forward recently in [74, 75], in the context of the
swampland program [88, 89, 90]. Conclusions and outlooks can be found in
Sect.6, whereas an appendix contains examples of consistent and inconsistent
geometric configurations.

##  2 The 8d USp(16) superstring and its SUSY breaking avatar

In this section, we review the construction of the supersymmetric USp(16)
orientifold model in 8 dimensions and then present its non-supersymmetric
version.

###  2.1 The type IIB torus amplitudes

Let us start by describing alternative viewpoints for deriving the
supersymmetric and non-supersymmetric torus amplitudes to be combined later on
with orientifold amplitudes.

The original orientifold models described in [91, 92] make use of a non-
trivial quantized background for the internal components of the antisymmetric
tensor field, $B_{ij}$. This field is odd under worldsheet parity and
therefore it is projected out by the orientifold projection $\Omega$ in type I
superstring. However, this still leaves the possibility to add a quantized
value $\frac{2}{\alpha^{\prime}}B_{ij}\in\mathbb{Z}$, where $\alpha^{\prime}$
is the string tension. In this case, the left and right momenta of closed-
string states are given, for a torus factorized into two circles, by

$p_{\rm
L,R}^{8}=\frac{m_{8}+n_{9}/2}{R_{8}}\pm\frac{n_{8}R_{8}}{\alpha^{\prime}}\
,\qquad p_{\rm
L,R}^{9}=\frac{m_{9}-n_{8}/2}{R_{9}}\pm\frac{n_{9}R_{9}}{\alpha^{\prime}}\ .$
(2.1)

The type IIB torus amplitude is given by

$\displaystyle{\mathcal{T}}=\int\frac{\text{d}^{2}\tau}{\tau_{2}^{5}}\\!\\!$
$\displaystyle\;\left[\Lambda_{m_{9},2n_{9}}\Lambda_{m_{8},2n_{8}}+\Lambda_{m_{9}+1/2,2n_{9}}\Lambda_{m_{8},2n_{8}+1}\right.$
$\displaystyle\\!\left.+\,\Lambda_{m_{9},2n_{9}+1}\Lambda_{m_{8}+1/2,2n_{8}}+\Lambda_{m_{9}+1/2,2n_{9}+1}\Lambda_{m_{8}+1/2,2n_{8}+1}\right]\left|\frac{V_{8}-S_{8}}{\eta^{8}}\right|^{2}\
,$ (2.2)

where $V_{8}$, $S_{8}$ (along with $O_{8}$, $C_{8}$) are the ${\rm SO}(8)$
affine characters and $\eta$ is the Dedeking function. They all depend on the
Teichmüller parameter $\tau$ of the genus-1 surface, whose imaginary part is
denoted $\tau_{2}$. Moreover, the lattices are expressed in terms of

$\Lambda_{m_{i},n_{i}}=q^{{\alpha^{\prime}\over 4}\big{(}{m_{i}\over
R_{i}}+n_{i}{R_{i}\over\alpha^{\prime}}\big{)}^{2}}\,\bar{q}^{{{\alpha^{\prime}}\over
4}\big{(}{m_{i}\over R_{i}}-n_{i}{R_{i}\over\alpha^{\prime}}\big{)}^{2}}\
,\qquad q=e^{2i\pi\tau}\ ,$ (2.3)

where $m_{i}$, $n_{i}$ are the momentum and winding numbers along direction
$X^{i}$.444Throughout our work, all discrete sums over integer $m_{i}$,
$n_{i}$ are implicit. The conventions used in partition functions are those
given e.g. in the reviews [85, 86]. Note that this amplitude is invariant
under the T-duality transformation
$(R_{8},R_{9})\to\big{(}\frac{\alpha^{\prime}}{2R_{8}},\frac{\alpha^{\prime}}{2R_{9}}\big{)}$.

There is another particularly useful way of constructing the torus amplitude
with non-trivial discrete antisymmetric tensor uncovered by Pradisi [93]. The
starting point is a freely-acting orbifold of type IIB with $B_{89}=0$ and
generator $g=\delta_{w_{8}}\delta_{p_{9}}$, where $\delta_{w_{8}}$ stands for
a winding shift along direction $X^{8}$ while $\delta_{p_{9}}$ denotes a
momentum shift along direction $X^{9}$. The action of this generator on the
lattice states is

$g|{\bf m},{\bf n}\rangle=(-1)^{n_{8}+m_{9}}|{\bf m},{\bf n}\rangle\ .$ (2.4)

The gauging of the theory with this generator implies the existence of four
contributions in the torus amplitude corresponding to the untwisted and
twisted sectors, both with or without insertion of the orbifold generator in
the traces. One obtains

$\mathcal{T}=\frac{1}{2}\int\frac{\text{d}^{2}\tau}{\tau_{2}^{5}}\\!\\!\;\left[1+(-1)^{n_{8}+m_{9}}\right]\left(\Lambda_{m_{8},n_{8}}\Lambda_{m_{9},n_{9}}+\Lambda_{m_{8}+\frac{1}{2},n_{8}}\Lambda_{m_{9},n_{9}+\frac{1}{2}}\right)\left|\frac{V_{8}-S_{8}}{\eta^{8}}\right|^{2}\
.$ (2.5)

A rescaling of the radius $R_{9}\to 2R_{9}$ then leads to the torus amplitude
with discrete antisymmetric tensor given in Eq. (2.2).

Note that another derivation can be obtained by applying the T-duality
transformation $R_{8}\to{\alpha^{\prime}\over R_{8}}=\tilde{R}_{8}$ on the
freely-acting orbifold of type IIB with $B_{89}=0$. In fact, the complex
coordinate $Z=\frac{\tilde{X}_{8}+iX_{9}}{2\pi\tilde{R}_{8}}$, where
$\tilde{X}_{8}$ is the T-dual coordinate, satisfies the identifications
$Z=Z+1=Z+i\frac{R_{9}}{\tilde{R}_{8}}$. Moreover, the orbifold generator
$g=\delta_{w_{8}}\delta_{p_{9}}$ is mapped to
${\tilde{g}}=\delta_{p_{8}}\delta_{p_{9}}$ defined as
$(\tilde{X}_{8},X_{9})=(\tilde{X}_{8}+\pi\tilde{R}_{8},X_{9}+\pi R_{9})$. We
have therefore three identifications, which can be encoded in the following
two:

$Z=Z+1\ ,\quad Z=Z+U\ ,\quad{\rm where}\quad
U=\frac{1}{2}+i\frac{R_{9}}{2\tilde{R}_{8}}\ .$ (2.6)

Hence, by rescaling $R_{9}\to 2R_{9}$, the coordinate $Z$ is that of a tilted
torus of complex structure $U=\frac{1}{2}+i\frac{R_{9}}{\tilde{R}_{8}}$.
However, it is known that the type IIA theory compactified on this tilted
torus is T-dual to the type IIB theory with antisymmetric background
$B_{89}={\alpha^{\prime}\over 2}$.

From the freely-acting orbifold perspective, it is now easy to build a non-
supersymmetric deformation of the type IIB model in a Scherk–Schwarz spirit.
It is obtained by replacing $g$ with the generator
$g^{\prime}=(-1)^{F}\delta_{w_{8}}\delta_{p_{9}}$, where $F$ denotes the
spacetime fermion number. The construction of the torus amplitude is
straightforward and the result is

$\displaystyle{\mathcal{T}}=$ $\displaystyle\;{1\over
2}\int\frac{\text{d}^{2}\tau}{\tau_{2}^{5}}\\!\\!\;\bigg{\\{}\Lambda_{m_{8},n_{8}}\Lambda_{m_{9},n_{9}}\left|V_{8}-S_{8}\right|^{2}+(-1)^{n_{8}+m_{9}}\Lambda_{m_{8},n_{8}}\Lambda_{m_{9},n_{9}}\left|V_{8}+S_{8}\right|^{2}$
$\displaystyle+\Lambda_{m_{8}+\frac{1}{2},n_{8}}\Lambda_{m_{9},n_{9}+\frac{1}{2}}\left|O_{8}-C_{8}\right|^{2}+(-1)^{n_{8}+m_{9}}\Lambda_{m_{8}+\frac{1}{2},n_{8}}\Lambda_{m_{9},n_{9}+\frac{1}{2}}\left|O_{8}+C_{8}\right|^{2}\bigg{\\}}\frac{1}{|\eta^{8}|^{2}}\
.$ (2.7)

The rescaling of the radius $R_{9}\to 2R_{9}$ then leads to

$\displaystyle{\mathcal{T}}=\int\frac{\text{d}^{2}\tau}{\tau_{2}^{5}}$
$\displaystyle\\!\\!\;\bigg{\\{}\;\left(\Lambda_{m_{8},2n_{8}}\Lambda_{m_{9},2n_{9}}+\Lambda_{m_{8},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}}\right)\left(|V_{8}|^{2}+|S_{8}|^{2}\right)$
$\displaystyle-\left(\Lambda_{m_{8},2n_{8}+1}\Lambda_{m_{9},2n_{9}}+\Lambda_{m_{8},2n_{8}}\Lambda_{m_{9}+\frac{1}{2},2n_{9}}\right)\left(V_{8}\overline{S}_{8}+\overline{V}_{8}S_{8}\right)$
$\displaystyle+\left(\Lambda_{m_{8}+\frac{1}{2},2n_{8}}\Lambda_{m_{9},2n_{9}+1}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}+1}\right)\left(|O_{8}|^{2}+|C_{8}|^{2}\right)$
$\displaystyle-\left(\Lambda_{m_{8}+\frac{1}{2},2n_{8}+1}\Lambda_{m_{9},2n_{9}+1}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}}\Lambda_{m_{9}+\frac{1}{2},2n_{9}+1}\right)\left(O_{8}\overline{C}_{8}+\overline{O}_{8}C_{8}\right)\bigg{\\}}\frac{1}{|\eta^{8}|^{2}}\
.$ (2.8)

The type IIB gravitinos acquire masses

$M_{1}=\frac{R_{8}}{\alpha^{\prime}}\qquad\text{or}\qquad
M_{2}=\frac{1}{2R_{9}}\ ,$ (2.9)

which vanish in the supersymmetric limits $R_{8}\to 0$ and/or
$R_{9}\to\infty$. Moreover, as usual with a Scherk–Schwarz mechanism, a scalar
in the twisted sector becomes tachyonic when the radii satisfy

$\frac{1}{4R_{8}^{2}}+\frac{R_{9}^{2}}{\alpha^{\prime
2}}<\frac{2}{\alpha^{\prime}}\ .$ (2.10)

Notice that, unlike its supersymmetric version (2.2), the torus amplitude
(2.8) is not invariant under the T-duality
$(R_{8},R_{9})\to\big{(}\frac{\alpha^{\prime}}{2R_{8}},\frac{\alpha^{\prime}}{2R_{9}}\big{)}$.
Indeed, this transformation amounts to exchanging the lattice sums of the two
directions and thus switching $X^{8}\leftrightarrow X^{9}$, leading to the new
amplitude

$\displaystyle{\tilde{\mathcal{T}}}=\int\frac{\text{d}^{2}\tau}{\tau_{2}^{5}}$
$\displaystyle\\!\\!\;\bigg{\\{}\;\left(\Lambda_{m_{8},2n_{8}}\Lambda_{m_{9},2n_{9}}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}}\Lambda_{m_{9},2n_{9}+1}\right)\left(|V_{8}|^{2}+|S_{8}|^{2}\right)$
$\displaystyle-\left(\Lambda_{m_{8},2n_{8}}\Lambda_{m_{9},2n_{9}+1}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}}\Lambda_{m_{9},2n_{9}}\right)\left(V_{8}\overline{S}_{8}+\overline{V}_{8}S_{8}\right)$
$\displaystyle+\left(\Lambda_{m_{8},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}+1}\right)\left(|O_{8}|^{2}+|C_{8}|^{2}\right)$
$\displaystyle-\left(\Lambda_{m_{8},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}+1}+\Lambda_{m_{8}+\frac{1}{2},2n_{8}+1}\Lambda_{m_{9}+\frac{1}{2},2n_{9}}\right)\left(O_{8}\overline{C}_{8}+\overline{O}_{8}C_{8}\right)\bigg{\\}}\frac{1}{|\eta^{8}|^{2}}\
.$ (2.11)

The latter can therefore be obtained by a free action generated by
$g^{\prime\prime}=(-1)^{F}\delta_{p_{8}}\delta_{w_{9}}$, followed by the
rescaling $R_{8}\to 2R_{8}$. In this case, the masses of the gravitinos are

$M_{1}=\frac{1}{2R_{8}}\qquad\text{or}\qquad
M_{2}=\frac{R_{9}}{\alpha^{\prime}}$ (2.12)

and supersymmetry is restored in the limits $R_{8}\to\infty$ and/or $R_{9}\to
0$.

###  2.2 The supersymmetric orientifold amplitudes

In eight dimensions, the gauge group in supersymmetric orientifold models has
rank $16$, $8$ or 0.555We refer only to the gauge group arising from the open-
string/D-brane sector. For rank $8$, the gauge group of maximal dimension,
i.e. in the absence of Wilson lines, is $\text{USp}(16)$. This 8d model was
first constructed by Bianchi, Pradisi and Sagnotti in terms of D9-branes and
an O$9_{-}$-plane [91, 92]. It also has a dual description in terms of CHL
strings [94]. Moreover, it admits a geometrical T-dual picture understood
later on by Witten, which we will consider hereafter [95].

In the case of the standard $\text{SO}(32)$ type I superstring, i.e. with
$B_{89}=0$, the standard T-duality transformation
$(R_{8},R_{9})\to({\alpha^{\prime}\over R_{8}},{\alpha^{\prime}\over R_{9}})$
turns the $\text{O}9_{-}$-plane wrapping the torus into four
$\text{O}7_{-}$-planes located at the orientifold fixed points of the
generator $\Omega^{\prime}=\Omega\Pi_{8}\Pi_{9}(-1)^{F_{\rm L}}$. In our
notations, $\Pi_{i}$ is a parity operation $X^{i}\to-X^{i}$ and $F_{\rm L}$ is
the left-handed fermion number. This geometry is depicted in Fig. 1(a) and the
resulting model contains 16 D7-branes666Or equivalently 32 “half-branes” of
type IIB organized as 16 mirror pairs referred to as 16 “branes” in the
orientifold theory. in order to cancel the RR tadpole.

(a) Geometry of the standard $\text{SO}(32)$ superstring. There is an ${\rm
O}7_{-}$-plane at each of the four fixed points.

(b) Geometry of the supersymmetric $\text{USp}(16)$ model. There is 1 ${\rm
O}7_{+}$-plane at $(0,0)$ and 3 ${\rm O}7_{-}$-planes at $(\pi R_{8},0)$,
$(0,\pi R_{9})$ and $(\pi R_{8},\pi R_{9})$.

(c) Geometry of the non-supersymmetric $\text{USp}(16)$ model. There is 1
$\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$-plane at $(0,0)$, 1 $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{-}$-plane at $(\pi R_{8},0)$ and 2 ${\rm
O}7_{-}$-planes at $(0,\pi R_{9})$ and $(\pi R_{8},\pi R_{9})$.

Figure 1: Eight dimensional T-dual geometries: The standard $\text{SO}(32)$
superstring theory, the $\text{USp}(16)$ supersymmetric theory and its non-
supersymmetric version.

On the other hand, it has been shown that for the $\text{USp}(16)$ theory,
i.e. with $B_{89}={\alpha^{\prime}\over 2}$, the T-duality transformation
$(R_{8},R_{9})\to({\alpha^{\prime}\over 2R_{8}},{\alpha^{\prime}\over
2R_{9}})$ turns the original $\text{O}9_{-}$-plane into three
$\text{O}7_{-}$-planes and one $\text{O}7_{+}$-plane [95]. While an
$\text{O}7_{-}$-plane has charge (and tension) equal to $-4$ in units of a
regular D7-brane charge, an $\text{O}7_{+}$-plane has charge (and tension)
equal to $+4$. The geometry is depicted in Fig. 1(b), where $R_{8}$, $R_{9}$
now refer to the radii in the T-dual theory. The switch
$\text{O}7_{-}\to\text{O}7_{+}$ has the overall effect of halving the RR
tadpole, a fact that requires the addition of only eight D7-branes (16 half-
branes). The rank of the gauge group is thus reduced to 8. Furthermore, while
D7-branes on top of an $\text{O}7_{-}$-plane lead to an orthogonal (SO) gauge
group, D7-branes on top of an $\text{O}7_{+}$-plane lead to a symplectic (USp)
gauge factor. Therefore, the configuration with all the D7-branes sitting on
top of the $\text{O}7_{+}$-plane yields the gauge symmetry $\text{USp}(16)$.

From now on, the description of the $\text{USp}(16)$ theory we choose is that
of the type IIB theory with orientifold projection
$\Omega^{\prime}=\Omega\Pi_{8}\Pi_{9}(-1)^{F_{\rm L}}$, which involves
$\text{O}7_{\pm}$-planes and D7-branes. The spectrum is encoded in the
partition functions which can be worked out using standard methods. The torus
contribution is given by half that given in Eq. (2.2).777Remind that the type
IIB amplitude (2.2) is self-dual. Hence, it can be used in the orientifold
theory obtained by modding with $\Omega$ [91, 92] or that obtained with
$\Omega^{\prime}$. Being T-dual, they are physically equivalent. Moreover, the
Klein, cylinder and Möbius amplitudes are

$\displaystyle\mathcal{K}$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{2n_{9}}W_{2n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}2i\tau_{2}\big{)}\
,$ $\displaystyle\mathcal{A}$
$\displaystyle=\frac{N^{2}}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{n_{9}}W_{n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}\genfrac{}{}{}{1}{i\tau_{2}}{2}\big{)}\
,$ $\displaystyle\mathcal{M}$
$\displaystyle=\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{n_{9}}\big{[}(-1)^{n_{9}}W_{2n_{8}}-W_{2n_{8}+1}\big{]}\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
,$ (2.13)

where $N$ is the number of half-D7-branes and the lattices of winding modes
are defined as

$W_{n_{i}}=e^{-\pi\tau_{2}n_{i}^{2}\frac{R_{i}^{2}}{\alpha^{\prime}}}\ .$
(2.14)

The “field-theory” open-string spectrum is encoded in

$\displaystyle(\mathcal{A}+\mathcal{M})|_{\rm
FT}=\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}$
$\displaystyle\left[\frac{N(N+1)}{2}W_{2n_{9}}W_{2n_{8}}\right.$
$\displaystyle\left.+\frac{N(N-1)}{2}\big{(}W_{2n_{9}+1}W_{2n_{8}}+W_{n_{9}}W_{2n_{8}+1}\big{)}\right]\left.\frac{V_{8}-S_{8}}{\eta^{8}}\right|_{0}\
,$ (2.15)

where the index $0$ stands for the constant mode of the characters. It is
manifestly supersymmetric and describes a $\text{USp}(N)$ gauge symmetry. The
value $N=16$ is found by imposing the RR tadpole condition, which can be
derived from the amplitudes in the tree-level channel,

$\displaystyle\mathcal{\tilde{K}}$
$\displaystyle=\frac{2^{5}\alpha^{\prime}}{8R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,P_{m_{9}}P_{m_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\
,$ $\displaystyle\mathcal{\tilde{A}}$
$\displaystyle=\frac{2^{-5}N^{2}\alpha^{\prime}}{2R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,P_{m_{9}}P_{m_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\
,$ $\displaystyle\mathcal{\tilde{M}}$
$\displaystyle=\frac{N\alpha^{\prime}}{2R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,\big{[}P_{2m_{9}+1}-(-1)^{m_{8}}P_{2m_{9}}\big{]}P_{m_{8}}\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})\
,$ (2.16)

where the lattices of momentum states are given by

$P_{m_{i}}=e^{-\pi\frac{l}{2}m_{i}^{2}\frac{\alpha^{\prime}}{R_{i}^{2}}}\ .$
(2.17)

The tree-level amplitudes encode uniquely the geometry of the D-branes and
O-planes. Indeed, the geometry can in general be revealed by remembering that
the tree-level channel amplitudes capture the propagation of closed strings
between orientifold planes and/or D-branes. As an example, a generic Klein-
bottle amplitude can be formally written as

$\mathcal{\tilde{K}}=\sum_{a,{\bf m}}\sum_{A,B}(-1)^{F_{\rm
L}}C_{aA}\,C_{aB}\,G_{a{\bf m}}({\bf x}_{A},{\bf x}_{B})\ ,$ (2.18)

where $a$ labels the NS-NS and RR closed-string degrees of freedom in ten
dimensions and ${\bf m}=(m_{8},m_{9})$ (in 8d models as above) are the
internal momenta of their Kaluza–Klein (KK) modes. Moreover, $G_{a{\bf
m}}({\bf x}_{A},{\bf x}_{B})$ is the tree-level scalar propagator transverse
to the O-planes for a flat internal space, while $C_{aA}$ captures the
coupling to the O-plane $A$ located at ${\bf x}_{A}=(x^{8}_{A},x^{9}_{A})$. In
our examples, $C_{aA}\propto T_{A}$ for the NS-NS states and $C_{aA}\propto
Q_{A}$ for the RR ones, where $T_{A}$ and $Q_{A}$ denote the tension and
charge of the O-plane $A$ and the proportionality constants are equal.
Actually, the closed-string states $a,{\bf m}$ propagating in
$\mathcal{\tilde{K}}$, which are bosons arising in the NS-NS and RR sector,
have different Lorentz structures: For instance the dilaton is a scalar, the
graviton is a tensor, etc. Hence, they have different propagators and
couplings to branes and orientifolds. Contracting the couplings and the
propagators, one obtains the effective couplings $C_{aA}$ and a scalar
propagator $G_{a{\bf m}}({\bf x}_{A},{\bf x}_{B})$ for each closed-string
state. Explicitly, we have

$\displaystyle G_{a{\bf m}}({\bf x}_{A},{\bf x}_{B})$ $\displaystyle=e^{i{\bf
m}({\bf x}_{A}-{\bf
x}_{B})}\frac{1}{p_{\parallel}^{2}+M_{a}^{2}+\sum_{i}\frac{m_{i}^{2}}{R_{i}^{2}}}$
$\displaystyle=\frac{\pi\alpha^{\prime}}{2}e^{i{\bf m}({\bf x}_{A}-{\bf
x}_{B})}\int_{0}^{\infty}\text{d}l\
e^{-\pi\frac{l}{2}\alpha^{\prime}\big{(}M_{a}^{2}+\sum_{i}\frac{m_{i}^{2}}{R_{i}^{2}}\big{)}}\
,$ (2.19)

where by convention the variables $x^{i}_{A}$ take values in the range
$[-\pi,\pi]$ and the internal coordinates are defined as $X^{i}=x^{i}R_{i}$.
The closed-string channel Klein-bottle amplitude is therefore given by

$\mathcal{\tilde{K}}=\frac{\pi\alpha^{\prime}}{2}\sum_{a,{\bf
m}}\sum_{A,B}C_{aA}C_{aB}\int_{0}^{\infty}\text{d}l\ e^{i{\bf m}({\bf
x}_{A}-{\bf
x}_{B})-\pi\frac{l}{2}\alpha^{\prime}\big{(}M_{a}^{2}+\sum_{i}\frac{m_{i}^{2}}{R_{i}^{2}}\big{)}}\
.$ (2.20)

The factors $e^{i{\bf m}({\bf x}_{A}-{\bf x}_{B})}$ capture the locations of
the O-planes and display the products of the wavefunctions of a closed-string
Kaluza–Klein mode $a,{\bf m}$ respectively located on the O-planes $A$ and
$B$.

In the 8d examples of this section, there are four orientifold fixed points
$(0,0)$, $(0,\pi R_{9})$,$(\pi R_{8},0)$, $(\pi R_{8},\pi R_{9})$, where the
O7-planes sit. The phases $e^{i{\bf m}({\bf x}_{A}-{\bf x}_{B})}$ encoding the
propagation between the four O-planes take values $1$, $(-1)^{m_{9}}$,
$(-1)^{m_{8}}$ or $(-1)^{m_{9}+m_{8}}$. Once dressed by the signs given by the
tensions and charges, they produce projectors in the tree-level channel
amplitude. In the SO($32$) type IIB orientifold case, which contains 4 ${\rm
O}7_{-}$-planes, the projector in the tree-level Klein-bottle amplitude is

$\Pi_{\mathcal{\tilde{K}}}\propto 4[1+(-1)^{m_{9}}][1+(-1)^{m_{8}}]\ ,$ (2.21)

which projects onto even KK states. In contrast, the corresponding one for the
supersymmetric $\text{USp}(16)$ type IIB orientifold satisfies

$\displaystyle\Pi_{\mathcal{\tilde{K}}}$ $\displaystyle\propto
4-2(-1)^{m_{9}}+2(-1)^{m_{9}}-2(-1)^{m_{8}}+2(-1)^{m_{8}}-2(-1)^{m_{9}+m_{8}}+2(-1)^{m_{9}+m_{8}}$
$\displaystyle\propto 4\ ,$ (2.22)

leading to no projection of the KK states.

The tree-level channel cylinder and Möbius amplitudes take similar formal
expressions. In the former case, the objects $A$ and $B$ are D-branes while in
the latter case they are a D-brane and an O-plane. For instance, in the ${\rm
USp}(16)$ model, the lattices in the Möbius amplitude involve all momentum
states subject to the projector

$\Pi=\frac{1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}\ ,$ (2.23)

which is consistent with the geometry of one stack of $8$ (regular) D7-branes
coincident with the O$7_{+}$-plane, whereas the three other orientifold fixed
points are occupied by standard O$7_{-}$-planes. Moving all the D7-branes on
top of one of the O$7_{-}$-planes lead to an $\text{SO}(16)$ gauge group,
whereas moving all of them into the bulk in one stack leads to a $\text{U}(8)$
open-string gauge group.

###  2.3 The non-supersymmetric orientifold amplitudes

Let us now turn to the implementation of supersymmetry breaking in the Klein,
cylinder and Möbius amplitudes, without introducing perturbative instabilities
or tadpoles. The allowed form of the corresponding torus amplitude will be
determined in the next subsection.

Geometrically, the mechanism of supersymmetry breaking is the following. A
pair of O$7_{+}$-plane and O$7_{-}$-plane have globally zero tension and RR
charge. From a string-theory viewpoint, it is possible to replace such a pair
by an $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$ and $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$ pair, which also has vanishing total tension and charge. However,
the second option breaks supersymmetry in the presence of the $8$ D7-branes
needed to cancel the tadpoles. This geometry is depicted in Fig. 1(c).
Supersymmetry breaking is not visible in the cylinder amplitude, which
describes D7-D7 amplitudes. It is less obvious but true that the orientifold
configuration with 2 O$7_{-}$, 1 $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{+}$, 1 $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{-}$ and the supersymmetric one with 1 O$7_{+}$
and 3 O$7_{-}$-planes lead to identical Klein-bottle amplitudes. Indeed, in
the former case we have

$\displaystyle\mathcal{\tilde{K}}$
$\displaystyle\propto\Bigl{\\{}[4-2(-1)^{m_{8}}+2(-1)^{m_{8}}](V_{8}-S_{8})+$
$\displaystyle\ \ \ \ \
\;\,[-2(-1)^{m_{9}}+2(-1)^{m_{9}}-2(-1)^{m_{9}+m_{8}}+2(-1)^{m_{9}+m_{8}}](V_{8}+S_{8})\Big{\\}}P_{m_{8}}P_{m_{9}}$
$\displaystyle\propto 4(V_{8}-S_{8})P_{m_{8}}P_{m_{9}}\ ,$ (2.24)

where the character $V_{8}-S_{8}$ describes the tree-level propagation of
closed strings between mutually BPS O7-planes (O$7_{-}$-O$7_{-}$, $\mkern
1.5mu\overline{\mkern-1.5mu\rm O7\mkern-1.5mu}\mkern 1.5mu_{\pm}$-$\mkern
1.5mu\overline{\mkern-1.5mu\rm O7\mkern-1.5mu}\mkern 1.5mu_{\pm}$), whereas
$V_{8}+S_{8}$ describes the tree-level propagation of closed strings between
mutually non-BPS O7-planes (O$7_{-}$-$\mkern 1.5mu\overline{\mkern-1.5mu\rm
O7\mkern-1.5mu}\mkern 1.5mu_{\pm}$, $\mkern 1.5mu\overline{\mkern-1.5mu\rm
O7\mkern-1.5mu}\mkern 1.5mu_{\pm}$-O$7_{-}$). The phases reflect the geometry
of O-planes and lead to a cancellation of the non-BPS terms, leaving the
unprojected supersymmetric sum over all KK states as in the supersymmetric
case (see Eqs. (2.16) and (2.22)).

As will be shown later, the configuration where the $8$ D7-branes are
coincident with the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{+}$-plane is the only stable configuration at
one loop. Using the above given geometrical interpretation of O-planes, it is
easy to check that the tree-level channel amplitudes are given by

$\displaystyle\mathcal{\tilde{K}}$
$\displaystyle=\frac{2^{5}\alpha^{\prime}}{8R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,P_{m_{9}}P_{m_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\
,$ $\displaystyle\mathcal{\tilde{A}}$
$\displaystyle=\frac{2^{-5}N^{2}\alpha^{\prime}}{2R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,P_{m_{9}}P_{m_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\
,$ $\displaystyle\mathcal{\tilde{M}}$
$\displaystyle=\frac{N\alpha^{\prime}}{2R_{9}R_{8}}\int_{0}^{\infty}\text{d}l\,\big{[}P_{2m_{9}+1}-(-1)^{m_{8}}P_{2m_{9}}\big{]}P_{m_{8}}\frac{{\hat{V}}_{8}-(-1)^{m_{8}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})\
.$ (2.25)

Notice that the only change in (2.25) compared to the supersymmetric case
(2.16) is the extra phase $(-1)^{m_{8}}$ in the RR couplings of the closed
strings propagating between the D7-branes and the O7-planes in the Möbius
amplitude. The projector in the RR sector is thus transformed accordingly,

$\displaystyle\Pi_{\rm NSNS}$
$\displaystyle=\frac{+1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}$
$\displaystyle\Pi_{\rm RR}=(-1)^{m_{8}}\Pi_{\rm NSNS}$
$\displaystyle=\frac{-1-(-1)^{m_{9}}+(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}\ ,$
(2.26)

where $\Pi_{\rm NSNS}$ is identical to that of the supersymmetric case, Eq.
(2.23). The above projectors encode all the information about the geometry. In
fact, with all D-branes located at the origin, the change of signs of the RR
couplings at $(0,0)$ and $(\pi R_{8},0)$ tells us that the orientifold planes
located there preserve opposite supersymmetry as compared to the D7-branes and
are therefore $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$ and $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$.

The loop-channel amplitudes can be worked out by the usual methods, leading to

$\displaystyle\mathcal{K}=\frac{1}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{2n_{9}}W_{2n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}2i\tau_{2}\big{)}\
,$
$\displaystyle\mathcal{A}=\frac{N^{2}}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{n_{9}}W_{n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}\genfrac{}{}{}{1}{i\tau_{2}}{2}\big{)}\
,$
$\displaystyle\mathcal{M}=\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{n_{9}}\big{[}(-1)^{n_{9}}W_{2n_{8}}-W_{2n_{8}+1}\big{]}\frac{{\hat{V}}_{8}+(-1)^{n_{9}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
.$ (2.27)

Comparing the Möbius amplitude with its supersymmetric counterpart (2.13)
reveals a supersymmetry breaking orientifold projection

$\Omega^{\prime\prime}=\Omega\Pi_{8}\Pi_{9}(-1)^{F_{\rm
L}}(-\delta_{w_{9}})^{F}\ ,$ (2.28)

where, as before, $\Pi_{i}$ is the parity operation $X^{i}\to-X^{i}$, $F_{\rm
L}$ is the left-moving fermion number, $F$ is the spacetime fermion number and
$\delta_{w_{9}}$ generates a winding shift in the coordinate $X^{9}$. The
latter acts on the zero-modes as $|{\bf m},{\bf n}\rangle\to(-1)^{n_{9}}|{\bf
m},{\bf n}\rangle$, as follows from a left-right asymmetric action $X_{\rm
L}^{9}\to X_{\rm L}^{9}+\frac{\pi R_{9}}{2}$, $X_{\rm R}^{9}\to X_{\rm
R}^{9}-\frac{\pi R_{9}}{2}$ on the left- and right-moving parts of the
coordinate. Notice that since there is no fermion propagating in the Klein
bottle, the supersymmetry breaking deformation $(-\delta_{w_{9}})^{F}$ has no
effect in this amplitude.

The massless field-theory open-string spectrum is captured by

$(\mathcal{A}+\mathcal{M})|_{\rm
FT}=\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\left[\frac{N(N+1)}{2}\left.{V_{8}\over\eta^{8}}\right|_{0}-\frac{N(N-1)}{2}\left.{S_{8}\over\eta^{8}}\right|_{0}\right]\
$ (2.29)

and displays supersymmetry breaking at the string scale, of the brane
supersymmetry breaking type. The gauge group is $\text{USp}(16)$ as before.
The vector bosons are thus in the symmetric representation, but the fermions
are in the antisymmetric representation, which contains in particular the
gauge-singlet goldstino. This is the basic indication of the nonlinear
realization of supersymmetry where the D7-branes sit. It is then easy to move
D7-branes in the internal two-torus and derive the resulting spectrum. Putting
all D-branes on top of the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{-}$-plane leads to an $\text{SO}(16)$ gauge
group with massless fermions in the symmetric representation. The latter
contains the singlet goldstino implying again a nonlinear supersymmetry and a
supersymmetry breaking at the string scale. Putting all D7-branes on top of
one of the two remaining ${{\rm O}7_{-}}$-planes leads to a supersymmetric
massless spectrum with $\text{SO}(16)$ gauge group and a supersymmetry
breaking at the massive level due to the far-away presence of the two $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern 1.5mu$-planes.

Let us stress again that despite the fact that the O7-planes are of types
orientifold and anti-orientifold, the Klein bottle amplitude is exactly the
same as in the supersymmetric case, due to the cancellation of the
supersymmetry-breaking contributions.888We will discuss in the next subsection
the issue of supersymmetry in the closed-string spectrum. The O-planes and
anti O-planes are mutually non-BPS. The cancellation of the non-supersymmetric
contributions in the Klein bottle does not mean that the closed-string sector
is supersymmetric, even ignoring the supersymmetry breaking transmission from
the open sector. As we will see, the most plausible possibility is that the
tree-level closed-string spectrum has softly broken supersymmetry. Another
possibility, which we consider however unlikely, is that the closed-string
spectrum is supersymmetric but the interactions are not. Another insight about
this issue is the gravitino masses: A supersymmetric closed-string spectrum
would be in contradiction with the presence of orientifold and anti-
orientifold planes, which impose opposite boundary conditions for the
gravitinos. At one loop, only the Möbius amplitude “knows” about supersymmetry
breaking, without however generating NS-NS tadpoles (of course, RR tadpoles
are non-negotiable and always have to cancel).

The precursor paper of Angelantonj and Cardella contains a model equivalent to
the one presented above [77].999In the model constructed in Sect. 3 of [77],
the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$-$\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$ pair sits on the diagonal of the two-torus, which corresponds to a
different choice of the projector in the Möbius, $\Pi_{\rm
RR}=(-1)^{m_{9}+m_{8}}\Pi_{\rm NSNS}$. The 8 D7-branes were separated into two
stacks of four branes, sitting on top of the anti-orientifolds. The attraction
of the D7-branes on top of $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{+}$ cancels the repulsion of the D7-branes on
top of $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$. However, as already known by the authors of [77] and obvious from
the discussion below, this configuration is unstable, since the D7-branes on
top of $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$ are repelled by $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{-}$ and attracted towards the $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern 1.5mu_{+}$ -plane,
leading to the stable configuration with one stack of eight coincident
D7-branes, negative potential and $\text{USp}(16)$ gauge group discussed
above. In our work, we provide a microscopic geometrical interpretation of the
source of supersymmetry breaking in terms of the replacement of an
O$7_{+}$-O$7_{-}$ pair by an $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{+}$-$\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{-}$ pair, which leads to a novel form of brane
supersymmetry breaking without tadpoles. Moreover, as argued in the next
subsection, we believe that the closed-string sector is not supersymmetric at
tree-level but features spontaneously broken supersymmetry.

Let us now make considerations of energetics. The O-planes have no dynamical
positions. The D7-branes, on the other hand, do have dynamical positions. To
find which configuration is expected to minimize the one-loop effective
potential, recall that the D7-branes are mutually BPS and have therefore no
net interactions with the O${7_{+}}$ and O${7_{-}}$-planes. On the contrary,
they are attracted by the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}7\mkern-1.5mu}\mkern 1.5mu_{+}$-plane and repelled by the $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern 1.5mu_{-}$-plane.
Hence, the only stable configuration is obtained by putting all D7-branes on
top of the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$-plane, leading to a $\text{USp}(16)$ gauge group and breaking SUSY
at the string scale, as explained above. At first sight, one could think that
a second option would be to put some stuck (or rigid) half-D7-branes on top of
O${7_{-}}$ or $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{-}$-planes, with no gauge group (but a $\mathbb{Z}_{2}$ global
symmetry). As will be seen in Sect. 4, such configurations are however
inconsistent, a fact that can be checked by adding probe D3-branes.

To confirm these expectations, we write the Möbius amplitude for arbitrary
brane positions along $X^{9}$ and $X^{8}$. To this end, we introduce vectors
$\vec{a}_{\alpha}=(a_{\alpha}^{8},a_{\alpha}^{9})$ such that the position of
the half-brane $\alpha\in\\{1,\dots,16\\}$ along direction $X^{i}$ is $2\pi
a_{\alpha}^{i}R_{i}$. In both channels, we obtain

$\displaystyle\begin{split}{\cal
M}&=\frac{1}{2}\sum_{\alpha}\int\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,\Bigg{\\{}\,\left[(-1)^{n_{9}}W_{2n_{8}+2a_{\alpha}^{2}}-W_{2n_{8}+1+2a_{\alpha}^{2}}\right]\frac{\hat{V}_{8}}{\hat{\eta}^{8}}\\\
&\hskip
79.6678pt-\left[(-1)^{n_{9}}W_{2n_{8}+1+2a_{\alpha}^{2}}-W_{2n_{8}+2a_{\alpha}^{2}}\right]\frac{\hat{S}_{8}}{\hat{\eta}^{8}}\Bigg{\\}}\,W_{n_{9}+2a_{\alpha}^{1}}\
,\\\ \tilde{{\cal
M}}&=\frac{\alpha^{\prime}}{2R_{9}R_{8}}\sum_{\alpha}\int\text{d}l\,e^{4i\pi
m_{9}a_{\alpha}^{1}\,}e^{2i\pi
m_{8}a_{\alpha}^{2}}\,P_{m_{8}}\,\Bigg{\\{}\,\left[e^{2i\pi
a_{\alpha}^{1}}P_{2m_{9}+1}-(-1)^{m_{8}}P_{2m_{9}}\right]\frac{\hat{V}_{8}}{\hat{\eta}^{8}}\\\
&\hskip 203.43703pt-\left[e^{2i\pi
a_{\alpha}^{1}}(-1)^{m_{8}}P_{2m_{9}+1}-P_{2m_{9}}\right]\frac{\hat{S}_{8}}{\hat{\eta}^{8}}\Bigg{\\}}\,.\end{split}$
(2.30)

The dependance of the one-loop effective potential ${\cal V}$ on the
independent positions can be derived solely from ${\cal M}$ and $\tilde{\cal
M}$ in various regimes of the internal radii. Among the $16$ vectors
$\vec{a}_{\alpha}$, at most $8$ are dynamical degrees of freedom since the
half-branes move by pairs and unpaired half-branes stuck at a fixed point are
not dynamical. We will label the dynamical ones by an index $r$. In Fig. 2, we
display the vector field $\big{(}\\!\\!-\\!\\!\frac{\partial{\cal V}}{\partial
a_{r}^{8}},-\frac{\partial{\cal V}}{\partial a_{r}^{9}}\big{)}$ obtained
numerically for a given brane $r$ of arbitrary position. As anticipated
before, the minimum of the potential is reached when the branes sit at the
origin, on the $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}7\mkern-1.5mu}\mkern
1.5mu_{+}$-plane.

Figure 2:  Example of vector field $\big{(}\\!\\!-\\!\\!\frac{\partial{\cal
V}}{\partial a_{r}^{8}},-\frac{\partial{\cal V}}{\partial a_{r}^{9}}\big{)}$
obtained numerically. The lighter the color is, the longer the vector norm is.

###  2.4 Consistent pairing of torus and orientifold amplitudes

At first sight, one may think that the supersymmetric torus amplitude (2.2) as
well as the non-supersymmetic ones (2.8) and (2.11) are all consistent with
the supersymmetric orientifold amplitudes (2.13) and their non-supersymmetric
deformation (2.27). If true, this would yield six different orientifold
models. There are however arguments based on the understanding of the
orientifold projections as well as on the effective field theories that
suggest that only two options are consistent.

Let us combine the non-supersymmetric torus amplitude (2.8) with the non-
supersymmetric orientifold amplitudes (2.27). The torus amplitude can be
constructed as an orbifold generated by
$g^{\prime}=(-1)^{F}\delta_{w_{8}}\delta_{p_{9}}$, while the orientifold
amplitudes are obtained from the action of
$\Omega^{\prime\prime}=\Pi_{8}\Pi_{9}(-1)^{F_{\rm L}}(-\delta_{w_{9}})^{F}$.
Hence, the (anti-)orientifold planes are located at the fixed points of
$\Omega^{\prime\prime}$ and $\Omega^{\prime\prime}g^{\prime}$. To be specific,
$\Omega^{\prime\prime}$ fixes $(0,0)$ as well as $(\pi R_{8},0)$, thanks to
the $2\pi R_{8}$ periodicity. Moreover $\Omega^{\prime\prime}g^{\prime}$ fixes
$(0,\pi R_{9})$ because $g^{\prime}$ contains a factor $\delta_{p_{9}}$ acting
as $X_{9}\to X_{9}+2\pi R_{9}$ after rescaling of the radius $R_{9}\to
2R_{9}$. It also fixes $(\pi R_{8},\pi R_{9})$ thanks to the $2\pi R_{8}$
periodicity. Notice that the presence of a factor $(-1)^{F}$ in such a group
element changes the orientifolds into anti-orientifold planes, as can be seen
in the Möbius amplitude where $(-1)^{F}$ changes the sign of the RR coupling.
As a result, the fixed points of $\Omega^{\prime\prime}$ are anti-orientifold
planes, while those of $\Omega^{\prime\prime}g^{\prime}$ are orientifold
planes due to the cancellation of the factors $(-1)^{F}$. Therefore, the
nature of the O-planes derived from the non-supersymmetric amplitudes (2.27)
and shown in Fig. 1(c) are in agreement with the non-supersymmetric torus
amplitude (2.8).

As shown in Sect. 2.1, the second non-supersymmetric torus amplitude (2.11) is
equivalent to (2.8) under the interchange of the coordinates
$X^{8}\leftrightarrow X^{9}$. Therefore, a consistent orientifold model is
obtained by applying the same operation on the non-supersymmetric orientifold
amplitudes (2.27) and orientifold action (2.28). The corresponding geometry of
O-planes is like in Fig. 1(c), with the ${\overline{\rm O7}_{-}}$-plane now
located at $(0,\pi R_{9})$.

Finally, reasoning as above with the generators
${g}=\delta_{w_{8}}\delta_{p_{9}}$ and
$\Omega^{\prime}=\Omega\Pi_{8}\Pi_{9}(-1)^{F_{\rm L}}$, one concludes that the
supersymmetric orientifold amplitudes (2.13) are compatible with the
supersymmetric torus amplitude (2.2).

In fact, the reason why the four other combinations of torus and orientifold
amplitudes are inconsistent is that the orientifold projections are not
symmetries of the closed-string spectrum. For instance, the supersymmetric
torus amplitude (2.2) does not seem to be compatible with the orientifold
projection (2.28). Indeed, because the factor $(-\delta_{w_{9}})^{F}$ in
$\Omega^{\prime\prime}$ is equivalent to $(-1)^{n_{9}F}$, fermions with even
and odd winding numbers $n_{9}$ are projected differently. Due to
interactions, the same conclusion should apply for bosons. However, there is
no such selection rule in (2.2), as opposed to the non-supersymmetric torus
amplitude (2.8), where fermions with even $n_{9}$ are gravitinos while those
with odd $n_{9}$ are spin $\frac{1}{2}$ particles.

Moreover, any attempt to combine the supersymmetric torus amplitude (2.2) with
the non-supersymmetric orientifold amplitudes (2.27) is unlikely to be
consistent, since the former implies the existence of massless gravitinos that
would be difficult to explain from the point of view of the effective field
theory. Instead, there seems to be no obstruction, from the point of view of
the effective supergravity, to couple the non-supersymmetric torus amplitude
(2.8) with the orientifold ones (2.27), since in this case all gravitinos are
massive.

To understand a little better how the nature of (anti-)orientifold planes may
or may not induce gravitino masses, let us consider first the orientifold
projection $\Omega^{\prime}$. In this case, ${\rm O}_{\pm}$-planes, which
preserve the same supersymmetry, are located at the four fixed points of
$T^{2}$. The action on the two gravitinos $\psi_{\mu}$ and
$\tilde{\psi}_{\mu}$ of the 10-dimensional type IIB is given by

$\displaystyle\Omega^{\prime}\psi_{\mu}(X^{8},X^{9})(\Omega^{\prime})^{-1}=\psi_{\mu}(-X^{8},-X^{9})=\Gamma_{8}\Gamma_{9}\tilde{\psi}_{\mu}(X^{8},X^{9})\
,$
$\displaystyle\Omega^{\prime}\tilde{\psi}_{\mu}(X^{8},X^{9})(\Omega^{\prime})^{-1}=\tilde{\psi}_{\mu}(-X^{8},-X^{9})=-\Gamma_{8}\Gamma_{9}\psi_{\mu}(X^{8},X^{9})\
,$ (2.31)

which preserves one linear combination. Let us now consider a geometry where
anti-orientifold planes are located at the four fixed points. The action of
the corresponding orientifold projection denoted $\tilde{\Omega}^{\prime}$
preserves the orthogonal combination of gravitinos. Hence, we have

$\displaystyle\tilde{\Omega}^{\prime}\psi_{\mu}(X^{8},X^{9})(\tilde{\Omega}^{\prime})^{-1}=\psi_{\mu}(-X^{8},-X^{9})=-\Gamma_{8}\Gamma_{9}\tilde{\psi}_{\mu}(X^{8},X^{9})\
,$
$\displaystyle\tilde{\Omega}^{\prime}\tilde{\psi}_{\mu}(X^{8},X^{9})(\tilde{\Omega}^{\prime})^{-1}=\tilde{\psi}_{\mu}(-X^{8},-X^{9})=\Gamma_{8}\Gamma_{9}\psi_{\mu}(X^{8},X^{9})\
.$ (2.32)

We have seen that the geometry corresponding to the non-supersymmetric
orientifold amplitudes (2.27) involves both orientifold and anti-orientifold
planes, as shown in Fig. 1(c). Therefore, the boundary conditions of the
gravitinos at $X^{9}=0$ are of the type (2.32), whereas at $X^{9}=\pi R_{9}$
they are of the type (2.31). Overall, one obtains a shift in the KK spectrum
of gravitinos $m_{9}/R_{9}\to(m_{9}+\frac{1}{2})/R_{9}$, which is precisely
what features the non-supersymmetric torus amplitude (2.8) (see Eq. (2.9)).

##  3 Lower dimensional compactifications

In this section, we extend the mechanism of supersymmetry breaking to models
in even dimension $d\leq 6$.

In type IIB, a change of basis can always put an arbitrary discrete background
for the antisymmetric tensor $B_{ij}$ into a block-diagonal form, with
$2\times 2$ antisymmetric matrices,

$B=\alpha^{\prime}\begin{pmatrix}0&\lambda_{d}&&&\\\ -\lambda_{d}&0&&(0)&\\\
&&\ddots&&\\\ &(0)&&0&\lambda_{9}\\\ &&&-\lambda_{9}&0\end{pmatrix}\
,\quad\lambda_{i}\in\Big{\\{}0,\frac{1}{2}\Big{\\}}\ .$ (3.33)

The rank of the tensor $B_{ij}$ is twice the number of non-zero
$\lambda_{i}$’s. Since they play no significant role in the sequel, we choose
to set to zero the off-diagonal elements of the symmetric tensor $G_{ij}$. The
internal space is thus a Cartesian product of circles of radii $R_{i}$.

In the supersymmetric case, one can switch on some $\lambda_{i}$’s by
implementing a free-orbifold action on the background where $B_{ij}=0$. For
instance, $\lambda_{9}=\lambda_{8}=\frac{1}{2}$ in 6d is achieved by
considering the orbifold generated by $g_{1}=\delta_{w_{8}}\delta_{p_{9}}$ and
$g_{2}=\delta_{w_{6}}\delta_{p_{7}}$. In 4d, for $\lambda_{7}=\frac{1}{2}$,
one simply adds an extra generator $g_{3}=\delta_{w_{4}}\delta_{p_{5}}$. By
considering an orientifold action involving parities in all internal
directions, one obtains a model involving $2^{10-d}$ ${\rm
O}(d-1)_{\pm}$-planes. It is then allowed to turn some ${\rm O}(d-1)_{+}$ \-
${\rm O}(d-1)_{-}$ pairs into $\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}\mkern-1.5mu}\mkern 1.5mu(d-1)_{+}$ \- $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}\mkern-1.5mu}\mkern 1.5mu(d-1)_{-}$ ones in
order to break supersymmetry.

In the following, we consider various configurations of orientifold planes of
this type and provide the corresponding Klein, cylinder and Möbius amplitudes.
In orbifold language, the corresponding type IIB backgrounds can be realized
by including operators $(-1)^{F}$ in the definition(s) of one or several of
the generators $g_{i}$.

###  3.1 Geometry description

The geometry of a model is given by the precise locations of the various ${\rm
O}_{\pm}$ and $\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}\mkern-1.5mu}\mkern
1.5mu_{\pm}$-planes. Since pictorial representations become involved when the
number of internal dimensions increases, it is useful to consider another way
to describe a generic geometry. If one specifies an ordering for the labelling
of the fixed points, the geometry can simply be given by the list of O-plane
types following this ordering. In the $10-d$ dimensional internal space, a
fixed point can be represented by a $(10-d)$-vector with components $0$ or $1$
that indicate if it is located at the origin or at $\pi R_{i}$ in each
direction $X^{i}$, $i\in\\{d,\dots,9\\}$. For example, in 6d the fixed point
located at $(X^{6},X^{7},X^{8},X^{9})=(0,\pi R_{7},\pi R_{8},0)$ is
represented by $(0,1,1,0)$.

In practice, let us label the fixed points by an index
$A\in\\{0,\dots,2^{(10-d)}-1\\}$. With this convention, their positions are
given by $A$ written as a binary number. For instance in 6d, the fixed points
are labelled as follows,

$\displaystyle A=0=(0,0,0,0)\ ,$ $\displaystyle A=1=(0,0,0,1)\ ,$
$\displaystyle A=2=(0,0,1,0)\ ,$ $\displaystyle A=3=(0,0,1,1)\ ,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdots\,$ $\displaystyle
A=15=(1,1,1,1)\ .$ (3.34)

###  3.2 Models in six dimensions

Projectors on the momenta in the Klein-bottle and Möbius amplitude can either
be factorized in the two internal $T^{2}$’s, or non-factorized. However, to
obtain fully consistent models, compatibility of the projectors with the RR
tadpole condition turns out not to be sufficient. Indeed, we give in the
appendix examples of non-factorizable projectors, where one is consistent and
another one is not. In the following, we will consider only consistent
factorizable projectors. Supersymmetry breaking in the orientifold amplitudes
will be implemented by choosing different projectors for the NS-NS and RR
closed-string states propagating between the D-branes and the O-planes in the
Möbius amplitude. On the contrary, the Klein-bottle and cylinder amplitudes
will take forms identical to those in the SUSY cases.

In six dimensions, a non-trivial antisymmetric tensor can have $\text{rank}\
B=2$ or $4$.

$\bullet$ ${\rm rank}\ B=2$: 8d model compactified on $T^{2}$ (T-dualized)

By compactifying the 8d model on an extra $T^{2}$ and T-dualizing both of its
coordinates, one finds a 6d model with D5-branes and 16 O5-planes. In the
supersymmetric case, one would have 12 O$5_{-}$ and 4 O$5_{+}$-planes. In the
non-SUSY case, one obtains a configuration with 8 O$5_{-}$ and
$4\times({\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern
1.5mu_{-}}+{\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern
1.5mu_{+}})$-planes, where the geometry is simply the 8d one duplicated along
the new compact dimensions. The rank of the gauge group is $8$ and, depending
on the location of the stacks of D5-branes, one finds for a single stack
$\text{USp}(16)$ if the D5-branes are on top of one ${\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{+}}$ or ${\rm
O}5_{+}$-plane, $\text{SO}(16)$ if the D5-branes are on top of one ${\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{-}}$-plane or
O$5_{-}$-plane, or $\text{U}(8)$ if the stack of D5-branes is in the bulk.
Supersymmetry is broken at the string scale (nonlinearly realized) if the
D5-branes are coincident with anti-orientifolds, and broken only at the
massive level (due to the separation in the internal space from the source of
supersymmetry breaking) if the D5-branes are coincident with orientifold
planes.

When the D5-branes are put at the origin, the corresponding projectors on the
momentum states running in the Möbius amplitude are

$\displaystyle\Pi_{\rm NSNS}$
$\displaystyle=\frac{1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}\times\prod_{i=6}^{7}\frac{1+(-1)^{m_{i}}}{2}\
,$ $\displaystyle\Pi_{\rm RR}$ $\displaystyle=(-1)^{m_{8}}\Pi_{\rm NSNS}\ .$
(3.35)

The torus amplitude can be constructed as a free-orbifold generated by
$g^{\prime}_{1}=(-1)^{F}\delta_{w_{8}}\delta_{p_{9}}$.

$\bullet$ ${\rm rank}\ B=4$:

Following the ordering of the fixed points given in Eq. (3.1), the list of
O±-planes of the SUSY model we discuss here is

$\displaystyle({\rm O}5_{+},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm
O}5_{+},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},$
$\displaystyle\hskip 170.71652pt{\rm O}5_{+},{\rm O}5_{+},{\rm O}5_{+},{\rm
O}5_{+},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-})\ ,$ (3.36)

with a total of 10 O$5_{-}$ and 6 O$5_{+}$-planes. There are several possible
consistent SUSY breaking deformations. One example corresponds to a
configuration containing 8 O$5_{-}$ and $4\times({\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{-}}+{\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{+}})$-planes as
follows,

$\displaystyle(\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}5_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}5\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}5_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}5_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern
1.5mu_{-},{\rm O}5_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}5\mkern-1.5mu}\mkern 1.5mu_{-},$ $\displaystyle\hskip 170.71652pt{\rm
O}5_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}5_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}5\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}5_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}5_{-})\ .$ (3.37)

The rank of the gauge group is $4$ and, depending on the location of the
stacks of D5-branes, one finds for a single stack $\text{USp}(8)$ if the
D5-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}5\mkern-1.5mu}\mkern 1.5mu_{+}}$ or O$5_{+}$ -plane, $\text{SO}(8)$ if the
D5-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}5\mkern-1.5mu}\mkern 1.5mu_{-}}$-plane or O$5_{-}$-plane, and $\text{U}(4)$
if the D5-brane stack is in the bulk. Supersymmetry breaking pattern is of
course the same as in the ${\rm rank}\ B=2$ case discussed above.

When the 4 D5-branes (8 half-D5-branes) are coincident with the $\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}5\mkern-1.5mu}\mkern 1.5mu_{+}$-plane at the
origin of the internal space, the projectors on the momentum states in the
tree-level channel Möbius amplitude are

$\displaystyle\Pi_{\rm NSNS}$
$\displaystyle=\frac{1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}\times\frac{1+(-1)^{m_{7}}-(-1)^{m_{6}}+(-1)^{m_{7}+m_{6}}}{2}\
,$ $\displaystyle\Pi_{\rm RR}$ $\displaystyle=(-1)^{m_{8}}\Pi_{\rm NSNS}\ .$
(3.38)

In order to write the orientifold amplitudes, it is convenient to denote
lattices and volume factors as folows,

$P_{\bf m}^{(10-d)}=\prod_{i=d}^{9}P_{m_{i}}\ ,\qquad W_{\bf
n}^{(10-d)}=\prod_{i=d}^{9}W_{n_{i}}\ ,\qquad v_{10-d}=\prod_{i=d}^{9}R_{i}\
.$ (3.39)

In these notations, the tree-level channel amplitudes are given by

$\displaystyle\mathcal{\tilde{K}}=$
$\displaystyle\;\frac{(\alpha^{\prime})^{2}}{v_{4}}\int_{0}^{\infty}\text{d}l\,P_{\bf
m}^{(4)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\ ,$
$\displaystyle\mathcal{\tilde{A}}=$
$\displaystyle\;\frac{2^{-5}N^{2}(\alpha^{\prime})^{2}}{2v_{4}}\int_{0}^{\infty}\text{d}l\,P_{\bf
m}^{(4)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\ ,$
$\displaystyle\mathcal{\tilde{M}}=$
$\displaystyle\,\frac{N(\alpha^{\prime})^{2}}{4v_{4}}\int_{0}^{\infty}\text{d}l\,\big{[}P_{2m_{9}+1}-(-1)^{m_{8}}P_{2m_{9}}\big{]}P_{m_{8}}\big{[}P_{2m_{7}}-(-1)^{m_{6}}P_{2m_{7}+1}\big{]}P_{m_{6}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{{\hat{V}}_{8}-(-1)^{m_{8}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})\
,$ (3.40)

while in the loop-channel they become

$\displaystyle\mathcal{K}=$
$\displaystyle\;\frac{1}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,W_{2{\bf
n}}^{(4)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}2i\tau_{2}\big{)}\ ,$
$\displaystyle\mathcal{A}=$
$\displaystyle\;\frac{N^{2}}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,W_{\bf
n}^{(4)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}\genfrac{}{}{}{1}{i\tau_{2}}{2}\big{)}\
,$ $\displaystyle\mathcal{M}=$
$\displaystyle\;\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,W_{n_{9}}\big{[}W_{2n_{8}}-(-1)^{n_{9}}W_{2n_{8}+1}\big{]}W_{n_{7}}\big{[}W_{2n_{6}}-(-1)^{n_{7}}W_{2n_{6}+1}\big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\times\frac{(-1)^{n_{9}}{\hat{V}}_{8}+{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
.$ (3.41)

The torus amplitude can be constructed as a free-orbifold generated by
$g^{\prime}_{1}=(-1)^{F}\delta_{w_{8}}\delta_{p_{9}}$ and
$g_{2}=\delta_{w_{6}}\delta_{p_{7}}$. The presence of the factor $(-1)^{F}$ in
$g^{\prime}_{1}$ can be understood from the difference between the NS-NS and
RR projectors in Eq. (3.38), which is the same as in 8d examples. Actually,
the same will be true for all models we construct in what follows, only
$g^{\prime}_{1}$ will contain the supersymmetry breaking deformation
$(-1)^{F}$.

###  3.3 Models in four dimensions

In four dimensions, a non-trivial antisymmetric tensor can have $\text{rank}\
B=2$, $4$ or $6$.

$\bullet$ ${\rm rank}\ B=2$: 8d model compactified on $T^{4}$ (T-dualized)

By compactifying the 8d model on $T^{4}$ and T-dualizing the four extra
compact directions, one finds a 4d model with D3-branes and 64 O3-planes. In
the supersymmetric case, one has $48\ {\rm O}3_{-}$ and $16\ {\rm
O}3_{+}$-planes. In the non-SUSY case, one obtains a configuration with $32\
{\rm O}3_{-}$ and $16\times({\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-}}+{\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+}})$-planes. Like in the six dimensional ${\rm
rank}\ B=2$ case, the geometry is simply the one of the 8d model duplicated
along the new compact directions. Since the model is T-dual to the 8d model
compactified on an extra $T^{4}$, the rank of the gauge group is 8. For a
single stack of D3-branes, the gauge symmetry is $\text{USp}(16)$ if the
D3-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+}}$ or O$3_{+}$-plane, $\text{SO}(16)$ if the
D3-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-}}$-plane or O$3_{-}$-plane, and $\text{U}(8)$
if the D3-brane stack is in the bulk. Supersymmetry is broken at the string
scale (nonlinearly realized) if the D3-branes are coincident with anti-
orientifolds, and broken only at the massive level (due to the separation in
the internal space from the source of supersymmetry breaking) if the D3-branes
are coincident with orientifold planes.

When the D3-branes are put at the origin, the corresponding projectors on the
momentum states running in the Möbius amplitude are

$\displaystyle\Pi_{\rm NSNS}$
$\displaystyle=\frac{1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}\times\prod_{i=4}^{7}\frac{1+(-1)^{m_{i}}}{2}\
,$ $\displaystyle\Pi_{\rm RR}$ $\displaystyle=(-1)^{m_{8}}\Pi_{\rm NSNS}\ .$
(3.42)

$\bullet$ ${\rm rank}\ B=4$: 6d model compactified on $T^{2}$ (T-dualized)

By compactifying the 6d model on $T^{2}$ and T-dualizing the two extra compact
directions, one finds a 4d model where, in the supersymmetric case, one has
$40\ {\rm O}3_{-}$ and $24\ {\rm O}3_{+}$-planes. In the non-SUSY case, one
finds a configuration with $24\ {\rm O}3_{-}$, $8\ {\rm O}3_{+}$ and
$16\times({\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-}}+{\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+}})$-planes. The geometry is the one of the 6d model duplicated along
the two new dimensions. Since the model is T-dual to the 6d model compactified
on $T^{2}$, the rank of the gauge group is $4$. For a single stack of
D3-branes, the gauge symmetry is $\text{USp}(8)$ if the D3-branes are on top
of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+}}$ or ${\rm O}3_{+}$-plane, $\text{SO}(8)$ if the D3-branes are on
top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-}}$-plane or O$3_{-}$-plane, and $\text{U}(4)$ if the D3-brane stack
is in the bulk. The supersymmetry breaking pattern is the same as in the
previous cases.

The corresponding projectors on the momentum states in the Möbius for
D3-branes put at the origin are

$\displaystyle\Pi_{\rm NSNS}=$
$\displaystyle\;\frac{1-(-1)^{m_{9}}-(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}}{2}$
$\displaystyle\;\times\frac{1+(-1)^{m_{7}}-(-1)^{m_{6}}+(-1)^{m_{7}+m_{6}}}{2}\times\prod_{i=4}^{5}\frac{1+(-1)^{m_{i}}}{2}\
,$ $\displaystyle\Pi_{\rm RR}=$ $\displaystyle\;(-1)^{m_{8}}\Pi_{\rm NSNS}\ .$
(3.43)

$\bullet$ ${\rm rank}\ B=6$:

Following our binary ordering, the geometry of the SUSY model discussed here
is

$\displaystyle\\!\\!({\rm O}3_{+},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm
O}3_{-},{\rm O}3_{+},{\rm O}3_{+},{\rm O}3_{+},{\rm O}3_{-},{\rm O}3_{+},{\rm
O}3_{+},{\rm O}3_{+},{\rm O}3_{-},$ $\displaystyle{\rm O}3_{+},{\rm
O}3_{+},{\rm O}3_{+},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{+},{\rm O}3_{+},{\rm
O}3_{+},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{-},$
$\displaystyle{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{-},{\rm
O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{+},{\rm O}3_{+},{\rm
O}3_{+},{\rm O}3_{-},{\rm O}3_{-},$ $\displaystyle{\rm O}3_{-},{\rm
O}3_{+},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{-},{\rm
O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{+},{\rm O}3_{+},$
$\displaystyle{\rm O}3_{+},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm
O}3_{+},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{-},{\rm O}3_{+},{\rm O}3_{-},{\rm
O}3_{-},{\rm O}3_{-})\ ,$ (3.44)

with a total of $36$ O$3_{-}$ and 28 O$3_{+}$-planes. Again, there are several
SUSY breaking deformations that are possible. One example is a configuration
with 20 O$3_{-}$, 12 O$3_{+}$ and $16\times({\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-}}+{\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+}})$-planes as
follows

$\displaystyle(\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}3_{+},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-},$ $\displaystyle{\rm O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}3_{+},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{-},$ $\displaystyle\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}3_{+},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},$ $\displaystyle{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-},{\rm O}3_{+},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}3_{+},$ $\displaystyle\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{-},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+},{\rm O}3_{-},\mkern
1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm
O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm O}3\mkern-1.5mu}\mkern
1.5mu_{+},{\rm O}3_{-},\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-},{\rm O}3_{-})\ .$ (3.45)

The rank of the gauge group is $2$ and, depending on the location of the
stacks of D3-branes, one finds for a single stack $\text{USp}(4)$ if the
D3-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{+}}$ or O$3_{+}$-plane, $\text{SO}(4)$ if the
D3-branes are on top of one ${\mkern 1.5mu\overline{\mkern-1.5mu{\rm
O}3\mkern-1.5mu}\mkern 1.5mu_{-}}$-plane or O$3_{-}$-plane, and $\text{U}(2)$
if the D3-brane stack is in the bulk.

When the 2 D3-branes (4 half-D3-branes) are at the origin, the projectors on
the momentum states in the Möbius amplitude are

$\displaystyle\Pi_{\rm NSNS}$
$\displaystyle=\prod_{i=2}^{4}\frac{1-(-1)^{m_{2i+1}}-(-1)^{m_{2i}}-(-1)^{m_{2i+1}+m_{2i}}}{2}\
,$ $\displaystyle\Pi_{\rm RR}$ $\displaystyle=(-1)^{m_{8}}\Pi_{\rm NSNS}\ .$
(3.46)

The tree-level amplitudes are given by

$\displaystyle\mathcal{\tilde{K}}=$
$\displaystyle\;\frac{(\alpha^{\prime})^{3}}{4v_{6}}\int_{0}^{\infty}\text{d}l\,P_{\bf
m}^{(6)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\ ,$
$\displaystyle\mathcal{\tilde{A}}=$
$\displaystyle\;\frac{2^{-5}N^{2}(\alpha^{\prime})^{3}}{2v_{6}}\int_{0}^{\infty}\text{d}l\,P_{\bf
m}^{(6)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\ ,$
$\displaystyle\mathcal{\tilde{M}}=$
$\displaystyle-\frac{N(\alpha^{\prime})^{2}}{8v_{4}}\int_{0}^{\infty}\text{d}l\,\prod_{i=2}^{4}\big{[}(-1)^{m_{2i}}P_{2m_{2i+1}}-P_{2m_{2i+1}+1}\big{]}P_{m_{2i}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{{\hat{V}}_{8}-(-1)^{m_{8}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})\
,$ (3.47)

while in the loop-channel they become

$\displaystyle\mathcal{K}=$
$\displaystyle\;\frac{1}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{3}}\,W_{2{\bf
n}}^{(6)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}2i\tau_{2}\big{)}\ ,$
$\displaystyle\mathcal{A}=$
$\displaystyle\;\frac{N^{2}}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{3}}\,W_{\bf
n}^{(6)}\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}\genfrac{}{}{}{1}{i\tau_{2}}{2}\big{)}\
,$ $\displaystyle\mathcal{M}=$
$\displaystyle\;\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,\prod_{i=2}^{4}W_{n_{2i+1}}\big{[}(-1)^{n_{2i+1}}W_{2n_{2i}}-W_{2n_{2i}+1}\big{]}\,\frac{{\hat{V}}_{8}+(-1)^{n_{9}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
.$ (3.48)

The torus amplitude can be constructed as a free-orbifold generated by
$g^{\prime}_{1}=(-1)^{F}\delta_{w_{8}}\delta_{p_{9}}$,
$g_{2}=\delta_{w_{6}}\delta_{p_{7}}$ and $g_{3}=\delta_{w_{4}}\delta_{p_{5}}$.

##  4 Consistency conditions from probe branes

It is well-known that the standard consistency rules of open-string partition
functions are not enough to define a consistent string model. There are indeed
K-theory constraints [96, 97], which can also be understood in terms of the
Witten $\text{SU}(2)$ anomalies [98] on probe branes [99]. We are therefore
interested in probe branes with $\text{SU}(2)$ gauge group. Probe branes mean
D-branes that are not constrained by the RR tadpole cancelation. In the type I
string, background branes are D9 and the probe branes can be of D7, D5, D3 or
D1 types, where the D5 and D1-branes are BPS, whereas the D7 and D3 are non-
BPS. Since D1-branes lead to a 2d theory, whereas we are interested in Witten
four-dimensional $\text{SU}(2)$ anomalies, we will ignore D1-branes in what
follows. D7 and D3-branes in type I support unitary gauge groups
$\text{U}(M)$, D5-branes support $\text{USp}(M)$ gauge groups [100, 101] for
$B_{ij}=0$, whereas $\text{SO}(M)$ is also possible on D5-branes for
$B_{ij}\not=0$. The cases of interest for us are SU$(2)\subset{\rm U}(2)$ and
USp(2), which will be implicitly assumed in what follows. For Witten SU(2)
anomaly, only the strings stretched between the background D9-branes and the
probe branes, which transform in the fundamental representation of the SU(2)
probe-brane gauge group, are relevant. Since the spectrum of these
bifundamental strings are given entirely by the cylinder amplitude,
supersymmetry breaking by the orientifold projection is not affecting our
discussion below. In the following, we will first consider the corresponding
spectra in type I language and then perform T-dualities in all internal
directions. After T-duality, one obtains a geometry with O+, O-,
$\overline{\rm O}_{+}$, $\overline{\rm O}_{-}$-planes, but due to the argument
above one can restrict to configurations with O±-planes only. The only cases
giving constraints are when background branes are located on O--planes, which
is implicitly assumed in what follows.

###  4.1 Probe branes in eight dimensions

In the type I string compactified to eight dimensions on $T^{2}$, two
T-dualities switch the description into the type IIB$/\Omega^{\prime}$
orientifold framework, where $\Omega^{\prime}$ contains two parity operations
$(X_{8},X_{9})\to(-X_{8},-X_{9})$. One finds that:

D9-D5 states contain six-dimensional Majorana–Weyl fermions in the $(M,2)$ of
the gauge group $\text{SO}(M)_{9}\times\text{USp}(2)_{5}$. After T-duality, if
the D5 probe brane wraps $T^{2}$, the configuration becomes D7-D3 with four-
dimensional Weyl fermions in the bifundamental representation. Placing some D7
and D3 on the four O7-planes, we learn that at each orientifold fixed point,
$M$ should be even. Therefore stuck half-D7-branes i.e. without dynamical
positions, and in particular SO($1$) configurations, are not allowed.

Once this rule is satisfied, there are no further non-trivial constraints
coming from D5 probe branes wrapped differently, or from D7 and D3 probe
branes.

###  4.2 Probe branes in six dimensions

In type I string compactified to six-dimensions on $T^{4}$, the probe branes
of interest are D5-branes (which are not points in $T^{4}$), D7-branes whose
worldvolume wrap $T^{4}$ and D3-branes. The strongest constraints come from
D9-D5 strings, which become D5-D5 strings after four T-dualities, with the
probe D5-branes wrapping a $T^{2}$ in the compact internal space. The
consistent configurations with stacks of odd numbers of background
half-D5-branes on orientifold fixed points must contain even numbers of such
stacks in each $T^{2}$ in $T^{4}$.

###  4.3 Probe branes in four dimensions

In type I string compactified to four-dimensions on $T^{6}$, the only probe
branes of interest are D7-branes and D3-branes. The former give no constraint
since the number of D9-D7 Weyl fermions after reduction to four dimensions is
always even. Moreover, after six T-dualities, the probe D3-branes become
D9-branes which wrap the entire internal space, while the background D9-branes
become D3-branes. Hence, the probe D9-branes intersect all background
D3-branes, leading to no constraint.

###  4.4 Extra non-perturbative constraint

In addition to the Witten anomaly, another constraint on the allowed
configurations comes from imposing that, in any dimension, the Wilson lines
(or brane positions after T-duality) belong to $\text{SO}(N)$ (actually
$\text{Spin}(N)$) and not $\text{O}(N)$. This is because at a nonperturbative
level, the component of O$(N)$ disconnected from SO($N$) cannot be
defined[97]. This implies that all determinants of Wilson-line matrices must
equal one.

In practice, when the branes are located at fixed points, the matrix of Wilson
lines along a direction $X^{i}$ is diagonal, with entries $1$ or $-1$ only.
The number of $1$’s corresponds to the number of half-branes sitting at the
origin of direction $X^{i}$ while the number of $-1$’s is the number of half-
branes at position $X^{i}=\pi R_{i}$ [18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30]. For the determinant to be one, we thus conclude that the allowed
brane configurations are the ones for which the number of half-branes in each
hyperplan $X^{i}=0$ or $X^{i}=\pi R_{i}$ is even.

##  5 Gravitino mass versus SUSY breaking scale on D-branes

The main feature of the class of models constructed in this paper is the
existence of two supersymmetry breaking mass scales: One in the closed-string
sector, which is related to the compactification (KK or winding) scale, and
another one in the open-string (gauge) sector, which is either the winding
scale or the string scale, depending on which one is smaller. As mentioned in
the introduction, this was already achieved in [76]. Our construction, which
is motivated by the orientifold projection put forward in the pioneering paper
[77], allows one to avoid the open-string instabilities typically present in
such constructions. In the following, we discuss in some details the mass
scales and the limits where supersymmetry is restored in the non-SUSY 8d model
of Sect. 2.

In the geometry of O-planes shown in Fig. 1(c), the 8 D7-branes can be put in
a single stack located in the bulk or coincident with one of the four
orientifold planes. Let us consider the latter case.

$\bullet$ Putting the stack on top of the ${\rm O}7_{-}$-plane closer to the
${\overline{\rm O7}_{-}}$-plane, the Möbius contribution to the vacuum energy
is positive, since the D7-branes are repelled from the ${\overline{{\rm
O}7}_{-}}$ plane and attracted towards the ${\overline{{\rm O}7}_{+}}$-plane.
The Möbius amplitude takes the form

$\mathcal{M}=-\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\,W_{n_{9}}\big{[}W_{2n_{8}}+(-1)^{n_{9}}W_{2n_{8}+1}\big{]}\frac{{\hat{V}}_{8}-(-1)^{n_{9}}{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
,$ (5.49)

while the cylinder amplitude is still given by (2.27). The “field-theory”
open-string spectrum is encoded in

$\displaystyle(\mathcal{A}+$ $\displaystyle\mathcal{M})|_{\rm FT}=$
$\displaystyle\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\left\\{\frac{N(N-1)}{2}\left.\left[W_{2n_{9}}W_{n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}+W_{2n_{9}+1}\left(W_{2n_{8}}\frac{V_{8}}{\eta^{8}}-W_{2n_{8}+1}\frac{S_{8}}{\eta^{8}}\right)\right]\right|_{0}\right.$
$\displaystyle\left.\left.\hskip
39.83368pt+\frac{N(N+1)}{2}\,W_{2n_{9}+1}\left(W_{2n_{8}+1}\frac{V_{8}}{\eta^{8}}-W_{2n_{8}}\frac{S_{8}}{\eta^{8}}\right)\right|_{0}\right\\}\,,$
(5.50)

and is supersymmetric at the massless level. The gauge group is
$\text{SO}(N)$, where $N=16$ is fixed by the RR tadpole condition. Since the
closed-string spectrum becomes supersymmetric when $R_{8}\to 0$ and/or
$R_{9}\to\infty$ (in particular gravitinos become massless), it is interesting
to take these limits in the open (gauge) spectrum. The first limit $R_{8}\to
0$ leads to a supersymmetric spectrum on the D7-branes. Indeed, the winding
towers of bosons and fermions collapse to the same value. Supersymmetry is
broken only at the massive winding level and for
$R_{8}\leq\sqrt{\alpha^{\prime}}$ can therefore be considered as spontaneous
at the field-theory massless level, after including quantum corrections. In
the other limit $R_{9}\to\infty$, the open-string states featuring
supersymmetry breaking become infinitely heavy and decouple at low
energy.101010However, none of the limits has a purely perturbative orientifold
description. In the first case $R_{8}\to 0$ the open-string spectrum does not
have a 9d interpretation, whereas in the second case $R_{9}\to\infty$ there
are local charges and tensions that generate a strong backreaction (local
tadpoles are not cancelled).

$\bullet$ Putting all D7-branes on one stack coincident with the ${\rm
O}7_{-}$-plane closer to the $\overline{{\rm O}7}_{+}$-plane, the Möbius
contribution to the vacuum energy is negative. Otherwise there are no major
differences.

$\bullet$ Let us now consider the case where all D7-branes are coincident with
the $\overline{{\rm O}7}_{+}$-plane, whose amplitudes are displayed in Eqs.
(2.27) and (2.25). The “field-theory” open-string spectrum is encoded in

$\displaystyle(\mathcal{A}+$ $\displaystyle\mathcal{M})|_{\rm
FT}=\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{5}}\left\\{\frac{N(N+1)}{2}\,W_{2n_{9}}\left(W_{2n_{8}}\frac{V_{8}}{\eta^{8}}-W_{2n_{8}+1}\frac{S_{8}}{\eta^{8}}\right)\right|_{0}$
$\displaystyle\left.\left.+\frac{N(N-1)}{2}\left[W_{2n_{9}}\left(W_{2n_{8}+1}\frac{V_{8}}{\eta^{8}}-W_{2n_{8}}\frac{S_{8}}{\eta^{8}}\right)+W_{2n_{9}+1}W_{n_{8}}\frac{V_{8}-S_{8}}{\eta^{8}}\right]\right|_{0}\right\\}\,.$
(5.51)

The gauge group is $\text{USp}(N)$, where $N=16$ is fixed by the RR tadpole
condition. In this case, the pattern of supersymmetry breaking depends on the
value of $R_{8}$. If this radius is large (and in general when branes are
coincident with anti-orientifold planes) the breaking is at the string scale,
with nonlinearly realized supersymmetry. This interpretation is valid in the
regime of large $R_{8}$ and $R_{9}>\sqrt{\alpha^{\prime}}$, when there are
light gravitinos in the spectrum.

In the limit $R_{8}\to 0$, the spectrum becomes however supersymmetric. In
fact, when $R_{8}$ is small, supersymmetry can be interpreted as spontaneously
broken, since there is a shift in the fermion masses compared to the bosons in
the winding sector. Although this seems similar to the familiar Scherk–Schwarz
breaking, the mechanism has also features in common with Brane Supersymmetry
Breaking since the deformation does not affect the cylinder but acts in the
Möbius amplitude by exchanging symmetric with antisymmetric gauge-group
representations for fermions (compared to bosons) in the open-string spectrum.
Throughout the paper, we have used the terminology “supersymmetry breaking at
the string scale” for this situation, in order to distinguish it with the case
where the D-branes are on top of ${\rm O}_{\pm}$-planes.

When $R_{9}\to\infty$ at fixed $R_{8}$, the spectrum encoded in (5.51) does
not become supersymmetric, whereas the closed-string spectrum does. This is
interesting since one may think that an exact Brane Supersymmetry Breaking
Spectrum is realized in this limit. If true, this would also be a counter-
example of the gravitino mass conjecture put forward recently in [74, 75].
However, when $R_{9}\to\infty$, the local sources from D-branes and O-planes
generate local tadpoles and thus large backreactions responsible for the
breaking of the effective field theory description, as conjectured in [74,
75].111111 Note however that these references do not claim that a limit of
zero gravitino mass is not possible. There are known examples of
Scherk–Schwarz type where the whole spectrum and interactions (closed and open
strings in orientifolds, or only closed strings in heterotic and type II
strings) become supersymmetric in the decompactification limit. The claim is
that such a limit is not possible, within an effective field theory
description, if there is some sector breaking supersymmetry in the limit of
vanishing gravitino mass. On the other hand, the model shows that the value of
the gravitino mass $m_{3/2}$ can be decoupled from the size of the scalar
potential $V$, for fixed values of the moduli. In particular,
$|m_{3/2}|\ll|V|^{1/d}$ is possible. Hence, we do not see any fundamental
reason in quantum gravity to necessarily have a high gravitino mass compared
to the Hubble scale [102], as recently proposed in [103, 104].

$\bullet$ Finally, putting all D7-branes on one stack coincident with the
$\overline{{\rm O}7}_{-}$-plane, the Möbius contribution to the vacuum energy
is positive and the pattern of supersymmetry breaking is similar to that found
in the previous case.

##  6 Conclusions and open questions

We have constructed supersymmetry-breaking orientifold models where a certain
number $n$ of O- (negative tension, negative charge) - O+ (positive tension,
positive charge) orientifold-plane pairs are transformed into $n$ ${\mkern
1.5mu\overline{\mkern-1.5mu\rm O\mkern-1.5mu}\mkern 1.5mu_{-}}$ (negative
tension, positive charge) - ${\mkern 1.5mu\overline{\mkern-1.5mu\rm
O\mkern-1.5mu}\mkern 1.5mu_{+}}$ (positive tension, negative charge) pairs.
The anti-orientifold plane pairs preserve the other half of supersymmetries,
compared to the other ingredients of the background, which are O±-planes and
D-branes. In the open-string sector, supersymmetry is only broken in the
Möbius vacuum amplitude, whereas the closed-string sector has softly broken
supersymmetry.

The main interest of this mechanism is that both the original O- \- O+ pairs
and their SUSY breaking cousins ${\mkern 1.5mu\overline{\mkern-1.5mu\rm
O\mkern-1.5mu}\mkern 1.5mu_{-}}$ \- ${\mkern 1.5mu\overline{\mkern-1.5mu\rm
O\mkern-1.5mu}\mkern 1.5mu_{+}}$ have zero tension and charge, so that the
total tension and charge of the models are unchanged upon replacement.
Therefore there are neither NS-NS nor RR tadpoles generated in the process.

Depending on where the background D-branes sit in the internal space, their
massless spectrum is supersymmetric or not. In the latter case, which
corresponds to D-branes located on anti-orientifold planes, the pattern of
supersymmetry breaking depends on the value of a radius. If it is large, the
breaking is at the string scale, with nonlinearly realized supersymmetry. On
the contrary, if the radius is small, the same configuration describes a
spontaneous breaking of supersymmetry. In this regime, the winding states in
the D-brane spectrum are light and supersymmetry breaking can be interpreted
as a shift in the fermion masses compared to the bosons in the winding sector.
This seems similar to the more familiar breaking by compactification
(Scherk–Schwarz), but it differs in detail in that the brane-brane cylinder
amplitude is not subject to this shift.

Constructions of this type naturally stabilize open-string moduli. Indeed,
energetic considerations favor the D-branes to sit on top of ${\overline{\rm
O}_{+}}$-planes, where the scale of supersymmetry breaking on their
worldvolume is maximal.

An interesting issue in such models is their effective field-theory limits. At
first sight, as initially considered in [77], it seems possible that the
closed-string spectrum is supersymmetric at tree level. However, we have
provided arguments showing that this is not plausible, as it would be at odds
with the boundary conditions of the gravitinos imposed by the simultaneous
presence of orientifold and anti-orientifold planes, which suggests massive
gravitinos. In fact, we have given reasons in favor of a specific soft
supersymmetry-breaking deformation in the closed-string sector, rendering
massive the gravitinos. Moreover, if an exact supersymmetric closed-string
spectrum would be compatible with the orientifold amplitudes discussed in our
paper, one would obtain new models of Brane Supersymmetry Breaking type. This
would also give counter-examples of the gravitino mass conjecture [74, 75],
whereas the models constructed in our paper are in agreement with it. More
generally, existence of models with exact supersymmetry in the closed-string
sector and broken supersymmetry in the open-string sector would contradict the
conjecture in [74, 75]. Three more comments are in order here:

First of all, all known models of this type, which are of BSB type or with
internal magnetic fields and broken supersymmetry, have NS-NS disk tadpoles
that could plausibly trigger a breakdown of the effective field theory in the
perturbative vacuum, in agreement with [74, 75]. Secondly, the exact
supersymmetry in the closed-string sectors of such models is valid only at
tree-level, as it is broken by quantum corrections induced by the open-string
sector. It is unclear to us if the conjecture on the gravitino mass should
apply to the classical theory (tree-level spectrum) or to the quantum one.
Lastly, in the string models we have constructed, it is possible to decouple
the gravitino mass from the size of the scalar potential. In particular, the
gravitino can be much lighter than the scale determined by the magnitude of
the quantum potential. This fact could play a role in inflationary models of
the type studied in [105, 106, 103, 104, 102, 107].

Eventually, the string models we have considered are based on toroidal
compactifications and the fermionic spectrum, once reduced to four dimensions,
is non-chiral. It would be of course very interesting to construct chiral
four-dimensional models with supersymmetry breaking, by combining the
mechanism put forward in this paper with other ingredients producing
chirality, like orbifolds and/or fluxes. We hope to come back to this
interesting question in the near future.

## Acknowledgments

E.D. thanks Carlo Angelantonj, Jihad Mourad, Gianfranco Pradisi and Augusto
Sagnotti for valuable discussions and correspondence. E.D. was supported in
part by the “Agence Nationale de la Recherche” (ANR).

##  Appendix A Consistency of supersymmetric Möbius projectors

In the construction of models in even dimension lower than 8, we have chosen
factorized Möbius projectors for simplicity. However, this is not imposed by
the RR tadpole condition. In this appendix, we will confirm that non-
factorized projectors can be fully consistent, but we will also see that
imposing the RR tadpole condition is not enough to obtain a Möbius projector
fully consistent. In the latter case, the inconsistency can only be seen in
the direct Klein-bottle amplitude, while it is invisible in the open-string
sector. We will study supersymmetric examples in 6d with $\text{rank}\ B=4$.

###  A.1 A consistent non-factorized Möbius projector

In 6d, with all D-branes located at the origin, a generic Möbius projector
contains $16$ terms, one for each fixed point with appropriate phases. The RR
tadpole condition constrains the overall charge of the O-planes and thus the
number of ${\rm O}_{-}$ and ${\rm O}_{+}$-planes. In the projector, this
translates into a given number of minus and plus signs. In 6d with maximal
rank for $B_{ij}$, there are $10$ O$5_{-}$ and $6$ O$5_{+}$-planes. This
yields $10$ terms with a minus sign and $6$ terms with a plus sign in the
projector. The total number of possibilities that fulfils this requirement is
${16\choose 6}=8008$.

Let us look at the following projector,

$\displaystyle\Pi=\ $ $\displaystyle
1+(-1)^{m_{9}}+(-1)^{m_{8}}-(-1)^{m_{9}+m_{8}}+(-1)^{m_{7}}-(-1)^{m_{9}+m_{7}}-(-1)^{m_{8}+m_{7}}$
$\displaystyle-(-1)^{m_{9}+m_{8}+m_{7}}+(-1)^{m_{6}}-(-1)^{m_{9}+m_{6}}-(-1)^{m_{8}+m_{6}}-(-1)^{m_{9}+m_{8}+m_{6}}$
$\displaystyle-(-1)^{m_{7}+m_{6}}-(-1)^{m_{9}+m_{7}+m_{6}}-(-1)^{m_{8}+m_{7}+m_{6}}+(-1)^{m_{9}+m_{8}+m_{7}+m_{6}}\
.$ (A.1)

With all D-branes at the origin, we deduce the geometry of the model,

$\displaystyle({\rm O}5_{+},{\rm O}5_{+},{\rm O}5_{+},{\rm O}5_{-},{\rm
O}5_{+},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{+},$
$\displaystyle\hskip 170.71652pt{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm
O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{+})\ .$ (A.2)

It turns out to give no projection in the tree-level channel Klein-bottle
amplitude, as was the case in Eq. (3.40). This is thus consistent with the
supersymmetric torus amplitude generated by $g_{1}$ and $g_{2}$. The
transverse cylinder amplitude is also the one obtained in Eq. (3.40), while
the Möbius amplitudes, both in tree-level and loop channels, are

$\displaystyle\tilde{{\cal M}}=$
$\displaystyle-\frac{N(\alpha^{\prime})^{2}}{4v_{4}}\int_{0}^{\infty}\text{d}l\,\Big{[}\left(P_{2m_{9}}+P_{2m_{9}+1}(-1)^{m_{6}}\right)\left(P_{2m_{8}}P_{2m_{7}+1}+P_{2m_{8}+1}P_{2m_{7}}\right)$
$\displaystyle+\left(P_{2m_{9}+1}-P_{2m_{9}}(-1)^{m_{6}}\right)\left(P_{2m_{8}}P_{2m_{7}}-P_{2m_{8}+1}P_{2m_{7}+1}\right)\Big{]}P_{m_{6}}\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})$
$\displaystyle{\cal M}=$
$\displaystyle\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\Big{[}W_{n_{9}}\left(W_{2n_{8}}W_{2n_{7}}-W_{2n_{8}+1}W_{2n_{7}+1}\right)\left(W_{2n_{6}}+W_{2n_{6}+1}(-1)^{n_{9}}\right)$
$\displaystyle+\left(W_{2n_{8}}W_{2n_{7}+1}+W_{2n_{8}+1}W_{2n_{7}}\right)\left(W_{2n_{6}}(-1)^{n_{9}}-W_{2n_{6}+1}\right)\Big{]}\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2}).$
(A.3)

The tree-level channel Möbius amplitude contains all momentum states, just
like the tree-level Klein bottle and cylinder, so that the factorization
property of the amplitudes is satisfied. The loop-channel Möbius amplitude is
also consistent with the cylinder since it contains the contributions of the
same states with signs. We conclude that the projector (A.1) yields a fully
consistent model.

###  A.2 An inconsistent Möbius projector

Now consider the following projector, which has the correct number of signs to
satisfy the RR tadpole condition,

$\displaystyle\Pi=\ $ $\displaystyle
1+(-1)^{m_{9}}+(-1)^{m_{8}}+(-1)^{m_{9}+m_{8}}+(-1)^{m_{7}}+(-1)^{m_{9}+m_{7}}-(-1)^{m_{8}+m_{7}}$
$\displaystyle-(-1)^{m_{9}+m_{8}+m_{7}}-(-1)^{m_{6}}-(-1)^{m_{9}+m_{6}}-(-1)^{m_{8}+m_{6}}-(-1)^{m_{9}+m_{8}+m_{6}}$
$\displaystyle-(-1)^{m_{7}+m_{6}}-(-1)^{m_{9}+m_{7}+m_{6}}-(-1)^{m_{8}+m_{7}+m_{6}}-(-1)^{m_{9}+m_{8}+m_{7}+m_{6}}\
.$ (A.4)

With all D-branes at the origin, the distribution of O-planes is given by

$\displaystyle({\rm O}5_{+},{\rm O}5_{+},{\rm O}5_{+},{\rm O}5_{+},{\rm
O}5_{+},{\rm O}5_{+},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},$
$\displaystyle\hskip 170.71652pt{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm
O}5_{-},{\rm O}5_{-},{\rm O}5_{-},{\rm O}5_{-})\ .$ (A.5)

With this geometry, the tree-level channel Klein bottle is now different and
not all momentum states are present. The tree-level and loop-channel Klein-
bottle and Möbius amplitudes are

$\displaystyle\tilde{\mathcal{K}}$
$\displaystyle=\;\frac{(\alpha^{\prime})^{2}}{v_{4}}\int_{0}^{\infty}\text{d}l\,P_{2m_{9}}\left(P_{m_{8}}P_{m_{7}}P_{m_{6}}+8P_{2m_{8}}P_{2m_{7}}P_{2m_{6}+1}\right)\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}il\big{)}\
,$ $\displaystyle\mathcal{K}$
$\displaystyle=\;\frac{1}{4}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,W_{n_{9}}\left(W_{2n_{8}}W_{2n_{7}}W_{2n_{6}}+W_{n_{8}}W_{n_{7}}(-1)^{n_{6}}W_{n_{6}}\right)\frac{V_{8}-S_{8}}{\eta^{8}}\big{(}2i\tau_{2}\big{)}\
,$ $\displaystyle\tilde{\mathcal{M}}$
$\displaystyle=\,\frac{N(\alpha^{\prime})^{2}}{4v_{4}}\int_{0}^{\infty}\text{d}l\,P_{2m_{9}}\Big{\\{}P_{2m_{8}}\left[P_{m_{7}}P_{m_{6}}-2P_{2m_{7}}\left(P_{2m_{6}}-P_{2m_{6}+1}\right)\right]$
$\displaystyle\hskip
113.81102pt+P_{2m_{8}+1}\left(P_{2m_{7}}-P_{2m_{7}+1}\right)P_{m_{6}}\Big{\\}}\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(il+\genfrac{}{}{}{1}{1}{2})\
,$ $\displaystyle\mathcal{M}$
$\displaystyle=\frac{N}{2}\int_{0}^{\infty}\frac{\text{d}\tau_{2}}{\tau_{2}^{4}}\,W_{n_{9}}W_{n_{8}}\left(W_{2n_{7}}W_{2n_{6}}-W_{n_{7}}W_{2n_{6}+1}+(-1)^{n_{8}}W_{2n_{7}+1}W_{2n_{6}}\right)$
$\displaystyle\hskip
261.76526pt\times\frac{{\hat{V}}_{8}-{\hat{S}}_{8}}{{\hat{\eta}}^{8}}(\genfrac{}{}{}{1}{i\tau_{2}}{2}+\genfrac{}{}{}{1}{1}{2})\
.$ (A.6)

The loop-channel Möbius amplitude contains all states present in the cylinder.
Moreover, the tree-level Klein-bottle, cylinder and Möbius amplitudes respect
amplitude factorization. The only inconsistency comes from the Klein bottle in
the loop-channel, which contains states not present in the torus amplitude.
This means that the RR tadpole condition is not enough to produce a consistent
Möbius projector. The inconsistency can only be seen in the closed-string
sector and comes from the geometry of the model.

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