$\displaystyle+N_{\text{NS}3}\mathrm{d}y^{4}\wedge\mathrm{d}y^{5}\wedge\mathrm{d}y^{6}\big{]}\qquad
N_{\text{NS}1,2,3}\in\mathbb{Z},$ (303)

and we see that (285) keep on being satisfied. We now have $f=u=0$ and
$g=\frac{i\pi\sqrt{\alpha^{\prime}}}{R_{6}}e^{A}N_{\text{NS}3}\mathrm{d}y^{4}\wedge\mathrm{d}y^{5}$,
and the domain-wall BPSness violation (286) now reads

$g+if=-4\pi
r\sqrt{\alpha^{\prime}}e^{-2A}R_{4}R_{5}\mathrm{d}y^{4}\wedge\mathrm{d}y^{5}$
(304)

with

$r=-\frac{ie^{3A}N_{\text{NS}3}}{8\pi\sqrt{\alpha^{\prime}}R_{4}R_{5}R_{6}}.$
(305)

The additional DWSB contribution, which happens through the new components of
the NSNS-flux in (303), doesn’t modify the violation of the D-string BPSness:
(290) is again identically satisfied, while (291) reduces to

$e^{2A-\phi}H\wedge\text{Im}w=4\pi\sqrt{\alpha^{\prime}}e^{A}(R_{1}\hat{\alpha}_{1}\mathrm{d}y^{1}+R_{2}\hat{\alpha}_{2}\mathrm{d}y^{2})\wedge
j$ (306)

which is satisfied with

$\displaystyle\hat{\alpha}_{1}=$
$\displaystyle-\frac{e^{2A}N_{\text{NS}1}}{16\pi\sqrt{\alpha^{\prime}}R_{3}R_{4}R_{6}}$
(307) $\displaystyle\hat{\alpha}_{2}=$
$\displaystyle-\frac{e^{2A}N_{\text{NS}2}}{16\pi\sqrt{\alpha^{\prime}}R_{3}R_{5}R_{6}}.$
(308)

The gauge BPSness sets the RR-fluxes to

$\displaystyle F_{1}=$ $\displaystyle
e^{2A-\phi}\big{[}\frac{R_{4}N_{\text{NS}2}}{R_{3}R_{5}R_{6}}\mathrm{d}y^{4}-\frac{R_{5}N_{\text{NS}1}}{R_{3}R_{4}R_{6}}\mathrm{d}y^{5}-\frac{R_{3}N_{\text{NS}3}}{R_{4}R_{5}R_{6}}\mathrm{d}y^{3}\big{]}$
(309) $\displaystyle F_{3}=$ $\displaystyle
e^{2A-\phi}\ast_{\mathcal{B}_{4}}\mathrm{d}e^{-4A}$ (310)

The RR-flux quantisation implies that we must have

$\displaystyle
e^{2A-\phi}\frac{R_{5}N_{\text{NS}1}}{R_{3}R_{4}R_{6}}=N_{\text{R}1}\qquad
e^{2A-\phi}\frac{R_{4}N_{\text{NS}2}}{R_{3}R_{5}R_{6}}=N_{\text{R}2}\qquad
e^{2A-\phi}\frac{R_{3}N_{\text{NS}3}}{R_{4}R_{5}R_{6}}=N_{\text{R}3}$ (311)

with $N_{\text{R}1},N_{\text{R}2},N_{\text{R}3}\in\mathbb{Z}$. The RR Bianchi
identities then reduce to

$-\hat{\nabla}^{2}_{\mathcal{B}_{4}}e^{-4A}=\frac{1}{e^{2A-\phi}(2\pi)^{2}\alpha^{\prime}\Pi_{a=3}^{6}R_{a}}\big{[}\sum_{k=1}^{3}N_{\text{NS}k}N_{\text{R}k}+\sum_{i\in\text{D}5\text{'}s,\text{O}5\text{'}s}q_{i}\delta^{4}_{\mathcal{B}_{4}}(y_{i})\big{]}$
(312)

with $q_{\text{D}5}=-q_{\text{O}5}=1$. Integrating this condition on the base
yields the following tadpole condition

$\sum_{k=1}^{3}N_{\text{NS}k}N_{\text{R}k}+n_{\text{D}5}=16.$ (313)

Turning to the equations of motion, the conditions (293)$-$(296) are obeyed,
provided that $e^{2A-\phi}=\text{constant}$. The NSNS-field equation and
internal Einstein equations are therefore satisfied, as discussed above.

##  6 Conclusion

In this paper, we investigated some corners of the landscape of non-
supersymmetric flux vacua, in the light of Generalised Complex Geometry. In
Generalised Complex Geometry the conditions for warped backgrounds with
$\mathcal{N}=1$ supersymmetry can be expressed as differential equations on
polyforms, called pure spinors, which, in turn, are seen as generalised
calibration conditions for various types of D-branes wrapping cycles on the
internal manifold: D-string, domain-wall, or space-filling branes, from the
four-dimensional perspective.

This interpretation provides different controlled ways of breaking
supersymmetry in ten-dimensions by violating the BPSness of D-string, domain-
wall, or space-filling probe branes. The philosophy is to look at non-
supersymmetric solutions via a two-step procedure: one first solves the
deformed supersymmetry equations, which are still first order equations, and
then look at the extra contraints that the equations of motion impose.

This approach was first used in Lust:2008zd , where non-supersymmetric
solutions where find by deforming the domain-wall calibration condition. Our
work extends Lust:2008zd  to new classes of solutions obtained by deforming
the D-string BPS condition or both the domain-wall and D-string ones.

To solve the ten-dimensional equations of motion, it proves convenient to use
an effective four-dimensional potential obtained by integrating the ten-
dimensional supergravity action on the internal manifold. The effective
potential can be expressed in terms of pure spinors as an integral over the
compact space. Then the equations of motion are obtained by varying it with
respect to the physical fields and are also given in terms of the pure
spinors. The equations of motion are rather convoluted in the general case and
are still to be studied. However, by choosing specific classes of
supersymmetry breaking, they become tractable. We were able to solve them for
a variety of concrete type II SU$(2)$ and SU$(3)$ backgrounds respecting (68).

When studying the effective potential for the SSB backgrounds, we witnessed
the presence of some terms belonging in vector representations of the
SU$(3)\times$SU$(3)$ structure, which are believed to be massive modes from
the effective point of view. In that sense, our classes of backgrounds
describe a set of fully ten-dimensional non-supersymmetric solutions of type
II supergravity. It is not clear weather they admit a proper four-dimensional
interpretation. It would be interesting to apply the recently developed
exceptional generalised geometry techniques Josse  to study eventual
consistent truncations of such backgrounds.

However, we showed that our class of background shares a property with the
one-parameter DWSB class introduced in Lust:2008zd , namely the existence of a
‘truncation’ dictated by the geometry such that the ‘off-shell’ effective
potential is positive semi-definite, and vanishes at the solutions. By
‘truncation’ we meant that we only consider the off-shell deformations of the
potential which are compatible with the generalised foliation of the internal
space, so it is by no mean a rigorous truncation to a finite set of mode. We
therefore present this as an interesting property of our class of backgrounds,
but this is a weaker statement than arguing for the perturbative stability of
our backgrounds, since we have no control over the modes we are ‘truncating’
away and keeping in our ‘off-shell’ potential.

We carried the same analysis, from the effective potential and the derivation
of the equations of motion, to the construction of concrete backgrounds, for
another class of vacua, which have the same supersymmetry breaking term
violating the D-string BPSness, but supplemented with the one-parameter DWSB
contribution violating the domain-wall BPSness. Each supersymmetry breaking
term brings different contributions to the effective potential and therefore
to the equations of motion, which can be solved separately, which we did for
several explicit SU$(2)$ and SU$(3)$ background constructions.

One obvious extension of this work would be to consider different patterns of
the D-string BPSness violation, depending on the generalised current
associated to the background D-branes or not, and to look for explicit
solutions.

Another natural extension of the present work is to carry the constructions of
the analogous Heterotic backgrounds, where the building block of the
supersymmetry breaking term entering the modified D-string condition could now
be, for instance, the base volume-form of some elliptically fibered internal
manifolds like the ones discussed in DWSBheterotic .

Another interesting direction would be to consider solutions of ten-
dimensional supergravity violating the D-string BPSness, but without any
vector modes under the SU$(3)\times$SU$(3)$ structure, for which it would
probably be possible to get a deeper understanding of their associated
effective theories. In the case of vacua with an external space being
Minkowski, such a D-string BPSness violation should be accompanied by a
domain-wall BPSness violation, otherwise there would only be positive
contributions to the effective potential. One could then hope to interpret the
associated effective theories as, for instance, solutions of four-dimensional
$\mathcal{N}=1$ supergravity with non-vanishing D-, F-terms and
superpotential, extending the supersymmetric analysis of Martucci . We will
come back to these questions in Menet:2023rml .

Finally, the techniques from Exceptional Generalised Geometry can shed a
different light on the study of non-supersymmetric equations of motion: by
classifying the possible supersymmetry breaking terms in terms of different
representations of the torsion of some Generalised structure, one can
reformulate the equations of motion as first order differential conditions on
these torsion representations. We hope to come back to this idea in the near
future Menet:2024 .

Acknowledgements

It is a pleasure to thank Davide Cassani, Mariana Graña, Luca Martucci,
Michela Petrini and Dan Waldram for insightful discussions.

##  Appendix A Supergravity Conventions

###  A.1 Bosonic sector

Our supergravity conventions are identical to the ones of Lust:2008zd . We
introduce the following ten- and six-dimensional Hodge operators

$\displaystyle\tilde{\ast}_{10}=$ $\displaystyle\ast_{10}\circ\sigma$ (314)
$\displaystyle\tilde{\ast}_{6}=$ $\displaystyle\ast_{6}\circ\sigma$ (315)

with, for a $p$-form $\omega$

$\displaystyle\ast_{10}\omega_{p}=-\frac{1}{p!(10-p)!}\sqrt{-g}\
\epsilon_{M_{1}...M_{10}}\omega^{M_{11-p}...M_{10}}\mathrm{d}x^{M_{1}}\wedge...\wedge\mathrm{d}x^{M_{10-p}}$
(316) $\displaystyle\ast_{6}\omega_{p}=\frac{1}{p!(6-p)!}\sqrt{-g}\
\epsilon_{m_{1}...m_{6}}\omega^{m_{7-p}...m_{6}}\mathrm{d}y^{m_{1}}\wedge...\wedge\mathrm{d}y^{m_{6-p}}.$
(317)

The ten-dimensional RR-field strength self duality is then

$F^{10}=\tilde{\ast}_{10}F^{10}.$ (318)

The type II pseudo-action in democratic formalism is

$S=\frac{1}{2\kappa^{2}_{10}}\int\mathrm{d}^{10}x\sqrt{-g}\Big{\\{}e^{-2\phi}[R+4(\mathrm{d}\phi)^{2}-\frac{1}{2}H^{2}]-\frac{1}{4}F^{2}\Big{\\}}+S^{(\text{loc})},$
(319)

where $2\kappa^{2}_{10}=(2\pi)^{7}\alpha^{\prime 4}$ and for any real p-form
$\omega$ we define $\omega^{2}=\omega\cdot\omega$ with $\cdot$ defined as

$\omega\cdot\chi=\frac{1}{p!}\omega_{M_{1}...M_{p}}\chi^{M_{1}...M_{p}}.$
(320)

In the text, if $\omega$ is complex, then we consider
$|\omega|^{2}=\omega\cdot\bar{\omega}$. Varying this action and imposing the
self-duality conditions, we find the following equations of motion.

The dilaton equation

$\nabla^{2}\phi-(\mathrm{d}\phi)^{2}+\frac{1}{4}R-\frac{1}{8}H^{2}-\frac{1}{4}\frac{\kappa^{2}_{10}e^{2\phi}}{\sqrt{-g}}\frac{\delta
S^{(\text{loc})}}{\delta\phi}=0,$ (321)

the $B$-field equation

$-\mathrm{d}(e^{-2\phi}\ast_{10}H)+\frac{1}{2}[\ast_{10}F\wedge
F]_{8}+2\kappa^{2}_{10}\frac{\delta S^{(\text{loc})}}{\delta B}=0,$ (322)

the Einstein equation

$\displaystyle
e^{-2\phi}[g_{MN}+2g_{MN}\mathrm{d}\phi\cdot\mathrm{d}\phi-2g_{MN}\nabla^{2}\phi+2\nabla_{M}\nabla_{N}\phi$
(323)
$\displaystyle-\frac{1}{2}\iota_{M}H\cdot\iota_{N}H+\frac{1}{4}g_{MN}H\cdot
H)]-\frac{1}{4}\iota_{M}F\cdot\iota_{N}F-\kappa^{2}_{10}T^{(\text{loc})}_{MN}=0,$
(324)

with

$T^{(\text{loc})}_{MN}=-\frac{2}{\sqrt{-g}}\frac{\delta
S^{(\text{loc})}}{g^{MN}},$ (325)

and the RR-fluxes variation gives the Bianchi identities

$\mathrm{d}_{H}F=-j_{\text{source}}.$ (326)

Combining the dilaton equation of motion with the Einstein equations, one can
write the modified Einstein equations

$\displaystyle
R_{MN}+2\nabla_{M}\nabla_{N}\phi-\iota_{M}H\cdot\iota_{N}H-\frac{1}{4}e^{2\phi}\iota_{M}F\cdot\iota_{N}F$
(327)
$\displaystyle-\kappa^{2}_{10}e^{2\phi}\Big{(}T^{(\text{loc})}_{MN}+\frac{g_{MN}}{2\sqrt{-g}}\frac{\delta
S^{(\text{loc})}}{\delta\phi}\Big{)}=0.$ (328)

Finally, we define the Mukai pairing for a pair of polyforms $\omega$ and
$\chi$

$\braket{\omega,\chi}=\omega\wedge\sigma(\chi)|_{6},$ (329)

and more generally, we use throughout the paper

$\braket{\omega,\chi}_{k}=\omega\wedge\sigma(\chi)|_{k}.$ (330)

In the case of a six-dimensional manifold $\mathcal{M}$, the Mukai pairing
satisfies the following property

$\int_{\mathcal{M}}\braket{\mathrm{d}_{H}\omega,\chi}=\int_{\mathcal{M}}\braket{\omega,\mathrm{d}_{H}\chi}.$
(331)

###  A.2 Gamma matrices

We use a real representation of the ten-dimensional gamma matrices
$\Gamma_{M}$. The ten-dimensional chiral operator is

$\Gamma_{(10)}=\Gamma^{01...9}$ (332)

with flat ten-dimensional indices. For any p-form $\omega$, we denote its
image under the Clifford map $\not{\omega}$ with

$\omega\equiv\frac{1}{p!}\omega_{M_{1}...M_{p}}\mathrm{d}x^{M_{1}...M_{p}}\quad\longleftrightarrow\quad\not{\omega}=\frac{1}{p!}\omega_{M_{1}...M_{p}}\Gamma^{M_{1}...M_{p}}.$
(333)

We define the splitting of the ten-dimensional gamma matrices into four- and
six-dimensional gamma matrices $\hat{\gamma}^{\mu}$ and $\gamma^{m}$ as

$\Gamma^{\mu}=e^{-A}\hat{\gamma}^{\mu}\otimes\mathbb{1}\qquad\Gamma^{m}=\gamma_{(4)}\otimes\gamma^{m}.$
(334)

The $\hat{\gamma}^{\mu}$ are associated to the unwarped four-dimensional
metric, and $\gamma_{(4)}=i\hat{\gamma}^{0123}$ is the usual four-dimensional
chiral operator. The six-dimensional chiral operator is
$\gamma_{(6)}=-i\gamma^{123456}$ so we have
$\Gamma_{(10)}=\gamma_{(4)}\otimes\gamma_{(6)}$.

The chirality of the internal spinors is then

$\gamma_{(6)}\eta_{1}=\eta_{1}\qquad\gamma_{(6)}\eta_{2}=\mp\eta_{2}\qquad\text{in
type IIA/IIB}.$ (335)

##  Appendix B Some Generalised Complex Geometry elements

Generalised vectors are sections of the generalised tangent bundle, locally
defined as $E\simeq T\mathcal{M}\oplus T^{\ast}\mathcal{M}$. The generalised
tangent bundle admits a natural SO$(6,6)$ structure, defined by the inner
product on generalised vectors

$V\cdot W=\iota_{v}\rho+\iota_{w}\xi,$ (336)

with $V=v+\xi\in\Gamma(E)$ and $W=w+\rho\in\Gamma(E)$. Using a two-component
notation for the generalised vectors

$V=v+\xi\equiv\begin{pmatrix}v\\\ \xi\end{pmatrix},$ (337)

one can parametrise the SO$(6,6)$ generators as

$\mathcal{O}_{A}=\begin{pmatrix}\ A&0\\\ \
0&(A^{T})^{-1}\end{pmatrix}\qquad\mathcal{O}_{b}=\begin{pmatrix}\ \mathbb{1}&\
\ 0\ \\\ \ b&\ \ \mathbb{1}\
\end{pmatrix}\qquad\mathcal{O}_{\beta}=\begin{pmatrix}\ \mathbb{1}&\ \ \beta\
\\\ \ 0&\ \ \mathbb{1}\ \end{pmatrix},$ (338)

with $A$ any GL$(d,\mathbb{R})$ matrix, $b$ any two-form and $\beta$ any two-
vector. The SO$(6,6)$ adjoint action on generalised vectors is then

$\displaystyle V=v+\xi\quad\rightarrow\quad\mathcal{O}_{A}\cdot V=$
$\displaystyle\ Av+(A^{T})^{-1}\xi$ (339)
$\displaystyle\quad\mathcal{O}_{b}\cdot V=$ $\displaystyle\
v+(\xi-\iota_{v}b)$ (340) $\displaystyle\quad\mathcal{O}_{\beta}\cdot V=$
$\displaystyle\ (v-\iota_{\beta}\xi)+\xi.$ (341)

On another note, the generalised vectors obey the Cliff$(6,6)$ Clifford
algebra

$\\{V,W\\}=(V,W)\qquad V,\ W\in\Gamma(E),$ (342)

and the Spin$(6,6)$ spinors are sections of a spinor representation of
Cliff$(6,6)$, locally isomorphic to the space of polyforms. The action of a
generalised vector $V=v+\xi$ on such a spinor $\Psi$ is

$V\cdot\Psi=\iota_{v}\Psi+\xi\wedge\Psi.$ (343)

A line-bundle of pure spinors is in one-to-one correspondence with an (almost)
generalised complex structure, see for instance Gualtieri:2007ng  for more
details.

An almost generalised complex structure is a map

$\mathcal{J}\ :\ E\rightarrow E,$ (344)

respecting

$\displaystyle\mathcal{J}^{2}=$ $\displaystyle-\mathbb{1}$ (345)
$\displaystyle\mathcal{J}^{T}\mathcal{I}\mathcal{J}=$
$\displaystyle\mathcal{I}$ (346)

with

$\mathcal{I}\ :E\times E\rightarrow\mathbb{R}\quad\text{such
that}\quad\mathcal{I}(V,W)=(V,W)\qquad V,\ W\in\Gamma(E).$ (347)

It is then an integrable almost generalised complex structure or simply
generalised complex structure if its $+i-$eigenbundle is stable under the
following Courant bracket

$[V,W]_{C}=[v,w]+\mathcal{L}_{v}\rho-\mathcal{L}_{w}\xi-\frac{1}{2}\mathrm{d}(\iota_{v}\rho-\iota_{w}\xi),$
(348)

with $V=v+\xi\in\Gamma(E)$ and $W=w+\rho\in\Gamma(E)$. An almost generalised
complex structure is $H$-integrable if its $+i-$eigenbundle is stable under
the twisted Courant bracket

$[V,W]^{H}_{C}=[V,W]_{C}+\iota_{v}\iota_{w}H.$ (349)

When considering the Clifford action on a polyform $\omega$, one can write the
twisted Courant bracket as

$[V,W]^{H}_{C}\cdot\omega=[\\{V\cdot,\mathrm{d}_{H}\\},W\cdot]\omega.$ (350)

One can also define a natural action of (almost) generalised complex
structures on the space of differential forms, explicitly252525See Cavalcanti
for the formal details.:

$\mathcal{J}\cdot=\frac{1}{2}\big{(}J_{mn}\mathrm{d}y^{m}\wedge\mathrm{d}y^{n}\wedge+2{I^{m}}_{n}[\mathrm{d}y^{n},\iota_{m}]+P^{mn}\iota_{m}\iota_{n}\big{)}$
(351)

with

$\mathcal{J}=\begin{pmatrix}I&P\\\ J&-I^{T}\end{pmatrix}.$ (352)

One can then decompose the space of polyforms in terms of $\mathcal{J}$
eigenbundles

$\Lambda^{\bullet}T^{\ast}\mathcal{M}\otimes\mathbb{C}=\bigoplus_{k=-3}^{3}V_{k},$
(353)

with $k$ the eigenvalues of the sub-spaces $V_{k}$, which are representations
of the $SU(3,3)$ structure group associated with $\mathcal{J}$.

For example, in the case of the (almost) generalised complex structure
associated to the pure spinor $\Psi\propto\Omega$, its relation with the
standard Hodge decomposition is

$V_{k}=\bigoplus_{p}\Lambda^{3-p,3-k-p},$ (354)

where $\Lambda^{3-p,3-k-p}$ are sections of the $(p,q)$-forms defined by the
standard associated (almost) complex structure.

If the generalised manifold admits two compatible pure spinors, or
equivalently two commuting (almost) generalised complex structures
$\mathcal{J}_{1}$ and $\mathcal{J}_{2}$, one can further decompose the space
of polyforms in terms of SU$(3)\times$SU$(3)$ representations which are the
common $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ eigenbundles

$\Lambda^{\bullet}T^{\ast}\mathcal{M}\otimes\mathbb{C}=\bigoplus_{k=-3}^{3}\bigoplus_{l=-3}^{3}V_{k,l}.$
(355)

This decomposition is the generalised Hodge diamond decomposition, see for
instance Martucci  for more details.

##  Appendix C Supersymmetry breaking and pure spinors

We consider a ten-dimensional SU$(3)\times$SU$(3)$ background with a Minkowski
four-dimensional space and a ten-dimensional bispinor
$\epsilon=(\epsilon_{1},\epsilon_{2})^{T}$ as in (7). We give here the
parametrisation of the most general supersymmetry breaking, and we start by
writing down the non-vanishing supersymmetry variations

$\displaystyle\delta\psi^{(1)}_{\mu}=\frac{1}{2}e^{A}\hat{\gamma}_{\mu}\zeta\otimes\mathcal{V}_{1}+c.c.\qquad$
$\displaystyle\delta\psi^{(2)}_{\mu}=\frac{1}{2}e^{A}\hat{\gamma}_{\mu}\zeta\otimes\mathcal{V}_{2}+c.c.$
(356)
$\displaystyle\delta\psi^{(1)}_{m}=\zeta\otimes\mathcal{U}^{1}_{m}+c.c.\qquad$
$\displaystyle\delta\psi^{(2)}_{m}=\zeta\otimes\mathcal{U}^{2}_{m}+c.c.$ (357)
$\displaystyle\Delta\epsilon_{1}=\zeta\otimes\mathcal{S}_{1}+c.c.\qquad$
$\displaystyle\Delta\epsilon_{2}=\zeta\otimes\mathcal{S}_{2}+c.c.,$ (358)

where $\mathcal{V}_{1,2},\ \mathcal{U}^{1,2}_{m}$ and $\mathcal{S}_{1,2}$ are
internal spinors parametrising the supersymmetry breaking

$\displaystyle\mathcal{V}_{1}=$ $\displaystyle\
\not{\partial}A\eta_{1}+\frac{1}{4}e^{\phi}\gamma_{(6)}\not{F}\eta_{2}$
$\displaystyle\mathcal{V}_{2}=$ $\displaystyle\
\not{\partial}A\eta_{2}-\frac{1}{4}e^{\phi}\gamma_{(6)}\not{F}^{\dagger}\eta_{1}$
$\displaystyle\mathcal{S}_{1}=$ $\displaystyle\
(\not{\nabla}-\not{\partial}\phi+2\not{\partial}A+\frac{1}{4}\not{H})\eta_{1}$
$\displaystyle\mathcal{S}_{2}=$ $\displaystyle\
(\not{\nabla}-\not{\partial}\phi+2\not{\partial}A+\frac{1}{4}\not{H})\eta_{2}$
$\displaystyle\mathcal{U}^{1}_{m}=$ $\displaystyle\
(\nabla_{m}+\frac{1}{4}\iota_{m}\not{H})\eta_{1}+\frac{1}{8}e^{\phi}\not{F}\gamma_{m}\gamma_{(6)}\eta_{2}$
$\displaystyle\mathcal{U}^{2}_{m}=$ $\displaystyle\
(\nabla_{m}-\frac{1}{4}\iota_{m}\not{H})\eta_{2}-\frac{1}{8}e^{\phi}\not{F}\gamma_{m}\gamma_{(6)}\eta_{1}.$
(359)

Following Lust:2008zd , we expand them in terms of supersymmetry breaking
parameters in the following way

$\displaystyle\mathcal{V}_{1}=r_{1}\eta_{1}^{\ast}+s^{1}_{m}\gamma^{m}\eta_{1}$
$\displaystyle\qquad\mathcal{V}_{2}=r_{2}\eta_{2}^{\ast}+s^{2}_{m}\gamma^{m}\eta_{2}$
(360)
$\displaystyle\mathcal{S}_{1}=t_{1}\eta_{1}^{\ast}+u^{1}_{m}\gamma^{m}\eta_{1}$
$\displaystyle\qquad\mathcal{S}_{2}=t_{2}\eta_{2}^{\ast}+u^{2}_{m}\gamma^{m}\eta_{2}$
(361)
$\displaystyle\mathcal{U}^{1}_{m}=p^{1}_{m}\eta_{1}+q^{1}_{mn}\gamma^{n}\eta_{1}^{\ast}$
$\displaystyle\qquad\mathcal{U}^{2}_{m}=p^{2}_{m}\eta_{2}+q^{2}_{mn}\gamma^{n}\eta_{2}^{\ast}.$
(362)

It has been shown in Lust:2008zd  that these parameters do not mix under
T-duality. One can now rewrite the most general non-supersymmetric pure spinor
equations, expanded on the generalised Hodge diamond, in terms of these
supersymmetry breaking parameters

$\displaystyle
e^{-2A+\phi}\mathrm{d}_{H}(e^{2A-\phi}\Psi_{1})+2\mathrm{d}A\wedge\text{Re}\Psi_{1}-e^{\phi}\tilde{\ast}_{6}F=\Upsilon$
(363) $\displaystyle e^{-3A+\phi}\mathrm{d}_{H}(e^{3A-\phi}\Psi_{2})=K$ (364)

with

$\displaystyle\Upsilon=$
$\displaystyle\frac{1}{2}(-1)^{|\Psi_{1}|}(r_{1}^{\ast}+t_{2}^{\ast})\Psi_{2}+\frac{1}{2}(-1)^{|\Psi_{1}|}(r_{2}+t_{1})\bar{\Psi}_{2}+\frac{1}{2}(s^{1}_{m})^{\ast}\gamma^{m}\bar{\Psi}_{1}+\frac{1}{2}(-1)^{|\Psi_{1}|}s^{2}_{m}\bar{\Psi}_{1}\gamma^{m}$
$\displaystyle+\frac{1}{2}[u^{1}_{m}+(p^{2}_{m})^{\ast}]\gamma^{m}\Psi_{1}+\frac{1}{2}(-1)^{|\Psi_{1}|}[(u^{2}_{m})^{\ast}+p^{1}_{m}]\Psi_{1}\gamma^{m}$
$\displaystyle+\frac{1}{2}(q^{2}_{mn})^{\ast}\gamma^{m}\Psi_{2}\gamma^{n}-\frac{1}{2}q^{1}_{mn}\gamma^{n}\bar{\Psi}_{2}\gamma^{m}$
$\displaystyle K=$
$\displaystyle\frac{1}{2}(-1)^{|\Psi_{1}|}t_{2}\Psi_{2}-\frac{1}{2}(-1)^{|\Psi_{1}|}t_{1}\bar{\Psi}_{2}+\frac{1}{2}(u^{1}_{m}+p^{2}_{m})\gamma^{m}\Psi_{2}+\frac{1}{2}(-1)^{|\Psi_{2}|}(u^{2}_{m}+p_{m}^{1})\Psi_{2}\gamma^{m}$
$\displaystyle+\frac{1}{2}q^{1}_{mn}\gamma^{n}\bar{\Psi}_{1}\gamma^{m}-\frac{1}{2}q^{2}_{mn}\gamma^{m}\Psi_{1}\gamma^{n}.$
(365)

Let us now specify this expansion to the most general case of pure string-like
supersymmetry breaking. We first impose domain-wall BPSness, namely

$K=0,$ (366)

which gives the following constraints on the supersymmetry breaking parameters

$\displaystyle t_{1}=t_{2}=0$ (367) $\displaystyle
u^{1}_{m}=-\frac{1}{2}{(1+iJ_{1})^{k}}_{m}p^{2}_{k}$ (368) $\displaystyle
u^{2}_{m}=-\frac{1}{2}{(1+iJ_{2})^{k}}_{m}p^{1}_{k}$ (369)
$\displaystyle{(1+iJ_{2})^{k}}_{m}q^{1}_{kn}=0$ (370)
$\displaystyle{(1+iJ_{1})^{k}}_{m}q^{2}_{kn}=0,$ (371)

then we impose the gauge BPSness, which amounts to requiring $\Upsilon$ to be
purely imaginary, which yields

$\displaystyle r_{1}=-r_{2}\equiv r$ (372) $\displaystyle
q^{1}_{mn}=q^{2}_{nm}\equiv q_{mn}$ (373) $\displaystyle
s^{1}_{m}=\frac{1}{2}{(1+iJ_{1})^{k}}_{m}p^{2}_{k}-(p^{2}_{m})^{\ast}$ (374)
$\displaystyle
s^{2}_{m}=\frac{1}{2}{(1+iJ_{2})^{k}}_{m}p^{1}_{k}-(p^{1}_{m})^{\ast}.$ (375)

Imposing these constraints, the internal spinors describing the pure D-string
supersymmetry breaking read

$\displaystyle\mathcal{V}_{1}=r\eta_{1}^{\ast}+(p^{2}_{m}-(p^{2}_{m})^{\ast})\gamma^{m}\eta_{1}$
$\displaystyle\qquad\mathcal{V}_{2}=-r\eta_{2}^{\ast}+(p^{1}_{m}-(p^{1}_{m})^{\ast})\gamma^{m}\eta_{2}$
(376) $\displaystyle\mathcal{S}_{1}=-p^{2}_{m}\gamma^{m}\eta_{1}$
$\displaystyle\qquad\mathcal{S}_{2}=-p^{1}_{m}\gamma^{m}\eta_{2}$ (377)
$\displaystyle\mathcal{U}^{1}_{m}=p^{1}_{m}\eta_{1}+q_{mn}\gamma^{n}\eta_{1}^{\ast}$
$\displaystyle\qquad\mathcal{U}^{2}_{m}=p^{2}_{m}\eta_{2}+q_{nm}\gamma^{n}\eta_{2}^{\ast}.$
(378)

The most general D-string supersymmetry breaking term is therefore

$\displaystyle\Upsilon=$
$\displaystyle\frac{1}{2}(-1)^{|\Psi_{1}|}(r^{\ast}\Psi_{2}-r\bar{\Psi}_{2})+(\frac{1}{4}{(1-iJ_{1})^{k}}_{m}(p^{2}_{k})^{\ast}-\frac{1}{2}p^{2}_{m})\gamma^{m}\bar{\Psi}_{1}$
$\displaystyle+(-1)^{|\Psi_{1}|}(\frac{1}{4}{(1+iJ_{2})^{k}}_{m}p^{1}_{k}-\frac{1}{2}(p^{1}_{m})^{\ast})\bar{\Psi}_{1}\gamma^{m}$
$\displaystyle-(\frac{1}{4}{(1+iJ_{1})^{k}}_{m}p^{2}_{k}-\frac{1}{2}(p^{2}_{m})^{\ast})\gamma^{m}\Psi_{1}$
$\displaystyle-(-1)^{|\Psi_{1}|}(\frac{1}{4}{(1-iJ_{2})^{k}}_{m}p^{1}_{k}-\frac{1}{2}p^{1}_{m})\Psi_{1}\gamma^{m}$
$\displaystyle+\frac{1}{2}(q_{mn})^{\ast}\gamma^{m}\Psi_{2}\gamma^{n}-\frac{1}{2}q_{mn}\gamma^{m}\bar{\Psi}_{2}\gamma^{n}.$
(379)

The non-supersymmetric backgrounds presented in 3.2 have the following
supersymmetry breaking parameters

$\displaystyle r=$ $\displaystyle\ 0$ (380) $\displaystyle p^{1}_{m}=$
$\displaystyle\
e^{A}(-1)^{|\Psi_{1}|+1}\left[\frac{1}{2}{J_{2}^{k}}_{m}(-\delta^{n}_{k}+\frac{1}{2}{\Lambda^{n}}_{k})\alpha_{n}+\frac{1}{4}\Lambda_{nm}\alpha_{q}J_{1}^{qn}\right.$
(381)
$\displaystyle\left.+\frac{3i}{2}(-\delta^{n}_{m}+\frac{1}{2}{\Lambda^{n}}_{m})\alpha_{n}+\frac{i}{4}{J_{2}^{k}}_{m}\Lambda_{nk}\alpha_{q}J_{1}^{qn}\right]$
(382) $\displaystyle p^{2}_{m}=$ $\displaystyle\
e^{A}\left[(-1)^{|\Psi_{1}|+1}{J_{1}^{k}}_{m}\Lambda_{kn}\alpha_{q}J_{2}^{nq}-\frac{1}{2}((-1)^{|\Psi_{2}|}\delta_{m}^{n}+(-1)^{|\Psi_{1}|}\frac{1}{2}{\Lambda_{m}}^{n})\alpha_{n}\right.$
(383)
$\displaystyle\left.-3i(-1)^{|\Psi_{1}|}\Lambda_{mn}\alpha_{q}J_{2}^{nq}-\frac{i}{2}{J_{1}^{k}}_{m}((-1)^{|\Psi_{2}|}\delta_{k}^{n}+(-1)^{|\Psi_{1}|}\frac{1}{2}{\Lambda_{k}}^{n})\alpha_{n}\right]$
(384) $\displaystyle q_{mn}=$
$\displaystyle-\frac{e^{A}}{2}\alpha_{p}\left[(-1)^{|\Psi_{1}|}\Lambda_{mq}({\Omega_{2}^{qp}}_{n})^{\ast}-\Lambda_{qn}{\Omega_{1}^{pq}}_{m}\right],$
(385)

where $J_{1,2}$ are the (almost) complex structures defined by $\eta_{1}$ and
$\eta_{2}$

${{J_{1,2}}^{m}}_{n}=\frac{i}{\lVert\eta_{1,2}\rVert}\eta_{1,2}^{\dagger}{\gamma^{m}}_{n}\eta_{1,2},$
(386)

and $\Omega_{1,2mnp}$ are the $(3,0)-$forms with respect to the (almost)
complex structures defined by $\eta_{1}$ and $\eta_{2}$:

$\Omega_{1,2mnp}=\frac{1}{\lVert\eta_{1,2}\rVert}\eta_{1,2}^{T}\gamma_{mnp}\eta_{1,2}.$
(387)

In order to write down the supersymmetry breaking parameters of the non-
supersymmetric backgrounds with both SSB and DWSB contributions presented in
3.3, one just have to add the pure DWSB parameters given in Appendix B of
Lust:2008zd  to the one above.

The above supersymmetry breaking parameters correspond to the following
decomposition of the supersymmetry breaking term in (68) on the
SU$(3)\times$SU$(3)$ structure defined by the two pure spinors

$\displaystyle\mathrm{d}_{H}(\text{e}^{2A-\phi}\text{Im}\Psi_{1})=$
$\displaystyle\text{e}^{3A-\phi}\left(\gamma^{m}\Psi_{1}\left[(-1)^{|j|+1}\Lambda_{mn}\alpha_{p}J_{2}^{np}-\frac{i}{2}\alpha_{n}((-1)^{|\Psi_{2}|}\delta^{n}_{m}+(-1)^{|j|}\frac{1}{2}{\Lambda_{m}}^{n})\right]\right.$
$\displaystyle+\gamma^{m}\bar{\Psi}_{1}\left[(-1)^{|j|+1}\Lambda_{mn}\alpha_{p}J_{2}^{np}+\frac{i}{2}\alpha_{n}((-1)^{|\Psi_{2}|}\delta^{n}_{m}+(-1)^{|j|}\frac{1}{2}{\Lambda_{m}}^{n})\right]$
$\displaystyle+\Psi_{1}\gamma^{m}\left[-\frac{1}{4}\Lambda_{nm}\alpha_{p}J_{1}^{pn}-\frac{i}{2}\alpha_{n}(-\delta^{n}_{m}+\frac{1}{2}{\Lambda^{n}}_{m})\right]$
$\displaystyle+\bar{\Psi}_{1}\gamma^{m}\left[-\frac{1}{4}\Lambda_{nm}\alpha_{p}J_{1}^{pn}+\frac{i}{2}\alpha_{n}(-\delta^{n}_{m}+\frac{1}{2}{\Lambda^{n}}_{m})\right]$
$\displaystyle+\frac{i}{4}\gamma^{m}\Psi_{2}\gamma^{n}\alpha_{p}\left[(-1)^{|j|}\Lambda_{mq}({\Omega_{2}^{qp}}_{n})^{\ast}-\Lambda_{qn}{\Omega_{1}^{pq}}_{m}\right]$
$\displaystyle\left.-\frac{i}{4}\gamma^{m}\bar{\Psi}_{2}\gamma^{n}\alpha_{p}\left[(-1)^{|j|}\Lambda_{mq}{\Omega_{2}^{qp}}_{n}-\Lambda_{qn}({\Omega_{1}^{pq}}_{m})^{\ast}\right]\right).$
(388)

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