DMUS-MP-22-04

Heterotic de-Sitter Solutions

D. Farotti
Department of Mathematics, University of Surrey
Guildford, GU2 7XH, UK.
<EMAIL_ADDRESS>

Abstract

We classify all warped product de-Sitter backgrounds in heterotic
supergravity, up to two loops. We find that warped $dS_{n}$ backgrounds, with
$n\geq 3$, are $\mathbb{R}^{1,n}\times M_{9-n}$, where $M_{9-n}$ is a
$(9-n)$-dimensional Riemannian manifold. Moreover, we establish that warped
$dS_{2}$ backgrounds are $AdS_{3}\times M_{7}$, where $M_{7}$ is a
7-dimensional Riemannian manifold.

##  1 Introduction

Cosmological observations in the nineties revealed that at the present time
the universe possesses a small positive cosmological constant [1]. Moreover,
shortly after the big bang, there is good reason to think that the universe
underwent a period of exponential expansion, called inflation [2]. Hence de-
Sitter space seems to play an important role in the understanding of our past
and present universe. Within the framework of string theory, a quantum gravity
description of de-Sitter space presents various difficulties [3]. Moreover, in
the classical limit, there are no go-theorems stating that there are no smooth
compactifications of ten or eleven-dimensional supergravity to warped de-
Sitter space of any dimension [4, 5, 6]. Taking these issues into account, it
has been shown that there exist vacua with positive cosmological constant
within the string landscape [7]. However, these vacua are meta-stable and
decay to the true vacuum [8]. In the context of holography, one of the most
notable proposed dual descriptions of de-Sitter space is the dS-CFT
correspondence, which directly tries to generalize the original AdS-CFT
correspondence [9] to de-Sitter space [10, 11, 12].

To make progress towards the classification of de-Sitter backgrounds in string
theory, heterotic supergravity is a natural place to start. Indeed,
supersymmetric heterotic supergravity backgrounds have been thoroughly
investigated using spinorial geometry techniques: the Killing spinor equations
(KSE) have been solved in all cases and the geometry of these backgrounds has
been determined [13, 14, 15]. Moreover, the $\alpha^{{}^{\prime}}$ corrections
to heterotic backgrounds can be studied in the sigma model approach [16].
According to this scheme, the field equations correspond to the vanishing of
the sigma model beta functions, which are an infinite perturbation series in
the string tension $\alpha^{{}^{\prime}}$. In particular, the Bianchi
identities are modified: the Green-Schwarz anomaly-cancellation mechanism
requires that at one-loop $dH$ is equal to the difference of the Pontrjagin
forms of spacetime and the vector bundle of the gauge sector [17]. Consistency
of the theory then requires to take into account corrections up to two loops
in the bosonic field equations and the KSE; such corrections have a simpler
form compared to type II supergravity theories [18, 19, 20], hence anomaly
corrected heterotic backgrounds are more tractable. Anomaly corrected
backgrounds in heterotic theory have been studied in [21], in the context of
supersymmetric near-horizon geometries and in [22], in the context of the
classification of supersymmetric warped $AdS$ backgrounds.

In this paper we analyze warped de-Sitter backgrounds
$dS_{n}\times_{w}M_{10-n}$ in heterotic supergravity, including
$\alpha^{{}^{\prime}}$ corrections up to two loops in sigma model perturbation
theory. In particular, we require that when $\alpha^{{}^{\prime}}\to 0$, the
geometry of the spacetime is exactly $dS_{n}\times_{w}M_{10-n}$. This
corresponds to take the zero-th order term $k^{0}$ in the
$\alpha^{{}^{\prime}}$ expansion of the cosmological constant $k$ to be non-
zero, that is $k^{0}=\ell^{-2}$. As stated before, there are many no-go
theorems which rule out de-Sitter solutions in supergravity; these no-go
theorems assume that the warp factor and fluxes are smooth, and that the
internal manifold is smooth and compact without boundary. Thus, our motivation
is to construct a systematic classification of supersymmetric de Sitter
solutions in supergravity theories from a purely local perspective. This
evades the no-go theorems mentioned before. In particular, in this work we do
not assume the warp factor to be smooth, nor the internal manifold $M_{10-n}$
to be either smooth or compact. Our local analysis makes use only of the
bosonic field equations, the Bianchi identities and the KSE of the theory.
Furthermore, we do not assume the Killing spinors to factorize, as it is known
that such factorizations can produce a miscounting of supersymmetries [23]. We
find that warped product de Sitter solutions in heterotic supergravity are
very restricted. Indeed, we show that warped product $dS_{2}$ solutions are
direct product $AdS_{3}$ solutions and all warped product $dS_{n}$ solutions
for $n\geq 3$ are $\mathbb{R}^{1,n}\times M_{9-n}$. These results complement
[24], where, utilizing world-sheet CFT arguments, it is shown that in
heterotic string theory $dS_{n}$ vacua with $n\geq 4$ are forbidden to all
orders in $\alpha^{{}^{\prime}}$.

It is useful to compare and contrast the heterotic results to the case of
$D=11$ supergravity. The warped product $dS_{n}$ solutions of $D=11$
supergravity have similar foliation properties for $n\geq 5$. In particular,
for $n=5$, $dS_{5}$ arises as a conformal foliation of $\mathbb{R}^{1,5}$,
corresponding to the directions along the M5-brane worldvolume [25]. In
contrast, the warped product $dS_{4}$ solutions of $D=11$ supergravity
preserving $N=8$ supersymmetry, which have been classified in [26] using
spinorial geometry techniques, do not have an analogous foliation into
$AdS_{5}$ or $\mathbb{R}^{1,4}$. Moreover, for warped product $dS_{3}$
backgrounds in $D=11$ supergravity, we exhibit a rich algebraic structure,
which also suggests that such solutions are not foliations of $dS_{3}$ into
$AdS_{4}$ or $\mathbb{R}^{1,3}$.

The paper is organized as follows. In section 2 we present the bosonic field
equations, the Bianchi identities and the KSE of heterotic perturbation
theory, up to two loops. Moreover, we formulate the ansatz for the bosonic
fields in the case of warped de-Sitter backgrounds $dS_{n}\times_{w}M_{10-n}$
and we reduce the 10-dimensional gravitino KSE along the internal manifold
$M_{10-n}$, for generic $n$. In section 3 we analyse warped product $dS_{n}$
backgrounds with $n\geq 4$, up to two loops. In particular, we decompose the
field equations, the Bianchi identities, the dilatino and the gaugino KSE
along $M_{10-n}$ and we utilize this to show that the 10-dimensional spacetime
is $\mathbb{R}^{1,n}\times M_{9-n}$, where $M_{9-n}$ is a $(9-n)$-dimensional
Riemannian manifold. Moreover, we show that the 1-form $d\Phi$, the 3-form $H$
and the non abelian field strength $F$ have non-vanishing components only
along $M_{9-n}$. In section 4 we study warped product $dS_{3}$ backgrounds, up
to two loops. In this case, we show that the 10-dimensional spacetime is
$\mathbb{R}^{1,3}\times M_{6}$, where $M_{6}$ is a $6$-dimensional Riemannian
manifold. Moreover, the 1-form $d\Phi$, the 3-form $H$ and the non abelian
field strength $F$ have non-vanishing components only along $M_{6}$. In
section 5 we investigate warped $dS_{2}$ backgrounds, up to two loops. After
decomposing the bosonic field equations, Bianchi identities and algebraic KSE
on $M_{8}$, we show that all these backgrounds are $AdS_{3}\times M_{7}$,
where $M_{7}$ is a 7-dimensional Riemannian manifold. In section 6 we study
the Killing superalgebra $\mathfrak{g}$ of warped $dS_{3}$ backgrounds in
$D=11$ supergravity and we show that there is a rich algebraic structure
underlying $\mathfrak{g}$. Some brief conclusions are presented in Section 7.
The paper ends with two appendices. Appendix A contains our conventions for
the curvature tensor of a connection and some useful formulae which are used
in the computations of the paper. Appendix B contains some details about the
Killing superalgebra of $D=11$ warped $dS_{3}$ backgrounds. In particular, we
list the spinorial Lie derivatives and the conditions arising from the super-
Jacobi identities.

##  2 Bosonic field equations and KSE

In this section we present the bosonic field equations and the Killing spinor
equations (KSE) of heterotic perturbation theory, up to two loops. The bosonic
field equations of heterotic supergravity up to two loops in sigma model
perturbation theory are given by [16]

$\displaystyle
R_{MN}-\frac{1}{4}H_{ML_{1}L_{2}}H_{N}^{~{}L_{1}L_{2}}+2\nabla_{M}\nabla_{N}\Phi$
$\displaystyle+\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\check{R}_{ML_{1},L_{2}L_{3}}\check{R}_{N}^{~{}~{}L_{1},L_{2}L_{3}}-F_{MLij}F_{N}^{~{}Lij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.1)
$\displaystyle\nabla^{M}(e^{-2\Phi}H_{MN_{1}N_{2}})=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.2)
$\displaystyle\nabla^{M}(e^{-2\Phi}F_{MN})+\frac{1}{2}e^{-2\Phi}H_{NL_{1}L_{2}}F^{L_{1}L_{2}}=\mathcal{O}(\alpha^{{}^{\prime}})$
(2.3)
$\displaystyle\nabla^{M}\nabla_{M}\Phi-2(\nabla\Phi)^{2}+\frac{1}{12}H_{MNR}H^{MNR}$
$\displaystyle-\frac{\alpha^{{}^{\prime}}}{16}\bigg{(}\check{R}_{N_{1}N_{2},N_{3}N_{4}}\check{R}^{N_{1}N_{2},N_{3}N_{4}}-F_{MNij}F^{MNij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.4)

where $M,N\dots$ are 10-dimensional frame indices, $i,j$ are gauge indices and
$\check{R}$ is the curvature tensor with respect to the connection
$\check{\nabla}:=\nabla-\frac{1}{2}H$. Our conventions for the curvature are
presented in Appendix A. In the following, gauge indices are understood.
Moreover, the Bianchi identities read

$\displaystyle
dH=-\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\textrm{tr}(\check{R}\wedge\check{R})-\textrm{tr}(F\wedge
F)\bigg{)}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (2.5)

We remark that the right hand side of (2.5) is required for anomaly
cancellations and is expressed as the difference of two Pontryagin forms, one
of the tangent space of spacetime and the other of the gauge sector bundle.
Furthermore, the KSEs are given by

$\displaystyle\hat{\nabla}_{M}\epsilon:=\nabla_{M}\epsilon-\frac{1}{8}H_{MN_{1}N_{2}}\Gamma^{N_{1}N_{2}}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.6)
$\displaystyle\bigg{(}\Gamma^{M}\nabla_{M}\Phi-\frac{1}{12}H_{N_{1}N_{2}N_{3}}\Gamma^{N_{1}N_{2}N_{3}}\bigg{)}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.7) $\displaystyle
F_{MN}\Gamma^{MN}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}})$ (2.8)

where $\hat{\nabla}:=\nabla+\frac{1}{2}H$. Notice that the KSEs have the same
form up to two loops in sigma model perturbation theory as that at the zero-th
order.
In our work, we analyze warped product $dS_{n}$ backgrounds in heterotic
supergravity, up to two loops. In the perturbation theory scheme, the bosonic
fields and the Killing spinors are expanded in $\alpha^{{}^{\prime}}$. The
metric on the $D=10$ spacetime $M_{10}$ is given by

$\displaystyle
ds^{2}(M_{10})=A^{2}ds^{2}(dS_{n})+ds^{2}(M_{10-n})+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.9)

where $A$ is a function of the co-ordinates of the Riemannian manifold
$M_{10-n}$ and

$\displaystyle
ds^{2}(dS_{n})=\frac{1}{(1+\frac{k}{4}|x|^{2})^{2}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}~{}~{}~{}~{}~{}\mu,\nu=0,1,\dots
n-1$ (2.10)

is the metric of $n$-dimensional de-Sitter spacetime, where
$|x|^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu}$ and $k=\frac{1}{\ell^{2}}$. Notice that
$k$ itself admits a perturbation series in $\alpha^{{}^{\prime}}$, that is

$\displaystyle
k=k^{0}+k^{1}\alpha^{{}^{\prime}}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(2.11)

In our analysis, we consider $k^{0}=\frac{1}{\ell^{2}}$, that is, in the limit
$\alpha^{{}^{\prime}}\to 0$ (2.9) reduces to $dS_{n}\times_{w}M^{10-n}$. Let
us introduce on $M_{10}$ the co-frame

$\displaystyle\textbf{e}^{\mu}=\frac{A}{\mathcal{U}}dx^{\mu}~{},~{}~{}~{}~{}~{}\textbf{e}^{a}=e^{a}_{~{}\alpha}(y)dy^{\alpha}$
(2.12)

where $a=n,n+1,\dots,9$, $y^{\alpha}$ denote the co-ordinates on $M_{10-n}$
and

$\displaystyle\mathcal{U}=1+\frac{k}{4}|x|^{2}~{}.$ (2.13)

In terms of the co-frame (2.12), the metric tensor (2.9) reads

$\displaystyle
ds^{2}(M_{10})=\eta_{\mu\nu}\textbf{e}^{\mu}\textbf{e}^{\nu}+ds^{2}(M_{10-n})+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.14)

where

$\displaystyle ds^{2}(M_{10-n})=\delta_{ab}\textbf{e}^{a}\textbf{e}^{b}~{}.$
(2.15)

Moreover, we assume that the dilaton $\Phi$ is a function of the co-ordinates
of $M_{10-n}$. The ansatz for the 3-form $H$ and the non-abelian field
strength $F$ depends on $n$. Assuming $H$ and $F$ to be invariant under the
isometry group of $dS_{n}$ up to two loops, it follows that

  * •

$n\geq 4$

$\displaystyle H=X+\mathcal{O}(\alpha^{{}^{\prime}2})$ (2.16) $\displaystyle
F=\mathcal{F}+\mathcal{O}(\alpha^{{}^{\prime}})$ (2.17)

where $X\in\Omega^{3}(M_{10-n})$ and $\mathcal{F}$ is a 2-form on $M_{10-n}$
taking values in $\mathfrak{g}$, where we denote by $\mathfrak{g}$ the Lie
algebra of the gauge group.

  * •

$n=3$

$\displaystyle
H=f~{}\textrm{dvol}(dS_{3})+X+\mathcal{O}(\alpha^{{}^{\prime}2})$ (2.18)
$\displaystyle F=\mathcal{F}+\mathcal{O}(\alpha^{{}^{\prime}})$ (2.19)

where $f\in\Omega^{0}(M_{7})$, $X\in\Omega^{3}(M_{7})$ and $\mathcal{F}$ is a
2-form on $M_{7}$ taking values in $\mathfrak{g}$.

  * •

$n=2$

$\displaystyle
H=B\wedge\textrm{dvol}(dS_{2})+X+\mathcal{O}(\alpha^{{}^{\prime}2})$ (2.20)
$\displaystyle
F=f\textrm{dvol}(dS_{2})+\mathcal{F}+\mathcal{O}(\alpha^{{}^{\prime}})$ (2.21)

where $B\in\Omega^{1}(M_{8})$, $X\in\Omega^{3}(M_{8})$,
$f\in\Omega^{0}(M_{8})$ and $\mathcal{F}$ is a 2-form on $M_{8}$ taking values
in $\mathfrak{g}$.

###  2.1 Reduction of the gravitino KSE on $M_{10-n}$

In this sub-section we reduce the gravitino KSE (2.6) on $M_{10-n}$, for
generic $n$. Using (A.2), (2.16), (2.18) and (2.20), the $M=\mu$ component of
(2.6) is given by

$\displaystyle\frac{\partial}{\partial
x^{\mu}}\epsilon=\frac{1}{\mathcal{U}}\bigg{(}-\frac{k}{4}x^{\nu}\Gamma_{\nu\mu}+\Gamma_{\mu}\mathcal{C}_{(n)}\bigg{)}\epsilon+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.22)

where we have defined111If $\omega\in\Omega^{p}(M_{10-n})$, then
$\not{\omega}=\omega_{a_{1}a_{2}\dots a_{p}}\Gamma^{a_{1}a_{2}\dots a_{p}}$.

$\displaystyle\mathcal{C}_{(n)}=\begin{cases}-\frac{1}{2}\widetilde{\not{\nabla}}A~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\geq
4\\\ \\\
-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}cA^{-2}\Gamma^{(3)}~{}~{}~{}~{}~{}~{}\,\,n=3\\\
\\\
-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}~{}~{}~{}~{}~{}~{}n=2\end{cases}$
(2.23)

with $\Gamma^{(n)}$ being the highest rank Gamma matrix on $dS_{n}$, that is

$\displaystyle\Gamma^{(n)}:=\frac{1}{n!}\epsilon_{\mu_{1}\mu_{2}\dots\mu_{n}}\Gamma^{\mu_{1}\mu_{2}\dots\mu_{n}}~{}.$
(2.24)

Equation (2.22) yields a partial differential equation for $\epsilon$

$\displaystyle\frac{\partial}{\partial x^{\mu}}\frac{\partial}{\partial
x^{\nu}}\epsilon+\frac{k}{4\mathcal{U}}\big{(}x_{\mu}\frac{\partial}{\partial
x^{\nu}}\epsilon+x_{\nu}\frac{\partial}{\partial
x^{\mu}}\epsilon\big{)}-\frac{k^{2}}{16\mathcal{U}^{2}}x_{\mu}x_{\nu}\epsilon+\frac{k}{4\mathcal{U}}\eta_{\mu\nu}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.25)

whose solution is given by

$\displaystyle\epsilon=\mathcal{U}^{-1/2}\big{(}\psi+x^{\mu}\tau_{\mu}\big{)}+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.26)

where $\psi$ and $\tau_{\mu}$ are 16-component chiral spinors which depend
only on the co-ordinates on $M_{10-n}$. Inserting (2.26) into (2.22), we get

$\displaystyle\tau_{\mu}=\Gamma_{\mu}\mathcal{C}_{(n)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.27)

and

$\displaystyle\bigg{(}\frac{k}{4}+\widehat{\mathcal{C}}_{(n)}\mathcal{C}_{(n)}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.28)

where $\widehat{\mathcal{C}}_{(n)}$ is defined by

$\displaystyle\widehat{\mathcal{C}}_{(n)}=\begin{cases}\frac{1}{2}\widetilde{\not{\nabla}}A~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\geq
4\\\ \\\
\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}cA^{-2}\Gamma^{(3)}~{}~{}~{}~{}~{}~{}n=3\\\
\\\
\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}~{}~{}~{}~{}~{}n=2\end{cases}$
(2.29)

Furthermore, substituting (2.27) into (2.26), we find

$\displaystyle\epsilon=\mathcal{U}^{-1/2}\bigg{(}1+x^{\mu}\Gamma_{\mu}\mathcal{C}_{(n)}\bigg{)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(2.30)

Moreover, the $M=a$ component of (2.6) reads

$\displaystyle\widetilde{\nabla}_{a}\epsilon=\sigma_{a(n)}\epsilon+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.31)

where $\sigma_{a(n)}$ is given by

$\displaystyle\sigma_{a(n)}=\begin{cases}\frac{1}{8}X_{abc}\Gamma^{bc}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\geq
3\\\ \\\
\frac{1}{8}X_{abc}\Gamma^{bc}+\frac{1}{4}A^{-2}B_{a}\Gamma^{(2)}~{}~{}~{}~{}n=2\end{cases}$
(2.32)

Substituting (2.30) into (2.31), we find

$\displaystyle\widetilde{\nabla}_{a}\psi=\sigma_{a(n)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.33)

and

$\displaystyle\bigg{(}\widetilde{\nabla}_{a}\mathcal{C}_{(n)}+\mathcal{C}_{(n)}\sigma_{a(n)}-\widehat{\sigma}_{a(n)}\mathcal{C}_{(n)}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(2.34)

where $\hat{\sigma}_{a(n)}$ is defined by

$\displaystyle\widehat{\sigma}_{a(n)}=\begin{cases}\frac{1}{8}X_{abc}\Gamma^{bc}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\geq
3\\\ \\\
\frac{1}{8}X_{abc}\Gamma^{bc}-\frac{1}{4}A^{-2}B_{a}\Gamma^{(2)}~{}~{}~{}~{}n=2\end{cases}$
(2.35)

Hence, the reduction of the gravitino KSEs (2.6) on $M_{10-n}$ yield the
gravitino KSE (2.33) on $M_{10-n}$, supplemented by the quadratic conditions
(2.28) and (2.34).

##  3 $dS_{n}$, with $n\geq 4$

In this section, we study warped $dS_{n}$ backgrounds, with $n\geq 4$. Using
(2.23) and (2.30), we find

$\displaystyle\epsilon=\mathcal{U}^{-1/2}\bigg{(}1-\frac{1}{2}x^{\mu}\Gamma_{\mu}\widetilde{\not{\nabla}}A\bigg{)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.1)

where $\mathcal{U}$ is defined by (2.13). Moreover, the reduction of the
gravitino KSE (2.6) on $M_{10-n}$ yields (2.33), that is

$\displaystyle\widetilde{\nabla}_{a}\psi=\frac{1}{8}X_{abc}\Gamma^{bc}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.2)

supplemented by the quadratic conditions (2.28) and (2.34), which imply

$\displaystyle k-(\widetilde{\nabla}A)^{2}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.3)

and

$\displaystyle\bigg{(}-\frac{1}{2}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A+\frac{1}{4}\widetilde{\nabla}^{c}AX_{cab}\bigg{)}\Gamma^{b}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.4)

respectively. Equation (3.4) yields

$\displaystyle-\frac{1}{2}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A+\frac{1}{4}\widetilde{\nabla}^{c}AX_{cab}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.5)

Splitting (3.5) into symmetric and anti-symmetric part in $(a,b)$, we get

$\displaystyle\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.6)

and

$\displaystyle\widetilde{\nabla}^{c}AX_{cab}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.7)

Furthermore, using (3.1), (2.16) and (2.17), the dilatino and the gaugino KSE
(2.7) and (2.8) decompose on $M_{10-n}$ as

$\displaystyle\bigg{(}\widetilde{\not{\nabla}}\Phi-\frac{1}{12}\not{X}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.8) $\displaystyle\not{\mathcal{F}}\psi=\mathcal{O}(\alpha^{{}^{\prime}})$
(3.9)

supplemented by the quadratic conditions

$\displaystyle\bigg{(}\widetilde{\not{\nabla}}\Phi-\frac{1}{12}\not{X}\bigg{)}\widetilde{\not{\nabla}}A\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.10)
$\displaystyle\not{\mathcal{F}}\widetilde{\not{\nabla}}A\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(3.11)

Combining (3.8) and (3.10) and using (3.7), we obtain

$\displaystyle\widetilde{\nabla}^{a}A\widetilde{\nabla}_{a}\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.12)

Moreover, combining (3.9) and (3.11), we get

$\displaystyle\widetilde{\nabla}^{a}A\mathcal{F}_{ab}=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(3.13)

Using (3.3), (3.6) and (3.7), a straightforward computation shows that the
only non-vanishing components of $\check{R}$ up to two loops are given by

$\displaystyle\tilde{\check{R}}_{ab,cd}:=\widetilde{R}_{abcd}-\frac{1}{2}\widetilde{\nabla}_{a}X_{cbd}+\frac{1}{2}\widetilde{\nabla}_{b}X_{cad}+\frac{1}{4}X_{cam}X^{m}_{~{}~{}bd}-\frac{1}{4}X_{cbm}X^{m}_{~{}~{}ad}~{}.$
(3.14)

In order to obtain (3.14), we have used (A.6), (A.2) and (A.3). Using the
previous results, the bosonic field equations (2.1)-(2.4) decompose on
$M_{10-n}$ as follows

$\displaystyle\widetilde{R}_{ab}-\frac{1}{4}X_{amn}X_{b}^{~{}mn}+2\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\Phi+\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\tilde{\check{R}}_{ac_{1},c_{2}c_{3}}\tilde{\check{R}}_{b}^{~{}~{}c_{1},c_{2}c_{3}}-\mathcal{F}_{acij}\mathcal{F}_{b}^{~{}cij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.15)
$\displaystyle(\widetilde{\nabla}^{c}X)_{cab}-2\widetilde{\nabla}^{c}\Phi
X_{cab}=\mathcal{O}(\alpha^{{}^{\prime}2})$ (3.16)
$\displaystyle\widetilde{\nabla}^{b}\mathcal{F}_{ba}-2\widetilde{\nabla}^{b}\Phi\mathcal{F}_{ba}+\frac{1}{2}X_{ab_{1}b_{2}}\mathcal{F}^{b_{1}b_{2}}=\mathcal{O}(\alpha^{{}^{\prime}})$
(3.17)
$\displaystyle\widetilde{\nabla}^{a}\widetilde{\nabla}_{a}\Phi-2(\widetilde{\nabla}\Phi)^{2}+\frac{1}{12}X^{2}-\frac{\alpha^{{}^{\prime}}}{16}\bigg{(}\tilde{\check{R}}_{a_{1}a_{2},a_{3}a_{4}}\tilde{\check{R}}^{a_{1}a_{2},a_{3}a_{4}}-\mathcal{F}_{ab,ij}\mathcal{F}^{ab,ij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.18)

where we have used (2.16), (2.17), (A.2) and (A.4). Moreover, the Bianchi
identities (2.5) imply

$\displaystyle\tilde{d}X=-\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\textrm{tr}(\tilde{\check{R}}\wedge\tilde{\check{R}})-\textrm{tr}(\mathcal{F}\wedge\mathcal{F})\bigg{)}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.19)

Let us introduce the 1-form on $M_{10-n}$

$\displaystyle V=\frac{1}{\sqrt{k}}dA~{}.$ (3.20)

Equation (3.6) implies

$\displaystyle\widetilde{\nabla}_{a}V_{b}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.21)

Moreover, (3.3) implies $V^{2}=1+\mathcal{O}(\alpha^{{}^{\prime}2})$. Hence

$\displaystyle ds^{2}(M_{10-n})=V\otimes
V+ds^{2}(M_{9-n})+\mathcal{O}(\alpha^{{}^{\prime}2})=\frac{1}{k}dA^{2}+ds^{2}(M_{9-n})+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.22)

Substituting (3.22) in (2.9), we find

$\displaystyle
ds^{2}(M_{10})=ds^{2}(M_{1,n})+ds^{2}(M_{9-n})+\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.23)

where we have defined

$\displaystyle ds^{2}(M_{1,n})=A^{2}ds^{2}(dS_{n})+\frac{1}{k}dA^{2}~{}.$
(3.24)

It is straightforward to check that the Riemann tensor of (3.24) vanishes,
hence $M_{1,n}\simeq\mathbb{R}^{1,n}$ and (3.23) yields

$\displaystyle
ds^{2}(M_{10})=ds^{2}(\mathbb{R}^{1,n})+ds^{2}(M_{9-n})+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.25)

Notice that (3.7), (3.12) and (3.13) imply that $X$, $\Phi$ and $\mathcal{F}$
have non-vanishing components only along $M_{9-n}$. Moreover, the vector field
$V$ dual to the 1-form (3.20)222We denote the vector field dual to the 1-form
(3.20) by the same symbol $V$. preserves $X$, $\Phi$ and $\mathcal{F}$, that
is

$\displaystyle\widetilde{\mathcal{L}}_{V}\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.26)
$\displaystyle\widetilde{\mathcal{L}}_{V}\mathcal{F}=\mathcal{O}(\alpha^{{}^{\prime}})$
(3.27)
$\displaystyle\widetilde{\mathcal{L}}_{V}X=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.28)

Equation (3.26) is a consequence of (3.12). Equation (3.27) follows from
(3.13), by fixing the gauge $i_{V}\mathcal{A}=0$ where $\mathcal{A}$ is the
gauge potential $\mathcal{A}$, defined by
$\mathcal{F}=d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}$. In order to show
(3.28), first of all notice that

$\displaystyle\mathcal{L}_{V}X=i_{V}\tilde{d}X+\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.29)

by means of (3.7). Moreover, implementing (3.1) in the 10-dimensional
gravitino integrability conditions

$\displaystyle\hat{R}_{MN,PQ}\Gamma^{PQ}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.30)

we find

$\displaystyle\hat{R}_{MN,PQ}\Gamma^{PQ}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.31)

and

$\displaystyle\hat{R}_{MN,PQ}\Gamma^{PQ}\not{V}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.32)

Taking $M,N$ to be internal indices in (3.31) and (3.32), we get

$\displaystyle\tilde{\hat{R}}_{mn,ab}\Gamma^{ab}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(3.33)

and

$\displaystyle\tilde{\hat{R}}_{mn,ab}\Gamma^{ab}\not{V}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(3.34)

In order to obtain (3.33) and (3.34), we have used
$\hat{R}_{mn,\mu\nu}=\hat{R}_{mn,\mu a}=\mathcal{O}(\alpha^{{}^{\prime}2})$
which follow from (A.7). Equations (3.33) and (3.34) yield

$\displaystyle
V^{a}\tilde{\hat{R}}_{mn,ab}=\mathcal{O}(\alpha^{{}^{\prime}2})$ (3.35)

hence, using (A.8), it follows that

$\displaystyle
V^{a}\tilde{\check{R}}_{ab,mn}=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$ (3.36)

Implementing (3.13) and (3.36) in the Bianchi identities (3.19), we find

$\displaystyle i_{V}\tilde{d}X=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (3.37)

Substituting (3.37) in (3.29), we recover (3.28). To summarize, in heterotic
perturbation theory, up to two loops, warped $dS_{n}$ backgrounds with $n\geq
4$ are direct product manifolds of $(n+1)$-Minkowski spacetime and a
Riemannian manifold $M_{9-n}$. Moreover, the 1-form $d\Phi$, the 3-form $H$
and the non abelian field strength $F$ have non vanishing components only
along $M_{9-n}$. Such backgrounds have been extensively studied in the
literature, see e.g. [27], [28] and references within.

##  4 $dS_{3}$ backgrounds

In this section, we analyze warped $dS_{3}$ backgrounds, up to two loops. In
this case, equation (2.28) yields

$\displaystyle\bigg{(}\frac{k}{4}-\frac{1}{4}(\widetilde{\nabla}A)^{2}+\frac{1}{4}fA^{-2}\widetilde{\not{\nabla}}A\Gamma^{(3)}+\frac{1}{16}f^{2}A^{-4}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(4.1)

Since $\Gamma^{a}\Gamma^{(3)}$ is anti-hermitian, equation (4.1) implies

$\displaystyle\frac{k}{4}-\frac{1}{4}(\widetilde{\nabla}A)^{2}+\frac{1}{16}f^{2}A^{-4}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(4.2)

and

$\displaystyle
f\widetilde{\not{\nabla}}A\Gamma^{(3)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(4.3)

Equation (4.3) yields

$\displaystyle f\tilde{d}A=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (4.4)

Moreover, the zero-th order term in $\alpha^{{}^{\prime}}$ in the perturbation
series of $\tilde{d}A$ is not vanishing, as a consequence of (4.2). Hence
(4.4) yields

$\displaystyle f=\mathcal{O}(\alpha^{{}^{\prime}2})$ (4.5)

that is $H=X+\mathcal{O}(\alpha^{{}^{\prime}2})$ by means of (2.18), which in
turn implies that the case $n=3$ can be treated exactly as $n\geq 4$. In
particular, equation (3.25) still holds for $n=3$, that is

$\displaystyle
ds^{2}(M_{10})=ds^{2}(\mathbb{R}^{1,3})+ds^{2}(M_{6})+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(4.6)

Moreover, equations (3.7), (3.12) and (3.13) still hold, i.e.

$\displaystyle
i_{V}X=\mathcal{O}(\alpha^{{}^{\prime}2})~{},~{}~{}~{}i_{V}d\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})~{},~{}~{}~{}i_{V}\mathcal{F}=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(4.7)

To summarize, in heterotic perturbation theory, up to two loops, warped
$dS_{3}$ backgrounds are $\mathbb{R}^{1,3}\times M_{6}$. Moreover, the 1-form
$d\Phi$, the 3-form $H$ and the non abelian field strength $F$ have non
vanishing components only along $M_{6}$.

##  5 $dS_{2}$

In this section, we study warped product $dS_{2}$ backgrounds, up to two
loops. First of all, equation (2.30) reads

$\displaystyle\epsilon=\mathcal{U}^{-1/2}\bigg{(}1+x^{\mu}\Gamma_{\mu}\big{(}-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}\big{)}\bigg{)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.1)

Moreover, the reduction of the gravitino KSE (2.6) on $M_{10-n}$ yields
(2.33), that is

$\displaystyle\widetilde{\nabla}_{a}\psi=\bigg{(}\frac{1}{8}X_{abc}\Gamma^{bc}+\frac{1}{4}A^{-2}B_{a}\Gamma^{(2)}\bigg{)}\psi+\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.2)

supplemented by the quadratic constraints (2.28) and (2.34), which are
equivalent to the conditions

$\displaystyle\bigg{(}\frac{k}{4}-\frac{1}{4}(\widetilde{\nabla}A)^{2}+\frac{1}{4}A^{-1}\widetilde{\nabla}_{a}AB_{b}\Gamma^{ab}\Gamma^{(2)}+\frac{1}{16}A^{-2}B^{2}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.3)

and

$\displaystyle\bigg{(}-\frac{1}{2}\Gamma^{b}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A-\frac{1}{4}A^{-2}\widetilde{\nabla}_{a}AB_{b}\Gamma^{b}\Gamma^{(2)}+\frac{1}{4}A^{-1}\widetilde{\nabla}_{a}B_{b}\Gamma^{b}\Gamma^{(2)}+\frac{1}{4}\widetilde{\nabla}^{d}AX_{dab}\Gamma^{b}$
$\displaystyle-\frac{1}{4}A^{-2}B_{a}\widetilde{\nabla}_{b}A\Gamma^{b}\Gamma^{(2)}-\frac{1}{8}A^{-1}B^{d}X_{dab}\Gamma^{b}\Gamma^{(2)}+\frac{1}{8}A^{-3}B_{a}B_{b}\Gamma^{b}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.4)

respectively. Furthermore, using (5.1), (2.20) and (2.21), the dilatino and
the gaugino KSE (2.7) and (2.8) decompose on $M_{10-n}$ as

$\displaystyle\bigg{(}\widetilde{\not{\nabla}}\Phi-\frac{1}{12}\not{X}-\frac{1}{2}A^{-2}\not{B}\Gamma^{(2)}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.5)

and

$\displaystyle
2fA^{-2}\Gamma^{(2)}\psi+\not{\mathcal{F}}\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.6)

Notice that equation (5.6) is equivalent to

$\displaystyle
2fA^{-2}\psi+\Gamma^{(2)}\not{\mathcal{F}}\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.7)

Equation (5.7) implies $f=\mathcal{O}(\alpha^{{}^{\prime}})$, that is
$F=\mathcal{F}+\mathcal{O}(\alpha^{{}^{\prime}})$ and

$\displaystyle\not{\mathcal{F}}\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.8)

Moreover, we find the quadratic constraints

$\displaystyle\bigg{(}\widetilde{\not{\nabla}}\Phi-\frac{1}{12}\not{X}+\frac{1}{2}A^{-2}\not{B}\Gamma^{(2)}\bigg{)}\bigg{(}-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.9)
$\displaystyle\not{\mathcal{F}}\bigg{(}-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}\bigg{)}\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.10)

Equation (5.3) yields

$\displaystyle\frac{k}{4}-\frac{1}{4}(\widetilde{\nabla}A)^{2}+\frac{1}{16}A^{-2}B^{2}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.11)

and

$\displaystyle\widetilde{\nabla}_{a}AB_{b}\Gamma^{ab}\psi=\mathcal{O}(\alpha^{{}^{\prime}2}).$
(5.12)

Equation (5.11) implies that $dA$ is not vanishing, at zero-th order in
$\alpha^{{}^{\prime}}$. Hence, point-wise, we can choose a frame such that, at
zero-th order in $\alpha^{{}^{\prime}}$, $\widetilde{\nabla}_{2}A\neq 0$,
$\widetilde{\nabla}_{3}A=0,\dots,\widetilde{\nabla}_{9}A=0$, where we have
denoted by $2,3,\dots 9$, the directions along $M_{8}$. Thus (5.12) yields

$\displaystyle B_{i}\Gamma^{i}\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$ (5.13)

where $i=3,4,\dots 9$. Equation (5.13) yields
$B_{i}=\mathcal{O}(\alpha^{{}^{\prime}2})$, which means that

$\displaystyle B=\mathcal{G}\tilde{d}A+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.14)

for some function $\mathcal{G}$. The integrability conditions of the
10-dimensional gravitino KSE (2.6) read

$\displaystyle\hat{R}_{MN,RS}\Gamma^{RS}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.15)

Taking $M,N$ to be internal indices in (5.15), we obtain

$\displaystyle\hat{R}_{ab,\mu\nu}\Gamma^{\mu\nu}\epsilon+2\hat{R}_{ab,\nu
c}\Gamma^{\nu
c}\epsilon+\hat{R}_{ab,cd}\Gamma^{cd}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.16)

Using (A.7), a direct computation yields $\hat{R}_{ab,\nu
c}=\mathcal{O}(\alpha^{{}^{\prime}2})$, hence (5.16) reads

$\displaystyle\hat{R}_{ab,\mu\nu}\epsilon^{\mu\nu}\epsilon+\hat{R}_{ab,cd}\Gamma^{(2)}\Gamma^{cd}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.17)

Equation (5.17) implies

$\displaystyle\hat{R}_{ab,\mu\nu}=\mathcal{O}(\alpha^{{}^{\prime}2})$ (5.18)

which is equivalent to

$\displaystyle-\frac{1}{2}A^{-2}(\tilde{d}B)_{ab}+A^{-3}(dA\wedge
B)_{ab}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (5.19)

Using (5.14), we have that $\tilde{d}A\wedge
B=\mathcal{O}(\alpha^{{}^{\prime}2})$, hence (5.19) yields

$\displaystyle\tilde{d}B=\mathcal{O}(\alpha^{{}^{\prime}2})$ (5.20)

hence $\mathcal{G}=\mathcal{G}(A)+\mathcal{O}(\alpha^{{}^{\prime}2})$. Notice
that if $\psi$ is Killing, i.e. if it satisfies (5.2), (5.5), (5.8), (5.3),
(5.4), (5.9) and (5.10), then $\Gamma^{(2)}\psi$ is Killing, since
$[\Gamma^{(2)},\Gamma^{a}]=0$. Moreover $(\Gamma^{(2)})=1$, hence w.l.o.g. we
can set

$\displaystyle\Gamma^{(2)}\psi=\psi~{}.$ (5.21)

Substituting (5.21) and (5.14) in (5.4), we find

$\displaystyle-\frac{1}{2}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A-\frac{1}{2}A^{-2}\mathcal{G}\widetilde{\nabla}_{a}A\widetilde{\nabla}_{b}A+\frac{1}{4}A^{-1}\widetilde{\nabla}_{a}B_{b}+\frac{1}{4}\widetilde{\nabla}^{d}AX_{dab}$
$\displaystyle-\frac{1}{8}A^{-1}\mathcal{G}\widetilde{\nabla}^{d}AX_{dab}+\frac{1}{8}A^{-3}\mathcal{G}^{2}\widetilde{\nabla}_{a}A\widetilde{\nabla}_{b}A=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.22)

In the following, we define the function $g$ by $\mathcal{G}:=A^{2}g$, that is

$\displaystyle B=A^{2}gdA+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (5.23)

Notice that $g=g(A)+\mathcal{O}(\alpha^{{}^{\prime}2})$ by means of (5.20).
Using (5.20), the anti-symmetric part of (5.22) yields

$\displaystyle\bigg{(}1-\frac{1}{2}Ag\bigg{)}\widetilde{\nabla}^{d}AX_{dab}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.24)

Notice that (5.11) implies

$\displaystyle\frac{k}{4}-\frac{1}{4}\bigg{(}1-\frac{1}{4}A^{2}g^{2}\bigg{)}(\widetilde{\nabla}A)^{2}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.25)

If $1-\frac{1}{2}Ag=\mathcal{O}(\alpha^{{}^{\prime}})$, then
$(1+\frac{1}{2}Ag)(1-\frac{1}{2}Ag)=1-\frac{1}{4}A^{2}g^{2}=\mathcal{O}(\alpha^{{}^{\prime}})$,
which is in contradiction with respect to (5.25). Hence $1-\frac{1}{2}Ag$ has
non-vanishing zero-th term in $\alpha^{{}^{\prime}}$ and (5.24) yields

$\displaystyle\widetilde{\nabla}^{d}AX_{dab}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.26)

Moreover, inserting (5.23) in (5.22), the symmetric part of (5.22) yields

$\displaystyle-\frac{1}{2}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A+\frac{1}{4}A\widetilde{\nabla}_{a}(g\widetilde{\nabla}_{b}A)+\frac{1}{8}Ag^{2}\widetilde{\nabla}_{a}A\widetilde{\nabla}_{b}A=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.27)

We will come back to (5.27) later. Next, notice that equations (5.23) and
(5.21) imply

$\displaystyle\bigg{(}-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{1}{4}A^{-1}\not{B}\Gamma^{(2)}\bigg{)}\psi=-\frac{1}{2}\big{(}1-\frac{1}{2}Ag\big{)}\widetilde{\not{\nabla}}A\psi$
(5.28)

hence (5.9) and (5.10) can be rewritten as

$\displaystyle\bigg{(}\widetilde{\not{\nabla}}\Phi-\frac{1}{12}\not{X}+\frac{1}{2}A^{-2}\not{B}\bigg{)}\widetilde{\not{\nabla}}A\psi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.29)

and

$\displaystyle\not{\mathcal{F}}\widetilde{\not{\nabla}}A\psi=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.30)

Using (5.8) and (5.30), we find

$\displaystyle\widetilde{\nabla}^{b}A\mathcal{F}_{ab}=\mathcal{O}(\alpha^{{}^{\prime}})~{}.$
(5.31)

Furthermore, utilizing (5.5) and (5.29), we find

$\displaystyle\widetilde{\nabla}^{a}A\widetilde{\nabla}_{a}\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.32)

where we have used (5.26). Implementing (5.11), (5.20), (5.23), (5.26) and
(5.27) in (A.6), we find that the non-vanishing components of $\check{R}$ are
given by

$\displaystyle\check{R}_{\mu
a,b\rho}=\frac{1}{2}\big{(}\eta_{\mu\rho}+\epsilon_{\mu\rho}\big{)}\widetilde{\nabla}_{a}(g\widetilde{\nabla}_{b}A)$
(5.33)

and

$\displaystyle\check{R}_{ab,cd}=\tilde{\check{R}}_{ab,cd}~{}.$ (5.34)

Now consider the $\alpha^{{}^{\prime}}$ terms in the $(\mu,\nu)$ component of
the Einstein equations (2.1). First of all

$\displaystyle F_{\mu
Lij}F_{\nu}^{~{}Lij}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$ (5.35)

Moreover, using (5.33), compute333On $dS_{2}$
$\epsilon_{\mu\sigma}\epsilon_{\nu}^{~{}\sigma}=-\eta_{\mu\nu}$.

$\displaystyle\check{R}_{\mu
L_{1},L_{2}L_{3}}\check{R}_{\nu}^{~{}L_{1},L_{2}L_{3}}$ $\displaystyle=$
$\displaystyle 2\check{R}_{\mu a,\sigma b}\check{R}_{\nu}^{~{}a,\sigma b}$
(5.36) $\displaystyle=$
$\displaystyle(\eta_{\mu\sigma}+\epsilon_{\mu\sigma})(\delta_{\nu}^{~{}\sigma}+\epsilon_{\nu}^{~{}\sigma})C_{ab}C^{ab}+\mathcal{O}(\alpha^{{}^{\prime}2})$
$\displaystyle=$ $\displaystyle\mathcal{O}(\alpha^{{}^{\prime}2})$

where we have defined
$C_{ab}:=\frac{1}{2}\widetilde{\nabla}_{a}(g\widetilde{\nabla}_{b}A)$. Using
(5.35) and (5.36), the $(\mu,\nu)$ component of the Einstein equations (2.1)
reads

$\displaystyle
kA^{-2}-A^{-1}\widetilde{\nabla}^{a}\widetilde{\nabla}_{a}A-A^{-2}(\widetilde{\nabla}A)^{2}+\frac{1}{2}g^{2}(\widetilde{\nabla}A)^{2}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.37)

where we have used (5.32). Moreover, the gauge field equations (2.2) yield

$\displaystyle\widetilde{\nabla}^{a}B_{a}-2A^{-1}\widetilde{\nabla}^{a}AB_{a}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.38)

by implementing (5.32). Inserting (5.23) in (5.38), we have

$\displaystyle
A^{2}\frac{dg}{dA}(\widetilde{\nabla}A)^{2}+A^{2}g\widetilde{\nabla}^{a}\widetilde{\nabla}_{a}A=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.39)

Combining (5.37) and (5.39) with (5.25), we find that $g(A)$ satisfies the ODE

$\displaystyle\frac{dg}{dA}+\frac{1}{4}g^{3}A=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.40)

whose solution is given by

$\displaystyle
g=\frac{\eta}{\sqrt{q+\frac{A^{2}}{4}}}+\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.41)

where $\eta^{2}=1$ and $q$ is a constant up to two loops. Notice that $q$ is
not vanishing, as a consequence of (5.25). Inserting (5.40) into (5.27), we
find

$\displaystyle\widetilde{\nabla}_{a}W_{b}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.42)

where we have defined $W:=gdA$. Notice that (5.42) and (5.33) imply that the
only non-vanishing component of $\check{R}$ up to two loops is (5.34).
Moreover, using (5.41) and (5.25), we find

$\displaystyle W^{2}=\frac{k}{q}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.43)

Equation (5.43) implies that $q$ is positive. Inserting (5.42) in (5.27), we
find

$\displaystyle-2A^{-1}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A+\frac{1}{2}g^{2}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.44)

Using (5.44), the $(a,b)$-component of the Einstein equations (2.1) read

$\displaystyle\widetilde{R}_{ab}-\frac{1}{4}X_{amn}X_{b}^{~{}mn}+2\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\Phi+\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\tilde{\check{R}}_{ac_{1},c_{2}c_{3}}\tilde{\check{R}}_{b}^{~{}~{}c_{1},c_{2}c_{3}}-\mathcal{F}_{acij}\mathcal{F}_{b}^{~{}cij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.45)

Moreover, the $(a,b)$-component of the gauge field equations (2.2) yields

$\displaystyle(\widetilde{\nabla}^{c}X)_{cab}-2\widetilde{\nabla}^{c}\Phi
X_{cab}=\mathcal{O}(\alpha^{{}^{\prime}2})$ (5.46)

The non-trivial component of (2.3) yields

$\displaystyle\widetilde{\nabla}^{b}\mathcal{F}_{ba}-2\widetilde{\nabla}^{b}\Phi\mathcal{F}_{ba}+\frac{1}{2}X_{ab_{1}b_{2}}\mathcal{F}^{b_{1}b_{2}}=\mathcal{O}(\alpha^{{}^{\prime}})$
(5.47)

and the dilaton equation (2.4) is

$\displaystyle\widetilde{\nabla}^{a}\widetilde{\nabla}_{a}\Phi-2(\widetilde{\nabla}\Phi)^{2}+\frac{1}{12}X^{2}-\frac{\alpha^{{}^{\prime}}}{16}\bigg{(}\tilde{\check{R}}_{a_{1}a_{2},a_{3}a_{4}}\tilde{\check{R}}^{a_{1}a_{2},a_{3}a_{4}}-\mathcal{F}_{ab,ij}\mathcal{F}^{ab,ij}\bigg{)}=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.48)

Moreover, the Bianchi identities (2.5) read

$\displaystyle\tilde{d}X=-\frac{\alpha^{{}^{\prime}}}{4}\bigg{(}\textrm{tr}(\tilde{\check{R}}\wedge\tilde{\check{R}})-\textrm{tr}(\mathcal{F}\wedge\mathcal{F})\bigg{)}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.49)

Equation (5.42) implies that

$\displaystyle ds^{2}(M_{8})$ $\displaystyle=$
$\displaystyle\frac{1}{W^{2}}W\otimes
W+ds^{2}(M_{7})+\mathcal{O}(\alpha^{{}^{\prime}2})$ (5.50) $\displaystyle=$
$\displaystyle\frac{q}{k}\frac{dA^{2}}{q+\frac{A^{2}}{4}}+ds^{2}(M_{7})+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$

Substituting (5.50) in (2.9), we get

$\displaystyle
ds^{2}(M_{10})=ds^{2}(M_{3})+ds^{2}(M_{7})+\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.51)

where

$\displaystyle
ds^{2}(M_{3})=A^{2}\frac{\eta_{\mu\nu}dx^{\mu}dx^{\nu}}{\big{(}1+\frac{k}{4}|x|^{2}\big{)}^{2}}+\frac{q}{k}\frac{dA^{2}}{q+\frac{A^{2}}{4}}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.52)

Define $y=\sqrt{k}x$, $A=\sqrt{q}\tilde{A}$ and $c^{2}=\frac{q}{k}$. Then
(5.52) can be rewritten solely in terms of $c^{2}$ as follows

$\displaystyle
ds^{2}(M_{3})=c^{2}\tilde{A}^{2}\frac{\eta_{\mu\nu}dy^{\mu}dy^{\nu}}{\big{(}1+\frac{1}{4}|y|^{2}\big{)}^{2}}+c^{2}\frac{d\tilde{A}^{2}}{1+\frac{\tilde{A}^{2}}{4}}+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.53)

The Riemann tensor of (5.53) is

$\displaystyle
R_{\alpha\beta\gamma\delta}(M_{3})=-\frac{1}{4c^{2}}\big{(}g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma}\big{)}+\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.54)

where $\alpha\in\\{y^{\mu},\tilde{A}\\}$. Hence $M_{3}\simeq
AdS_{3}+\mathcal{O}(\alpha^{{}^{\prime}2})$ and (5.51) yields

$\displaystyle
ds^{2}(M_{10})=ds^{2}(AdS_{3})+ds^{2}(M_{7})+\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.55)

Moreover, equations (5.26), (5.31) and (5.32) read

$\displaystyle
i_{W}F=\mathcal{O}(\alpha^{{}^{\prime}})~{},~{}~{}~{}i_{W}d\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})~{},~{}~{}~{}i_{W}X=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.56)

where we have denoted by $W$ the vector field dual to the 1-form $W=gdA$.
Following an argument similar to that one in Section 3, we find

$\displaystyle\mathcal{L}_{W}\Phi=\mathcal{O}(\alpha^{{}^{\prime}2})~{},~{}~{}~{}\mathcal{L}_{W}F=\mathcal{O}(\alpha^{{}^{\prime}})~{},~{}~{}~{}\mathcal{L}_{W}X=\mathcal{O}(\alpha^{{}^{\prime}2})$
(5.57)

Notice that (2.20) and (5.52) imply that

$\displaystyle
H=\sqrt{\frac{k}{q}}\textrm{dvol}(AdS_{3})+X+\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(5.58)

Hence, we have found that warped $dS_{2}$ backgrounds in heterotic
perturbation theory up to two loops are direct product manifolds of $AdS_{3}$
and an internal Riemannian manifold $M_{7}$. Such backgrounds have been
classified in [22].

##  6 $D=11$ $dS_{3}$ Killing superalgebra

In the previous sections, we have shown that for heterotic warped product de
Sitter geometries, the warped product $dS_{2}$ solutions are $AdS_{3}\times
M_{7}$ and all warped product $dS_{n}$ solutions for $n\geq 3$ are
$\mathbb{R}^{1,n}\times M_{9-n}$. Hence, de Sitter geometries are very
restricted locally in heterotic supergravity. As is shown in [26], the
conditions imposed by supersymmetry on $N=8$, $D=11$ warped $dS_{4}$
backgrounds are much weaker, and there is no foliation of $dS_{4}$ into
$AdS_{5}$ or $\mathbb{R}^{1,4}$ appearing. In this section, we analyze warped
$dS_{3}$ backgrounds in $D=11$ supergravity from the point of view of their
Killing superalgebra $\mathfrak{g}$. In particular, we find that there is a
rich algebraic structure arising from the super-Jacobi identities of
$\mathfrak{g}$. This suggests that such backgrounds are not generically
foliations of $dS_{3}$ into $AdS_{4}$ or $\mathbb{R}^{1,3}$. In this section
we will refer to some formulae of section 2; from now on, it is understood
that these formulae hold with $\alpha^{{}^{\prime}}=0$.

Let us consider $dS_{3}\times_{w}M_{8}$ backgrounds in $D=11$ supergravity,
where the metric is as in (2.9). Moreover, we assume the 4-form $F$ to be
invariant under the isometry group $O(3,1)$ of $dS_{3}$, hence

$\displaystyle F=B\wedge\textrm{dvol}(dS_{3})+X$ (6.1)

where $B$ is a 1-form on $M_{8}$ and $X$ is a 4-form on $M_{8}$. Notice that
$B$ and $X$ are closed forms on $M_{8}$, as a consequence of the Bianchi
identities $dF=0$. We introduce on $M_{11}$ the co-frame (2.12), where
$a=3,4,\dots,\sharp$ and $y^{\alpha}$ are the co-ordinates on $M_{8}$. The
Killing spinor equations of $D=11$ supergravity

$\displaystyle\bigg{(}\nabla_{A}-\frac{1}{288}\Gamma_{A}^{~{}~{}B_{1}\dots
B_{4}}F_{B_{1}\dots
B_{4}}+\frac{1}{36}F_{AB_{1}B_{2}B_{3}}\Gamma^{B_{1}B_{2}B_{3}}\bigg{)}\epsilon=0$
(6.2)

can be integrated along $dS_{3}$, yielding (2.30), where

$\displaystyle\mathcal{C}_{(3)}=-\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{A}{288}\not{X}-\frac{1}{6}A^{-2}\not{B}\Gamma^{(3)}~{}.$
(6.3)

In (6.3), we have denoted
$\Gamma^{(3)}:=\frac{1}{3!}\epsilon_{\mu_{1}\mu_{2}\mu_{3}}\Gamma^{\mu_{1}\mu_{2}\mu_{3}}$
the highest rank Gamma matrix on $dS_{3}$. Moreover, $\psi$ is a 32-component
Majorana spinor on $M_{8}$ which satisfies (2.33), where

$\displaystyle\sigma_{a(3)}=\frac{1}{288}\Gamma_{a}^{~{}~{}b_{1}\dots
b_{4}}X_{b_{1}\dots
b_{4}}-\frac{1}{36}X_{ab_{1}b_{2}b_{3}}\Gamma^{b_{1}b_{2}b_{3}}+\frac{1}{12}A^{-3}\Gamma_{a}^{~{}~{}b}B_{b}\Gamma^{(3)}-\frac{1}{6}A^{-3}B_{a}\Gamma^{(3)}$

supplemented by the quadratic conditions (2.28) and (2.34), where

$\displaystyle\widehat{\mathcal{C}}_{(3)}=\frac{1}{2}\widetilde{\not{\nabla}}A+\frac{A}{288}\not{X}+\frac{1}{6}A^{-2}\not{B}\Gamma^{(3)}~{},$
$\displaystyle\widehat{\sigma}_{a(3)}=-\frac{1}{288}\Gamma_{a}^{~{}~{}b_{1}\dots
b_{4}}X_{b_{1}\dots
b_{4}}+\frac{1}{36}X_{ab_{1}b_{2}b_{3}}\Gamma^{b_{1}b_{2}b_{3}}+\frac{1}{12}A^{-3}\Gamma_{a}^{~{}~{}b}B_{b}\Gamma^{(3)}-\frac{1}{6}A^{-3}B_{a}\Gamma^{(3)}~{}.$

Equations (2.33) and (2.34) imply that if $\psi$ satisfies (2.33), so do
$\Gamma_{\mu}\mathcal{C}_{(3)}\psi$ and $\Gamma_{\mu\nu}\psi$. In the
following it is useful to introduce the light-cone vielbein
$\sqrt{2}\textbf{e}^{\pm}:=\textbf{e}^{1}\pm\textbf{e}^{0}$. In terms of this
frame, the 11-dimensional metric reads

$\displaystyle
ds^{2}(M_{11})=2\textbf{e}^{+}\textbf{e}^{-}+(\textbf{e}^{2})^{2}+ds^{2}(M_{8})~{}.$
(6.6)

Without loss of generality, we take $\psi=\psi_{+}$, where
$\Gamma_{+}\psi_{+}=0$. Let us assume there are $\mathfrak{n}$ linearly
independent Killing spinors $\psi_{+}^{r}$, $r=1,2,\dots,\mathfrak{n}$, that
is they satisfy (2.33). In other words, we take the vector space $V_{+}$
spanned by the Killing spinors $\psi_{+}^{r}$ to be
$\mathfrak{n}$-dimensional. Then, since
$\Gamma_{2}\mathcal{C}_{(3)}\psi_{+}^{r}$ are Killing spinors, it follows that

$\displaystyle\Gamma_{2}\mathcal{C}_{(3)}\psi_{+}^{r}=a^{r}_{~{}s}\psi_{+}^{s}$
(6.7)

where $a^{r}_{~{}s}$ is a constant $\mathfrak{n}\times\mathfrak{n}$ matrix.
Using (6.7) and the integrability conditions (2.28), it is straightforward to
show that

$\displaystyle a^{2}=-\frac{k}{4}\mathbb{I}_{\mathfrak{n}}$ (6.8)

hence, as $k>0$, it follows that
$\textrm{dim}~{}V_{+}=\mathfrak{n}=2\mathfrak{m}$. Furthermore, equation (6.8)
implies that there exists a basis
$\\{\psi_{+}^{2\hat{r}-1},\psi_{+}^{2\hat{r}}~{},~{}\hat{r}=1,2,\dots,\mathfrak{m}\\}$
of $V_{+}$ such that

$\displaystyle\Gamma_{2}\mathcal{C}_{(3)}\psi_{+}^{2\hat{r}-1}=\frac{\sqrt{k}}{2}\psi_{+}^{2\hat{r}}~{},~{}~{}~{}~{}\Gamma_{2}\mathcal{C}_{(3)}\psi_{+}^{2\hat{r}}=-\frac{\sqrt{k}}{2}\psi_{+}^{2\hat{r}-1}~{}.$
(6.9)

The previous discussion implies that the linearly independent Killing spinors
on $M_{8}$ are $\\{\psi_{\pm}^{2\hat{r}}~{},~{}\psi_{\pm}^{2\hat{r}-1}\\}$,
where $\psi_{-}^{2\hat{r}}$ and $\psi_{-}^{2\hat{r}-1}$ are given by

$\displaystyle\psi_{-}^{2\hat{r}-1}:=\Gamma_{-2}\psi_{+}^{2\hat{r}-1}~{},~{}~{}~{}~{}~{}\psi_{-}^{2\hat{r}}:=\Gamma_{-2}\psi_{+}^{2\hat{r}}~{}.$
(6.10)

Hence the number of supersymmetries $N$ preserved by warped $dS_{3}$
backgrounds in $D=11$ supergravity is $N=4\mathfrak{m}$ with
$\mathfrak{m}=1,2,\dots,8$. In order to simplify the notation we denote
$r:=\hat{r}$, hence in what follows $r=1,2,\dots,\frac{\mathfrak{n}}{2}$.
Using (2.30), define

$\displaystyle\epsilon_{1}^{2r}:=\mathcal{U}^{-1/2}\big{(}1+x^{\mu}\Gamma_{\mu}\mathcal{C}_{(3)}\big{)}\psi_{+}^{2r}~{},~{}~{}~{}~{}\epsilon_{1}^{2r-1}:=\mathcal{U}^{-1/2}\big{(}1+x^{\mu}\Gamma_{\mu}\mathcal{C}_{(3)}\big{)}\psi_{+}^{2r-1}$
$\displaystyle\epsilon_{2}^{2r}:=\mathcal{U}^{-1/2}\big{(}1+x^{\mu}\Gamma_{\mu}\mathcal{C}_{(3)}\big{)}\psi_{-}^{2r}~{},~{}~{}~{}~{}\epsilon_{2}^{2r-1}:=\mathcal{U}^{-1/2}\big{(}1+x^{\mu}\Gamma_{\mu}\mathcal{C}_{(3)}\big{)}\psi_{-}^{2r-1}~{}.$
(6.11)

Inserting (6.9) in (6.11), one obtains

$\displaystyle\epsilon_{1}^{2r}=\mathcal{U}^{-1/2}\big{(}\psi_{+}^{2r}-\frac{\sqrt{k}}{2}x^{-}\psi_{-}^{2r-1}-\frac{\sqrt{k}}{2}x^{2}\psi_{+}^{2r-1}\big{)}$
$\displaystyle\epsilon_{1}^{2r-1}=\mathcal{U}^{-1/2}\big{(}\psi_{+}^{2r-1}+\frac{\sqrt{k}}{2}x^{-}\psi_{-}^{2r}+\frac{\sqrt{k}}{2}x^{2}\psi_{+}^{2r}\big{)}$
$\displaystyle\epsilon_{2}^{2r}=\mathcal{U}^{-1/2}\big{(}\psi_{-}^{2r}-\sqrt{k}x^{+}\psi_{+}^{2r-1}+\frac{\sqrt{k}}{2}x^{2}\psi_{-}^{2r-1}\big{)}$
$\displaystyle\epsilon_{2}^{2r-1}=\mathcal{U}^{-1/2}\big{(}\psi_{-}^{2r-1}+\sqrt{k}x^{+}\psi_{+}^{2r}-\frac{\sqrt{k}}{2}x^{2}\psi_{-}^{2r}\big{)}~{}.$
(6.12)

The first step in constructing the $dS_{3}$ Killing superalgebra consists of
computing the 11-dimensional bilinears

$\displaystyle
K(\epsilon_{1},\epsilon_{2}):=\langle(\Gamma_{+}-\Gamma_{-})\epsilon_{1},\Gamma_{A}\epsilon_{2}\rangle\textbf{e}^{A}$
(6.13)

where $\epsilon_{1}$ and $\epsilon_{2}$ are Killing spinors. It is a general
property of $D=11$ Killing superalgebras that the vector fields dual to (6.13)
- which will be denoted by the same name $K(\epsilon_{1},\epsilon_{2})$ \- are
Killing vectors [29], satisfying

$\displaystyle\mathcal{L}_{K(\epsilon_{1},\epsilon_{2})}g=0~{},~{}~{}~{}~{}\mathcal{L}_{K(\epsilon_{1},\epsilon_{2})}F=0~{}.$
(6.14)

In the following, we denote
$K_{1}^{rs}:=K(\epsilon_{1}^{r},\epsilon_{1}^{s})$,
$K_{2}^{rs}:=K(\epsilon_{2}^{r},\epsilon_{2}^{s})$ and
$K_{3}^{rs}:=K(\epsilon_{1}^{r},\epsilon_{2}^{s})$. Notice that
$K_{1}^{[rs]}=0$ and $K_{2}^{[rs]}=0$ as a consequence of (6.13). In the
context of $AdS_{3}$ Killing superalgebras in $D=11$ supergravity, it can be
shown via global arguments that the even sub-algebra $\mathfrak{g}_{0}$ of the
Killing superalgebra $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ is
a direct sum of the isometry algebra of $AdS_{3}$ and the isometry algebra of
the internal space [30]. In the following, we argue that this is not the case
for $dS_{3}$, that is a vector field $K$ satisfying (6.14) cannot be written
as $K=AK^{\textrm{dS}}+K^{\textrm{int}}$, where $K^{\textrm{dS}}$ is a
$dS_{3}$ Killing vector and $K^{\textrm{int}}$ is a Killing vector along
$M_{8}$. Substituting (6.12) in (6.13), a long computation shows that the
bilinears are given by

$\displaystyle
2K_{1}^{2r~{}2s}=V_{1}^{rs}+W_{1}^{rs}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2K_{2}^{2r~{}2s}=V_{3}^{rs}+W_{2}^{rs}$
$\displaystyle
2K_{1}^{2r-1~{}2s-1}=-V_{1}^{rs}+W_{1}^{rs}~{},~{}~{}~{}~{}~{}~{}2K_{2}^{2r-1~{}2s-1}=-V_{3}^{rs}+W_{2}^{rs}$
$\displaystyle
2K_{1}^{2r~{}2s-1}=V_{2}^{rs}-W_{5}^{rs}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2K_{2}^{2r~{}2s-1}=V_{4}^{rs}-W_{6}^{rs}$
(6.15) $\displaystyle
2K_{3}^{(2r~{}2s)}=V_{5}^{rs}+W_{3}^{rs}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2K_{3}^{[2r~{}2s]}=U_{1}^{rs}+W_{8}^{rs}$
$\displaystyle
2K_{3}^{(2r-1~{}2s-1)}=-V_{5}^{rs}+W_{3}^{rs}~{},~{}~{}~{}~{}~{}~{}2K_{3}^{[2r-1~{}2s-1]}=-U_{1}^{rs}+W_{8}^{rs}$
$\displaystyle
2K_{3}^{(2r~{}2s-1)}=V_{6}^{rs}-W_{7}^{rs}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2K_{3}^{[2r~{}2s-1]}=U_{2}^{rs}+W_{4}^{rs}~{}.$
(6.16)

In (6.15) and (6.16), the vector fields $\\{V_{i}^{rs}\\}_{i=1}^{6}$ are
defined by

$\displaystyle
V_{1}^{rs}:=\sqrt{2k}~{}d^{rs}(L^{02}-L^{12})+\sqrt{2}~{}c^{rs}(W^{0}-W^{1})$
$\displaystyle
V_{2}^{rs}:=\sqrt{2}~{}d^{rs}(W^{0}-W^{1})-\sqrt{2k}~{}c^{rs}(L^{02}-L^{12})$
$\displaystyle
V_{3}^{rs}:=2\sqrt{2}~{}c^{rs}(W^{0}+W^{1})-2\sqrt{2k}~{}d^{rs}(L^{02}+L^{12})$
$\displaystyle
V_{4}^{rs}:=2\sqrt{2}~{}d^{rs}(W^{0}+W^{1})+2\sqrt{2k}~{}c^{rs}(L^{02}+L^{12})$
$\displaystyle V_{5}^{rs}:=2\sqrt{k}~{}d^{rs}L^{01}+2c^{rs}W^{2}$
$\displaystyle V_{6}^{rs}:=-2\sqrt{k}~{}c^{rs}L^{01}+2d^{rs}W^{2}$ (6.17)

where $c^{rs}$ and $d^{rs}$ are constant444The constancy of (6.18), that is
$\widetilde{\nabla}_{a}c^{rs}=\widetilde{\nabla}_{a}d^{rs}=0$, follows
straightforwardly from the KSE (2.33). symmetric
$\frac{\mathfrak{n}}{2}\times\frac{\mathfrak{n}}{2}$ matrices given by

$\displaystyle
c^{rs}:=A^{-1}\big{(}\langle\psi_{+}^{2r},\psi_{+}^{2s}\rangle-\langle\psi_{+}^{2r-1},\psi_{+}^{2s-1}\rangle\big{)}~{},~{}~{}d^{rs}:=A^{-1}\big{(}\langle\psi_{+}^{2s},\psi_{+}^{2r-1}\rangle+\langle\psi_{+}^{2r},\psi_{+}^{2s-1}\rangle\big{)}$
(6.18)

and the vector fields $L^{\alpha\beta}$ and $W^{\alpha}$, with
$\alpha,\beta=0,1,2$ are $dS_{3}$ Killing vector fields, defined by

$\displaystyle(L^{\alpha\beta})^{\mu}:=x^{\alpha}\eta^{\mu\beta}-x^{\beta}\eta^{\mu\alpha}~{},~{}~{}~{}(W^{\alpha})^{\mu}:=\big{(}1-\frac{k}{4}|x|^{2}\big{)}\eta^{\mu\alpha}+\frac{k}{2}x^{\alpha}x^{\mu}~{}.$
(6.19)

The vector fields (6.17) are $dS_{3}$ Killing vectors, i.e.

$\displaystyle\mathcal{L}_{V_{i}^{rs}}g_{(3)}=0~{}~{}~{}~{}~{}~{}i=1,2,\dots,6$
(6.20)

where we have decomposed the 11-dimensional metric as
$g=A^{2}g_{(3)}+g_{(8)}$, where $g_{(3)}$ is the $dS_{3}$ metric and $g_{(8)}$
is the internal metric. Moreover, in (6.16), the vector fields
$\\{U_{i}^{rs}\\}_{i=1}^{2}$ are defined by

$\displaystyle
U_{1}^{rs}:=2\bigg{(}\langle\psi_{+}^{2r},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s}\rangle-\langle\psi_{+}^{2r-1},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s-1}\rangle\bigg{)}\textbf{e}_{a}$
$\displaystyle
U_{2}^{rs}:=2\bigg{(}\langle\psi_{+}^{2r},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s-1}\rangle-\langle\psi_{+}^{2s},\Gamma^{a}\Gamma_{2}\psi_{+}^{2r-1}\rangle\bigg{)}\textbf{e}_{a}~{}.$
(6.21)

The vector fields (6.21) are internal Killing vectors, i.e.

$\displaystyle\widetilde{\mathcal{L}}_{U_{i}^{rs}}g_{(8)}=0~{}~{}~{}~{}~{}i=1,2$
(6.22)

which preserve the warp factor and the bosonic fluxes

$\displaystyle\widetilde{\mathcal{L}}_{U_{i}^{rs}}A=0~{},~{}~{}~{}\widetilde{\mathcal{L}}_{U_{i}^{rs}}B=0~{},~{}~{}~{}\widetilde{\mathcal{L}}_{U_{i}^{rs}}X=0~{}~{}~{}~{}~{}~{}~{}~{}i=1,2~{}.$
(6.23)

The obstruction for $\mathfrak{g}_{0}$ to be a direct sum of the isometry
algebra of $dS_{3}$ and the internal algebra is encoded in the vector fields
$W_{i}^{rs}$ in (6.15) and (6.16), which are defined by

$\displaystyle
W_{i}^{rs}:=f^{rs}\xi_{i}+h_{i}~{}Z^{rs}~{},~{}~{}~{}W_{i+4}^{rs}:=g^{rs}\xi_{i}+h_{i}~{}Y^{rs}~{}~{}~{}~{}~{}i=1,2,3,4~{}.$
(6.24)

In (6.24), $f^{rs}$ and $g^{rs}$ are functions on $M_{8}$ given by

$\displaystyle
f^{rs}:=A^{-1}\big{(}\langle\psi_{+}^{2r},\psi_{+}^{2s}\rangle+\langle\psi_{+}^{2r-1},\psi_{+}^{2s-1}\rangle\big{)}~{},~{}~{}~{}g^{rs}:=A^{-1}\big{(}\langle\psi_{+}^{2r-1},\psi_{+}^{2s}\rangle-\langle\psi_{+}^{2s-1},\psi_{+}^{2r}\rangle\big{)}~{},$
(6.25)

$Z^{rs}$ and $Y^{rs}$ are vector fields on $M_{8}$ defined as

$\displaystyle
Z^{rs}:=\bigg{(}\langle\psi_{+}^{2r},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s-1}\rangle+\langle\psi_{+}^{2s},\Gamma^{a}\Gamma_{2}\psi_{+}^{2r-1}\rangle\bigg{)}\textbf{e}_{a}$
$\displaystyle
Y^{rs}:=\bigg{(}\langle\psi_{+}^{2r},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s}\rangle+\langle\psi_{+}^{2r-1},\Gamma^{a}\Gamma_{2}\psi_{+}^{2s-1}\rangle\bigg{)}\textbf{e}_{a}~{}.$
(6.26)

Moreover

$\displaystyle\xi_{1}:=-2\sqrt{2}(P^{1}-P^{0})-\sqrt{2}(W^{0}-W^{1})~{},~{}~{}~{}\xi_{2}:=4\sqrt{2}(P^{1}+P^{0})-2\sqrt{2}(W^{0}+W^{1})$
$\displaystyle\xi_{3}:=4P^{2}-2W^{2}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\xi_{4}:=-2\sqrt{k}D$
(6.27)

where in (6.27) $P^{0}$, $P^{1}$, $P^{2}$ and $D$ are $dS_{3}$ conformal
Killing vectors given by

$\displaystyle(P^{\alpha})^{\mu}:=\eta^{\alpha\mu}~{},~{}~{}~{}D^{\mu}:=x^{\mu}$
(6.28)

and555We define the light-cone co-ordinates
$x^{\pm}:=\frac{1}{\sqrt{2}}(x^{1}\pm x^{0})$. Notice that
$|x|^{2}=2x^{+}x^{-}+(x^{2})^{2}$.

$\displaystyle
h_{1}:=-2\mathcal{U}^{-1}\sqrt{k}x^{-}~{},~{}~{}h_{2}:=4\mathcal{U}^{-1}\sqrt{k}x^{+}~{},~{}~{}h_{3}:=2\mathcal{U}^{-1}\sqrt{k}x^{2}~{},~{}~{}h_{4}:=2\mathcal{U}^{-1}\big{(}1-\frac{k}{4}|x|^{2}\big{)}~{}.$
(6.29)

Decomposing (6.14) along $g=A^{2}g_{(3)}+g_{(8)}$ and using the integrability
conditions (2.28), we find the following conditions ($i=1,2,3,4$)

$\displaystyle\widetilde{\mathcal{L}}_{Z^{rs}}g_{(8)}=0~{},~{}~{}~{}\widetilde{\mathcal{L}}_{Y^{rs}}g_{(8)}=0$
(6.30)
$\displaystyle\mathcal{L}_{\xi_{i}}g_{(3)}=-2\sqrt{k}~{}h_{i}g_{(3)}~{},~{}~{}~{}dh_{i}=\sqrt{k}\xi_{i}~{},$
(6.31) $\displaystyle
A^{-1}\widetilde{\mathcal{L}}_{Z^{rs}}A=\sqrt{k}f^{rs}~{},~{}~{}~{}A^{-1}\widetilde{\mathcal{L}}_{Y^{rs}}A=\sqrt{k}g^{rs}$
(6.32) $\displaystyle
df^{rs}=-\sqrt{k}A^{-2}Z^{rs}~{},~{}~{}~{}dg^{rs}=-\sqrt{k}A^{-2}Y^{rs}$
(6.33)
$\displaystyle\widetilde{\mathcal{L}}_{Z^{rs}}B=3\sqrt{k}f^{rs}B~{},~{}~{}~{}\widetilde{\mathcal{L}}_{Y^{rs}}B=3\sqrt{k}g^{rs}B$
(6.34)
$\displaystyle\widetilde{\mathcal{L}}_{Z^{rs}}X=0~{},~{}~{}~{}\widetilde{\mathcal{L}}_{Y^{rs}}X=0$
(6.35)

where in (6.31), we have denoted by $\xi_{i}$ the 1-form dual to the vector
field $\xi_{i}$ (6.27). In particular, equations (6.30), (6.32), (6.34) and
(6.35) imply that $Z^{rs}$ and $Y^{rs}$ are internal Killing vector fields,
which preserve $X$, but do not preserve either the warp factor $A$ or $B$. The
appearance of the vector fields (6.24) in (6.15) and (6.16) implies that the
even sub-algebra $\mathfrak{g}_{0}$ of the Killing superalgebra
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ is not a direct sum of
the isometry algebra of $dS_{3}$ and the isometry algebra of the internal
space. Indeed, the vector fields (6.24) have components along the internal
space $M_{8}$, which depend on the co-ordinates of $dS_{3}$.

The next step in the construction of the $dS_{3}$ Killing superalgebra
consists of computing the spinorial Lie derivatives

$\displaystyle\mathcal{L}_{K}\epsilon=\nabla_{K}\epsilon+\frac{1}{8}\not{dK}\epsilon$
(6.36)

where in (6.36) $K:=K(\epsilon_{1},\epsilon_{2})$ is the vector field dual to
(6.13), $\nabla$ is the Levi-Civita connection on $M_{11}$ and $\epsilon$ is a
Killing spinor. It is a general result of $D=11$ Killing superalgebras that
the spinor (6.36) is Killing [29]. In Appendix B we have listed all the
spinorial Lie derivatives (6.36) and some additional algebraic conditions
arising from (6.36), see equations (B.1)-(B.14). Moreover, it is well known
that the Killing superalgebra $\mathfrak{g}$ of $D=11$ supergravity satisfies
the super-Jacobi identities [29]

$\displaystyle[[x,y]_{\mathfrak{g}},z]_{\mathfrak{g}}+(-)^{|z|(|x|+|y|)}[[z,x]_{\mathfrak{g}},y]_{\mathfrak{g}}+(-)^{|x|(|y|+|z|)}[[y,z]_{\mathfrak{g}},x]_{\mathfrak{g}}=0$
(6.37)

where $x,y,z\in\mathfrak{g}$, $|x|=0$ if $x\in\mathfrak{g}_{0}$, $|x|=1$ if
$x\in\mathfrak{g}_{1}$ and

$\displaystyle[x,y]_{\mathfrak{g}}:=x\cdot y-(-)^{|x||y|}y\cdot x~{}.$ (6.38)

Thus, it is important to check whether the super-Jacobi identities (6.37) of
the $dS_{3}$ Killing superalgebra are automatically satisfied or provide some
additional conditions. It turns out that there are indeed some non-trivial
conditions arising from (6.37). They are presented in Appendix B, see
equations (B.15)-(B.38). A quite rich and complicated algebraic structure
emerges, which strongly suggests that warped $dS_{3}$ backgrounds are not
generically foliations into $AdS_{4}$ or $\mathbb{R}^{1,3}$. As a final
comment, we remark that equations (B.29)-(B.34) are reminiscent of the
quadratic conditions obeyed by the structure constants of a (complex)
3-algebra [31]; however, by carefully comparing (B.29)-(B.34) with equation
(2) of [31], we find that the index structure does not match, hence there is
no simple interpretation of $D=11$ $dS_{3}$ Killing superalgebras in terms of
3-algebras.

##  7 Conclusion

In this work, we have classified all warped product $dS_{n}$ solutions of
heterotic perturbation theory, up to two loops. We have found that

  * •

For $n\geq 3$, all such solutions are $\mathbb{R}^{1,n-1}\times M_{9-n}$,
where $M_{9-n}$ is a $(9-n)$-dimensional Riemannian manifold. Moreover, the
1-form $d\Phi$, the 3-form $H$ and the non-abelian field strength $F$ have
non-vanishing components only along $M_{9-n}$. Backgrounds of these form have
been studied in [27] and [28].

  * •

For $n=2$, all such solutions are $AdS_{3}\times M_{7}$, where $M_{7}$ is a
7-dimensional Riemannian manifold. These backgrounds have been classified in
[22].

The analysis done in this paper can be straightforwardly adapted to
investigate the case $k=0$, which corresponds to warped product Minkowski
solutions $\mathbb{R}^{1,n-1}\times_{w}M_{10-n}$. Notice that such backgrounds
can be thought of as the limit $\alpha^{{}^{\prime}}\to 0$ of (2.9) and (2.10)
when $k^{0}=0$, that is
$k=k^{1}\alpha^{{}^{\prime}}+\mathcal{O}(\alpha^{{}^{\prime}2})$. We find that
warped product $\mathbb{R}^{1,n-1}$ Minkowski backgrounds are
$\mathbb{R}^{1,n-1}\times M_{10-n}$, for $n\geq 3$. Moreover, in the case
$n=2$, there are two different classes:

  * •

$AdS_{3}\times M_{7}$, where $M_{7}$ is a 7-dimensional Riemannian manifold.

  * •

near-horizon heterotic black hole solutions, with $dh=0$, where $h=-d\log
A^{2}$ [32], [33].

Perturbative $\alpha{{}^{\prime}}$ corrections to the cosmological constant
$k$ have also been studied in the context of warped heterotic
compactifications. In [34], assuming the internal manifold to be compact and
without boundary, it is shown that de-Sitter backgrounds are excluded. The
result is established taking into account leading $\alpha^{{}^{\prime}}$
effects at tree-level and does not utilize supersymmetry. Moreover, in [35] it
is found that heterotic string compactifications at tree-level yields
4-dimensional Minkowski as the only maximally supersymmetric solution, to all
orders in $\alpha^{{}^{\prime}}$. The work in [34, 35] is extended and
complemented in [24], where, using a world-sheet CFT argument, it is shown
that the heterotic string essentially rules out $dS_{n}$ vacua with $n\geq 4$,
to all orders in $\alpha^{{}^{\prime}}$.

In Section 6, we have shown that warped product $dS_{3}$ backgrounds in $D=11$
supergravity are less restricted than in heterotic supergravity. Indeed, by
analyzing the Killing superalgebra of these backgrounds, a rich algebraic
structure emerges, which suggests that such backgrounds are not generically
foliations into $AdS_{4}$ or $\mathbb{R}^{1,4}$. A similar situation occurs
for warped $dS_{4}$ backgrounds in $D=11$ supergravity: indeed, in [26], by
analyzing the constraints on the geometry imposed by minimal $N=8$
supersymmetry, it is shown that warped $dS_{4}$ backgrounds are not
generically foliations into $AdS_{5}$ or $\mathbb{R}^{1,5}$.

## Appendix A  Curvature: conventions and useful formulae

In this Appendix, we list our conventions for the curvature tensor of a
connection. The curvature of a generic connection $\Gamma$ is given by [21]

$\displaystyle
R_{AB,~{}~{}D}^{~{}~{}~{}~{}C}=\textbf{e}_{A}\Gamma^{C}_{~{}BD}-\textbf{e}_{B}\Gamma^{C}_{~{}AD}+\Gamma^{C}_{~{}AN}\Gamma^{N}_{~{}BD}-\Gamma^{C}_{~{}BN}\Gamma^{N}_{~{}AD}~{}.$
(A.1)

Let $\Gamma$ be the Levi-Civita connection of the 10-dimensional spacetime
(2.14). Then, the non-vanishing components of the spin connection in the frame
(2.12) are given by

$\displaystyle\Omega_{\mu,\nu\rho}$ $\displaystyle=$ $\displaystyle
kA^{-1}x_{[\nu}\eta_{\rho]\mu}$ $\displaystyle\Omega_{\mu,a\nu}$
$\displaystyle=$ $\displaystyle-\eta_{\mu\nu}A^{-1}{\tilde{\nabla}}_{a}A$
$\displaystyle\Omega_{a,bc}$ $\displaystyle=$
$\displaystyle\widetilde{\Omega}_{a,bc}$ (A.2)

where $\widetilde{\Omega}_{a,bc}$ is the spin connection on $M_{10-n}$.
Utilizing (A.1) and (A.2), the non-vanishing components of the 10-dimensional
Riemann tensor are

$\displaystyle R_{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\big{(}\eta_{\mu\rho}\eta_{\nu\sigma}-\eta_{\mu\sigma}\eta_{\nu\rho}\big{)}A^{-2}\big{(}k-(\widetilde{\nabla}A)^{2}\big{)}$
$\displaystyle R_{a\mu b\nu}$ $\displaystyle=$
$\displaystyle-\eta_{\mu\nu}A^{-1}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A$
$\displaystyle R_{abcd}$ $\displaystyle=$ $\displaystyle\widetilde{R}_{abcd}$
(A.3)

where $\widetilde{R}_{abcd}$ denotes the Riemann tensor on $M_{10-n}$.
Implementing (A.3), we compute the 10-dimensional Ricci tensor, whose non-
vanishing components are

$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta_{\mu\nu}\bigg{(}k(n-1)A^{-2}-A^{-1}\widetilde{\nabla}^{a}\widetilde{\nabla}_{a}A-(n-1)A^{-2}(\widetilde{\nabla}A)^{2}\bigg{)}$
$\displaystyle R_{ab}$ $\displaystyle=$
$\displaystyle\widetilde{R}_{ab}-nA^{-1}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}A$
(A.4)

where $\widetilde{R}_{ab}$ is the Ricci tensor on $M_{10-n}$.
Let us introduce the connections with torsion $\hat{\nabla}$ and
$\check{\nabla}$

$\displaystyle\check{\nabla}_{M}\xi^{N}=\nabla_{M}\xi^{N}-\frac{1}{2}H^{N}_{~{}~{}ML}\xi^{L}~{},~{}~{}~{}~{}\hat{\nabla}_{M}\xi^{N}=\nabla_{M}\xi^{N}+\frac{1}{2}H^{N}_{~{}~{}ML}\xi^{L}$
(A.5)

where $\nabla$ is the Levi-Civita connection and $\xi$ is a vector field.
Using (A.1) and (A.5), it follows that

$\displaystyle\check{R}_{AB,CD}=R_{ABCD}-\frac{1}{2}\nabla_{A}H_{CBD}+\frac{1}{2}\nabla_{B}H_{CAD}+\frac{1}{4}H_{CAN}H^{N}_{~{}~{}BD}-\frac{1}{4}H_{CBN}H^{N}_{~{}~{}AD}$
(A.6)

and

$\displaystyle\hat{R}_{AB,CD}=R_{ABCD}+\frac{1}{2}\nabla_{A}H_{CBD}-\frac{1}{2}\nabla_{B}H_{CAD}+\frac{1}{4}H_{CAN}H^{N}_{~{}~{}BD}-\frac{1}{4}H_{CBN}H^{N}_{~{}~{}AD}~{}.$
(A.7)

Notice that $\check{R}_{AB,CD}$ and $\hat{R}_{AB,CD}$ are anti-symmetric in
the first two indices and the last two indices. Moreover, equations (A.6) and
(A.7) imply the identity

$\displaystyle\check{R}_{AB,CD}-\hat{R}_{CD,AB}=\frac{1}{2}(dH)_{ABCD}~{}.$
(A.8)

Furthermore, notice that the integrability conditions of the gravitino KSE
(2.6)

$\displaystyle[\hat{\nabla}_{M},\hat{\nabla}_{N}]\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})$
(A.9)

can be expressed in terms of $\hat{R}$ as follows

$\displaystyle\hat{R}_{MN,RS}\Gamma^{RS}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(A.10)

Using (A.8), we can rewrite (A.10) as

$\displaystyle\bigg{(}\check{R}_{RS,MN}-\frac{1}{2}(dH)_{MNRS}\bigg{)}\Gamma^{RS}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}2})~{}.$
(A.11)

Since $dH=\mathcal{O}(\alpha^{{}^{\prime}})$ by means of the Bianchi
identities (2.5), it follows that (A.11) is equivalent to

$\displaystyle\check{R}_{RS,MN}\Gamma^{RS}\epsilon=\mathcal{O}(\alpha^{{}^{\prime}})$
(A.12)

which is the same condition as that satisfied by the curvature of the gauge
sector $F$ in the gaugino KSE (2.8).

## Appendix B  $dS_{3}$ superalgebras

Let us start this appendix by listing the spinorial Lie derivatives (6.36).
Using (6.12), (6.17), (6.21) and (6.24), a lengthy computation yields

$\displaystyle\mathcal{L}_{V_{1}^{mn}}\epsilon_{1}^{2r}=0~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{V_{1}^{mn}}\epsilon_{1}^{2r-1}=0$
$\displaystyle\mathcal{L}_{V_{1}^{mn}}\epsilon_{2}^{2r}=2\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r}+2\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r-1}~{},~{}~{}~{}\mathcal{L}_{V_{1}^{mn}}\epsilon_{2}^{2r-1}=2\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r-1}-2\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r}$
(B.1)

$\displaystyle\mathcal{L}_{V_{2}^{mn}}\epsilon_{1}^{2r}=0~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{V_{2}^{mn}}\epsilon_{1}^{2r-1}=0$
$\displaystyle\mathcal{L}_{V_{2}^{mn}}\epsilon_{2}^{2r}=2\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r-1}-2\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r}~{},~{}~{}~{}\mathcal{L}_{V_{2}^{mn}}\epsilon_{2}^{2r-1}=-2\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r}-2\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r-1}$

$\displaystyle\mathcal{L}_{V_{3}^{mn}}\epsilon_{1}^{2r}=-2\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r-1}-2\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r}~{},~{}~{}~{}\mathcal{L}_{V_{3}^{mn}}\epsilon_{1}^{2r-1}=2\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r}-2\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r-1}$
$\displaystyle\mathcal{L}_{V_{3}^{mn}}\epsilon_{2}^{2r}=0~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{V_{3}^{mn}}\epsilon_{2}^{2r-1}=0$
(B.3)

$\displaystyle\mathcal{L}_{V_{4}^{mn}}\epsilon_{1}^{2r}=-2\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r-1}+2\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r}~{},~{}~{}~{}\mathcal{L}_{V_{4}^{mn}}\epsilon_{1}^{2r-1}=2\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r}+2\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r-1}$
$\displaystyle\mathcal{L}_{V_{4}^{mn}}\epsilon_{2}^{2r}=0~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{V_{4}^{mn}}\epsilon_{2}^{2r-1}=0$
(B.4)

$\displaystyle\mathcal{L}_{V_{5}^{mn}}\epsilon_{1}^{2r}=-\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r}-\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r-1}~{},~{}\mathcal{L}_{V_{5}^{mn}}\epsilon_{1}^{2r-1}=-\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r-1}+\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r}$
$\displaystyle\mathcal{L}_{V_{5}^{mn}}\epsilon_{2}^{2r}=\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r}+\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r-1}~{},~{}~{}~{}\mathcal{L}_{V_{5}^{mn}}\epsilon_{2}^{2r-1}=\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r-1}-\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r}$

$\displaystyle\mathcal{L}_{V_{6}^{mn}}\epsilon_{1}^{2r}=\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r}-\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r-1}~{},~{}~{}~{}\mathcal{L}_{V_{6}^{mn}}\epsilon_{1}^{2r-1}=\sqrt{k}~{}c^{mn}\epsilon_{1}^{2r-1}+\sqrt{k}~{}d^{mn}\epsilon_{1}^{2r}$
$\displaystyle\mathcal{L}_{V_{6}^{mn}}\epsilon_{2}^{2r}=-\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r}+\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r-1}~{},~{}\mathcal{L}_{V_{6}^{mn}}\epsilon_{2}^{2r-1}=-\sqrt{k}~{}c^{mn}\epsilon_{2}^{2r-1}-\sqrt{k}~{}d^{mn}\epsilon_{2}^{2r}$
(B.6)

where $c^{mn}$ and $d^{mn}$ are the constants given by (6.18). Moreover

$\displaystyle\mathcal{L}_{U_{i}^{mn}}\epsilon_{1}^{2r}=(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}\mathcal{L}_{U_{i}^{mn}}\epsilon_{1}^{2r-1}=-(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle\mathcal{L}_{U_{i}^{mn}}\epsilon_{2}^{2r}=(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}\mathcal{L}_{U_{i}^{mn}}\epsilon_{2}^{2r-1}=-(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$
(B.7)

where $i=1,2$ and
$(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p},(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}$
are constants which are anti-symmetric in $m,n$. Equations (B.7) yield

$\displaystyle\widetilde{\mathcal{L}}_{U_{i}^{mn}}\psi_{+}^{2r}=(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}+(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}~{},~{}~{}~{}\widetilde{\mathcal{L}}_{U_{i}^{mn}}\psi_{+}^{2r-1}=(\alpha_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}-(\beta_{i})^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}~{}.$
(B.8)

Moreover

$\displaystyle\mathcal{L}_{W_{1}^{mn}}\epsilon_{1}^{2r}=\mathcal{L}_{W_{1}^{mn}}\epsilon_{1}^{2r-1}=\mathcal{L}_{W_{2}^{mn}}\epsilon_{2}^{2r}=\mathcal{L}_{W_{2}^{mn}}\epsilon_{2}^{2r-1}=0$
$\displaystyle\mathcal{L}_{W_{1}^{mn}}\epsilon_{2}^{2r}=A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}~{}~{}\mathcal{L}_{W_{1}^{mn}}\epsilon_{2}^{2r-1}=B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}-A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle\mathcal{L}_{W_{2}^{mn}}\epsilon_{1}^{2r}=-A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}\mathcal{L}_{W_{2}^{mn}}\epsilon_{1}^{2r-1}=-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$
(B.9) $\displaystyle
2\mathcal{L}_{W_{3}^{mn}}\epsilon_{1}^{2r}=-A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{3}^{mn}}\epsilon_{1}^{2r-1}=-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{3}^{mn}}\epsilon_{2}^{2r}=A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}~{}~{}2\mathcal{L}_{W_{3}^{mn}}\epsilon_{2}^{2r-1}=B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}-A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{4}^{mn}}\epsilon_{1}^{2r}=-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{4}^{mn}}\epsilon_{1}^{2r-1}=A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{4}^{mn}}\epsilon_{2}^{2r}=-B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{4}^{mn}}\epsilon_{2}^{2r-1}=A^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$

$\displaystyle\mathcal{L}_{W_{5}^{mn}}\epsilon_{1}^{2r}=\mathcal{L}_{W_{5}^{mn}}\epsilon_{1}^{2r-1}=\mathcal{L}_{W_{6}^{mn}}\epsilon_{2}^{2r}=\mathcal{L}_{W_{6}^{mn}}\epsilon_{2}^{2r-1}=0$
$\displaystyle\mathcal{L}_{W_{5}^{mn}}\epsilon_{2}^{2r}=F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}~{}~{}\mathcal{L}_{W_{5}^{mn}}\epsilon_{2}^{2r-1}=G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}-F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle\mathcal{L}_{W_{6}^{mn}}\epsilon_{1}^{2r}=-F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}\mathcal{L}_{W_{6}^{mn}}\epsilon_{1}^{2r-1}=-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$

$\displaystyle
2\mathcal{L}_{W_{7}^{mn}}\epsilon_{1}^{2r}=-F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{7}^{mn}}\epsilon_{1}^{2r-1}=-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{7}^{mn}}\epsilon_{2}^{2r}=F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}~{}~{}2\mathcal{L}_{W_{7}^{mn}}\epsilon_{2}^{2r-1}=G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}-F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{8}^{mn}}\epsilon_{1}^{2r}=-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{8}^{mn}}\epsilon_{1}^{2r-1}=F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{1}^{2p-1}$
$\displaystyle
2\mathcal{L}_{W_{8}^{mn}}\epsilon_{2}^{2r}=-G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}~{},~{}~{}~{}2\mathcal{L}_{W_{8}^{mn}}\epsilon_{2}^{2r-1}=F^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\epsilon_{2}^{2p-1}$
(B.12)

where $A^{pqr}_{~{}~{}~{}~{}l}$, $B^{pqr}_{~{}~{}~{}~{}l}$,
$F^{pqr}_{~{}~{}~{}~{}l}$ and $G^{pqr}_{~{}~{}~{}~{}l}$ are constants. In
particular $A^{pqr}_{~{}~{}~{}~{}l}$ and $B^{pqr}_{~{}~{}~{}~{}l}$ are
symmetric in $p,q$ while $F^{pqr}_{~{}~{}~{}~{}l}$ and
$G^{pqr}_{~{}~{}~{}~{}l}$ are anti-symmetric in $p,q$. Moreover

$\displaystyle
4\widetilde{\mathcal{L}}_{Z^{mn}}\psi_{+}^{2r}=A^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}-B^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}~{},~{}~{}~{}4\widetilde{\mathcal{L}}_{Z^{mn}}\psi_{+}^{2r-1}=A^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}$
$\displaystyle
4\widetilde{\mathcal{L}}_{Y^{mn}}\psi_{+}^{2r}=F^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}-G^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}~{},~{}~{}~{}4\widetilde{\mathcal{L}}_{Y^{mn}}\psi_{+}^{2r-1}=F^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}$
(B.13)

Furthermore, the following algebraic conditions hold

$\displaystyle
2\sqrt{k}\big{(}f^{mn}\psi_{+}^{2r-1}+A^{-1}\not{Z}^{mn}\Gamma_{2}\psi_{+}^{2r}\big{)}=A^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}+B^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}$
$\displaystyle
2\sqrt{k}\big{(}f^{mn}\psi_{+}^{2r}-A^{-1}\not{Z}^{mn}\Gamma_{2}\psi_{+}^{2r-1}\big{)}=A^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}-B^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}$
$\displaystyle
2\sqrt{k}\big{(}g^{mn}\psi_{+}^{2r-1}+A^{-1}\not{Y}^{mn}\Gamma_{2}\psi_{+}^{2r}\big{)}=F^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}+G^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}$
$\displaystyle
2\sqrt{k}\big{(}g^{mn}\psi_{+}^{2r}-A^{-1}\not{Y}^{mn}\Gamma_{2}\psi_{+}^{2r-1}\big{)}=F^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p-1}-G^{mnr}_{~{}~{}~{}~{}~{}p}\psi_{+}^{2p}$
(B.14)

In the final part of this Appendix, we list the Super-Jacobi identities (6.37)
of the $dS_{3}$ Killing superalgebra. Using (6.12), (6.17), (6.21) and (6.24),
we find the following results

  * •

$(\mathfrak{g}_{0},\mathfrak{g}_{0},\mathfrak{g}_{0})$: these conditions,
which correspond to the ordinary Jacobi identities of a Lie algebra, are
automatically satisfied.

  * •

$(\mathfrak{g}_{0},\mathfrak{g}_{0},\mathfrak{g}_{1})$ the non-trivial
conditions are given by

$\displaystyle-A^{mn[p}_{~{}~{}~{}~{}~{}~{}l}B^{q]lr}_{~{}~{}~{}~{}t}+B^{mn[p}_{~{}~{}~{}~{}~{}~{}l}G^{q]lr}_{~{}~{}~{}~{}t}-(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}B^{mnl}_{~{}~{}~{}~{}t}+(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}A^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle+A^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.15) $\displaystyle
A^{mn[p}_{~{}~{}~{}~{}~{}~{}l}A^{q]lr}_{~{}~{}~{}~{}t}-B^{mn[p}_{~{}~{}~{}~{}~{}~{}l}F^{q]lr}_{~{}~{}~{}~{}t}+(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}A^{mnl}_{~{}~{}~{}~{}t}+(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}B^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle-A^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.16)

$\displaystyle-A^{mn[p}_{~{}~{}~{}~{}~{}~{}l}G^{q]lr}_{~{}~{}~{}~{}t}-B^{mn[p}_{~{}~{}~{}~{}~{}~{}l}B^{q]lr}_{~{}~{}~{}~{}t}-(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}B^{mnl}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}A^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle+A^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.17) $\displaystyle
A^{mn[p}_{~{}~{}~{}~{}~{}~{}l}F^{q]lr}_{~{}~{}~{}~{}t}+B^{mn[p}_{~{}~{}~{}~{}~{}~{}l}A^{q]lr}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}B^{mnl}_{~{}~{}~{}~{}t}-A^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}A^{mnl}_{~{}~{}~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.18)

$\displaystyle-F^{mn[p}_{~{}~{}~{}~{}~{}~{}l}B^{q]lr}_{~{}~{}~{}~{}t}+G^{mn[p}_{~{}~{}~{}~{}~{}~{}l}G^{q]lr}_{~{}~{}~{}~{}t}-(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}G^{mnl}_{~{}~{}~{}~{}t}+(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}F^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle+F^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}+G^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.19) $\displaystyle
F^{mn[p}_{~{}~{}~{}~{}~{}~{}l}A^{q]lr}_{~{}~{}~{}~{}t}-G^{mn[p}_{~{}~{}~{}~{}~{}~{}l}F^{q]lr}_{~{}~{}~{}~{}t}+(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}F^{mnl}_{~{}~{}~{}~{}t}+(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}G^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle-F^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}+G^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.20)

$\displaystyle-F^{mn[p}_{~{}~{}~{}~{}~{}~{}l}G^{q]lr}_{~{}~{}~{}~{}t}-G^{mn[p}_{~{}~{}~{}~{}~{}~{}l}B^{q]lr}_{~{}~{}~{}~{}t}-(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}G^{mnl}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}F^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle+F^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}+G^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.21) $\displaystyle
F^{mn[p}_{~{}~{}~{}~{}~{}~{}l}F^{q]lr}_{~{}~{}~{}~{}t}+G^{mn[p}_{~{}~{}~{}~{}~{}~{}l}A^{q]lr}_{~{}~{}~{}~{}t}+(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}F^{mnl}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}G^{mnl}_{~{}~{}~{}~{}t}$
$\displaystyle-F^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}+G^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}=0~{},$
(B.22)

$\displaystyle
A^{pqr}_{~{}~{}~{}~{}s}B^{mns}_{~{}~{}~{}~{}t}-B^{pqr}_{~{}~{}~{}~{}~{}s}A^{mns}_{~{}~{}~{}~{}t}+A^{mnr}_{~{}~{}~{}~{}~{}s}B^{pqs}_{~{}~{}~{}~{}t}-B^{mnr}_{~{}~{}~{}~{}~{}s}A^{pqs}_{~{}~{}~{}~{}t}=0~{},$
(B.23) $\displaystyle
A^{pqr}_{~{}~{}~{}~{}s}A^{mns}_{~{}~{}~{}~{}t}+B^{pqr}_{~{}~{}~{}~{}~{}s}B^{mns}_{~{}~{}~{}~{}t}+A^{mnr}_{~{}~{}~{}~{}~{}s}A^{pqs}_{~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}~{}s}B^{pqs}_{~{}~{}~{}~{}t}$
$\displaystyle+4\sqrt{k}\delta^{r}_{~{}t}\bigg{(}-A^{mn(p}_{~{}~{}~{}~{}~{}~{}l}d^{q)l}+B^{mn(p}_{~{}~{}~{}~{}~{}~{}~{}l}c^{q)l}\bigg{)}=0~{},$
(B.24)

$\displaystyle
F^{pqr}_{~{}~{}~{}~{}s}B^{mns}_{~{}~{}~{}~{}t}-G^{pqr}_{~{}~{}~{}~{}~{}s}A^{mns}_{~{}~{}~{}~{}t}+A^{mnr}_{~{}~{}~{}~{}~{}s}G^{pqs}_{~{}~{}~{}~{}t}-B^{mnr}_{~{}~{}~{}~{}~{}s}F^{pqs}_{~{}~{}~{}~{}t}=0~{},$
(B.25) $\displaystyle
F^{pqr}_{~{}~{}~{}~{}s}A^{mns}_{~{}~{}~{}~{}t}+G^{pqr}_{~{}~{}~{}~{}~{}s}B^{mns}_{~{}~{}~{}~{}t}+A^{mnr}_{~{}~{}~{}~{}~{}s}F^{pqs}_{~{}~{}~{}~{}t}+B^{mnr}_{~{}~{}~{}~{}~{}s}G^{pqs}_{~{}~{}~{}~{}t}$
$\displaystyle-4\sqrt{k}\delta^{r}_{~{}t}\bigg{(}A^{mn[p}_{~{}~{}~{}~{}~{}~{}l}c^{q]l}+B^{mn[p}_{~{}~{}~{}~{}~{}~{}~{}l}d^{q]l}\bigg{)}=0~{},$
(B.26)

$\displaystyle
F^{pqr}_{~{}~{}~{}~{}s}G^{mns}_{~{}~{}~{}~{}t}-G^{pqr}_{~{}~{}~{}~{}~{}s}F^{mns}_{~{}~{}~{}~{}t}+F^{mnr}_{~{}~{}~{}~{}~{}s}G^{pqs}_{~{}~{}~{}~{}t}-G^{mnr}_{~{}~{}~{}~{}~{}s}F^{pqs}_{~{}~{}~{}~{}t}=0~{},$
(B.27) $\displaystyle
F^{pqr}_{~{}~{}~{}~{}s}F^{mns}_{~{}~{}~{}~{}t}+G^{pqr}_{~{}~{}~{}~{}~{}s}G^{mns}_{~{}~{}~{}~{}t}+F^{mnr}_{~{}~{}~{}~{}~{}s}F^{pqs}_{~{}~{}~{}~{}t}+G^{mnr}_{~{}~{}~{}~{}~{}s}G^{pqs}_{~{}~{}~{}~{}t}$
$\displaystyle-4\sqrt{k}\delta^{r}_{~{}t}\bigg{(}F^{mn[p}_{~{}~{}~{}~{}~{}~{}l}c^{q]l}+G^{mn[p}_{~{}~{}~{}~{}~{}~{}~{}l}d^{q]l}\bigg{)}=0~{},$
(B.28)

$\displaystyle(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{1})^{mnl}_{~{}~{}~{}~{}t}-(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}(\beta_{1})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{1})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}+(\beta_{1})^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\alpha_{1})^{q]lr}_{~{}~{}~{}~{}t}+2(\beta_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\alpha_{2})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.29)

$\displaystyle(\alpha_{1})^{pqr}_{~{}~{}~{}~{}l}(\beta_{1})^{mnl}_{~{}~{}~{}~{}t}+(\beta_{1})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{1})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{1})^{mnr}_{~{}~{}~{}~{}l}(\beta_{1})^{pql}_{~{}~{}~{}~{}t}-(\beta_{1})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{1})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{1})^{q]lr}_{~{}~{}~{}~{}t}+2(\beta_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{2})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.30)

$\displaystyle(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{1})^{mnl}_{~{}~{}~{}~{}t}-(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}(\beta_{1})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{1})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}+(\beta_{1})^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\alpha_{2})^{q]lr}_{~{}~{}~{}~{}t}-2(\beta_{1})^{mn[p}_{~{}~{}~{}~{}~{}l}(\alpha_{1})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.31)

$\displaystyle(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}(\beta_{1})^{mnl}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{1})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{1})^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}-(\beta_{1})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{2})^{q]lr}_{~{}~{}~{}~{}t}-2(\beta_{1})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{1})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.32)

$\displaystyle(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{2})^{mnl}_{~{}~{}~{}~{}t}-(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}(\beta_{2})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{2})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}+(\beta_{2})^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{2})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\alpha_{2})^{q]lr}_{~{}~{}~{}~{}t}-2(\beta_{2})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\alpha_{1})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.33)

$\displaystyle(\alpha_{2})^{pqr}_{~{}~{}~{}~{}l}(\beta_{2})^{mnl}_{~{}~{}~{}~{}t}+(\beta_{2})^{pqr}_{~{}~{}~{}~{}l}(\alpha_{2})^{mnl}_{~{}~{}~{}~{}t}-(\alpha_{2})^{mnr}_{~{}~{}~{}~{}l}(\beta_{2})^{pql}_{~{}~{}~{}~{}t}-(\beta_{2})^{mnr}_{~{}~{}~{}~{}l}(\alpha_{2})^{pql}_{~{}~{}~{}~{}t}$
$\displaystyle+2(\alpha_{2})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{2})^{q]lr}_{~{}~{}~{}~{}t}-2(\beta_{2})^{mn[p}_{~{}~{}~{}~{}~{}~{}l}(\beta_{1})^{q]lr}_{~{}~{}~{}~{}t}=0~{},$
(B.34)

  * •

$(\mathfrak{g}_{0},\mathfrak{g}_{1},\mathfrak{g}_{1})$: the non-trivial
conditions are666Note that the commutators of the internal Killing vectors
(6.21) are fixed by the super-Jacobi identities via (B.35).

$\displaystyle[U_{1}^{mn},U_{1}^{rs}]=-2(\alpha_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{1})^{s]l}-2(\beta_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{2})^{s]l}$
$\displaystyle[U_{2}^{mn},U_{2}^{rs}]=-2(\alpha_{2})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{2})^{s]l}+2(\beta_{2})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{1})^{s]l}$
$\displaystyle[U_{1}^{mn},U_{2}^{rs}]=-2(\alpha_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{2})^{s]l}+2(\beta_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{1})^{s]l}$
$\displaystyle(\alpha_{2})^{rs[m}_{~{}~{}~{}~{}~{}l}(U_{1})^{n]l}+(\beta_{2})^{rs[m}_{~{}~{}~{}~{}~{}l}(U_{2})^{n]l}=-(\alpha_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{2})^{s]l}+(\beta_{1})^{mn[r}_{~{}~{}~{}~{}~{}l}(U_{1})^{s]l}$
(B.35)

$\displaystyle
2(\alpha_{1})^{rs(m}_{~{}~{}~{}~{}~{}l}f^{n)l}-2(\beta_{1})^{rs(m}_{~{}~{}~{}~{}~{}l}g^{n)l}=A^{mn[r}_{~{}~{}~{}~{}~{}l}f^{s]l}-B^{mn[r}_{~{}~{}~{}~{}~{}l}g^{s]l}$
$\displaystyle
2(\alpha_{2})^{rs(m}_{~{}~{}~{}~{}~{}l}f^{n)l}-2(\beta_{2})^{rs(m}_{~{}~{}~{}~{}~{}l}g^{n)l}=A^{mn[r}_{~{}~{}~{}~{}~{}l}g^{l]s}+B^{mn[r}_{~{}~{}~{}~{}~{}l}f^{s]l}$
$\displaystyle-2(\alpha_{1})^{rs[m}_{~{}~{}~{}~{}~{}l}g^{n]l}-2(\beta_{1})^{rs[m}_{~{}~{}~{}~{}~{}l}f^{n]l}=F^{mn[r}_{~{}~{}~{}~{}~{}l}f^{s]l}-G^{mn[r}_{~{}~{}~{}~{}~{}l}g^{s]l}$
$\displaystyle-2(\alpha_{2})^{rs[m}_{~{}~{}~{}~{}~{}l}g^{n]l}-2(\beta_{2})^{rs[m}_{~{}~{}~{}~{}~{}l}f^{n]l}=F^{mn[r}_{~{}~{}~{}~{}~{}l}g^{s]l}+G^{mn[r}_{~{}~{}~{}~{}~{}l}f^{s]l}$
(B.36)

$\displaystyle[Z^{rs},Z^{pq}]=\frac{1}{4}B^{rs(p}_{~{}~{}~{}~{}l}(U_{2})^{q)l}+\frac{1}{4}A^{rs(p}_{~{}~{}~{}~{}l}(U_{1})^{q)l}$
$\displaystyle[Z^{rs},Y^{pq}]=\frac{1}{4}B^{rs[p}_{~{}~{}~{}~{}l}(U_{1})^{q]l}-\frac{1}{4}A^{rs[p}_{~{}~{}~{}~{}l}(U_{2})^{q]l}$
$\displaystyle[Y^{rs},Y^{pq}]=-\frac{1}{4}F^{rs[p}_{~{}~{}~{}~{}l}(U_{2})^{q]l}+\frac{1}{4}G^{rs[p}_{~{}~{}~{}~{}l}(U_{1})^{q]l}$
$\displaystyle
G^{pq(r}_{~{}~{}~{}~{}~{}l}(U_{2})^{s)l}+F^{pq(r}_{~{}~{}~{}~{}~{}l}(U_{1})^{s)l}=-B^{rs[p}_{~{}~{}~{}~{}~{}l}(U_{1})^{q]l}+A^{rs[p}_{~{}~{}~{}~{}~{}l}(U_{2})^{q]l}$
(B.37)

  * •

$(\mathfrak{g}_{1},\mathfrak{g}_{1},\mathfrak{g}_{1})$: the non-trivial
conditions read

$\displaystyle
2\sqrt{k}d^{rs}\delta^{t}_{q}-2\sqrt{k}d^{t(r}\delta^{s)}_{q}-(\alpha_{1})^{t(rs)}_{~{}~{}~{}~{}~{}q}-(\beta_{2})^{t(rs)}_{~{}~{}~{}~{}~{}q}=0$
$\displaystyle
A^{rst}_{~{}~{}~{}q}-A^{t(rs)}_{~{}~{}~{}~{}q}+G^{t(rs)}_{~{}~{}~{}~{}q}-(\alpha_{1})^{t(rs)}_{~{}~{}~{}~{}~{}q}+(\beta_{2})^{t(rs)}_{~{}~{}~{}~{}~{}q}=0$
$\displaystyle
B^{rst}_{~{}~{}~{}q}-B^{t(rs)}_{~{}~{}~{}~{}q}-F^{t(rs)}_{~{}~{}~{}~{}q}-(\alpha_{2})^{t(rs)}_{~{}~{}~{}~{}~{}q}-(\beta_{1})^{t(rs)}_{~{}~{}~{}~{}~{}q}=0$
$\displaystyle-2\sqrt{k}c^{rs}\delta^{t}_{q}+2\sqrt{k}c^{t(r}\delta^{s)}_{q}-(\alpha_{2})^{t(rs)}_{~{}~{}~{}~{}~{}q}+(\beta_{1})^{t(rs)}_{~{}~{}~{}~{}~{}q}=0$
$\displaystyle-G^{rst}_{~{}~{}~{}~{}q}-G^{t[rs]}_{~{}~{}~{}~{}q}+A^{t[rs]}_{~{}~{}~{}~{}q}-(\alpha_{1})^{t[rs]}_{~{}~{}~{}~{}~{}q}+(\beta_{2})^{t[rs]}_{~{}~{}~{}~{}~{}q}=0$
$\displaystyle-F^{rst}_{~{}~{}~{}~{}q}-F^{t[rs]}_{~{}~{}~{}~{}q}-B^{t[rs]}_{~{}~{}~{}~{}q}+(\beta_{1})^{t[rs]}_{~{}~{}~{}~{}~{}q}+(\alpha_{2})^{t[rs]}_{~{}~{}~{}~{}~{}q}=0~{}.$
(B.38)

## Acknowledgments

DF is partially supported by the STFC DTP Grant ST/S505742. DF would like to
thank Jan Gutowski for useful discussions.

## Data Management

No additional research data beyond the data presented and cited in this work
are needed to validate the research findings in this work.

## References

  * [1] S. Perlmutter et al. [Supernova Cosmology Project], Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae Astrophys. J. 517 (1999), 565-586 [arXiv:astro-ph/9812133 [astro-ph]].
  * [2] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems Phys. Rev. D 23 (1981), 347-356
  * [3] E. Witten, Quantum gravity in de Sitter space, [arXiv:hep-th/0106109 [hep-th]].
  * [4] G. W. Gibbons, Aspects of Supergravity Theories, Supersymmetry, Supergravity and Related Topics, F. del Aguila, J. A. de Azcárraga and L. E. Ibánẽz, eds., 346–351. World Scientific (1985).
  * [5] B. de Wit, D. J. Smit and N. D. Hari Dass, Residual Supersymmetry of Compactified D=10 Supergravity, Nucl. Phys. B 283 (1987), 165.
  * [6] J. M. Maldacena and C. Nunez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001), 822; arXiv:hep-th/0007018 [hep-th].
  * [7] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003), 046005
  * [8] N. Goheer, M. Kleban and L. Susskind, The Trouble with de Sitter space JHEP 07 (2003), 056 [arXiv:hep-th/0212209 [hep-th]].
  * [9] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231; arXiv:hep-th/9711200 [hep-th].
  * [10] A. Strominger, The dS / CFT correspondence, JHEP 10 (2001), 034; arXiv:hep-th/0106113 [hep-th].
  * [11] A. Strominger, Inflation and the dS / CFT correspondence, JHEP 11 (2001), 049; arXiv:hep-th/0110087 [hep-th].
  * [12] J. M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 0305 (2003) 013, [astro-ph/0210603].
  * [13] U. Gran, P. Lohrmann and G. Papadopoulos, The spinorial geometry of supersymmetric heterotic string backgrounds, JHEP 02 (2006) 063 [hep-th/0510176].
  * [14] U. Gran, G. Papadopoulos, D. Roest and P. Sloane, Geometry of all supersymmetric type-I backgrounds, JHEP 08 (2007) 074 [hep-th/0703143].
  * [15] G. Papadopoulos, Heterotic supersymmetric backgrounds with compact holonomy revisited, Class. Quant. Grav. 27 (2010) 125008
  * [16] C. M. Hull and P. K. Townsend, The two loop beta function for sigma models with torsion, Phys. Lett. B 191 (1987) 115
  * [17] M. B. Green and J. H. Schwarz, Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory Phys. Lett. B 149 (1984), 117-122
  * [18] J. Gillard, G. Papadopoulos and D. Tsimpis, Anomaly, fluxes and (2,0) heterotic string compactifications, JHEP 06 (2003) 035 [hep-th/0304126].
  * [19] C. G. Callan Jr., J. A. Harvey and A. Strominger, Supersymmetric string solitons, hep-th/9112030.
  * [20] P. S. Howe and G. Papadopoulos, Finiteness and anomalies in (4,0) supersymmetric $\sigma$-models, Nucl. Phys. B 381 (1992) 360 [hep-th/9203070].
  * [21] A. Fontanella, J. B. Gutowski and G. Papadopoulos, Anomaly Corrected Heterotic Horizons, JHEP 10 (2016), 121 [arXiv:1605.05635 [hep-th]].
  * [22] S. W. Beck, J. B. Gutowski and G. Papadopoulos, Geometry and supersymmetry of heterotic warped flux AdS backgrounds JHEP 07 (2015), 152, [arXiv:1505.01693 [hep-th]].
  * [23] U. Gran, J. B. Gutowski and G. Papadopoulos, On supersymmetric Anti-de-Sitter, de-Sitter and Minkowski flux backgrounds, Class. Quant. Grav. 35 (2018) no.6, 065016 [arXiv:1607.00191 [hep-th]].
  * [24] D. Kutasov, T. Maxfield, I. Melnikov and S. Sethi, Constraining de Sitter Space in String Theory, Phys. Rev. Lett. 115 (2015) no.7, 071305 [arXiv:1504.00056 [hep-th]].
  * [25] D. Farotti and J. Gutowski, Supersymmetric $dS_{n}$ solutions for $n\geq 5$ in $D=11$ supergravity [arXiv:2204.11903 [hep-th]].
  * [26] M. Di Gioia and J. Gutowski, Supersymmetric dS4 solutions in D=11 supergravity [arXiv:2207.01532 [hep-th]].
  * [27] T. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math, 2002 International Press Vol. 6, No. 2, pp. 303-336
  * [28] M. Fernandez, S. Ivanov, L. Ugarte and R. Villacampa, Compact supersymmetric solutions of the heterotic equations of motion in dimension 5, Nucl. Phys. B 820 (2009), 483-502 [arXiv:0811.2137 [math.DG]].
  * [29] J. M. Figueroa-O’Farrill, P. Meessen and S. Philip, Supersymmetry and homogeneity of M-theory backgrounds, Class. Quant. Grav. 22 (2005), 207-226 [arXiv:hep-th/0409170 [hep-th]]
  * [30] S. Beck, U. Gran, J. Gutowski and G. Papadopoulos, All Killing Superalgebras for Warped AdS Backgrounds, JHEP 12 (2018), 047 [arXiv:1710.03713 [hep-th]].
  * [31] J. Bagger and N. Lambert, Three-Algebras and N=6 Chern-Simons Gauge Theories Phys. Rev. D 79 (2009), 025002 [arXiv:0807.0163 [hep-th]].
  * [32] J. Gutowski and G. Papadopoulos, Heterotic Black Horizons, JHEP 07 (2010), 011 [arXiv:0912.3472 [hep-th]].
  * [33] U. Gran, J. Gutowski and G. Papadopoulos, AdS backgrounds from black hole horizons, Class. Quant. Grav. 30 (2013), 055014 [arXiv:1110.0479 [hep-th]].
  * [34] S. R. Green, E. J. Martinec, C. Quigley and S. Sethi, Constraints on String Cosmology, Class. Quant. Grav. 29 (2012), 075006 [arXiv:1110.0545 [hep-th]].
  * [35] F. F. Gautason, D. Junghans and M. Zagermann, On Cosmological Constants from $\alpha^{{}^{\prime}}$-Corrections, JHEP 06 (2012), 029, [arXiv:1204.0807 [hep-th]].