Compactifications of Type II Supergravities in Superspace

Osvaldo Chandia♣ and Brenno Carlini Vallilo♠

♣ Departamento de Ciencias, Facultad de Artes Liberales
Universidad Adolfo Ibáñez, Diagonal Las Torres 2640, Peñalolén, Chile

♠ Departamento de Ciencias Físicas, Facultad de Ciencias Exactas
Universidad Andrés Bello, Sazié 2212, Santiago, Chile

Abstract

We propose a way to describe compactifications of Type II supergravities with
fluxes directly from superspace. The superspace used is the one that arises
naturally from the pure spinor superstring. We show how previous results of
flux compactifications can be obtained from our method.

†† ♣<EMAIL_ADDRESS>♠<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction and motivation
  2. 2 Ten-dimensional Type II supergravities form the pure spinor string
    1. 2.1 Redefining the connections
  3. 3 Superspace compactification
    1. 3.1 Factorization and the Pure Spinor Equations
    2. 3.2 The cosmological constant
    3. 3.3 Closure of the algebra
    4. 3.4 Summary
  4. 4 A simple flux background
  5. 5 Conclusions and prospects
  6. A Ten dimensional $\gamma$ matrices
  7. B The calculation of $[\nabla_{\alpha},S_{\beta J}\nabla_{\dot{\beta}}{}^{J}+\widetilde{S}_{\beta\widetilde{J}}\widetilde{\nabla}_{\dot{\beta}}{}^{\widetilde{J}}+S_{\dot{\beta}}{}^{J}\nabla_{\beta J}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\nabla}_{\beta\widetilde{J}}]$ and other identities
  8. C Sigma matrices for $SU(3)$ Structure

##  1 Introduction and motivation

Supersymmetry is pivotal in superstring theory, underlying many of its most
intriguing properties. Supersymmetry enables exact computations and nontrivial
checks of the web of dualities connecting different string theories. Just as
space-time provides the natural geometric framework for formulating Lorentz
and generally covariant theories, superspace is the ideal mathematical setting
for describing supersymmetric field theories and supergravity models. The
additional fermionic dimensions of superspace elegantly incorporate
supersymmetric transformations. As such, superspace would be an invaluable
tool for constructing, analyzing, and connecting supersymmetric string vacua.
Much as space-time geometry provided vital insights leading to general
relativity, the geometry of superspace has the potential to furnish critical
clues needed for a better understanding of string theory.

Nonetheless, despite its long history, the use of superspace in string theory
has been minimal. One technical difficulty is that spinor representations have
different properties for different space-time dimensions. Related to that, the
superspace description of supersymmetric multiplets varies widely depending on
the number of supersymmetries. Perhaps the most critical obstacle is obtaining
an off-shell formulation for theories with sixteen or more supersymmetries.

The use of superspace becomes challenging to avoid when considering Ramond-
Ramond backgrounds. In the RNS formulation, they couple to the world-sheet
vertex operators containing spin fields, which are operators that change the
RNS fermions’ boundary conditions and also have non-zero picture numbers.
These two properties render a world-sheet description of finite RR backgrounds
rather strenuous [1, 2]. One way to partially solve these problems comes in
the form of the so-called hybrid formalism [3]. Suppose one is interested in
only maintaining Poincaré covariance in lower dimensions. In that case, it is
possible to choose the world-sheet super-charges such that their anti-
commutators give the translation operator in the zero picture. After some
field redefinitions, the world-sheet variables will transform linearly under
supersymmetries. The hybrid formalism was used in the description of RR
background in e.g., [4, 5, 6, 7, 8]. A promising way to describe RR
backgrounds using the RNS formalism is superstring field theory (SSFT) [9, 10,
11, 12]. This approach represents the background as a classical solution of
the SSFT equations of motion. As far as we know, the first paper that explored
this idea for Ramond-Ramond backgrounds was [13]. More recently, a very
detailed work describing a case of flux compactification appeared [14]. A
different approach to studying flux backgrounds using superspace was discussed
in [15].

In this work, we study compactifications of Type II supergravity theories that
preserve $N=1$ supersymmetry in four dimensions working directly from
superspace. The literature on the subject is too vast to give a proper list of
references. The reviews [16, 17] provide an excellent (although not up-to-
date) summary of the subject and contain an extensive list of references.
Nevertheless, we will cite works directly related to some of our conclusions
in the body of the paper. The curved superspace formulation we use comes from
the pure spinor superstring [18]. This formulation is non-standard; it has two
connections and a scaling symmetry that acts only on the spinors. Of course,
it can be mapped to standard descriptions, but since we want to apply the
results to the pure spinor superstring, we will use the more adequate
formalism.

The usual way to study111Our notation and more precise definitions, together
with the proof of some claims, will be given in the next section. symmetries
in a curved (super)space to introduce a Killing supervector field

$\displaystyle K=\xi^{A}\nabla_{A}+\frac{1}{2}\Lambda^{ab}M_{ab}+\sigma
S+\widetilde{\sigma}\widetilde{S},$ (1.1)

where $M_{ab}$ are the Lorentz generators and $(S,\widetilde{S})$ are scale
generators that act only on spinors. The defining condition the Killing
supervector satisfies is

$\displaystyle[K,\nabla_{A}]=0.$ (1.2)

If the theory is purely geometrical, this is sufficient. However, ten-
dimensional supergravity theories have additional matter fields. Another
characteristic of these theories is that they have a scalar superfield $\Phi$
whose first component is the dilaton and that almost all gauge covariant
tensors (the only exception being the Ramond-Ramond five form in Type IIB) can
be found at higher $\theta$ components. Therefore, for a general supergravity
background to be invariant under the transformation generated by $K$, it is
also necessary to require

$\displaystyle\delta\Phi=K(\Phi)=0.$ (1.3)

In a previous work [19]222See also [20, 21] for earlier works., we studied
heterotic string backgrounds with $N=1$ supersymmetry in four dimensions
directly from the ten-dimensional superspace using (1.2) and (1.3). Usually,
given a fixed background, the Killing supervectors are found from (1.2).
Instead, we wanted to find the conditions on superspaces if we demand that the
Killing supervectors have some desired form. For example, for the space to
have translation invariance in some of its dimensions, we should require that
$\xi^{a}|_{\theta=0}\neq 0$ for and $\nabla_{A}\xi^{a}|_{\theta=0}=0$
$a=0,1,2,3$. Then, the higher $\theta$ components of $\xi^{A}$ can be found
solving (1.2) in terms of the general background. This process will also fix
to zero the $|_{\theta}=0$ projection of some background tensors. The same
steps can be applied to Type II supergravities.

Although this process will help understand the curved superspace of
compactifications, it will not help study the first important set of
observables: the massless physical states. In the pure spinor formalism the
physical states are described by operators with ghost number $2$ in the
cohomology of the BRST operator $Q=\int
d\sigma(\lambda^{\alpha}d_{\alpha}+\widetilde{\lambda}^{\widetilde{\alpha}}\widetilde{d}_{\widetilde{\alpha}})$,
where $(\lambda,\widetilde{\lambda})$ are the pure spinor ghosts and
$(d_{\alpha},\widetilde{d}_{\widetilde{\alpha}})$ are the conjugate momenta
for the odd superspace variables $(\theta,\widetilde{\theta})$. At the lowest
order in $\alpha^{\prime}$, the action of $Q$ can be described by the
differential operator

$\displaystyle
Q=\lambda^{\alpha}\nabla_{\alpha}+\widetilde{\lambda}^{\widetilde{\alpha}}\widetilde{\nabla}_{\widetilde{\alpha}}.$
(1.4)

where $(\nabla_{\alpha},\widetilde{\nabla}_{\widetilde{\alpha}})$ are the odd
superspace covariant derivatives. The massless vertex operators can be written
as

$\displaystyle{\mathcal{V}}=\lambda^{\alpha}\widetilde{\lambda}^{\widetilde{\alpha}}A_{\alpha\widetilde{\alpha}}(Z),$
(1.5)

where $A_{\alpha\widetilde{\alpha}}(Z)$ is a tensor superfield satisfying the
equations coming from $Q{\mathcal{V}}=0$ and the gauge transformations from
$\delta{\mathcal{V}}=Q\xi$, for some appropriate $\xi$. In the case of a
compactification preserving $N=1$ supersymmetry in four dimensions, the fields
describing the fluctuations of the internal geometry are organized into four-
dimensional chiral and anti-chiral multiplets. It is natural to ask if
describing these multiplets in a standard four-dimensional superfield is
possible. In other words, the vertex operator should factorize333We will use
the $\mathfrak{sl}(2,\mathbb{C})+\mathfrak{su}(4)$ decomposition for the ten-
dimensional spinors. For a longer explanation of the notation used in this
paper, see the end of the present section 1.

$\displaystyle{\mathcal{V}}=\sum_{\Gamma}\left(\Phi^{\Gamma}_{\alpha\widetilde{\alpha}}(Z_{d=4})\Psi^{\Gamma}_{I\widetilde{I}}(Y)\lambda^{\alpha
I}\widetilde{\lambda}^{\widetilde{\alpha I}}+{\rm c.c.}\right),$ (1.6)

where $\Gamma$ is a generic index for the different compactification moduli,
$\Psi^{\Gamma}_{I\widetilde{I}}(Y)$ is some internal “superspace harmonic
form” and $\Phi^{\Gamma}_{\alpha\widetilde{\alpha}}(Z_{d=4})$ is the desired
four dimensional superfield. The equations of motion for the fields
$\Phi^{\Gamma}$ should also have the standard superspace form. For that to be
possible, we must describe how the usual four-dimensional odd covariant
derivatives appear inside $Q$. In the
$\mathfrak{sl}(2,\mathbb{C})+\mathfrak{su}(4)$ notation, the BRST differential
(1.4) is written as

$\displaystyle Q=\lambda^{\alpha I}\nabla_{\alpha
I}+\lambda^{\dot{\alpha}}{}_{I}\nabla_{\dot{\alpha}}{}^{I}+\widetilde{\lambda}^{\alpha\widetilde{I}}\widetilde{\nabla}_{\alpha\widetilde{I}}+\widetilde{\lambda}^{\dot{\alpha}}{}_{\widetilde{I}}\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{I}}.$
(1.7)

The position of the index $\widetilde{I}$ depends on whether we choose type
IIA or IIB. The first step for our goal is to introduce a pair of complex
bosonic $SU(4)$ spinors $(\chi^{I},\widetilde{\chi}^{\widetilde{I}})$
superfields and define the following covariant derivatives

$\displaystyle\nabla_{\alpha}=\chi^{I}\nabla_{\alpha
I}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\nabla}_{\alpha\widetilde{I}},$
(1.8) $\displaystyle\Delta_{\alpha}=\chi^{I}\nabla_{\alpha
I}-\widetilde{\chi}^{\widetilde{I}}\widetilde{\nabla}_{\alpha\widetilde{I}},\quad\Delta_{\alpha
k}=\chi_{I}\sigma_{k}^{IJ}\nabla_{\alpha J},\quad\widetilde{\Delta}_{\alpha
k}=\widetilde{\chi}_{\widetilde{I}}\widetilde{\sigma}_{k}^{\widetilde{I}\widetilde{J}}\widetilde{\nabla}_{\alpha\widetilde{J}},$
(1.9)

and their complex conjugates. The full covariant derivatives can be recovered
as

$\displaystyle\nabla_{\alpha
I}=\frac{1}{2|\chi|^{2}}\left(\chi_{I}(\nabla_{\alpha}+\Delta_{\alpha})-\sigma^{k}_{IJ}\chi^{J}\Delta_{\alpha{k}}\right),$
(1.10)
$\displaystyle\widetilde{\nabla}_{\alpha\widetilde{I}}=\frac{1}{2|\widetilde{\chi}|^{2}}\left(\widetilde{\chi}_{I}(\nabla_{\alpha}-\Delta_{\alpha})-\widetilde{\sigma}^{k}_{\widetilde{I}\widetilde{J}}\widetilde{\chi}^{\widetilde{J}}\widetilde{\Delta}_{\alpha
k}\right).$ (1.11)

Which can be checked using the the identities
$\sigma_{k}^{IJ}\sigma^{k}_{KL}=-2(\delta^{I}_{K}\delta^{J}_{L}-\delta^{I}_{L}\delta^{J}_{K})$
and its analog for $\widetilde{\sigma}^{\mathsf{k}}$. The set (1.8) has
sixteen complex linearly independent derivatives with the choices
$\sigma^{\bar{\mathsf{k}}}_{IJ}\chi^{J}=\widetilde{\sigma}^{\bar{\mathsf{k}}}_{\widetilde{I}\widetilde{J}}\widetilde{\chi}^{\widetilde{J}}=0$.

We are going to define $(\nabla_{\alpha},\nabla_{\dot{\alpha}})$ as the “four-
dimensional” covariant derivatives and
$\allowbreak(\Delta_{\alpha},\allowbreak\Delta_{\dot{\alpha}},\allowbreak\Delta_{\alpha
k},\allowbreak\Delta_{\dot{\alpha}k},\allowbreak\widetilde{\Delta}_{\alpha
k},\widetilde{\Delta}_{\dot{\alpha}k})$ as the “mixed” covariant derivatives.
This distinction is because it is impossible to eliminate the four-dimensional
derivative on the left-hand side of the anti-commutators of the mixed
covariant derivatives. In this approach, an “internal” superfield will be
defined as being annihilated by all “four-dimensional” derivatives and Lorentz
generators.

If we define

$\displaystyle\lambda^{\alpha}=\frac{1}{2|\chi|^{2}}\left(\chi_{I}\lambda^{\alpha
I}+\widetilde{\chi}_{\widetilde{I}}\widetilde{\lambda}^{\alpha\widetilde{I}}\right),\quad\Lambda^{\alpha}=\frac{1}{2|\widetilde{\chi}|^{2}}\left(\chi_{I}\lambda^{\alpha
I}-\widetilde{\chi}_{\widetilde{I}}\widetilde{\lambda}^{\alpha\widetilde{I}}\right),$
(1.12)
$\displaystyle\Lambda^{\alpha}_{{{k}}}=-\frac{1}{2|\chi|^{2}}(\sigma_{{{k}}})_{KL}\chi^{K}\lambda^{\alpha
L},\quad\widetilde{\Lambda}^{\alpha}_{{{k}}}=-\frac{1}{2|\widetilde{\chi}|^{2}}(\sigma_{{{k}}})_{\widetilde{K}\widetilde{L}}\widetilde{\chi}^{\widetilde{K}}\widetilde{\lambda}^{\alpha\widetilde{L}},$
(1.13)

we can then write the BRST charge as

$\displaystyle
Q=\lambda^{\alpha}\nabla_{\alpha}+\Lambda^{\alpha}\Delta_{\alpha}+\Lambda^{\alpha}_{k}\nabla_{\alpha{k}}+\widetilde{\Lambda}^{\alpha}_{k}\widetilde{\nabla}_{\alpha
k}+{\rm c.c.},$ (1.14)

The idea is to impose that the four-dimensional super-covariant derivatives
generate the covariant derivative algebra of either flat space of $AdS_{4}$.
$(\nabla_{\alpha},\nabla_{\dot{\alpha}})$ are not the most general
possibility; other terms could be added. However, we will find that
$(\chi^{I},\widetilde{\chi}^{\widetilde{I}})$ superfields provide conditions
that are general enough.

This work is organized as follows. In Section 2, we review ten dimensional
Type II supergravities in the pure spinor formulation. We describe how various
tensors are determined by the Berkovits-Howe constraints. In Section 3, we
derive the conditions imposed on the curved supergeometry by requiring $N=1$
four-dimensional supersymmetry in $AdS_{4}$ or flat space. In Section 4, we
discuss how the conditions derived in Section 3 can be solved with a
nontrivial compactification. We conclude the paper and discuss future work in
Section 5.

#### A warning about notation

This work deals with various types of indices. We decided to use different
conventions in different sections to reduce the number of index decorations.
In Section 2, there are only ten-dimensional tensors and spinors; lowercase
Latin letters at the beginning of the alphabet $a,b,c,\cdots$ are ten-
dimensional local Lorentz indices. Lowercase Greek letters at the beginning of
the alphabet $\alpha,\beta,\gamma,\cdots$ are ten-dimensional spinor indices
from $1$ to $16$. The meaning of the spinor indices with a tilde depends on
whether it is Type IIA or Type IIB. Tildes will also differentiate left and
right-moving sectors in the world-sheet variables , e.g.
$(\partial,\widetilde{\partial})$. In Sections 3 and 4, we need indices to
describe the four-dimensional superspace and indices to describe the internal
six-dimensional space. In these sections, we use lowercase Latin letters at
the beginning of the alphabet $a,b,c,\cdots$ for four-dimensional local
Lorentz indices. Furthermore, undotted and dotted lowercase Greek letters at
the beginning of the alphabet $\alpha,\beta,\gamma,\cdots$ are four-
dimensional chiral and anti-chiral spinor indices from $1$ to $2$; uppercase
Latin letter in the middle of the alphabet $I,J,K,\cdots$ are six-dimensional
spinor indices from $1$ to $4$. Real tangent space indices for the internal
space will be $i,j,k,\cdots$ from $1$ to $6$, and complex internal indices
will be unbarred and barred lowercase Latin letters in upright San Serif font
$\mathsf{k},\mathsf{l},\mathsf{m},\cdots$ from $1$ to $3$. Finally, when
referring to the full ten-dimensional set of tangent space-time indices is
necessary, we will use an underline as $\underline{a}$. Some useful identities
are listed in Appendix A.

##  2 Ten-dimensional Type II supergravities form the pure spinor string

Consider the type II superspace in ten dimensions used in the pure spinor
formalism. The geometry of this superspace is constrained by the pure spinor
BRST symmetry, which is generated by the charges

$\displaystyle Q=\oint
d\sigma\lambda^{\alpha}d_{\alpha},\quad{\widetilde{Q}}=\oint
d\sigma{\widetilde{\lambda}}^{\widetilde{\alpha}}{\widetilde{d}}_{\widetilde{\alpha}},$
(2.1)

where $(\lambda,{\widetilde{\lambda}})$ are the pure spinor variables and
$(d,{\widetilde{d}})$ are the world-sheet generators of translations in
superspace. The BRST invariant world-sheet action is given by

$\displaystyle S=\int d^{2}\sigma$
$\displaystyle\left(\frac{1}{2}\Pi_{a}\widetilde{\Pi}^{a}+\frac{1}{2}\Pi^{A}\widetilde{\Pi}^{B}B_{BA}+d_{\alpha}\widetilde{\Pi}^{\alpha}+\widetilde{d}_{\widetilde{\alpha}}\Pi^{\widetilde{\alpha}}+\omega_{\alpha}{\widetilde{\partial}}\lambda^{\alpha}+{\widetilde{\omega}}_{\widetilde{\alpha}}\partial{\widetilde{\lambda}}^{\widetilde{\alpha}}\right.$
(2.2)
$\displaystyle\left.+\lambda^{\beta}\widetilde{\partial}Z^{M}\Omega_{M\beta}{}^{\alpha}\omega_{\alpha}+\widetilde{\lambda}^{\widetilde{\beta}}\partial
Z^{M}\widehat{\Omega}_{M\widetilde{\beta}}{}^{\widetilde{\alpha}}{\widetilde{\omega}}_{\widetilde{\alpha}}+d_{\alpha}{\widetilde{d}}_{\widetilde{\beta}}P^{\alpha\widetilde{\beta}}\right.$
(2.3)
$\displaystyle\left.+\lambda^{\alpha}\omega_{\beta}{\widetilde{d}}_{\widetilde{\gamma}}C_{\alpha}{}^{\beta\widetilde{\gamma}}+{\widetilde{\lambda}}^{\widetilde{\alpha}}{\widetilde{\omega}}_{\widetilde{\beta}}d_{\gamma}C_{\widetilde{\alpha}}{}^{\widetilde{\beta}\gamma}+\lambda^{\alpha}\omega_{\beta}{\widetilde{\lambda}}^{\widetilde{\gamma}}{\widetilde{\omega}}_{\widetilde{\rho}}S_{\alpha\widetilde{\gamma}}{}^{\beta\widetilde{\rho}}\right),$
(2.4)

where the background fields are functions of the superspace coordinates are
$Z^{M}=(X^{m},\theta^{\mu},\widetilde{\theta}^{\widetilde{\mu}})$ where
$m=0,\dots 9\;;\mu,\widetilde{\mu}=1,\dots 16$ are curved superspace indices.
The fermionic indices $\mu$ and $\widetilde{\mu}$ have the opposite chirality
for the type IIA case and have the same chirality for type IIB case. This
means that $\widetilde{\theta}^{\widetilde{\mu}}=\widetilde{\theta}_{\mu}$ for
the type IIA case and
$\widetilde{\theta}^{\widetilde{\mu}}=\widetilde{\theta}^{\mu}$ for the type
IIB case. The super-coordinates enter in the world-sheet action in the
combinations $\Pi^{A}=\partial Z^{M}E_{M}{}^{A}$ and
$\widetilde{\Pi}^{A}=\widetilde{\partial}Z^{M}E_{M}{}^{A}$ where
$\partial,\widetilde{\partial}$ are the derivatives respect to the world-sheet
coordinates and $E_{M}{}^{A}$ is the super-vielbein and rotates the super-
space index $M$ into the local super-space index $A$. This local index ranges
in $(a,\alpha,\widetilde{\alpha})$. The world-sheet action includes the
covariant derivatives of the pure spinor variables $\lambda^{\alpha}$ and
$\widetilde{\lambda}^{\widetilde{\alpha}}$, they are

$\displaystyle{\widetilde{D}}\lambda^{\alpha}=\widetilde{\partial}\lambda^{\alpha}+\lambda^{\beta}\widetilde{\partial}Z^{M}\Omega_{M\beta}{}^{\alpha},\quad
D\widetilde{\lambda}^{\widetilde{\alpha}}=\partial\widetilde{\lambda}^{\widetilde{\alpha}}+\widetilde{\lambda}^{\widetilde{\beta}}\partial
Z^{M}\widehat{\Omega}_{M\widetilde{\beta}}{}^{\widetilde{\alpha}},$ (2.5)

where $\Omega_{M\alpha}{}^{\beta}$ and
$\widehat{\Omega}_{M\widetilde{\beta}}{}^{\widetilde{\alpha}}$ are the
background connections. To respect a symmetry involving the conjugate variable
of $\lambda$ and the conjugate variable of $\widetilde{\lambda}$, these
connections have the form

$\displaystyle\Omega_{M\alpha}{}^{\beta}=\delta_{\alpha}^{\beta}\Omega_{M}+\frac{1}{4}(\gamma^{ab})_{\alpha}{}^{\beta}\Omega_{Mab},\quad\widehat{\Omega}_{M\widetilde{\alpha}}{}^{\widetilde{\beta}}=\delta_{\widetilde{\alpha}}^{\widetilde{\beta}}\widehat{\Omega}_{M}+\frac{1}{4}(\gamma^{ab})_{\widetilde{\alpha}}{}^{\widetilde{\beta}}\widehat{\Omega}_{Mab},$
(2.6)

where $\gamma^{ab}$ is the antisymmetric product of two symmetric $16\times
16$ $\gamma$-matrices. Usually, there is only one connection, the Lorentz
connection $\Omega_{Mab}$. It turns out that $\Omega_{M}$ and
$\widehat{\Omega}_{M}$ are related to the background dilaton superfield $\Phi$
and $\widehat{\Omega}_{Mab}$ is a function of $\Omega_{Mab}$ and torsion
components, as we show below. This reduction only happens after gauge fixing
[18] the three independent local Lorentz symmetries that the action (2.2)
suffers. These five local gauge symmetries generate some apparent puzzles; the
fact that there are two different curvatures and that gamma matrices are not
invariant tensors. This last fact implies that the covariant derivatives
(defined with the connections directly from (2.2)) of the gamma matrices do
not vanish [19].

The nilpotency of the BRST charges (2.1) together with the holomorphicity of
$\lambda^{\alpha}d_{\alpha}$ and the anti-holomorphicity of
${\widetilde{\lambda}}^{\widetilde{\alpha}}{\widetilde{d}}_{\widetilde{\alpha}}$
put the background fields to satisfy the constraints of type II supergravity
in ten dimensions [18]. These constraints are expressed in terms of torsion
and curvature components. As usual, we define the super one forms
$E^{A}=dZ^{M}E_{M}{}^{A}$,
$\Omega_{\alpha}{}^{\beta}=dZ^{M}\Omega_{M\alpha}{}^{\beta}$,
$\widehat{\Omega}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}=dZ^{M}\widehat{\Omega}_{M\widetilde{\alpha}}{}^{\widetilde{\beta}}$.
From this, we define the curvature and torsion 2-forms. The curvatures are

$\displaystyle
R_{\alpha}{}^{\beta}=d\Omega_{\alpha}{}^{\beta}+\Omega_{\alpha}{}^{\gamma}\Omega_{\gamma}{}^{\alpha},\quad\widehat{R}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}=d\widehat{\Omega}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}+\widehat{\Omega}_{\widetilde{\alpha}}{}^{\widetilde{\gamma}}\widehat{\Omega}_{\widetilde{\gamma}}{}^{\widetilde{\beta}},$
(2.7)

here the wedge product is assumed. The torsion is the covariant derivative of
the vielbein one-form, for $T^{\alpha}$ and $T^{\widetilde{\alpha}}$ we have

$\displaystyle T^{\alpha}=\nabla
E^{\alpha}=dE^{\alpha}+E^{\beta}\Omega_{\beta}{}^{\alpha},\quad
T^{\widetilde{\alpha}}=\nabla
E^{\widetilde{\alpha}}=dE^{\widetilde{\alpha}}+E^{\widetilde{\beta}}\widehat{\Omega}_{\widetilde{\beta}}{}^{\widetilde{\alpha}},$
(2.8)

where, again, the wedge product is assumed. There are two possible definitions
of torsion as the covariant derivative of $E^{a}$. The first choice uses
$\Omega^{ab}$ as the connection, and the second uses $\widehat{\Omega}^{ab}$
as a connection. Then we have

$\displaystyle T^{a}=\nabla
E^{a}=dE^{a}+E^{b}\Omega_{b}{}^{a},\quad\widehat{T}^{a}=\widehat{\nabla}E^{a}=dE^{a}+E^{b}\widehat{\Omega}_{b}{}^{a}.$
(2.9)

Note that $\widehat{T}_{AB}{}^{c}=T_{AB}{}^{c}$ unless $A$ or $B$ takes the
value $a$ or $b$. Similarly, two possible curvatures are defined with
$\Omega^{ab}$ or $\widehat{\Omega}^{ab}$. They are

$\displaystyle
R_{a}{}^{b}=d\Omega_{a}{}^{b}+\Omega_{a}{}^{c}\Omega_{c}{}^{b},\quad\widehat{R}_{a}{}^{b}=d\widehat{\Omega}_{a}{}^{b}+\widehat{\Omega}_{a}{}^{c}\widehat{\Omega}_{c}{}^{b}.$
(2.10)

The torsions of (2.7) satisfy the Bianchi identities, they are

$\displaystyle\nabla T^{\alpha}=E^{\beta}R_{\beta}{}^{\alpha},\quad\nabla
T^{\widetilde{\alpha}}=E^{\widetilde{\beta}}\widehat{R}_{\widetilde{\beta}}{}^{\widetilde{\alpha}},$
(2.11)

and the torsions in (2.9) satisfy the identities

$\displaystyle\nabla
T^{a}=E^{b}R_{b}{}^{a},\quad\widehat{\nabla}\widehat{T}^{a}=E^{b}\widehat{R}_{b}{}^{a}.$
(2.12)

From the superspace point of view, the curvature $\widehat{R}_{a}{}^{b}$ and
torsion $\widehat{T}^{a}$ are superfluous; they do not contain any new
geometric information. Below, we will show how they relate to other torsion
and curvature components. Some constraints to have an on-shell superspace are
simpler when written in terms of $\widehat{R}_{a}{}^{b}$ and
$\widehat{T}^{a}$. Furthermore, they are also essential since some sigma model
couplings are given by these tensors in a background field expansion [22]
[23]. Another background field is the Kalb-Ramond two-form $B$. Its exterior
derivative is $H=dB$, and the Bianchi identity is $dH=0$.

BRST symmetry imposes constraints on background torsion and $H$-components. We
have $T_{AB}{}^{C}=H_{ABC}=0$ for $(A,B,C)\in(\alpha,\widetilde{\alpha})$ and

$\displaystyle
T_{\alpha\widetilde{\beta}}{}^{a}=H_{a\alpha\widetilde{\alpha}}=H_{ab\alpha}=H_{ab\widetilde{\alpha}}=T_{a\alpha}{}^{\beta}=T_{a\widetilde{\alpha}}{}^{\widetilde{\beta}}=0$
(2.13) $\displaystyle T_{\alpha\beta}{}^{a}=-\gamma^{a}_{\alpha\beta},\quad
T_{\widetilde{\alpha}\widetilde{\beta}}{}^{a}=-\gamma^{a}_{\widetilde{\alpha}\widetilde{\beta}},\quad
H_{a\alpha\beta}=-(\gamma_{a})_{\alpha\beta},\quad
H_{a\widetilde{\alpha}\widetilde{\beta}}=(\gamma_{a})_{\widetilde{\alpha}\widetilde{\beta}}.$
(2.14)

We now use these constraints in Bianchi identities to obtain constraints on
other torsion components. As it was done in [18], Bianchi identities imply

$\displaystyle T_{\alpha
ab}=2(\gamma_{ab}\Omega)_{\alpha},\quad\widehat{T}_{\widetilde{\alpha}ab}=2(\gamma_{ab}\widehat{\Omega})_{\widetilde{\alpha}},$
(2.15)
$\displaystyle\Omega_{\widetilde{\alpha}}=\widehat{\Omega}_{\alpha}=\Omega_{a}=\widehat{\Omega}_{a}=T_{\widetilde{\alpha}ab}=\widehat{T}_{\alpha
ab}=0.$ (2.16)

Consider the Bianchi identity $dH=0$. We have two options: they are

$\displaystyle\nabla_{[A}H_{BCD]}+\frac{3}{2}T_{[AB}{}^{E}H_{ECD]}=0.$ (2.17)

or

$\displaystyle\widehat{\nabla}_{[A}H_{BCD]}+\frac{3}{2}\widehat{T}_{[AB}{}^{E}H_{ECD]}=0.$
(2.18)

which are equivalent. These identities with
$(A,B,C,D)\in(\alpha,\widetilde{\alpha})$ and
$(A,B,C,D)\in(a,\alpha,\widetilde{\alpha})$ are trivially satisfied. Consider
the identity (2.17) with $(A,B,C,D)\in(a,b,\alpha,\beta)$, it implies that
$T_{abc}+H_{abc}=0$. Similarly, the identity (2.18) with
$(A,B,C,D)\in(a,b,\widetilde{\alpha},\widetilde{\beta})$, it implies that
$\widehat{T}_{abc}-H_{abc}=0$. At this point $\widehat{\Omega}_{a}{}^{b}$ is
given by $\Omega_{a}{}^{b}$ and torsion components. In fact,

$\displaystyle\widehat{\Omega}_{ca}{}^{b}=\Omega_{ca}{}^{b}-T_{ca}{}^{b},\quad\widehat{\Omega}_{\alpha
a}{}^{b}=\Omega_{\alpha a}{}^{b}-T_{\alpha
a}{}^{b},\quad\widehat{\Omega}_{\widetilde{\alpha}a}{}^{b}=\Omega_{\widetilde{\alpha}a}{}^{b}+\widehat{T}_{\widetilde{\alpha}a}{}^{b}.$
(2.19)

We now add the remaining constraints for the background superfields. They are

$\displaystyle
R_{\widetilde{\alpha}\widetilde{\beta}\alpha}{}^{\beta}=\widehat{R}_{\alpha\beta\widetilde{\alpha}}{}^{\widetilde{\beta}}=0,\quad
T_{a\alpha}{}^{\widetilde{\alpha}}=(\gamma_{a})_{\alpha\beta}{}P^{\beta\widetilde{\alpha}},\quad
T_{a\widetilde{\alpha}}{}^{\alpha}=-(\gamma_{a})_{\widetilde{\alpha}\widetilde{\beta}}P^{\alpha\widetilde{\beta}},$
(2.20) $\displaystyle
C_{\alpha}{}^{\beta\widetilde{\alpha}}=-\nabla_{\alpha}P^{\beta\widetilde{\alpha}},\quad
C_{\widetilde{\alpha}}{}^{\widetilde{\beta}\alpha}=\nabla_{\widetilde{\alpha}}P^{\alpha\widetilde{\beta}},$
(2.21) $\displaystyle
R_{a\widetilde{\alpha}\alpha}{}^{\beta}=-(\gamma_{a})_{\widetilde{\alpha}\widetilde{\beta}}C_{\alpha}{}^{\beta\widetilde{\beta}},\quad\widehat{R}_{a\alpha\widetilde{\alpha}}{}^{\widetilde{\beta}}=-(\gamma_{a})_{\alpha\beta}C_{\widetilde{\alpha}}{}^{\widetilde{\beta}\alpha},$
(2.22) $\displaystyle
S_{\alpha\widetilde{\alpha}}{}^{\beta\widetilde{\beta}}=\nabla_{\alpha}C_{\widetilde{\alpha}}{}^{\beta\widetilde{\beta}}+\widehat{R}_{\alpha\widetilde{\gamma}\widetilde{\alpha}}{}^{\widetilde{\beta}}P^{\beta\widetilde{\gamma}}=\nabla_{\widetilde{\alpha}}C_{\alpha}{}^{\beta\widetilde{\beta}}-R_{\widetilde{\alpha}\gamma\alpha}{}^{\beta}P^{\gamma\widetilde{\beta}},$
(2.23)

where $P$ is the RR field-strength.

As it was shown in [18] (see also [23]),

$\displaystyle\Omega_{\alpha}=4\nabla_{\alpha}\Phi,\quad\widehat{\Omega}_{\widetilde{\alpha}}=4\nabla_{\widetilde{\alpha}}\Phi,$
(2.24)

where $\Phi$ is the dilaton superfield. Even derivatives of $\Omega_{\alpha}$
and $\widehat{\Omega}_{\widetilde{\alpha}}$ are connected to torsion
components according to [23]

$\displaystyle\gamma^{b}_{\alpha\beta}{}T_{ab}{}^{\beta}=8\nabla_{a}\Omega_{\alpha},\quad\gamma^{b}_{\widetilde{\alpha}\widetilde{\beta}}T_{ab}{}^{\widetilde{\beta}}=8\nabla_{a}\widehat{\Omega}_{\widetilde{\alpha}}.$
(2.25)

The fermionic derivatives of $\Omega_{\alpha}$ and
$\Omega_{\widetilde{\alpha}}$ are related to the RR field-strength superfield
as

$\displaystyle\nabla_{\widetilde{\alpha}}\Omega_{\alpha}=\frac{1}{8}(\gamma_{a}P\gamma^{a})_{\alpha\widetilde{\alpha}},\quad\nabla_{\alpha}\widehat{\Omega}_{\widetilde{\alpha}}=-\frac{1}{8}(\gamma_{a}P\gamma^{a})_{\alpha\widetilde{\alpha}}.$
(2.26)

These equations come from the Bianchi identities involving
$R_{\widetilde{\alpha}(\alpha\beta)}{}^{\beta}$ and
$\widehat{R}_{\alpha(\widetilde{\alpha}\widetilde{\beta})}{}^{\widetilde{\beta}}$.

The torsion component $T_{abc}$ is related to $\Phi$. Mixing the Bianchi
identity $R_{(\alpha\beta\gamma)}{}^{\rho}=0$ and the Bianchi identity
involving $R_{\alpha\beta ab}$, together with the Bianchi identity involving
$\nabla_{\alpha}H_{abc}$ and the constraint $H_{abc}+T_{abc}=0$, we obtain the
equation

$\displaystyle
R_{(\alpha\beta}\delta_{\gamma)}^{\rho}-\gamma_{a}^{\rho\sigma}\gamma^{a}_{(\alpha\beta}\nabla_{\gamma)}\Omega_{\sigma}+\frac{1}{16}(\gamma^{ab})_{(\alpha}{}^{\rho}\gamma^{c}_{\beta\gamma)}T_{abc}=0,$
(2.27)

where $R_{\alpha\beta}=\nabla_{(\alpha}\Omega_{\beta)}$. This implies

$\displaystyle\nabla_{\alpha}\Omega_{\beta}=\frac{1}{4}\nabla_{\alpha}\nabla_{\beta}\Phi=\frac{1}{8}\gamma^{a}_{\alpha\beta}\nabla_{a}\Phi-\frac{1}{96}\gamma^{abc}_{\alpha\beta}H_{abc}.$
(2.28)

Similarly, the Bianchi identity
$\widehat{R}_{(\widetilde{\alpha}\widetilde{\beta}\widetilde{\gamma})}{}^{\widetilde{\rho}}=0$,
the Bianchi identities involving
$\widehat{R}_{\widetilde{\alpha}\widetilde{\beta}ab}$ and
$\widehat{\nabla}_{\widetilde{\alpha}}H_{abc}$, together with the constraint
$H_{abc}-\widehat{T}_{abc}=0$ imply the equation

$\displaystyle\nabla_{\widetilde{\alpha}}\widehat{\Omega}_{\widetilde{\beta}}=\frac{1}{4}\nabla_{\widetilde{\alpha}}\nabla_{\widetilde{\beta}}\Phi=\frac{1}{8}\gamma^{a}_{\widetilde{\alpha}\widetilde{\beta}}\nabla_{a}\Phi+\frac{1}{96}\gamma^{abc}_{\widetilde{\alpha}\widetilde{\beta}}H_{abc}.$
(2.29)

###  2.1 Redefining the connections

We now redefine the connections to have only one connection for vectors. This
will be useful in the next section. As we reviewed above, the Berkovits-Howe
constraints for type II supergravity imply that

$\displaystyle\Omega=E^{\alpha}\Omega_{\alpha}=\frac{1}{4}E^{\alpha}\nabla_{\alpha}\Phi,\quad\widehat{\Omega}=E^{\widetilde{\alpha}}\widehat{\Omega}_{\widetilde{\alpha}}=\frac{1}{4}E^{\widetilde{\alpha}}\nabla_{\widetilde{\alpha}}\Phi,$
(2.30)

and $\Omega_{ab},\widehat{\Omega}_{ab}$ are related according to [24]

$\displaystyle\widehat{\Omega}_{cab}-\Omega_{cab}=-T_{cab}=H_{cab},\quad\widehat{\Omega}_{\alpha
ab}-\Omega_{\alpha ab}=-T_{\alpha
ab},\quad\widehat{\Omega}_{\widetilde{\alpha}ab}-\Omega_{\widetilde{\alpha}ab}=\widehat{T}_{\widetilde{\alpha}ab},$
(2.31)

or written in terms of forms

$\displaystyle\widehat{\Omega}_{ab}-\Omega_{ab}=F_{ab},$ (2.32)

where

$\displaystyle F_{ab}=E^{c}H_{cab}-E^{\alpha}T_{\alpha
ab}+E^{\widetilde{\alpha}}\widehat{T}_{\widetilde{\alpha}ab}.$ (2.33)

We now redefine the connections to have only one connection for vectors. The
redefinition is

$\displaystyle\Omega_{\alpha}{}^{\beta}=\Omega^{\prime}_{\alpha}{}^{\beta}+x\delta_{\alpha}^{\beta}\Omega-\frac{1}{8}(\gamma^{ab})_{\alpha}{}^{\beta}F_{ab},\quad\widehat{\Omega}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}=\widehat{\Omega}^{\prime}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}+x\delta_{\widetilde{\alpha}}^{\widetilde{\beta}}\widehat{\Omega}+\frac{1}{8}(\gamma^{ab})_{\widetilde{\alpha}}{}^{\widetilde{\beta}}F_{ab},$
(2.34)

where $x$ is a free parameter that can be chosen later for convenience. From
here, one obtains

$\displaystyle\Omega^{\prime}_{\alpha}{}^{\beta}=(1-x)\Omega\delta_{\alpha}^{\beta}+\frac{1}{4}(\gamma^{ab})_{\alpha}{}^{\beta}\left(\Omega_{ab}+\frac{1}{2}F_{ab}\right),$
(2.35)
$\displaystyle\widehat{\Omega}^{\prime}_{\widetilde{\alpha}}{}^{\widetilde{\beta}}=(1-x)\widehat{\Omega}\delta_{\widetilde{\alpha}}^{\widetilde{\beta}}+\frac{1}{4}(\gamma^{ab})_{\widetilde{\alpha}}{}^{\widetilde{\beta}}\left(\widehat{\Omega}_{ab}-\frac{1}{2}F_{ab}\right).$
(2.36)

Note that
$\Omega_{ab}+\frac{1}{2}F_{ab}=\widehat{\Omega}_{ab}-\frac{1}{2}F_{ab}$
because (2.32). The new torsions can be calculated as follows

$\displaystyle T^{\prime\alpha}=\nabla^{\prime}E^{\alpha}=\nabla
E^{\alpha}-xE^{\alpha}\Omega+{\textstyle{1\over{\raise
0.3014pt\hbox{$\scriptstyle{2}$}}}}E^{\beta}F_{\beta}{}^{\alpha},$ (2.37)
$\displaystyle T^{\prime a}=\nabla^{\prime}E^{a}=\nabla
E^{a}+{\textstyle{1\over{\raise
0.3014pt\hbox{$\scriptstyle{2}$}}}}E^{b}F_{b}{}^{a},$ (2.38) $\displaystyle
T^{\prime\widetilde{\alpha}}=\nabla^{\prime}E^{\widetilde{\alpha}}=\nabla
E^{\widetilde{\alpha}}-xE^{\widetilde{\alpha}}\widehat{\Omega}-{\textstyle{1\over{\raise
0.3014pt\hbox{$\scriptstyle{2}$}}}}E^{\widetilde{\beta}}F_{\widetilde{\beta}}{}^{\widetilde{\alpha}}.$
(2.39)

The components of $T^{\prime\alpha}$ are

$\displaystyle T^{\prime}_{ab}{}^{\alpha}=T_{ab}{}^{\alpha},\quad
T^{\prime}_{a\widetilde{\beta}}{}^{\alpha}=T_{a\widetilde{\beta}}{}^{\alpha},\quad
T^{\prime}_{\widetilde{\gamma}\widetilde{\beta}}{}^{\alpha}=T_{\widetilde{\gamma}\widetilde{\beta}}{}^{\alpha}=0,$
(2.40) $\displaystyle
T^{\prime}_{\gamma\beta}{}^{\alpha}=\left(\frac{5}{2}-x\right)\delta^{\alpha}_{(\gamma}\Omega_{\beta)}-2\gamma^{a}_{\gamma\beta}(\gamma_{a}\Omega)^{\alpha},$
(2.41) $\displaystyle
T^{\prime}_{a\beta}{}^{\alpha}=\frac{1}{8}(\gamma^{bc})_{\beta}{}^{\alpha}H_{abc},\quad
T^{\prime}_{\widetilde{\gamma}\beta}{}^{\alpha}=\frac{1}{4}(\gamma^{ab})_{\beta}{}^{\alpha}(\gamma_{ab}\widehat{\Omega})_{\widetilde{\gamma}}.$
(2.42)

The components of $T^{\prime\widetilde{\alpha}}$ are

$\displaystyle
T^{\prime}_{ab}{}^{\widetilde{\alpha}}=T_{ab}{}^{\widetilde{\alpha}},\quad
T^{\prime}_{a\beta}{}^{\widetilde{\alpha}}=T_{a\beta}{}^{\widetilde{\alpha}},\quad
T^{\prime}_{\gamma\beta}{}^{\widetilde{\alpha}}=T_{\gamma\beta}{}^{\widetilde{\alpha}}=0,$
(2.43) $\displaystyle
T^{\prime}_{\widetilde{\gamma}\widetilde{\beta}}{}^{\widetilde{\alpha}}=\left(\frac{5}{2}-x\right)\delta^{\widetilde{\alpha}}_{(\widetilde{\gamma}}\widehat{\Omega}_{\widetilde{\beta})}-2\gamma^{a}_{\widetilde{\gamma}\widetilde{\beta}}(\gamma_{a}\widehat{\Omega})^{\widetilde{\alpha}},$
(2.44) $\displaystyle
T^{\prime}_{a\widetilde{\beta}}{}^{\widetilde{\alpha}}=-\frac{1}{8}(\gamma^{bc})_{\widetilde{\beta}}{}^{\widetilde{\alpha}}H_{abc},\quad
T^{\prime}_{\gamma\widetilde{\beta}}{}^{\widetilde{\alpha}}=\frac{1}{4}(\gamma^{ab})_{\widetilde{\beta}}{}^{\widetilde{\alpha}}(\gamma_{ab}\Omega)_{\gamma}.$
(2.45)

And the components of $T^{\prime a}$ are

$\displaystyle T^{\prime}_{cb}{}^{a}=T_{cb}{}^{a}+H_{cb}{}^{a}=0,\quad
T^{\prime}_{\alpha\beta}{}^{a}=T_{\alpha\beta}{}^{a}=-\gamma^{a}_{\alpha\beta},\quad
T^{\prime}_{\widetilde{\alpha}\widetilde{\beta}}{}^{a}=T_{\widetilde{\alpha}\widetilde{\beta}}{}^{a}=-\gamma^{a}_{\widetilde{\alpha}\widetilde{\beta}}$
(2.46) $\displaystyle T^{\prime}_{\alpha
b}{}^{a}=(\gamma_{b}{}^{a}\Omega)_{\alpha},\quad
T^{\prime}_{\widetilde{\alpha}b}{}^{a}=(\gamma_{b}{}^{a}\widehat{\Omega})_{\widetilde{\alpha}},\quad
T^{\prime}_{\alpha\widetilde{\beta}}{}^{a}=T_{\alpha\widetilde{\beta}}{}^{a}=0.$
(2.47)

The new curvature two-form becomes

$\displaystyle R^{\prime}_{ab}=R_{ab}+\frac{1}{2}\nabla
F_{ab}+\frac{1}{4}F_{a}{}^{c}F_{cb},$ (2.48)

when it is expressed in components, we obtain

$\displaystyle
R^{\prime}_{ABab}=R_{ABab}+\frac{1}{2}\nabla_{[A}F_{B]ab}+\frac{1}{2}T_{AB}{}^{C}F_{Cab}-\frac{1}{4}F_{[Aa}{}^{c}F_{B]cb}.$
(2.49)

In particular,

$\displaystyle R^{\prime}_{\alpha\beta ab}=$
$\displaystyle\frac{1}{4}(\gamma_{[a})_{\alpha\beta}\nabla_{b]}\Phi+\frac{3}{8}\gamma^{c}_{\alpha\beta}\left(H_{abc}-6(\Omega\gamma_{abc}\Omega)\right)+\frac{1}{48}(\gamma_{abcde})_{\alpha\beta}\left(H^{cde}-6(\Omega\gamma^{cde}\Omega)\right),$
(2.50) $\displaystyle R^{\prime}_{\widetilde{\alpha}\widetilde{\beta}ab}=$
$\displaystyle\frac{1}{4}(\gamma_{[a})_{\widetilde{\alpha}\widetilde{\beta}}\nabla_{b]}\Phi-\frac{3}{8}\gamma^{c}_{\widetilde{\alpha}\widetilde{\beta}}\left(H_{abc}+6(\widehat{\Omega}\gamma_{abc}\widehat{\Omega})\right)-\frac{1}{48}(\gamma_{abcde})_{\widetilde{\alpha}\widetilde{\beta}}\left(H^{cde}+6(\widehat{\Omega}\gamma^{cde}\widehat{\Omega})\right),$
(2.51) $\displaystyle R^{\prime}_{\alpha\widetilde{\beta}ab}=$
$\displaystyle\frac{1}{8}(\gamma_{ab}\gamma_{c}P\gamma^{c})_{\alpha\widetilde{\beta}}+\frac{1}{8}(\gamma^{c}P\gamma_{c}\gamma_{ab})_{\alpha\widetilde{\beta}}-(\gamma_{[a}P\gamma_{b]})_{\alpha\widetilde{\beta}}+(\gamma_{[a}{}^{c}\Omega)_{\alpha}(\gamma_{b]c}\widehat{\Omega})_{\widetilde{\beta}}.$
(2.52)

Interestingly, with this choice of connection, the RR fields and $H$ appear
similarly in the torsions. We want to stress that the redefinition does not
change how the BRST charge acts in vertex operators in the lowest order of
$\alpha^{\prime}$. We will drop all ′s from torsions and curvatures in the
following sections. The scaling connections
$(\Omega_{\alpha},\widehat{\Omega}_{\widetilde{\alpha}})$, when they appear in
expressions below, are shorthand for
$(\frac{1}{4}\nabla_{\alpha}\Phi,\frac{1}{4}\nabla_{\widetilde{\alpha}}\Phi)$
respectively.

##  3 Superspace compactification

In this section, we will describe in more detail what we mean by superspace
compactification described in the introduction. Using the the
$\mathfrak{sl}(2,\mathbb{C})+\mathfrak{su}(4)$ decomposition the basic
operators acting on general tensor superfields are
$(\nabla_{a},\nabla_{i},\nabla_{\alpha
I},\widetilde{\nabla}_{\alpha\widetilde{I}},\nabla_{\dot{\alpha}}{}^{I},\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{I}},M_{ab},M_{ai},M_{ij})$.
The position of the index $\widetilde{I}$ depends on whether we are discussing
Type IIA or IIB. For Type IIB $\widetilde{\nabla}_{\alpha\widetilde{I}}$ means
$\widetilde{\nabla}_{\alpha I}$ and
$\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{I}}$ means
$\widetilde{\nabla}_{\dot{\alpha}}{}^{I}$ and for Type IIA
$\widetilde{\nabla}_{\alpha\widetilde{I}}$ means
$\widetilde{\nabla}_{\alpha}{}^{I}$ and
$\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{I}}$ means
$\widetilde{\nabla}_{\dot{\alpha}I}$.

A central concept in compactifications is the existence of nowhere-vanishing
spinors in the compactification manifold. This work will suppose two such
spinors exist and promote them to bosonic, complex superfields. We will denote
them $(\chi^{I},\widetilde{\chi}^{\widetilde{I}})$. The constraints they
satisfy will be derived throughout this paper. For Type IIB
$\widetilde{\chi}^{\widetilde{I}}=\widetilde{\chi}^{I}$ and for Type IIA
$\widetilde{\chi}^{\widetilde{I}}=\widetilde{\chi}_{I}$. Furthermore, they
satisfy the reality conditions $\left(\chi^{I}\right)^{\dagger}=\chi_{I}$ and
$\left(\widetilde{\chi}^{\widetilde{I}}\right)^{\dagger}=\widetilde{\chi}_{\widetilde{I}}$.
We define the following covariant derivatives

$\displaystyle\nabla_{\alpha}=\chi^{I}\nabla_{\alpha
I}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\nabla}_{\alpha\widetilde{I}},\quad\nabla_{\dot{\alpha}}=\chi_{I}\nabla_{\dot{\alpha}}{}^{I}+\widetilde{\chi}_{\widetilde{I}}\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{I}}.$
(3.1)

The set of operators
$(\nabla_{\alpha},\nabla_{\dot{\alpha}},\nabla_{\alpha\dot{\alpha}}\equiv\\{\nabla_{\alpha},\nabla_{\dot{\alpha}}\\},M_{ab})$
will be called four dimensional or space-time operators. The set composed by
$(\nabla_{i},M_{ij})$ will be called internal operators. Finally, the set
$(\Delta_{\alpha},\Delta_{\dot{\alpha}},\Delta_{\alpha
i},\Delta_{\dot{\alpha}i},\widetilde{\Delta}_{\alpha
i},\widetilde{\Delta}_{\dot{\alpha}i},M_{ai})$ defined in (1.9) will be called
mixed operators. For the next definition, we will use $\circ$ in place of any
combination of four-dimensional Lorentz indices $(a,\alpha,\dot{\alpha})$ and
$\bullet$ for any combination of local internal tangent space indices $(i,I)$.
A four dimensional tensor superfield ${\mathcal{T}}^{\circ}$ will satisfy

$\displaystyle\nabla_{i}{\mathcal{T}}^{\circ}=M_{ij}{\mathcal{T}}^{\circ}=0.$
(3.2)

Similarly, an internal tensor superfield will satisfy

$\displaystyle\nabla_{\alpha}{\mathcal{T}}^{\bullet}=\nabla_{\dot{\alpha}}{\mathcal{T}}^{\bullet}=M_{ab}{\mathcal{T}}^{\bullet}=0.$
(3.3)

Mixed operators will not be required to act trivially in any tensor
superfield. In the same way, mixed tensor superfields
${\mathcal{T}}^{\circ\bullet}$ will not be annihilated by any type of
operator. The first example of an internal tensor superfield is the dilaton
superfield $\Phi$. Since we want to preserve four-dimensional Poincaré or
$SO(2,4)$ covariance, it is natural to assume that it is an internal
superfield and satisfy

$\displaystyle\nabla_{\alpha}\Phi=\nabla_{\dot{\alpha}}\Phi=0.$ (3.4)

We will also see how this condition appears again later. Note that using
(2.24) the equation (3.4) can be written as

$\displaystyle\chi^{I}\Omega_{\alpha
I}+\widetilde{\chi}^{\widetilde{I}}\widehat{\Omega}_{\alpha\widetilde{I}}=0.$
(3.5)

Doing a Lorentz rotation with $M_{ai}$ we also have the following condition

$\displaystyle\chi^{I}\sigma_{IJ}^{i}\Omega_{\dot{\alpha}}{}^{J}+\widetilde{\chi}^{\widetilde{I}}\sigma^{i}_{\widetilde{I}\widetilde{J}}\widetilde{\Omega}_{\dot{\alpha}}{}^{\widetilde{J}}=0.$
(3.6)

The commutator $[M_{ai},M_{bj}]$ gives no further constraints if $\chi$ and
$\widetilde{\chi}$ are proportional to each other, but it will give a new
constraint otherwise. There are also the conjugates of the equations above.

The spinor superfields $(\chi,\widetilde{\chi})$ are further examples of
internal superfields

$\displaystyle\nabla_{\alpha}\chi^{I}=\nabla_{\dot{\alpha}}\chi^{I}=\nabla_{\alpha}\widetilde{\chi}^{\widetilde{I}}=\nabla_{\dot{\alpha}}\widetilde{\chi}^{\widetilde{I}}=0.$
(3.7)

Furthermore, we will require that the four-dimensional operators satisfy an
$AdS_{4}$ (or $d=4$ Poincaré) superalgebra

$\displaystyle\\{\nabla_{\alpha},\nabla_{\beta}\\}=FM_{\alpha\beta},\quad\\{\nabla_{\dot{\alpha}},\nabla_{\dot{\beta}}\\}=-F^{\dagger}M_{\dot{\alpha}\dot{\beta}},\quad\\{\nabla_{\alpha},\nabla_{\dot{\alpha}}\\}=\nabla_{\alpha\dot{\alpha}},$
(3.8)
$\displaystyle[\nabla_{\alpha},\nabla_{\beta\dot{\beta}}]=-\frac{1}{2}\epsilon_{\alpha\beta}F\nabla_{\dot{\beta}},\quad[\nabla_{\dot{\alpha}},\nabla_{\beta\dot{\beta}}]=\frac{1}{2}\epsilon_{\dot{\alpha}\dot{\beta}}F^{\dagger}\nabla_{\beta},$
(3.9)
$\displaystyle[\nabla_{\alpha\dot{\alpha}},\nabla_{\beta\dot{\beta}}]=\frac{1}{2}|F|^{2}\left(\epsilon_{\dot{\alpha}\dot{\beta}}M_{\alpha\beta}+\epsilon_{\alpha\beta}M_{\dot{\alpha}\dot{\beta}}\right),$
(3.10)

where $F$ is a bosonic complex superfield and
$(M_{\alpha\beta},M_{\dot{\alpha}\dot{\beta}})$ are the four dimensional
Lorentz generators in spinor basis. In the case of a flat four-dimensional
compactification, we will require $F=0$. The third equation in (3.8) should be
interpreted as the definition of $\nabla_{\alpha\dot{\alpha}}$. Once the anti-
commutators in (3.8) are defined, the closure of the algebra fixes the
remaining equations. For that to happen, it is necessary to impose that $F$ is
an internal superfield

$\displaystyle\nabla_{\alpha}F=\nabla_{\dot{\alpha}}F=0.$ (3.11)

Usually $\nabla_{\alpha\dot{\alpha}}$ is defined with an $i$, which explains
the unusual sign in (3.10). One of our goals is to compute the superfield $F$
in terms of the torsions and curvatures implied by the Berkovits-Howe
supergravity constraints presented in previous sections and
$(\chi^{I},\widetilde{\chi}^{\widetilde{I}})$.

###  3.1 Factorization and the Pure Spinor Equations

From the definitions above, it is natural to assume that space-time operators
map space-time tensors to themselves and analogously to internal operators.
There are possible nontrivial commutators among the super-translations in the
two sets. Since $[\nabla_{i},\nabla_{\alpha}]$ act trivially on both
${\mathcal{T}}^{\circ}$ and ${\mathcal{T}}^{\bullet}$ we will require that

$\displaystyle[\nabla_{i},\nabla_{\alpha}]=0,$ (3.12)

The Bianchi identity with
$[\nabla_{i},\\{\nabla_{\alpha},\nabla_{\dot{\alpha}}\\}]$ implies that
$[\nabla_{i},\nabla_{\alpha\dot{\alpha}}]$ also vanishes. The condition (3.12)
implies many expected restrictions, a few of which are already implied by
(3.4). Expanding the left hand side of (3.12) we have

$\displaystyle[\nabla_{i},\nabla_{\alpha}]=$
$\displaystyle\;(\nabla_{i}\chi^{I})\nabla_{\alpha
I}+(\nabla_{i}\widetilde{\chi}^{\widetilde{I}})\widetilde{\nabla}_{\alpha\widetilde{I}}+\chi^{I}[\nabla_{i},\nabla_{\alpha
I}]+\widetilde{\chi}^{\widetilde{I}}[\nabla_{i},\widetilde{\nabla}_{\alpha\widetilde{I}}]$
(3.13) $\displaystyle=$
$\displaystyle\;\left(\left(\nabla_{i}\chi^{I}-\frac{1}{8}\chi^{J}(\sigma_{jk})_{J}{}^{I}H_{ijk}\right)\delta^{\beta}_{\alpha}-\widetilde{\chi}^{\widetilde{J}}(\sigma_{i})_{\widetilde{J}\widetilde{K}}P^{\beta
I}{}_{\alpha}{}^{\widetilde{K}}-\frac{1}{8}(\sigma^{ab})_{\alpha}{}^{\beta}\chi^{I}H_{iab}\right)\nabla_{\beta
I}$ (3.14) $\displaystyle+$
$\displaystyle\;\left(\left(\nabla_{i}\widetilde{\chi}^{\widetilde{I}}+\frac{1}{8}\widetilde{\chi}^{\widetilde{J}}(\sigma_{jk})_{\widetilde{J}}{}^{\widetilde{I}}H_{ijk}\right)\delta^{\beta}_{\alpha}+\chi^{J}(\sigma_{i})_{JK}P_{\alpha}{}^{K\beta\widetilde{I}}+\frac{1}{8}(\sigma^{ab})_{\alpha}{}^{\beta}\widetilde{\chi}^{\widetilde{I}}H_{iab}\right)\widetilde{\nabla}_{\beta\widetilde{I}}$
(3.15) $\displaystyle-$
$\displaystyle\left(\chi^{I}(\sigma_{i})_{IJ}\Omega^{\dot{\beta}J}+\widetilde{\chi}^{\widetilde{I}}(\sigma_{i})_{\widetilde{I}\widetilde{J}}\widehat{\Omega}^{\dot{\beta}\widetilde{J}}\right)(\sigma^{a})_{\alpha\dot{\beta}}\nabla_{a}-\left(\chi^{I}(\sigma_{ij})_{I}{}^{J}\Omega_{\alpha
J}+\widetilde{\chi}^{\widetilde{I}}(\sigma_{ij})_{\widetilde{I}}{}^{\widetilde{J}}\widehat{\Omega}_{\alpha\widetilde{J}}\right)\nabla_{j}$
(3.16) $\displaystyle-$
$\displaystyle\;\left(\frac{1}{8}\chi^{I}(\sigma_{j})_{IJ}(\sigma^{a})_{\alpha}{}^{\dot{\beta}}H_{ija}-\widetilde{\chi}^{\widetilde{I}}(\sigma_{i})_{\widetilde{I}\widetilde{K}}P^{\dot{\beta}}{}_{J}{}_{\alpha}{}^{\widetilde{K}}\right)\nabla_{\dot{\beta}}{}^{J}$
(3.17) $\displaystyle-$
$\displaystyle\;\left(-\frac{1}{8}\widetilde{\chi}^{\widetilde{I}}(\sigma_{j})_{\widetilde{I}\widetilde{J}}(\sigma^{a})_{\alpha}{}^{\dot{\beta}}H_{ija}+\chi^{I}(\sigma_{i})_{IJ}P^{\alpha
J\dot{\beta}}{}_{\widetilde{J}}\right)\widetilde{\nabla}_{\dot{\beta}}{}^{\widetilde{J}}$
(3.18) $\displaystyle+$ $\displaystyle\;\frac{1}{2}\left(\chi^{I}R_{i\alpha
I}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}R_{i\alpha\widetilde{I}}{}^{\underline{ab}}\right)M_{\underline{ab}}.$
(3.19)

We will now discuss the implication of each line of the second equality in
(3.13). The last line provides constraint equations involving the curvature,
namely

$\displaystyle\chi^{I}R_{i\alpha
I}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}R_{i\alpha\widetilde{I}}{}^{\underline{ab}}=0.$
(3.20)

This condition has mass dimension $3/2$; its first non-vanishing contribution
in the covariant $\theta$-expansion has dimension two. The first term in the
third line, with $\nabla_{a}$, vanish (3.6). The second term, with
$\nabla_{j}$, implies that

$\displaystyle\chi^{I}(\sigma_{ij})_{I}{}^{J}\Omega_{\alpha
J}+\widetilde{\chi}^{\widetilde{I}}(\sigma_{ij})_{\widetilde{I}}{}^{\widetilde{J}}\widehat{\Omega}_{\alpha\widetilde{J}}=0$
(3.21)

This is not an independent condition if $\widetilde{\chi}$ is proportional to
$\chi$, as explained after (3.6). The fourth and fifth lines imply relations
between $H$ and the RR field strengths. However, demanding that any background
is an internal superfield, we should also require that

$\displaystyle H_{ijc}=P^{\alpha I\dot{\beta}}{}_{\widetilde{J}}=0.$ (3.22)

This requirement also demands that

$\displaystyle H_{iab}=0,$ (3.23)

and that

$\displaystyle P^{\alpha
I}{}_{\beta}{}^{\widetilde{J}}=\delta^{\alpha}{}_{\beta}F^{I\widetilde{J}},$
(3.24)

since this is sufficient for $[M_{ab},P^{\alpha
I}{}_{\beta}{}^{\widetilde{J}}]=0$.

Finally, the factorization condition (3.1) implies that the normalizable
spinors satisfy

$\displaystyle\nabla_{i}\chi^{I}-\frac{1}{8}\chi^{J}(\sigma_{jk})_{J}{}^{I}H_{ijk}+\widetilde{\chi}^{\widetilde{J}}(\sigma_{i})_{\widetilde{J}\widetilde{K}}F^{I\widetilde{K}}=0,$
(3.25)
$\displaystyle\nabla_{i}\widetilde{\chi}^{\widetilde{I}}+\frac{1}{8}\widetilde{\chi}^{\widetilde{J}}(\sigma_{jk})_{\widetilde{J}}{}^{\widetilde{I}}H_{ijk}+\chi^{J}(\sigma_{i})_{JK}F^{K\widetilde{I}}=0.$
(3.26)

Defining bi-spinors superfields with $(\chi,\widetilde{\chi})$, e.g.
$\Psi^{IJ}=\chi^{I}\chi^{J}$, they will satisfy the tangent space version of
the so-called pure spinor equations [25, 26] up to a scale factor in
$(\chi,\widetilde{\chi})$. The exponential of the dilaton that appears in the
literature is contained in $F^{I\widetilde{J}}$, since Ramond-Ramond vertex
operators have this factor [27]. However, a fundamental difference exists;
(3.25) is a superspace condition. It implies higher mass dimension equations
in the covariant $\theta$ expansion.

###  3.2 The cosmological constant

We now compute the algebra (3.8) and imposing its form will imply further
constraints on the background geometry. Recall the definition

$\displaystyle\nabla_{\alpha}=\chi^{I}\nabla_{\alpha
I}+\widetilde{\chi}^{\widetilde{I}}\nabla_{\alpha\widetilde{I}}.$ (3.27)

Consider $\\{\nabla_{\alpha},\nabla_{\beta}\\}$. It becomes

$\displaystyle\\{\nabla_{\alpha},\nabla_{\beta}\\}$
$\displaystyle=\chi^{I}\chi^{J}\\{\nabla_{\alpha I},\nabla_{\beta
J}\\}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}\\{\nabla_{\alpha\widetilde{I}},\nabla_{\beta\widetilde{J}}\\}$
(3.28)
$\displaystyle+\chi^{I}\widetilde{\chi}^{\widetilde{J}}\left(\\{\nabla_{\alpha
I},\nabla_{\beta\widetilde{J}}\\}+\\{\nabla_{\beta
I},\nabla_{\alpha\widetilde{J}}\\}\right).$ (3.29)

We use the torsion $T^{\prime}$ and curvature $R^{\prime}$ components defined
in the previous section (equations (2.40), (2.43), (2.46), (2.48)) to compute
the anticommutators. We obtain

$\displaystyle\\{\nabla_{\alpha},\nabla_{\beta}\\}=\left(x-\frac{5}{2}\right)\chi^{I}\chi^{J}\Omega_{(\alpha
I}\nabla_{\beta)J}+\left(x-\frac{5}{2}\right)\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}\widehat{\Omega}_{(\alpha\widetilde{I}}\nabla_{\beta)\widetilde{J}}$
(3.30)
$\displaystyle+\chi^{I}\widetilde{\chi}^{\widetilde{J}}\left(\Omega_{(\alpha
I}\nabla_{\beta)\widetilde{J}}+\widehat{\Omega}_{(\alpha\widetilde{J}}\nabla_{\beta)I}-\frac{1}{4}(\sigma_{ij})_{I}{}^{K}(\sigma_{ij})_{\widetilde{J}}{}^{\widetilde{L}}\left(\Omega_{(\alpha
K}\nabla_{\beta)\widetilde{L}}+\widehat{\Omega}_{(\alpha\widetilde{L}}\nabla_{\beta)K}\right)\right)$
(3.31)
$\displaystyle+\frac{1}{2}U_{\alpha\beta}{}^{\underline{ab}}M_{\underline{ab}},$
(3.32)

where

$\displaystyle U_{\alpha\beta}{}^{\underline{ab}}=\chi^{I}\chi^{J}R_{\alpha
I\beta
J}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}R_{\alpha\widetilde{I}\beta\widetilde{J}}{}^{\underline{ab}}+\chi^{I}\widetilde{\chi}^{\widetilde{J}}R_{(\alpha
I\beta)\widetilde{J}}{}^{\underline{ab}}.$ (3.33)

The vanishing of the derivative terms imposes that

$\displaystyle\left(x-\frac{7}{2}\right)\chi^{J}\chi^{I}\Omega_{\alpha
I}-\frac{1}{4}(\chi\sigma_{ij})^{J}(\widetilde{\chi}\sigma_{ij}\widehat{\Omega}_{\alpha})=0,$
(3.34)
$\displaystyle\left(x-\frac{7}{2}\right)\widetilde{\chi}^{\widetilde{J}}\widetilde{\chi}^{\widetilde{I}}\widehat{\Omega}_{\alpha\widetilde{I}}-\frac{1}{4}(\widetilde{\chi}\sigma_{ij})^{\widetilde{J}}(\chi\sigma_{ij}\Omega_{\alpha})=0,$
(3.35)

together with their complex conjugates. We have used (3.5) to simplify the
expressions above. We will discuss solving this in section 4. The term with
$M_{ab}$ will give the $F$ of the first anti-commutator in (3.8). The terms
with $M_{ai}$ and $M_{ij}$ should vanish

$\displaystyle U_{\alpha\beta}{}^{ai}=\chi^{I}\chi^{J}R_{\alpha I\beta
J}{}^{{ai}}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}R_{\alpha\widetilde{I}\beta\widetilde{J}}{}^{{ai}}+\chi^{I}\chi^{\widetilde{J}}R_{\alpha
I\beta\widetilde{J}}{}^{{ai}}+\chi^{I}\widetilde{\chi}^{\widetilde{J}}R_{\beta
I\alpha\widetilde{J}}{}^{{ai}}=0,$ (3.36) $\displaystyle
U_{\alpha\beta}{}^{ij}=\chi^{I}\chi^{J}R_{\alpha I\beta
J}{}^{{ij}}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}R_{\alpha\widetilde{I}\beta\widetilde{J}}{}^{{ij}}+\chi^{I}\chi^{\widetilde{J}}R_{\alpha
I\beta\widetilde{J}}{}^{{ij}}+\chi^{I}\widetilde{\chi}^{\widetilde{J}}R_{\beta
I\alpha\widetilde{J}}{}^{{ij}}=0.$ (3.37)

With the assumptions above, we have

$\displaystyle\\{\nabla_{\alpha},\nabla_{\beta}\\}$
$\displaystyle=\frac{1}{2}\left(\chi^{I}\chi^{J}R_{\alpha I\beta
J}{}^{{ab}}+\widetilde{\chi}^{\widetilde{I}}\chi^{\widetilde{J}}R_{\alpha\widetilde{I}\beta\widetilde{J}}{}^{{ab}}+\chi^{I}\widetilde{\chi}^{\widetilde{J}}(R_{\alpha
I\beta\widetilde{J}}{}^{{ab}}+R_{\beta
I\alpha\widetilde{J}}{}^{{ab}})\right)M_{{ab}}.$ (3.38)

And finally, we obtain that

$\displaystyle
F=\frac{1}{24}(\sigma_{ab})^{\alpha\beta}\left(\chi^{I}\chi^{J}R_{\alpha
I\beta
J}{}^{{ab}}+\widetilde{\chi}^{\widetilde{I}}\chi^{\widetilde{J}}R_{\alpha\widetilde{I}\beta\widetilde{J}}{}^{{ab}}+\chi^{I}\widetilde{\chi}^{\widetilde{J}}(R_{\alpha
I\beta\widetilde{J}}{}^{{ab}}+R_{\beta
I\alpha\widetilde{J}}{}^{{ab}})\right).$ (3.39)

The coefficient of the four-dimensional Lorentz generator $M_{ab}$ is
symmetric in $\alpha\beta$, so there are no further constraints coming from
(3.38). Later, it will be checked if this is the same internal superfield in
the first anti-commutator (3.9).

###  3.3 Closure of the algebra

The next step is to define
$\nabla_{\alpha\dot{\alpha}}=\\{\nabla_{\alpha},\nabla_{\dot{\alpha}}\\}$.
Using
$\nabla_{\dot{\alpha}}\chi^{I}=\nabla_{\dot{\alpha}}\widetilde{\chi}^{\widetilde{I}}=\nabla_{\alpha}\chi_{I}=\nabla_{\alpha}\widetilde{\chi}_{\widetilde{I}}=0$
we obtain

$\displaystyle\nabla_{\alpha\dot{\alpha}}$
$\displaystyle=|\chi|^{2}(\sigma^{a})_{\alpha\dot{\alpha}}\nabla_{a}+|\widetilde{\chi}|^{2}(\sigma^{a})_{\alpha\dot{\alpha}}\nabla_{a}+\frac{1}{2}U_{\alpha\dot{\alpha}}{}^{\underline{ab}}M_{\underline{ab}}$
(3.40) $\displaystyle+S_{\alpha
J}\nabla_{\dot{\alpha}}{}^{J}+\widetilde{S}_{\alpha\widetilde{J}}\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{J}}+S_{\dot{\alpha}}{}^{J}\nabla_{\alpha
J}+\widetilde{S}_{\dot{\alpha}}{}^{\widetilde{J}}\widetilde{\nabla}_{\alpha\widetilde{J}}$
(3.41)

where

$\displaystyle S_{\alpha J}=(x-5/2)\chi^{I}\chi_{J}\Omega_{\alpha
I}+4|\chi|^{2}\Omega_{\alpha
J}+(\chi\sigma_{i})_{J}(\widetilde{\chi}\sigma_{i}\widehat{\Omega}_{\alpha})-\frac{1}{4}(\chi\sigma_{ij})_{J}(\widetilde{\chi}\sigma_{ij}\widehat{\Omega}_{\alpha}),$
(3.42)
$\displaystyle\widetilde{S}_{\alpha\widetilde{J}}=(x-5/2)\widetilde{\chi}^{I}\widetilde{\chi}_{J}\widehat{\Omega}_{\alpha\widetilde{I}}+4|\widetilde{\chi}|^{2}\widehat{\Omega}_{\alpha\widetilde{J}}+(\widetilde{\chi}\sigma_{i})_{\widetilde{J}}(\chi\sigma_{i}\Omega_{\alpha})-\frac{1}{4}(\widetilde{\chi}\sigma_{ij})_{\widetilde{J}}(\chi\sigma_{ij}\Omega_{\alpha}),$
(3.43) $\displaystyle
S_{\dot{\alpha}}{}^{J}=(x-5/2)\chi^{J}\chi_{I}\Omega_{\dot{\alpha}}{}^{I}+4|\chi|^{2}\Omega_{\dot{\alpha}}{}^{J}+(\chi\sigma_{i})^{J}(\widetilde{\chi}\sigma_{i}\widehat{\Omega}_{\dot{\alpha}})-\frac{1}{4}(\chi\sigma_{ij})^{J}(\widetilde{\chi}\sigma_{ij}\widehat{\Omega}_{\dot{\alpha}}),$
(3.44)
$\displaystyle\widetilde{S}_{\dot{\alpha}}{}^{\widetilde{J}}=(x-5/2)\widetilde{\chi}^{\widetilde{J}}\widetilde{\chi}_{\widetilde{I}}\widehat{\Omega}_{\dot{\alpha}}{}^{\widetilde{I}}+4|\widetilde{\chi}|^{2}\widehat{\Omega}_{\dot{\alpha}}{}^{\widetilde{J}}+(\widetilde{\chi}\sigma_{i})^{\widetilde{J}}(\chi\sigma_{i}\Omega_{\dot{\alpha}})-\frac{1}{4}(\widetilde{\chi}\sigma_{ij})^{\widetilde{J}}(\chi\sigma_{ij}\Omega_{\dot{\alpha}}),$
(3.45)

and,

$\displaystyle
U_{\alpha\dot{\alpha}}{}^{\underline{ab}}=\chi^{I}\chi_{J}R_{\alpha
I\dot{\alpha}^{J}}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}_{\widetilde{J}}R_{\alpha\widetilde{I}\dot{\alpha}^{\widetilde{J}}}{}^{\underline{ab}}+\chi^{I}\widetilde{\chi}_{\widetilde{J}}R_{\alpha
I\dot{\alpha}^{\widetilde{J}}}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}\chi_{J}R_{\alpha\widetilde{I}\dot{\alpha}^{J}}{}^{\underline{ab}}.$
(3.46)

We will not impose that any of these terms vanish; the whole expression (3.40)
defines $\nabla_{\alpha\dot{\alpha}}$.

Now we compute $[\nabla_{\alpha},\nabla_{\beta\dot{\beta}}]$ which should be
equal to $-\frac{1}{2}\epsilon_{\alpha\beta}F\nabla_{\dot{\beta}}$, where $F$
should be equal to the result given in (3.39). The curvatures in (3.46) are
given in (2.50). The computation can be broken down into a few smaller steps.
First, we can compute

$\displaystyle\frac{1}{2}[\nabla_{\alpha},U_{\beta\dot{\beta}}{}^{\underline{ab}}M_{\underline{ab}}]$
$\displaystyle=(\nabla_{\alpha}U_{\beta\dot{\beta}}{}^{\underline{ab}})M_{\underline{ab}}+\frac{1}{4}U_{\beta\dot{\beta}}{}^{ab}(\sigma_{ab})_{\alpha}{}^{\gamma}\nabla_{\gamma}$
(3.47)
$\displaystyle+\frac{1}{2}U_{\beta\dot{\beta}}{}^{ai}(\sigma_{a})_{\alpha\dot{\gamma}}(\sigma^{i})_{IJ}\chi^{I}\nabla^{\dot{\gamma}J}+\frac{1}{2}U_{\beta\dot{\beta}}{}^{ai}(\sigma_{a})_{\alpha\dot{\gamma}}(\sigma^{i})_{\tilde{I}\tilde{J}}\tilde{\chi}^{\tilde{I}}\tilde{\nabla}^{\dot{\gamma}\tilde{J}}.$
(3.48)

Note that $\nabla_{\alpha}$ and $\nabla_{\dot{\alpha}}$ should annihilate any
background field. Next, we add the contribution coming only from $\nabla_{a}$

$\displaystyle[\nabla_{\alpha},\nabla_{a}]$
$\displaystyle=-(\nabla_{a}\chi^{I})\nabla_{\alpha
I}-(\nabla_{a}\tilde{\chi}^{\tilde{I}})\nabla_{\alpha\tilde{I}}$ (3.49)
$\displaystyle+\chi^{I}\left((\sigma_{a}{}^{b})_{\alpha}{}^{\gamma}\Omega_{\gamma
I}\nabla_{b}+(\sigma_{a})_{\alpha\dot{\gamma}}(\sigma_{i})_{IJ}\Omega^{\dot{\gamma}J}\nabla_{i}+(\sigma_{a})_{\alpha\dot{\gamma}}P^{\dot{\gamma}}{}_{I}{}^{\dot{\rho}}{}_{\tilde{J}}\nabla_{\dot{\rho}}{}^{\tilde{J}}\right.$
(3.50) $\displaystyle\left.+(x-1)\nabla_{a}\Omega_{\alpha
I}{\mathcal{S}}+\frac{1}{2}R_{\alpha
Ia}{}^{\underline{bc}}M_{\underline{bc}}\right)$ (3.51)
$\displaystyle+\tilde{\chi}^{\tilde{I}}\left((\sigma_{a}{}^{b})_{\alpha}{}^{\gamma}\widehat{\Omega}_{\gamma\tilde{I}}\nabla_{b}+(\sigma_{a})_{\alpha\dot{\gamma}}(\sigma_{i})_{\tilde{I}\tilde{J}}\widehat{\Omega}^{\dot{\gamma}\tilde{J}}\nabla_{i}-(\sigma_{a})_{\alpha\dot{\gamma}}P^{\dot{\rho}}{}_{J}{}^{\dot{\gamma}}{}_{\tilde{I}}\nabla_{\dot{\rho}}{}^{J}\right.$
(3.52)
$\displaystyle\left.+(x-1)\nabla_{a}\widehat{\Omega}_{\alpha\tilde{I}}\tilde{\mathcal{S}}+\frac{1}{2}R_{\alpha\tilde{I}a}{}^{\underline{bc}}M_{\underline{bc}}\right),$
(3.53)

where $({\mathcal{S}},\tilde{\mathcal{S}})$ are the left and right scale
generators. Note that

$\displaystyle\nabla_{a}\Omega_{\alpha
I}=\frac{1}{4}\\{\nabla_{a},\nabla_{\alpha
I}\\}\Phi=-\frac{1}{4}(\sigma_{a})_{\alpha\dot{\gamma}}(\sigma_{i})_{IJ}\Omega^{\dot{\gamma}J}\nabla_{i}\Phi-\frac{1}{4}(\sigma_{a})_{\alpha\dot{\rho}}P^{\dot{\rho}}{}_{I}{}^{\dot{\gamma}}{}_{\tilde{J}}\nabla_{\dot{\gamma}}{}^{\tilde{J}}\Phi,$
(3.54)
$\displaystyle\nabla_{a}\widehat{\Omega}_{\beta\tilde{I}}=\frac{1}{4}\\{\nabla_{a},\nabla_{\beta\tilde{I}}\\}\Phi=-\frac{1}{4}(\sigma_{a})_{\beta\dot{\gamma}}(\sigma_{i})_{\tilde{I}\tilde{J}}\widehat{\Omega}^{\dot{\gamma}\tilde{J}}\nabla_{i}\Phi+\frac{1}{4}(\sigma_{a})_{\beta\dot{\rho}}P^{\dot{\gamma}}{}_{J}{}^{\dot{\rho}}{}_{\tilde{I}}\nabla_{\dot{\gamma}}{}^{J}\Phi.$
(3.55)

The final step is to compute $[\nabla_{\alpha},S_{\beta
J}\nabla_{\dot{\beta}}{}^{J}+\widetilde{S}_{\beta\widetilde{J}}\widetilde{\nabla}_{\dot{\beta}}{}^{\widetilde{J}}+S_{\dot{\beta}}{}^{J}\nabla_{\beta
J}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\nabla}_{\beta\widetilde{J}}]$
which is done in Appendix B. The final result is organized as

$\displaystyle[\nabla_{\alpha},\nabla_{\beta\dot{\beta}}]$
$\displaystyle=\Psi_{\alpha\beta\dot{\beta}}{}^{\dot{\gamma}}{}_{I}\nabla_{\dot{\gamma}}{}^{I}+\widetilde{\Psi}_{\alpha\beta\dot{\beta}}{}^{\dot{\gamma}}{}_{\widetilde{I}}\widetilde{\nabla}_{\dot{\gamma}}{}^{\widetilde{I}}+W^{a}_{\alpha\beta\dot{\beta}}\nabla_{a}+X^{i}_{\alpha\beta\dot{\beta}}\nabla_{i}$
(3.56) $\displaystyle+Y_{\alpha\beta\dot{\beta}}{}^{\gamma I}\nabla_{\gamma
I}+\widetilde{Y}_{\alpha\beta\dot{\beta}}{}^{\gamma\widetilde{I}}\widetilde{\nabla}_{\gamma\widetilde{I}}+Z_{\alpha\beta\dot{\beta}}{}^{\underline{ab}}M_{\underline{ab}}+Z^{\mathcal{S}}_{\alpha\beta\dot{\beta}}{\mathcal{S}}+Z^{\widetilde{\mathcal{S}}}_{\alpha\beta\dot{\beta}}\widetilde{\mathcal{S}},$
(3.57)

where444The expressions below do not include terms with $\Omega^{2}$,
$\widetilde{\Omega}^{2}$ and $\Omega\widetilde{\Omega}$. These terms are not
included since they need at least two Grassmann odd derivatives for a non-
vanishing contribution in the covariant $\theta$ expansion.

$\displaystyle\Psi_{\alpha\beta\dot{\beta}}{}^{\dot{\gamma}}{}_{I}$
$\displaystyle=-\frac{1}{2}U_{\beta\dot{\beta}}{}^{ai}(\sigma_{a})_{\alpha}{}^{\dot{\gamma}}\chi^{J}(\sigma^{i})_{JI}-2\epsilon_{\alpha\beta}(|\chi|^{2}+|\widetilde{\chi}|^{2})\widetilde{\chi}^{\widetilde{J}}\delta^{\dot{\gamma}}_{\dot{\beta}}F_{I\widetilde{J}}$
(3.58)
$\displaystyle+\epsilon_{\alpha\beta}\left(|\chi|^{2}(\chi\sigma_{i})_{I}\nabla_{i}\Phi-\frac{1}{4}(\chi\sigma_{i})_{I}(\widetilde{\chi}\sigma_{j}\sigma_{i}\widetilde{\chi})\nabla_{j}\Phi+\frac{1}{16}(\chi\sigma_{ij})_{I}(\widetilde{\chi}\sigma_{ijk}\widetilde{\chi})\nabla_{k}\Phi\right)\delta_{\dot{\beta}}^{\dot{\gamma}},$
(3.59)
$\displaystyle\widetilde{\Psi}_{\alpha\beta\dot{\beta}}{}^{\dot{\gamma}}{}_{\widetilde{I}}$
$\displaystyle=-\frac{1}{2}U_{\beta\dot{\beta}}{}^{ai}(\sigma_{a})_{\alpha}{}^{\dot{\gamma}}\widetilde{\chi}^{\widetilde{J}}(\sigma^{i})_{\widetilde{J}\widetilde{I}}-2\epsilon_{\alpha\beta}(|\chi|^{2}+|\widetilde{\chi}|^{2})\chi^{J}\delta^{\dot{\gamma}}_{\dot{\beta}}F_{J\widetilde{I}}$
(3.60)
$\displaystyle+\epsilon_{\alpha\beta}\left(|\widetilde{\chi}|^{2}(\widetilde{\chi}\sigma_{i})_{\widetilde{I}}\nabla_{i}\Phi-\frac{1}{4}(\widetilde{\chi}\sigma_{i})_{\widetilde{I}}(\chi\sigma_{j}\sigma_{i}\chi)\nabla_{j}\Phi+\frac{1}{16}(\widetilde{\chi}\sigma_{ij})_{\widetilde{I}}(\chi\sigma_{ijk}\chi)\nabla_{k}\Phi\right)\delta_{\dot{\beta}}^{\dot{\gamma}},$
(3.61) $\displaystyle W^{a}_{\alpha\beta\dot{\beta}}$
$\displaystyle=-(S_{\beta
I}\chi^{I}+\widetilde{S}_{\beta\widetilde{I}}\widetilde{\chi}^{\widetilde{I}})(\sigma^{a})_{\alpha\dot{\beta}},$
(3.62) $\displaystyle X^{i}_{\alpha\beta\dot{\beta}}$
$\displaystyle=-\epsilon_{\alpha\beta}(S_{\dot{\beta}}{}^{J}\chi^{I}(\sigma^{i})_{IJ}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\chi}^{\widetilde{I}}(\sigma^{i})_{\widetilde{I}\widetilde{J}}),$
(3.63) $\displaystyle Y_{\alpha\beta\dot{\beta}}{}^{\gamma I}$
$\displaystyle=\frac{1}{4}U_{\beta\dot{\beta}}{}^{ab}(\sigma_{ab})_{\alpha}{}^{\gamma}\chi^{I}+\delta^{\gamma}_{\alpha}\left(\frac{1}{2}U_{\beta\dot{\beta}}{}^{ij}(\chi\sigma_{ij})^{I}+(S\cdot\nabla)_{\beta\dot{\beta}}\chi^{I}\right),$
(3.64)
$\displaystyle\widetilde{Y}_{\alpha\beta\dot{\beta}}{}^{\gamma\widetilde{I}}$
$\displaystyle=\frac{1}{4}U_{\beta\dot{\beta}}{}^{ab}(\sigma_{ab})_{\alpha}{}^{\gamma}\widetilde{\chi}^{\widetilde{I}}+\delta^{\gamma}_{\alpha}\left(\frac{1}{2}U_{\beta\dot{\beta}}{}^{ij}(\widetilde{\chi}\sigma_{ij})^{\widetilde{I}}+(S\cdot\nabla)_{\beta\dot{\beta}}\widetilde{\chi}^{\widetilde{I}}\right),$
(3.65) $\displaystyle Z_{\alpha\beta\dot{\beta}}{}^{\underline{ab}}$
$\displaystyle=\frac{1}{2}(|\chi|^{2}+|\widetilde{\chi}|^{2})(\sigma^{a})_{\beta\dot{\beta}}\left(\chi^{I}R_{\alpha
Ia}{}^{\underline{ab}}+\widetilde{\chi}^{\widetilde{I}}R_{\alpha\widetilde{I}a}{}^{\underline{ab}}\right)+\frac{1}{2}\nabla_{\alpha}U_{\beta\dot{\beta}}{}^{\underline{ab}},$
(3.66) $\displaystyle Z^{\mathcal{S}}_{\alpha\beta\dot{\beta}}$
$\displaystyle=\left(\frac{1-x}{2}\right)\epsilon_{\alpha\beta}(|\chi|^{2}+|\widetilde{\chi}|^{2})\chi^{I}\left((\sigma_{i})_{IJ}\Omega_{\dot{\beta}}{}^{J}\nabla_{i}\Phi+P_{\dot{\beta}I}{}^{\dot{\gamma}}{}_{\widetilde{J}}\nabla_{\dot{\gamma}}{}^{\widetilde{J}}\Phi\right)$
(3.67)
$\displaystyle+\left(\frac{1-x}{4}\right)\epsilon_{\alpha\beta}S_{\dot{\beta}}{}^{J}\chi^{I}(\sigma_{i})_{IJ}\nabla_{i}\Phi$
(3.68) $\displaystyle+(1-x)\left(S_{\beta
J}\widetilde{\chi}^{\widetilde{I}}\nabla_{\alpha\widetilde{I}}\Omega_{\dot{\beta}}{}^{J}+\widetilde{S}_{\beta\widetilde{J}}\chi^{I}\nabla_{\dot{\beta}}{}^{\widetilde{J}}\Omega_{\alpha
I}+S_{\dot{\beta}}{}^{J}\widetilde{\chi}^{\widetilde{I}}\nabla_{\alpha\widetilde{I}}\Omega_{\beta
J}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\chi^{I}\nabla_{\beta\widetilde{J}}\Omega_{\alpha
I}\right),$ (3.69) $\displaystyle
Z^{\widetilde{\mathcal{S}}}_{\alpha\beta\dot{\beta}}$
$\displaystyle=\left(\frac{1-x}{2}\right)\epsilon_{\alpha\beta}(|\chi|^{2}+|\widetilde{\chi}|^{2})\widetilde{\chi}^{\widetilde{I}}\left((\sigma_{i})_{\widetilde{I}\widetilde{J}}\widehat{\Omega}_{\dot{\beta}}{}^{\widetilde{J}}\nabla_{i}\Phi-P^{\dot{\gamma}}{}_{J\dot{\beta}\widetilde{I}}\nabla_{\dot{\gamma}}{}^{J}\Phi\right)$
(3.70)
$\displaystyle+\left(\frac{1-x}{4}\right)\epsilon_{\alpha\beta}\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\chi}^{\widetilde{I}}(\sigma_{i})_{\widetilde{I}\widetilde{J}}\nabla_{i}\Phi$
(3.71) $\displaystyle+(1-x)\left(S_{\beta
J}\widetilde{\chi}^{\widetilde{I}}\nabla_{\dot{\beta}}{}^{J}\widehat{\Omega}_{\alpha\widetilde{I}}+\widetilde{S}_{\beta\widetilde{J}}\chi^{I}\nabla_{\alpha
I}\widehat{\Omega}_{\dot{\beta}}{}^{\widetilde{J}}+S_{\dot{\beta}}{}^{J}\widetilde{\chi}^{\widetilde{I}}\nabla_{\beta
J}\widehat{\Omega}_{\alpha\widetilde{I}}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\chi^{I}\nabla_{\alpha
I}\widehat{\Omega}_{\beta\widetilde{J}}\right).$ (3.72)

We used the notation

$\displaystyle(S\cdot\nabla)_{\alpha\dot{\alpha}}=S_{\alpha
J}\nabla_{\dot{\alpha}}{}^{J}+\widetilde{S}_{\alpha\widetilde{J}}\widetilde{\nabla}_{\dot{\alpha}}{}^{\widetilde{J}}+S_{\dot{\alpha}}{}^{J}\nabla_{\alpha
J}+\widetilde{S}_{\dot{\alpha}}{}^{\widetilde{J}}\widetilde{\nabla}_{\alpha\widetilde{J}},$
(3.73)

to have a shorter expression for the $Y$ tensors. Except for the first two
tensors above, all others should vanish. The last three tensors have a mass
dimension $3/2$, and the first nontrivial condition they imply has a mass
dimension of $2$. Of course, there is the obvious choice of fixing $x=1$. But
this is not the most convenient for a simple example. For the $Y$ tensors,
their $\theta^{0}$ component implies no four-dimension Lorentz breaking
background. The vanishing of $W$ and $X$ may imply interesting conditions that
will be discussed later.

The necessary conditions for the $\Psi$ tensors imply that

$\displaystyle
U_{\beta\dot{\beta}}{}^{ai}=V^{i}\sigma^{a}_{\beta\dot{\beta}},$ (3.74)

for some internal superfield $V^{i}$. Substituting this in $\Psi$ and imposing
(3.9) we have the following equations

$\displaystyle-\frac{1}{2}\chi_{I}F$
$\displaystyle=V^{i}\sigma^{i}_{JI}\chi^{J}-2(|\chi|^{2}+|\widetilde{\chi}|^{2})\widetilde{\chi}^{\widetilde{J}}F_{I\widetilde{J}}+|\chi|^{2}(\chi\sigma_{i})_{I}\nabla_{i}\Phi$
(3.75)
$\displaystyle-\frac{1}{4}(\chi\sigma_{i})_{I}(\widetilde{\chi}\sigma_{j}\sigma_{i}\widetilde{\chi})\nabla_{j}\Phi+\frac{1}{16}(\chi\sigma_{ij})_{I}(\widetilde{\chi}\sigma_{ijk}\widetilde{\chi})\nabla_{k}\Phi$
(3.76) $\displaystyle-\frac{1}{2}\widetilde{\chi}_{\widetilde{I}}F$
$\displaystyle=V^{i}\sigma^{i}_{\widetilde{J}\widetilde{I}}\widetilde{\chi}^{\widetilde{J}}-2(|\chi|^{2}+|\widetilde{\chi}|^{2})\chi^{J}F_{J\widetilde{I}}+|\widetilde{\chi}|^{2}(\widetilde{\chi}\sigma_{i})_{\widetilde{I}}\nabla_{i}\Phi$
(3.77)
$\displaystyle-\frac{1}{4}(\widetilde{\chi}\sigma_{i})_{\widetilde{I}}(\chi\sigma_{j}\sigma_{i}\chi)\nabla_{j}\Phi+\frac{1}{16}(\widetilde{\chi}\sigma_{ij})_{\widetilde{I}}(\chi\sigma_{ijk}\chi)\nabla_{k}\Phi$
(3.78)

Finally we calculate $F$ from the equations above contracting with $\chi$ and
$\widetilde{\chi}$ respectively

$\displaystyle F$
$\displaystyle=4\left(1+\frac{|\widetilde{\chi}|^{2}}{|\chi|^{2}}\right)\chi^{I}F_{I\widetilde{J}}\widetilde{\chi}^{\widetilde{J}}-\frac{1}{8|\chi|^{2}}(\chi\sigma_{ij}\chi)(\widetilde{\chi}\sigma_{ijk}\widetilde{\chi})\nabla_{k}\Phi$
(3.79)
$\displaystyle=4\left(1+\frac{|\chi|^{2}}{|\widetilde{\chi}|^{2}}\right)\chi^{I}F_{I\widetilde{J}}\widetilde{\chi}^{\widetilde{J}}-\frac{1}{8|\widetilde{\chi}|^{2}}(\widetilde{\chi}\sigma_{ij}\widetilde{\chi})(\chi\sigma_{ijk}\chi)\nabla_{k}\Phi.$
(3.80)

An $AdS_{4}$ solution is only possible if the above equations are equal to
(3.39).

###  3.4 Summary

Since this section describes numerous scattered conditions, discussing the
main assumptions and consequences is convenient. The starting assumption is
that it should be possible to construct a four-dimensional superspace
covariant derivative algebra (3.8,3.9,3.10) from the ten-dimensional one. We
also impose that the four-dimensional operators should annihilate internal
superfields. In particular, the dilaton superfield should satisfy

$\displaystyle\nabla_{\alpha}\Phi=0.$

The four-dimensional algebra imposes various conditions, and we pay particular
attention to the internal superfield $F$, whose first component corresponds to
the cosmological constant. We found different expressions for it: (3.39),
(3.79), and (3.80). If they are all equal, the background will be consistent
with four-dimensional supersymmetry. This will imply restrictions on the
various possible bosonic background fields. Finally, the factorization
discussed in the introduction also implies a familiar equation for the
normalizable spinors

$\displaystyle\nabla_{i}\chi^{I}-\frac{1}{8}\chi^{J}(\sigma_{jk})_{J}{}^{I}H_{ijk}+\widetilde{\chi}^{\widetilde{J}}(\sigma_{i})_{\widetilde{J}\widetilde{K}}F^{I\widetilde{K}}=0,$
(3.81)
$\displaystyle\nabla_{i}\widetilde{\chi}^{\widetilde{I}}+\frac{1}{8}\widetilde{\chi}^{\widetilde{J}}(\sigma_{jk})_{\widetilde{J}}{}^{\widetilde{I}}H_{ijk}+\chi^{J}(\sigma_{i})_{JK}F^{K\widetilde{I}}=0.$
(3.82)

In the next section, we will discuss the implications of the abovementioned
constraints.

##  4 A simple flux background

Let us now discuss a simple solution for the superspace constraints we derived
[26, 16]. We are going to assume Type IIB supergravity and that the whole
normalizable spinor superfields are equal

$\displaystyle\chi^{I}=\widetilde{\chi}^{I}.$ (4.1)

With this assumption, we can see that (3.79) is equal to (3.80) as required.
Furthermore, using (3.5) and (3.6) and some identities in Appendix A, one can
see that the $W$ and $X$ tensors in (3.62) and (3.63) are zero. The same can
be shown for (3.34) and (3.35) choosing $x=5$. Another simplification is that
with the choice $\chi^{I}=\widetilde{\chi}^{I}$ all contributions from
$H_{ijk}$ in the tensors $U_{\alpha\beta}{}^{\underline{ab}}$,
$U_{\dot{\alpha}\dot{\beta}}{}^{\underline{ab}}$ and
$U_{\alpha\dot{\beta}}{}^{\underline{ab}}$ drop.

With the assumption (4.1) the pure spinor equations (3.25) imply that

$\displaystyle\frac{1}{4}\chi^{J}(\sigma_{jk})_{J}{}^{I}H_{ijk}+\chi^{J}(\sigma_{i})_{JK}\left(F^{KI}-F^{IK}\right)=0.$
(4.2)

This can be solved in various ways. If there is only the Ramond-Ramond
$3$-form flux, the superfield $F^{KI}$ is symmetric, then we must have that

$\displaystyle\frac{1}{4}\chi^{J}(\sigma_{jk})_{J}{}^{I}H_{ijk}=0.$ (4.3)

Using the special sigma matrices basis discussed in Appendix C and the complex
decomposition of the Kalb-Ramond field-strength
$H_{ijk}=(H_{{\mathsf{i}}{\mathsf{j}}{\mathsf{k}}},H_{\bar{\mathsf{i}}\bar{\mathsf{j}}\bar{\mathsf{k}}},H_{\bar{\mathsf{i}}{\mathsf{j}}{\mathsf{k}}},H_{{\mathsf{i}}\bar{\mathsf{j}}\bar{\mathsf{k}}})$,
the last equation can be solved with

$\displaystyle
H_{{\mathsf{i}}{\mathsf{j}}{\mathsf{k}}}=H_{\bar{\mathsf{i}}\bar{\mathsf{j}}\bar{\mathsf{k}}}=\delta^{{\mathsf{j}}\bar{\mathsf{i}}}H_{\bar{\mathsf{i}}{\mathsf{j}}{\mathsf{k}}}=\delta^{{\mathsf{i}}\bar{\mathsf{j}}}H_{{\mathsf{i}}\bar{\mathsf{j}}\bar{\mathsf{k}}}=0.$
(4.4)

It is important to observe that it is not possible to set the whole superfield
$H_{ijk}=0$ since its $\theta^{2}$ terms contain the curvature.

The first non-trivial implication of (3.4) is, using (2.28), (B.15) and that
$\nabla_{\beta J}\chi^{I}\Big{|}_{\theta=0}=0$ we obtain

$\displaystyle\nabla_{\beta
J}\left(\chi^{I}\nabla_{\alpha}\Phi+\chi^{I}\widetilde{\nabla}_{\alpha
I}\Phi\right)\Big{|}_{\theta=0}=\frac{1}{8}\epsilon_{\beta\alpha}\chi^{I}\left(-\sigma^{i}_{IJ}\nabla_{i}\Phi+4F_{IJ}\right)\Big{|}_{\theta=0}=0.$
(4.5)

This equation can be solved non-trivially if the $SU(3)$ singlet part of the
Ramond-Ramond $3$-form vanishes

$\displaystyle\chi^{I}\chi^{K}F_{IK}=0,$ (4.6)

and the $\mathbf{3}$ is equal to the derivative of the dilaton. The
$\mathbf{6}$ representation of the RR field does not appear in the equation
above.

Computing $V_{i}$ with the conditions above, we find

$\displaystyle
V_{i}=\frac{1}{2}|\chi|^{2}\nabla_{i}\Phi-\chi^{I}\chi_{J}\left((\sigma_{i})_{IK}F^{KJ}+(\sigma_{i})^{JK}F_{KI}\right).$
(4.7)

If we use this information in equation (3.75) using $\chi=\widetilde{\chi}$ we
obtain

$\displaystyle-\frac{1}{2}\chi_{I}F=\frac{1}{2}|\chi|^{2}\chi^{K}(\sigma_{i})_{KI}\nabla_{i}\Phi-2|\chi|^{2}\chi^{K}F_{IK}.$
(4.8)

Contracting this last expression with $\chi^{I}$, we find that $F=0$. It is
straightforward to check that (3.39) also vanishes. Therefore, the fact that
the $SU(3)$ singlet of the RR $3$-form vanishes implies that the cosmological
constant is zero, up to terms with $\Omega^{2}$ and $\widetilde{\Omega}^{2}$.
This is a known result; having an $AdS_{4}$ compactification in Type IIB with
only one normalizable spinor is impossible [28, 29].

##  5 Conclusions and prospects

In this work, we have proposed a formalism to study compactifications of Type
II supergravity theories that preserve $N=1$ supersymmetry in four dimensions,
working directly in a curved superspace formulation inspired by the pure
spinor superstring.

The central hypothesis is the factorization of the massless vertex operators
into a product of four-dimensional superfields and internal “superspace
harmonic form.” These superspace harmonic forms can be considered the
generalization of the harmonic forms in a usual Calabi-Yau compactification.
This factorization relied on redefining the covariant derivatives in terms of
auxiliary bosonic spinor superfields $(\chi,\widetilde{\chi})$, separating the
four-dimensional derivatives from the mixed components. This decomposition
shows how the familiar four-dimensional superspace covariant derivatives
emerge naturally from the BRST differential.

Our superspace approach provides a geometric framework for constructing and
analyzing supersymmetric compactifications while maintaining manifest
supersymmetry and lower-dimensional covariance. It elegantly allows us to
directly implement symmetry conditions on the higher-dimensional backgrounds.
The factorized form of the massless fields is handy, as it connects directly
to the standard four-dimensional superfield formalism widely used in model-
building and phenomenology. Looking ahead, there are many potential
applications and extensions of this work. Our formalism can be applied to
study specific flux compactifications of interest. Furthermore, while our
focus has been on $N=1$ supersymmetry in four dimensions, the methods
developed here may also prove helpful for other amounts of global
supersymmetry and other space-time dimensions.

During our analysis, we found no condition that fixes the Grassmann odd
covariant derivatives of $(\chi,\widetilde{\chi})$ besides
$\nabla_{\alpha}\chi^{I}=0$. On purely dimensional grounds, it should have the
general form

$\displaystyle\nabla_{\alpha I}\chi^{J}=\Omega_{\alpha
K}H^{KJ}{}_{I}(\Phi,\chi,\widetilde{\chi})+\widehat{\Omega}_{\alpha\widetilde{K}}{\widetilde{H}}^{\widetilde{K}J}{}_{I}(\Phi,\chi,\widetilde{\chi}),$
(5.1)

where $(H,\widetilde{H})$ are internal superfields depending only on $\Phi$,
$\chi$ and $\widetilde{\chi}$. There are analogous expressions with other
derivatives. It may be possible to find the explicit expressions by demanding
consistency with (3.7) and (3.25). Related to this is the problem of finding
the minimal set of superspace equations necessary for the supersymmetry
algebra to hold.

Another crucial point is to add $D$-branes or other extended objects to the
curved background. It is well-known that compact solutions with Ramond-Ramond
fluxes cannot exist without source terms for extended objects [30, 31, 32].
These source terms will appear in some superspace Bianchi identities and will
change the solutions to these identities found in Section 2. Since the
Berkovits-Howe constraints are found from the world-sheet action, the presence
of extended objects may change the constraints themselves. This is still an
open problem in pure spinor formalism, and we plan to address it in the
future.

The most natural next step is to study vertex operators in the formalism
described here. One of the primary motivations for studying flux
compactifications is to generate masses for the moduli describing deformations
of the compactification manifold. Such mass terms should appear in the
equations of motion for the four-dimensional $N=1$ chiral multiplets. The ten-
dimensional covariant derivative algebra, with the fluxes appearing in them,
should provide such mass terms. A simple comparison with the ten-dimensional
case indicates that the scalar four-dimensional superfield in (1.6) is the
pre-potential of a chiral field. It would be interesting to verify these
statements.

A more ambitious goal is to find hidden world-sheet symmetries in the case of
a Calabi-Yau compactification. It is a textbook fact that the RNS formalism
possesses $N=2$ superconformal symmetry in the world-sheet in these
compactifications. This powerful symmetry is responsible for many
simplifications in perturbative calculations and non-perturbative effects. The
observable consequences of this symmetry should not depend on the formalism
used; therefore, they should manifest in some way in the pure spinor
formalism. A possible way for this to happen is that the composite $b$ [33,
34] ghost of the pure spinor formalism factories in a space-time plus
compactification and that the compactification part forms a closed world-sheet
current algebra with the compactification part of the BRST current.555This can
be defined in the same way we defined the space-time covariant derivatives in
the present work.

A formalism that combines a manifest supersymmetry formulation with the
preservation of world-sheet supersymmetry was recently introduced in [35]; it
incorporates characteristics of all previous superstring formalisms. This new
approach was studied in the curved background for the heterotic string case
[36] and the type II superstring case [37]. If it is possible to write the
action for a curved background so that the world-sheet supersymmetry is
manifest, the number of background couplings will be significantly smaller. If
that happens, analyzing the existence of global supersymmetries will be much
simpler. This a line of investigation that is worth pursuing.

Acknowledgments: We thank William D. Linch, III, for valuable comments and
suggestions. OC would like to thank the Kavli Institute for Theoretical
Physics (KITP), where part of this work was done. This work was partially
financed by FONDECYT Regular grants 1200342 and 1201550. This research was
partly supported by grant NSF PHY-2309135 to the Kavli Institute for
Theoretical Physics.

##  Appendix A Ten dimensional $\gamma$ matrices

A useful notation for spinorial indices is the following. Express the
$\bf{16}$ spinorial representation of $SO(1,9)$ as representations of
$SL(2,C)$ and vector representation $SU(4)$. Then, the superspace coordinate
$\theta^{\alpha}$ (here $\alpha=1,\dots,16)$ of ten dimensions is expressed as
$(\theta^{\alpha I},\theta^{\dot{\alpha}}_{I})$ (here $\alpha$ and
$\dot{\alpha}$ run from $1,2$, and $I=1,\dots,4$). The non-zero components of
the ten-dimensional gamma matrices,
$(\gamma^{a})_{\alpha\beta},(\widetilde{\gamma}^{a})^{\alpha\beta}$, are given
by (the ten-dimensional vector index is split in the four-dimensional vector
index $a$ and the six-dimensional vector index $i$)

$\displaystyle(\gamma^{a})_{{}_{\alpha
I},{{}_{\dot{\beta}}^{J}}}=(\sigma^{a})_{\alpha\dot{\beta}}\delta_{I}^{J},\quad(\gamma^{i})_{{}_{\alpha
I},{}_{\beta
J}}=(\sigma^{i})_{IJ}\epsilon_{\alpha\beta},\quad(\gamma^{i})_{{}_{\dot{\alpha}}^{I},{}_{\dot{\beta}}^{J}}=-(\sigma^{i})^{IJ}\epsilon_{\dot{\alpha}\dot{\beta}},$
(A.1) $\displaystyle(\widetilde{\gamma}^{a})^{{}_{\alpha
I},{}^{J}_{\dot{\beta}}}=-(\widetilde{\sigma}^{a})^{\dot{\beta}\alpha}\delta_{J}^{I},\quad(\widetilde{\gamma}^{i})^{{}_{\alpha
I},{}_{\beta
J}}=(\sigma^{i})^{IJ}\epsilon^{\alpha\beta},\quad(\widetilde{\gamma}^{i})^{{}_{\dot{\alpha}}^{I},{}_{\dot{\beta}}^{J}}=-(\sigma^{i})_{IJ}\epsilon^{\dot{\alpha}\dot{\beta}}.$
(A.2)

where $\sigma^{a},\widetilde{\sigma}^{a},\epsilon$’s are defined in [38], and
$\sigma^{i}$’s are defined to satisfy the Dirac algebra

$\displaystyle(\sigma_{i})_{IK}(\sigma_{j})^{KJ}+(\sigma_{j})_{IK}(\sigma_{i})^{KJ}=2\delta_{ij}\delta_{I}^{J}.$
(A.3)

Non-zero products of two gamma matrices are given by

$\displaystyle(\gamma_{ab})_{{}_{\alpha I}}{}^{{}_{\beta
J}}=-(\sigma_{ab})_{\alpha}{}^{\beta}\delta_{I}^{J},\quad(\gamma_{ab})_{{}_{\dot{\alpha}}^{I}}{}~{}^{{}_{\dot{\beta}}^{J}}=(\sigma_{ab})^{\dot{\beta}}{}_{\dot{\alpha}}\delta^{I}_{J},$
(A.4) $\displaystyle(\gamma_{ij})_{{}_{\alpha I}}{}^{{}_{\beta
J}}=(\sigma_{ij})_{I}{}^{J}\delta_{\alpha}^{\beta},\quad(\gamma_{ij})_{{}_{\dot{\alpha}}^{I}}{}~{}^{{}_{\dot{\beta}}^{J}}=(\sigma_{ij})^{I}{}_{J}\delta^{\dot{\beta}}_{\dot{\alpha}},$
(A.5) $\displaystyle(\gamma_{ai})_{{}_{\alpha
I}}{}~{}^{{}_{\dot{\beta}}^{J}}=-(\sigma_{a})_{\alpha\dot{\gamma}}\epsilon^{\dot{\gamma}\dot{\beta}}(\bar{\sigma}_{i})_{IJ},\quad(\gamma_{ai})_{{}_{\dot{\alpha}}^{I}}{}~{}^{{}_{\beta
J}}=-\epsilon^{\beta\gamma}(\sigma_{a})_{\gamma\dot{\alpha}}(\sigma_{i})^{IJ}.$
(A.6)

Non-zero products of three gammas

$\displaystyle(\gamma_{abc})^{{}_{\alpha
I}~{}{}_{\dot{\beta}}^{J}}=-i\epsilon_{abcd}(\widetilde{\sigma}^{d})^{\dot{\beta}\alpha}\delta^{I}_{J},\quad(\gamma_{abi})^{{}_{\alpha
I}~{}{}_{\beta
J}}=-\epsilon^{\beta\gamma}(\sigma_{ab})_{\gamma}{}^{\alpha}(\sigma_{i})^{IJ},$
(A.7)
$\displaystyle(\gamma_{abi})^{{}_{\dot{\alpha}}^{I}~{}{}_{\dot{\beta}}^{J}}=(\sigma_{ab})^{\dot{\alpha}}{}_{\dot{\gamma}}\epsilon^{\dot{\gamma}\dot{\beta}}(\bar{\sigma})_{IJ},\quad(\gamma_{aij})^{{}_{\alpha
I}~{}{}_{\dot{\beta}}^{J}}=-(\widetilde{\sigma}_{a})^{\dot{\beta}\alpha}(\sigma_{ij})^{I}{}_{J},$
(A.8) $\displaystyle(\gamma_{ijk})^{{}_{\alpha I}~{}{}_{\beta
J}}=\epsilon^{\alpha\beta}(\sigma_{ijk})^{IJ},\quad(\gamma_{ijk})^{{}_{\dot{\alpha}}^{I}~{}{}_{\dot{\beta}}^{J}}=-\epsilon^{\dot{\alpha}\dot{\beta}}(\sigma_{ijk})_{IJ}.$
(A.9)

Some useful identities involving four and six-dimensional $\sigma$ matrices
are

$\displaystyle(\sigma^{a})_{\alpha\dot{\beta}}(\sigma_{a})^{\dot{\gamma}\rho}=-2\delta_{\alpha}^{\rho}\delta_{\dot{\beta}}^{\dot{\gamma}},$
(A.10)
$\displaystyle(\sigma_{ab})_{\alpha}{}^{\beta}(\sigma^{ab})_{\gamma}{}^{\rho}=4\delta_{\alpha}^{\beta}\delta_{\gamma}^{\rho}-8\delta_{\gamma}^{\beta}\delta_{\alpha}^{\rho},$
(A.11)
$\displaystyle(\sigma_{ab})_{\alpha}{}^{\beta}(\sigma^{ab})^{\dot{\gamma}}{}_{\dot{\rho}}=0,$
(A.12)
$\displaystyle(\sigma^{i})^{IJ}(\sigma^{i})_{KL}=-2(\delta^{I}_{K}\delta^{J}_{L}-\delta^{I}_{L}\delta^{J}_{K}),$
(A.13) $\displaystyle(\sigma^{i})_{IJ}(\sigma^{i})_{KL}=-2\epsilon_{IJKL},$
(A.14)
$\displaystyle(\sigma^{ij})^{I}{}_{J}(\sigma^{ij})^{K}{}_{L}=2\delta^{I}_{J}\delta^{K}_{L}-8\delta^{I}_{L}\delta^{K}_{J}.$
(A.15)

##  Appendix B The calculation of $[\nabla_{\alpha},S_{\beta
J}\nabla_{\dot{\beta}}{}^{J}+\widetilde{S}_{\beta\widetilde{J}}\widetilde{\nabla}_{\dot{\beta}}{}^{\widetilde{J}}+S_{\dot{\beta}}{}^{J}\nabla_{\beta
J}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\nabla}_{\beta\widetilde{J}}]$
and other identities

In this appendix we include intermediate steps in the computation of
$[\nabla_{\alpha},S_{\beta
J}\nabla_{\dot{\beta}}{}^{J}+\widetilde{S}_{\beta\widetilde{J}}\widetilde{\nabla}_{\dot{\beta}}{}^{\widetilde{J}}+S_{\dot{\beta}}{}^{J}\nabla_{\beta
J}+\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\nabla}_{\beta\widetilde{J}}]$.
We begin with the following identities

$\displaystyle\nabla_{\alpha}\Omega_{\beta
J}=\frac{1}{4}\epsilon_{\alpha\beta}(\chi\sigma_{i})_{J}\nabla_{i}\Phi+(\nabla_{\beta
J}\chi^{I})\Omega_{\alpha I}+(\nabla_{\beta
J}\widetilde{\chi}^{\widetilde{I}})\widehat{\Omega}_{\alpha\widetilde{I}},$
(B.1)
$\displaystyle\nabla_{\alpha}\widehat{\Omega}_{\beta\widetilde{J}}=\frac{1}{4}\epsilon_{\alpha\beta}(\widetilde{\chi}\sigma_{i})_{\widetilde{J}}\nabla_{i}\Phi+(\nabla_{\beta\widetilde{J}}\chi^{I})\Omega_{\alpha
I}+(\nabla_{\beta\widetilde{J}}\widetilde{\chi}^{\widetilde{I}})\widehat{\Omega}_{\alpha\widetilde{I}},$
(B.2)
$\displaystyle\nabla_{\alpha}\Omega_{\dot{\beta}}{}^{J}=(\nabla_{\dot{\beta}}{}^{J}\chi^{I})\Omega_{\alpha
I}+(\nabla_{\dot{\beta}}{}^{J}\widetilde{\chi}^{\widetilde{I}})\widehat{\Omega}_{\alpha\widetilde{I}},$
(B.3)
$\displaystyle\nabla_{\alpha}\widehat{\Omega}_{\dot{\beta}}{}^{\widetilde{J}}=(\nabla_{\dot{\beta}}{}^{\widetilde{J}}\chi^{I})\Omega_{\alpha
I}+(\nabla_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\chi}^{\widetilde{I}})\widehat{\Omega}_{\alpha\widetilde{I}}.$
(B.4)

Using these results we obtain

$\displaystyle[\nabla_{\alpha},S_{\beta J}\nabla_{\dot{\beta}}{}^{J}]$
$\displaystyle=\epsilon_{\alpha\beta}\left(|\chi|^{2}(\chi\sigma_{i})_{J}\nabla_{i}\Phi-\frac{1}{4}(\chi\sigma_{i})_{J}\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}_{\widetilde{K}}(\sigma_{j}\sigma_{i})_{\widetilde{I}}{}^{\widetilde{K}}\nabla_{j}\Phi\right.$
(B.5)
$\displaystyle+\left.\frac{1}{16}(\chi\sigma_{ij})_{J}\widetilde{\chi}^{\widetilde{I}}\widetilde{\chi}^{\widetilde{K}}(\sigma_{ij}\sigma_{k})_{\widetilde{I}\widetilde{K}}\nabla_{k}\Phi\right)\nabla_{\dot{\beta}}{}^{J}$
(B.6) $\displaystyle+(1-x)S_{\beta
J}\widetilde{\chi}^{I}\left(\nabla_{\alpha\widetilde{I}}\Omega_{\dot{\beta}}{}^{J}S+\nabla_{\dot{\beta}}{}^{J}\widehat{\Omega}_{\alpha\widetilde{I}}\widetilde{S}\right)+\cdots,$
(B.7)
$\displaystyle[\nabla_{\alpha},\widetilde{S}_{\beta\widetilde{J}}\nabla_{\dot{\beta}}{}^{\widetilde{J}}]$
$\displaystyle=\epsilon_{\alpha\beta}\left(|\widetilde{\chi}|^{2}(\widetilde{\chi}\sigma_{i})_{J}\nabla_{i}\Phi-\frac{1}{4}(\widetilde{\chi}\sigma_{i})_{\widetilde{J}}\chi^{I}\chi_{K}(\sigma_{j}\sigma_{i})_{I}{}^{K}\nabla_{j}\Phi\right.$
(B.8)
$\displaystyle+\left.\frac{1}{16}(\widetilde{\chi}\sigma_{ij})_{\widetilde{J}}\chi^{I}\chi^{K}(\sigma_{ij}\sigma_{k})_{IK}\nabla_{k}\Phi\right)\nabla_{\dot{\beta}}{}^{\widetilde{J}}$
(B.9)
$\displaystyle+(1-x)\widetilde{S}_{\beta\widetilde{J}}\chi^{I}\left(\nabla_{\dot{\beta}}{}^{\widetilde{J}}\Omega_{\alpha
I}S+\nabla_{\alpha
I}\widehat{\Omega}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{S}\right)+\cdots,$
(B.10) $\displaystyle[\nabla_{\alpha},S_{\dot{\beta}}{}^{J}\nabla_{\beta J}]$
$\displaystyle=\frac{1}{4}(1-x)S_{\dot{\beta}}{}^{J}\chi^{I}\epsilon_{\alpha\beta}(\sigma_{i})_{IJ}\nabla_{i}\Phi
S$ (B.11)
$\displaystyle+(1-x)S_{\dot{\beta}}{}^{J}\widetilde{\chi}^{\widetilde{I}}\left(\nabla_{\alpha\widetilde{I}}\Omega_{\beta
J}S+\nabla_{\beta
J}\widehat{\Omega}_{\alpha\widetilde{I}}\widetilde{S}\right)+\cdots,$ (B.12)
$\displaystyle[\nabla_{\alpha},\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\nabla_{\beta\widetilde{J}}]$
$\displaystyle=\frac{1}{4}(1-x)\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\widetilde{\chi}^{\widetilde{I}}\epsilon_{\alpha\beta}(\sigma_{i})_{\widetilde{I}\widetilde{J}}\nabla_{i}\Phi\widetilde{S}$
(B.13)
$\displaystyle+(1-x)\widetilde{S}_{\dot{\beta}}{}^{\widetilde{J}}\chi^{I}\left(\nabla_{\alpha
I}\widehat{\Omega}_{\beta\widetilde{J}}\widetilde{S}+\nabla_{\beta\widetilde{J}}\Omega_{\alpha
I}S\right)+\cdots.$ (B.14)

The $\cdots$ in the commutators above include terms with $\Omega^{2}$,
$\widetilde{\Omega}^{2}$ and $\Omega\widetilde{\Omega}$. We do not include
them explicitly since these terms need at least two Grassmann odd derivatives
to have a non-vanishing contribution in the covariant $\theta$ expansion.

Other useful identifies used in some computations are

$\displaystyle(\gamma^{\underline{a}}P\gamma_{\underline{a}})_{\alpha
I\beta\tilde{J}}=2\epsilon_{\alpha\beta}P_{\dot{\gamma}I}{}^{\dot{\gamma}}{}_{\tilde{J}}+(\sigma^{i})_{IK}(\sigma^{i})_{\tilde{J}\tilde{L}}P_{\alpha}{}^{K}{}_{\beta}{}^{\tilde{L}}=-8\nabla_{\alpha
I}\widehat{\Omega}_{\beta\tilde{J}}=8\nabla_{\beta\tilde{J}}\Omega_{\alpha
I},$ (B.15)
$\displaystyle(\gamma^{\underline{a}}P\gamma_{\underline{a}})_{\alpha
I\dot{\beta}{}^{\tilde{J}}}=2P_{\dot{\beta}I\alpha}{}^{\tilde{J}}-(\sigma^{i})_{IK}(\sigma^{i})^{\tilde{J}\tilde{L}}P_{\alpha}{}^{K}{}_{\dot{\beta}\tilde{L}}=-8\nabla_{\alpha
I}\widehat{\Omega}_{\dot{\beta}}{}^{\tilde{J}}=8\nabla_{\dot{\beta}}{}^{\tilde{J}}\Omega_{\alpha
I},$ (B.16)
$\displaystyle(\gamma^{\underline{a}}P\gamma_{\underline{a}})_{\dot{\alpha}{}^{I}\beta\tilde{J}}=2P_{\beta}{}^{I}{}_{\dot{\alpha}\tilde{J}}-(\sigma^{i})^{IK}(\sigma^{i})_{\tilde{J}\tilde{L}}P_{\dot{\alpha}K\beta}{}^{\tilde{L}}=-8\nabla_{\dot{\alpha}}{}^{I}\widehat{\Omega}_{\beta\tilde{J}}=8\nabla_{\beta\tilde{J}}\Omega_{\dot{\alpha}}{}^{I},$
(B.17)
$\displaystyle(\gamma^{\underline{a}}P\gamma_{\underline{a}})_{\dot{\alpha}{}^{I}\dot{\beta}{}^{\tilde{J}}}=2\epsilon_{\dot{\alpha}\dot{\beta}}P_{\gamma}{}^{I\gamma\tilde{J}}+(\sigma^{i})^{IK}(\sigma^{i})^{\tilde{J}\tilde{L}}P_{\dot{\alpha}K\dot{\beta}\tilde{L}}=-8\nabla_{\dot{\alpha}}{}^{I}\widehat{\Omega}_{\dot{\beta}}{}^{\tilde{J}}=8\nabla_{\dot{\beta}}{}^{\tilde{J}}\Omega_{\dot{\alpha}}{}^{I}.$
(B.18)

##  Appendix C Sigma matrices for $SU(3)$ Structure

The case where the bosonic part of the compactification manifold has only one
normalizable spinor (in our work, this implies that $\widetilde{\chi}$ is
proportional to $\chi$) is commonly said that the space has $SU(3)$ structure.
This Appendix will describe a particular basis for the sigma matrices. We will
use complex indices for the internal directions and an $\mathfrak{su}(3)$
decomposition of the $\mathfrak{so}(6)$ algebra acting on the tangent space
indices of the internal superspace. The complex indices will range from $1$ to
$3$ and we will use
$\mathsf{k},\bar{\mathsf{k}},\mathsf{l},\bar{\mathsf{l}},\mathsf{m},\bar{\mathsf{m}}$
etc. and are raised and lowered with
$(\delta^{\mathsf{k}\bar{\mathsf{l}}},\delta_{\mathsf{k}\bar{\mathsf{l}}})$.
Index contraction can be written in different ways, as
$a^{i}b_{i}=\frac{1}{2}(\delta_{\mathsf{k}\bar{\mathsf{l}}}a^{\mathsf{k}}b^{\bar{\mathsf{l}}}+\delta_{\mathsf{k}\bar{\mathsf{l}}}b^{\mathsf{k}}a^{\bar{\mathsf{l}}})=\frac{1}{2}(a^{\mathsf{k}}b^{\bar{\mathsf{k}}}+b^{\mathsf{k}}a^{\bar{\mathsf{k}}})$.
The full $\mathfrak{so}(1,9)$ is then generated by
$(M_{ab},M_{a\mathsf{k}},M_{a\bar{\mathsf{k}}},M_{\mathsf{k}\bar{\mathsf{l}}},M_{\mathsf{k}\mathsf{l}},M_{\bar{\mathsf{k}}\bar{\mathsf{l}}},M_{\bf
1})$. The internal Lorentz generators are
$(M_{\mathsf{k}\bar{\mathsf{l}}},M_{\mathsf{k}\mathsf{l}},M_{\bar{\mathsf{k}}\bar{\mathsf{l}}},M_{\bf
1})$, where
$\delta^{\mathsf{k}\bar{\mathsf{l}}}M_{\mathsf{k}\bar{\mathsf{l}}}=0$.

In the case where $\widetilde{\chi}$ is proportional to $\chi$, an special
basis can be chosen for the $\sigma$-matrices such that

$\displaystyle(\sigma^{\bar{\mathsf{k}}})_{IJ}\chi^{J}=(\sigma^{\mathsf{k}})^{IJ}\chi_{J}=0.$
(C.1)

We also have the following properties

$\displaystyle
M^{\mathsf{k}\bar{\mathsf{l}}}\chi^{I}=M^{\mathsf{k}\bar{\mathsf{l}}}\chi_{I}=0,\quad
M_{\bf 1}\chi^{I}=\frac{1}{2}\chi^{I},\quad M_{\bf
1}\chi_{I}=-\frac{1}{2}\chi_{I},$ (C.2) $\displaystyle
M^{\mathsf{kl}}\chi^{I}=\frac{1}{2}(\sigma^{\mathsf{kl}})_{J}{}^{I}\chi^{J},\quad
M^{\mathsf{kl}}\chi_{I}=0,\quad
M^{\bar{\mathsf{k}}\bar{\mathsf{l}}}\chi^{I}=0,\quad
M^{\bar{\mathsf{k}}\bar{\mathsf{l}}}\chi_{I}=\frac{1}{2}(\sigma^{\bar{\mathsf{k}}\bar{\mathsf{l}}})_{I}{}^{J}\chi_{J}.$
(C.3)

The anti-symmetric products of two sigma matrices are

$\displaystyle(\sigma^{\mathsf{k}\bar{\mathsf{l}}})_{I}{}^{J}=\frac{1}{2}\left([\sigma^{\mathsf{k}},\sigma^{\bar{\mathsf{l}}}]\right)_{I}{}^{J},\quad(\sigma^{\mathsf{kl}})_{I}{}^{J}=\frac{1}{2}\left([\sigma^{\mathsf{k}},\sigma^{\mathsf{l}}]\right)_{I}{}^{J},\quad(\sigma^{\bar{\mathsf{k}}\bar{\mathsf{l}}})_{I}{}^{J}=\frac{1}{2}\left([\sigma^{\bar{\mathsf{k}}},\sigma^{\bar{\mathsf{l}}}]\right)_{I}{}^{J}.$
(C.4)

The spinor realizations of the internal Lorentz generators are

$\displaystyle\left(M_{\bf
1}\right)_{I}{}^{J}=\frac{1}{6}\delta_{\mathsf{k}\bar{\mathsf{l}}}\left(\sigma^{\mathsf{k}\bar{\mathsf{l}}}\right)_{I}{}^{J}\equiv\frac{1}{2}(\sigma_{\bf
1})_{I}{}^{J},\quad\left(M^{\mathsf{k}\bar{\mathsf{l}}}\right)_{I}{}^{J}=\frac{1}{2}\left((\sigma^{\mathsf{k}\bar{\mathsf{l}}})_{I}{}^{J}-\delta^{\mathsf{k}\bar{\mathsf{l}}}\left(\sigma_{\bf
1}\right)_{I}{}^{J}\right),$ (C.5)
$\displaystyle\left(M^{\mathsf{kl}}\right)_{I}{}^{J}=\frac{1}{2}(\sigma^{\mathsf{kl}})_{I}{}^{J},\quad\left(M^{\bar{\mathsf{k}}\bar{\mathsf{l}}}\right)_{I}{}^{J}=\frac{1}{2}(\sigma^{\bar{\mathsf{k}}\bar{\mathsf{l}}})_{I}{}^{J},$
(C.6)

so one can see that equations (C.2) are satisfied.

The anti-symmetric product of three sigma matrices with holomorphic and anti-
holomorphic indices will be defined as

$\displaystyle(\sigma^{\mathsf{klm}})_{IJ}=\epsilon^{\mathsf{klm}}(\sigma^{1}\sigma^{2}\sigma^{3})\equiv\epsilon^{\mathsf{klm}}(\sigma_{\Omega})_{IJ},$
(C.7)
$\displaystyle(\sigma^{\overline{\mathsf{klm}}})^{IJ}=\epsilon^{\overline{\mathsf{klm}}}(\sigma^{\bar{1}}\sigma^{\bar{2}}\sigma^{\bar{3}})\equiv\epsilon^{\overline{\mathsf{klm}}}(\sigma_{\bar{\Omega}})^{IJ}.$
(C.8)

Note that these matrices are symmetric in $IJ$ and that

$\displaystyle(\sigma_{\Omega})^{IJ}=\left((\sigma_{\bar{\Omega}})_{IJ}\right)^{\dagger},\quad(\sigma_{\bar{\Omega}})^{IJ}=\left((\sigma_{\Omega})_{IJ}\right)^{\dagger}$
(C.9)

Another property is that

$\displaystyle\chi_{I}=f\left(\sigma_{\Omega}\chi\right)_{I},\quad\chi^{I}=\bar{f}\left(\sigma_{\bar{\Omega}}\chi\right)^{I},$
(C.10)

for some internal superfield $f$. These also imply that

$\displaystyle\left(\sigma^{\bar{\mathsf{k}}}\chi\right)^{I}=\frac{f}{2}\epsilon^{\overline{\mathsf{klm}}}\left(\sigma_{\overline{\mathsf{lm}}}\chi\right)^{I},\quad\left(\sigma^{\mathsf{k}}\chi\right)_{I}=\frac{\bar{f}}{2}\epsilon^{\mathsf{klm}}\left(\sigma_{\mathsf{lm}}\chi\right)_{I}.$
(C.11)

The remaining linearly independent products of three sigma matrices are
$(\\{\sigma^{\mathsf{kl}},\sigma^{\bar{\mathsf{m}}}\\})_{IJ}$ and
$(\\{\sigma^{\overline{\mathsf{kl}}},\sigma^{\mathsf{m}}\\})_{IJ}$. They can
be decomposed into irreducible $\mathfrak{su}(3)$ representations using
$\delta_{\mathsf{k}\bar{\mathsf{l}}}$. We will define them as

$\displaystyle(\sigma^{\mathsf{k}}_{\mathbf{3}})_{IJ}=\frac{1}{2}\delta_{\mathsf{l}\bar{\mathsf{m}}}(\\{\sigma^{\mathsf{kl}},\sigma^{\bar{\mathsf{m}}}\\})_{IJ},\quad(\sigma^{\mathsf{kl}\bar{\mathsf{m}}}_{\mathbf{6}})_{IJ}=(\\{\sigma^{\mathsf{kl}},\sigma^{\bar{\mathsf{m}}}\\})_{IJ}-\frac{1}{2}(\sigma^{[\mathsf{k}}_{\mathbf{3}})_{IJ}\delta^{\mathsf{l}]\bar{\mathsf{m}}},$
(C.12)
$\displaystyle(\sigma^{\bar{\mathsf{k}}}_{\bar{\mathbf{3}}})_{IJ}=\frac{1}{2}\delta_{\mathsf{l}\bar{\mathsf{m}}}(\\{\sigma^{\overline{\mathsf{km}}},\sigma^{\mathsf{l}}\\})_{IJ},\quad(\sigma^{\overline{\mathsf{kl}}\mathsf{m}}_{\mathbf{6}})_{IJ}=(\\{\sigma^{\overline{\mathsf{kl}}},\sigma^{\mathsf{m}}\\})_{IJ}+\frac{1}{2}\delta^{\mathsf{m}[\bar{\mathsf{l}}}(\sigma^{\bar{\mathsf{k}}]}_{\bar{\mathbf{3}}})_{IJ}.$
(C.13)

$\sigma^{\mathsf{k}}_{\mathbf{3}}$ and
$\sigma^{\bar{\mathsf{k}}}_{\mathbf{3}}$ can also be written in terms of
$\\{\sigma^{\mathsf{k}},\sigma_{\mathbf{1}}\\}$ and
$\\{\sigma^{\bar{\mathsf{k}}},\sigma_{\mathbf{1}}\\}$. Note that

$\displaystyle(\chi\sigma^{\mathsf{k}}_{\mathbf{3}}\chi)=(\chi\sigma^{\mathsf{kl}\bar{\mathsf{m}}}_{\mathbf{6}}\chi)=(\chi\sigma^{\bar{\mathsf{k}}}_{\bar{\mathbf{3}}}\chi)=(\chi\sigma^{\overline{\mathsf{kl}}\mathsf{m}}_{\mathbf{6}}\chi)=0,$
(C.14)

so the only non-vanishing contraction with two $\chi^{I}$ is
$(\chi\sigma_{\Omega}\chi)$. Similarly, the only non-vanishing contraction
with $(\chi^{I},\chi_{J})$ is with powers of $\sigma_{\mathbf{1}}$.

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