%&plain
\magnification=\magstep1
% Here are some additional notes on spinor actions
% in the low dimensions that may be of some interest.
% Robert Bryant
% 27 May 1998
\font\eu=eufm10
\font\bigbf=cmbx12
\font\tenjap=msbm10\font\sevenjap=msbm7
\font\greekbold=eurb10
\def\ts{\textstyle }
\def\bbC{{\mathchoice{\hbox{\tenjap C}}{\hbox{\tenjap C}}
{\hbox{\sevenjap C}}{\hbox{\sevenjap C}}}}
\def\bbO{{\mathchoice{\hbox{\tenjap O}}{\hbox{\tenjap O}}
{\hbox{\sevenjap O}}{\hbox{\sevenjap O}}}}
\def\bbR{ {\mathchoice {\hbox{\tenjap R}} {\hbox{\tenjap R}}
{\hbox{\sevenjap R}} {\hbox{\sevenjap R}}}}
\def\bbS{{\mathchoice{\hbox{\tenjap S}}{\hbox{\tenjap S}}
{\hbox{\sevenjap S}}{\hbox{\sevenjap S}}}}
\def\bbZ{{\mathchoice{\hbox{\tenjap Z}}{\hbox{\tenjap Z}}
{\hbox{\sevenjap Z}}{\hbox{\sevenjap Z}}}}
\def\w{{\mathchoice{\,{\scriptstyle\wedge}\,}
{{\scriptstyle\wedge}}
{{\scriptscriptstyle\wedge}}{{\scriptscriptstyle\wedge}}}}
\def\xb{{\bf x}}\def\yb{{\bf y}}\def\wb{{\bf w}}\def\zb{{\bf z}}
\def\pb{{\bf p}}\def\qb{{\bf q}}\def\ub{{\bf u}}\def\vb{{\bf v}}
\def\ab{{\bf a}}\def\sb{{\bf s}}\def\hb{{\bf h}}
\def\zerob{{\bf 0}}\def\oneb{{\bf 1}}
\def\Cl{{{\rm C}\ell}}\def\Im{{\rm Im}}
\def\bT{{\bf T}}\def\bR{{\bf R}}\def\bF{{\bf F}}
\def\bH{{\bf H}}\def\bS{{\bf S}}
\def\la{\langle}\def\ra{\rangle}
\def\euk{\hbox{\eu k}}\def\eur{\hbox{\eu r}}
\def\eug{\hbox{\eu g}}\def\euspin{\hbox{\eu spin}}\def\euso{\hbox{\eu so}}
\def\eusu{\hbox{\eu su}}\def\euh{\hbox{\eu h}}
\def\eugl{\hbox{\eu gl}}\def\eusl{\hbox{\eu sl}}
\def\GL{{\rm GL}}\def\SL{{\rm SL}}\def\End{\mathop{\rm End}}
\def\Spin{{\rm Spin}}\def\SO{{\rm SO}}\def\SU{{\rm SU}}
\def\Sp{{\rm Sp}}\def\Hom{{\rm Hom}}
\def\bfalpha{\hbox{\greekbold\char11}}%bold Greek alpha
\def\bfeta{\hbox{\greekbold\char17}}%bold Greek eta
\def\bfomega{\hbox{\greekbold\char33}}%bold Greek omega
\def\bfsigma{\hbox{\greekbold\char27}}%bold Greek sigma
\def\bftheta{\hbox{\greekbold\char18}}%bold Greek theta
\def\bfTheta{\hbox{\greekbold\char02}}%bold Greek Theta
\def\bfphi{\hbox{\greekbold\char30}}%bold Greek phi
\def\bfPhi{\hbox{\greekbold\char08}}%bold Greek Phi
\def\bfpsi{\hbox{\greekbold\char32}}%bold Greek psi
\def\bfPsi{\hbox{\greekbold\char09}}%bold Greek Psi
\centerline{\bigbf Spin(10,1)-metrics with a parallel null spinor}
\centerline{\bigbf and maximal holonomy}
\bigskip
\bigskip
{\bf 0. Introduction.} The purpose of this addendum to the
earlier notes on spinors is to outline the construction of
Lorentzian metrics in $10{+}1$ dimensions that have a parallel
null spinor and whose holonomy is as large as possible. The
notation from the earlier note will be maintained here.
\bigskip
{\bf 1. The squaring map.} Consider the squaring
map~$\sigma:\bbO^4\to\bbR^{2+1}\oplus\bbO=\bbR^{10+1}$ that takes spinors
for~$\Spin(10,1)$ to vectors. This map~$\sigma$ is defined as follows:
$$
\sigma\left(\pmatrix{\xb_1\cr\yb_1\cr\xb_2\cr\yb_2\cr}\right)
= \pmatrix{|\xb_1|^2+|\yb_1|^2\cr
2\,\bigl(\xb_1\cdot\xb_2-\yb_1\cdot\yb_2\bigr)\cr
|\xb_2|^2+|\yb_2|^2\cr
2\,\bigl(\xb_1\,\yb_2+\xb_2\,\yb_1\bigr)\cr}.
$$
Define the inner product on vectors in~$\bbR^{2+1}\oplus\bbO$ by the rule
$$
\pmatrix{a_1\cr a_2\cr a_3\cr \xb}\cdot
\pmatrix{b_1\cr b_2\cr b_3\cr \yb}
= -2(a_1b_3+a_3b_1)+a_2b_2 + \xb\cdot\yb
$$
and let~$\SO(10,1)$ denote the subgroup of~$\SL(\bbR^{2+1}\oplus\bbO)$
that preserves this inner product. This group still has two components
of course, but only the identity component~$\SO^\uparrow(10,1)$ will be of
interest here. Let~$\rho:\Spin(10,1)\to\SO^\uparrow(10,1)$ be the homomorphism
whose induced map on Lie algebras is given by the isomorphism
$$
\rho'\left(
\pmatrix{
a_1 +x\,I_8 & C\,R_\xb & y\,I_8 & C\,R_\yb\cr
-C\,L_\xb & a_3 +x\,I_8 & C\,L_\yb & -y\,I_8 \cr
z\,I_8 & C\,R_\zb & a_1 -x\,I_8 & C\,R_\xb\cr
C\,L_\zb & -z\,I_8 & -C\,L_\xb & a_3 -x\,I_8 \cr}
\right)
=
\pmatrix{
2x & y & 0 & \overline{\yb}^*\cr
2z & 0 & 2y & 2\,\overline{\xb}^* \cr
0 & z & -2x & \overline{\zb}^*\cr
2\,\overline{\zb} & -2\,\overline{\xb} & 2\,\overline{\yb} & a_2\cr}.
$$
With these definitions, the squaring map~$\sigma$ is seen to have the
equivariance~$\sigma\bigl(g\,\zb\bigr) = \rho(g)\,\bigl(\sigma(\zb)\bigr)$
for all~$g$ in~$\Spin(10,1)$ and all~$\zb\in\bbO^4$.
With these definitions, the polynomial~$p$ has the expression~$p(\zb)
= -{\ts{1\over4}}\sigma(\zb)\cdot\sigma(\zb)$, from which its invariance
is immediate. Moreover, it follows from this that the squaring map
carries the orbits of~$\Spin(10,1)$ to the orbits of~$\SO^\uparrow(10,1)$
and that the image of~$\sigma$ is the union of the origin, the forward
light cone, and the future-directed timelike vectors.
\bigskip
{\bf 2. Parallel null spinors.} A non-zero spinor~$\zb\in\bbO^4$ will be
said to be {\it null} if~$p(\zb)=0$, or, equivalently, if~$\sigma(\zb)$ is
a null vector in~$\bbR^{10+1}$. The typical example is~$\zb_{1,0}$, whose
stabilizer subgroup~$H$ is the connected subgroup of~$\Spin(10,1)$ whose
Lie algebra is defined by the conditions~$x = z = \xb = \zb = 0$
and~$a\in\euk_1$. Consider the subgroup~$\rho(H)\subset\SO^\uparrow(10,1)$.
Its Lie algebra is given by
$$
\rho'(\euh) =
\left\{
\pmatrix{
0 & y & 0 & \overline{\yb}^*\cr
0 & 0 & 2y & 0 \cr
0 & 0 & 0 & 0\cr
0 & 0 & 2\,\overline{\yb} & a_2\cr}
\ \vrule\
\matrix{y\in\bbR,\cr\noalign{\vskip2pt} \yb\in\bbO,\cr
\noalign{\vskip2pt}\ a\in\euk_1}\ \right\}.
$$
The question is whether $\rho(H)$ can be the holonomy of a torsion-free
connection on an $11$-manifold.
The first thing to check is to see whether this subgroup satisfies the
Berger criteria. Suppose that $M$ were an $11$-manifold endowed with
a $\rho(H)$-structure~$B$ that is torsion-free. Then the Cartan structure
equations on~$B$ will be of the form
$$
\pmatrix{d\omega_1\cr d\omega_2\cr d\omega_3\cr d\bfomega\cr}
= -\pmatrix{
0 & \psi & 0 & {}^t\bfphi\cr
0 & 0 & 2\,\psi & 0 \cr
0 & 0 & 0 & 0\cr
0 & 0 & 2\,\bfphi & \bftheta \cr}\w
\pmatrix{\omega_1\cr \omega_2\cr \omega_3\cr \bfomega\cr}
$$
where~$\bfomega$ and $\bfphi$ take values in~$\bbO$ and~$\bftheta$ takes
values in the subalgebra~$\euspin(7)\subset\eugl(\bbO)$ that consists of
the elements of the form~$a_2$ with~$a\in\euk_1$. For such a
$\rho(H)$-structure, the Lorentzian metric~$g = -4\,\omega_1\,\omega_3
+{\omega_2}^2+\bfomega\cdot\bfomega$ has a parallel null spinor and~$B$
represents the structure reduction afforded by this parallel structure.
Note that the null 1-form~$\omega_3$ is parallel and well-defined on~$M$.
It (or, more properly, its metric dual vector field) is the square of
the parallel null spinor field.
Differentiating the Cartan structure equations yields the
first Bianchi identites:
$$
0
= \pmatrix{
0 & \Psi & 0 & {}^t\bfPhi\cr
0 & 0 & 2\,\Psi & 0 \cr
0 & 0 & 0 & 0\cr
0 & 0 & 2\,\bfPhi & \bfTheta \cr}\w
\pmatrix{\omega_1\cr \omega_2\cr \omega_3\cr \bfomega\cr}\,.
$$
where~$\Psi = d\psi$, $\bfPhi = d\bfphi + \bftheta\w\bfphi$, and
$\bfTheta = d\bftheta + \bftheta\w\bftheta$.
By the second line of this system, $\Psi\w\omega_3=0$, while the
first line implies that~$\Psi\w\omega_2\equiv0\bmod\bfomega$, so there must
be functions~$p$~and~$\qb$, with values in~$\bbR$ and $\bbO$ respectively,
so that
$$
\Psi = (p\,\omega_2 + \qb\cdot\bfomega)\w\omega_3\,.
$$
Substituting this into the first line of the system yields
$$
{}^t\bigl(\bfPhi - \qb\,\omega_2\w\omega_3\bigr)\w\bfomega = 0,
$$
so it follows that
$$
\bfPhi = \qb\,\omega_2\w\omega_3 + \bfsigma\w\bfomega\,,
$$
where~$\bfsigma = {}^t\bfsigma$ is some 1-form with values in the
symmetric part of~$\eugl(\bbO)$, which will be denoted~$S^2(\bbO)$
from now on. Substituting this last equation
into the last line of the Bianchi identities, yields
$$
2\,\bfsigma\w\bfomega\w\omega_3 + \bfTheta\w\bfomega = \zerob.
$$
In particular, this implies that~$\bfTheta\w\bfomega = \zerob\bmod\omega_3$,
so that~$\bfTheta \equiv\bR\bigl(\bfomega\w\bfomega\bigr)\bmod\omega_3$
where~$\bR$ is a function on~$B$ with values
in~${\cal K}\bigl(\euspin(7)\bigr)$, which is the irreducible~$\Spin(7)$
module of highest weight~$(0,2,0)$ and of (real) dimension~$168$. (This
uses the usual calculation of the curvature tensor of $\Spin(7)$-manifolds.)
Thus, set
$$
\bfTheta = \bR\bigl(\bfomega\w\bfomega\bigr) + 2\,\bfalpha\w\omega_3\,,
$$
where~$\bfalpha$ is a 1-form with values in $\euspin(7)$ whose entries can
be assumed, without loss of generality, to be linear combinations of
$\omega_1$, $\omega_2$, and the components of~$\bfomega$. Substituting
this last relation into the last line of the Bianchi identities now yields
$$
2\,\bfsigma\w\bfomega\w\omega_3 + 2\,(\bfalpha\w\omega_3)\w\bfomega = \zerob,
$$
which is equivalent to the condition
$$
\bfsigma\w\bfomega \equiv \bfalpha\w\bfomega \bmod\omega_3.
$$
In particular, this implies that~$\bfsigma-\bfalpha\equiv0\bmod\omega_3,
\bfomega$. Since~$\bfsigma$ and~$\bfalpha$ take values in~$S^2(\bbO)$ and
$\euspin(7)$ respectively, which are disjoint subspaces of~$\eugl(\bbO)$,
it follows that $\bfsigma\equiv\bfalpha\equiv0\bmod\omega_3,\bfomega$.
In particular, neither~$\omega_1$ nor~$\omega_2$ appear in the expressions
for~$\bfsigma$ and~$\bfalpha$. Recall that, by definition, $\omega_3$ does not
appear in the expression for~$\bfalpha$, so $\bfalpha$ must be a linear
combination of the components of~$\bfomega$ alone.
Now, from the above equation, it follows that
$$
\bfsigma\w\bfomega = \bfalpha\w\bfomega + \sb\,\omega_3\w\bfomega
$$
where~$\sb$ takes values in~$S^2(\bbO)$. Finally, the first line of
the Bianchi identities show that ${}^t\bfomega\w\bfalpha\w\bfomega = 0$,
so it follows that~$\bfalpha = \ab(\bfomega)$ where~$\ab$ is a function
on~$B$ that takes values in a subspace of~$\Hom\bigl(\bbO,\euspin(7)\bigr)$
that is of dimension~$8\cdot 21 - 56 = 112$. By the usual weights and
roots calculation, it follows that this subspace is irreducible, with
highest weight~$(0,1,1)$.
To summarize, the Bianchi identities show that the curvature of a
torsion-free~$\rho(H)$-structure~$B$ must have the form
$$\eqalign{
\bfPsi &=(p\,\omega_2 + \qb\cdot\bfomega)\w\omega_3\,,\cr
\bfPhi &= \qb\,\omega_2\w\omega_3
+ \sb\,\omega_3\w\bfomega + \ab(\bfomega)\w\bfomega \cr
\bfTheta &= \bR\bigl(\bfomega\w\bfomega\bigr) + 2\,\ab(\bfomega)\w\omega_3\cr
}
$$
where~$\bR$ takes values in~${\cal K}\bigl(\euspin(7)\bigr)$, the
irreducible $\Spin(7)$-representation of highest weight~$(0,2,0)$ (of
dimension~$168$), $\ab$ takes values in the irreducible
$\Spin(7)$-representation of highest weight~$(0,1,1)$ (of dimension~$112$),
$\sb$ takes values in~$S^2(\bbO)$ (the sum of a trivial representation
with an irreducible one of highest weight~$(0,0,2)$ and of dimension~$35$),
$\qb$ takes values in~$\bbO$, and $p$ takes values in~$\bbR$. Thus, the
curvature space ${\cal K}\bigl(\rho'(\euh)\bigr)$ has dimension~$325$.
By inspection, this curvature space passes Berger's first test (i.e.,
the generic element has the full~$\rho'(\euh)$ as its range).
Thus, a structure with the full holonomy is not ruled out by this method.
\bigskip
{\bf 3. Integrating the structure equations.} To go further in the
analysis, it will be useful to integrate the structure equations,
at least locally. This will be done by a series of observations.
To begin, notice that, since~$d\omega_3=0$, there exists, locally,
a function~$x_3$ on~$M$ so that $\omega_3=dx_3$. This function is
determined up to an additive constant, and can be defined on
any simply connected open subset~$U_0\subset M$.
Since~$d\omega_2 = -2\,\psi\w\omega_3 = -2\,\psi\w dx_3$, it follows that
any point of~$U_0$ has an open neighborhood~$U_1\subset U_0$ on which
there exists a function~$x_2$ for which~$\omega_2\w\omega_3 = dx_2\w dx_3$.
The function~$x_2$ is determined up to the addition of an arbitrary function
of~$x_3$. In consequence, there exists a function~$r$
on~$B_1 = \pi^{-1}(U_1)$ so that~$\omega_2 = dx_2 - 2r\,dx_3$.
It now follows from the structure equation for~$d\omega_2$
that~$\psi\w\omega_3 = dr\w dx_3$.
Consequently, there is a function~$f$ on~$B_1$ so that~$\psi = dr + f\,dx_3$.
Since~$\bfPsi = d\psi$ is $\pi$-basic, it follows that~$df\w dx_3$ is
well-defined on~$U_1$. Consequently,~$f$ is well-defined on~$U_1$
up to the addition of an arbitrary function of~$x_3$.
Now, since
$$
d\omega_1 = -\psi\w\omega_2 -{}^t\bfphi\w\bfomega
= -(dr + f\,dx_3)\w(dx_2 - 2r\,dx_3) -{}^t\bfphi\w\bfomega,
$$
it follows that
$$
d(\omega_1 + r\,dx_2 - r^2\,dx_3) = f\,dx_2\w dx_3 -{}^t\bfphi\w\bfomega.
$$
The fact that the 2-form on the right hand side is closed, together with
the fact that the system~$I$ of dimension~9 spanned by $dx_3$ and the
components of~$\bfomega$ is integrable (which follows from the structure
equations), implies that there are functions~$G$ and~$\bF$ on~$B$ so
that
$$
d(\omega_1 + r\,dx_2 - r^2\,dx_3) = d( G\,dx_3 - {}^t\bF\,\bfomega ),
$$
from which it follows that there is a function~$x_1$ on~$B$ so that
$$
\omega_1 = dx_1 - r\,dx_2 + r^2\,dx_3 + G\,dx_3 - {}^t\bF\,\bfomega\,.
$$
The function~$x_1$ is determined (once the choices of~$x_3$ and~$x_2$ are
made) up to an additive function that is constant on the leaves of the
system~$I$, i.e., up to the addition of an (arbitrary) function of 9
variables. Expanding $d( G\,dx_3 - {}^t\bF\,\bfomega )
= f\,dx_2\w dx_3 -{}^t\bfphi\w\bfomega$ via the structure equations and
reducing modulo~$dx_3$ yields
$$
{}^t\bigl(d\bF + \bftheta\,\bF)\w\bfomega
\equiv {}^t\bfphi\w\bfomega \bmod dx_3\,.
$$
so that there must exist functions $\bH$ and $\ub = {}^t\ub$ so that
$$
\bfphi = d\bF + \bftheta\,\bF + \bH\,dx_3 + \ub\,\bfomega\,.
$$
Substituting this back into the relation $d( G\,dx_3 - {}^t\bF\,\bfomega )
= f\,dx_2\w dx_3 -{}^t\bfphi\w\bfomega$ yields
$$
dG+2\,{}^t\bF\,d\bF-{}^t\bigl(\bH-2\ub\,\bF\bigr)\,\bfomega
\equiv f\,dx_2\bmod dx_3\,.
$$
Setting $G = g - \bF\cdot\bF$ and $\hb = \bH-2\ub\,\bF$, this becomes
$$
dg \equiv f\,dx_2 + {}^t\hb\,\bfomega \bmod dx_3\,,
$$
with the formulae
$$
\eqalign{
\omega_1
&=dx_1-r\,dx_2+r^2\,dx_3+(g{-}\bF\cdot\bF)\,dx_3-{}^t\bF\,\bfomega\,,\cr
\bfphi &= d\bF + \bftheta\,\bF + (\hb+2\ub\,\bF)\,dx_3 + \ub\,\bfomega\,.\cr
}
$$
Now the final structure equation becomes
$$
d\bfomega = - 2\bigl(d\bF + \bftheta\,\bF + \ub\,\bfomega\bigr)\w dx_3
- \bftheta\w\bfomega
$$
which can be rearranged to give
$$
d\bigl(\bfomega + 2\bF\,dx_3\bigr)
= -\bigl(\bftheta - 2\ub\,dx_3\bigr)\w \bigl(\bfomega + 2\bF\,dx_3\bigr)\,.
$$
This suggests setting $\bfeta = \bfomega + 2\bF\,dx_3$ and writing the
formulae found so far as
$$
\eqalign{
\omega_1&=dx_1-r\,dx_2+r^2\,dx_3+(g{+}\bF\cdot\bF)\,dx_3-{}^t\bF\,\bfeta\,,\cr
\omega_2&=dx_2 - 2r\,dx_3\,,\cr
\omega_3&=dx_3\,,\cr
\bfomega &= - 2\bF\,dx_3 + \bfeta\,,\cr
\noalign{\vskip5pt}
\psi &= dr + f\,dx_3\,,\cr
\bfphi &= d\bF + \bftheta\,\bF + \hb\,dx_3 + \ub\,\bfeta\,,\cr
\noalign{\vskip5pt}
dg &\equiv f\,dx_2 + {}^t\hb\,\bfeta \bmod dx_3\,,\cr
d\bfeta &= -\bigl(\bftheta - 2\ub\,dx_3\bigr)\w\bfeta \,.
}
$$
where, in these equations, $\bftheta$ takes values in~$\euspin(7)$ and
$\ub={}^t\ub$. Note that
$$
-4\,\omega_1\,\omega_3 + {\omega_2}^2 +\bfomega\cdot\bfomega
= -4\,dx_1\,dx_3 + {dx_2}^2 - 4g\,{dx_3}^2 + \bfeta\cdot\bfeta.
$$
\bigskip
{\bf 4. Interpreting the integration.}
I now want to describe how these formulae give a recipe for
writing down all of the solutions to our problem.
By the last of the
structure equations, the eight components of~$\bfeta$ describe an
integrable system of rank~$8$ that is (locally) defined on the
original 11-manifold. Let us restrict to a neighborhood where the
leaf space of~$\bfeta$ is simple, i.e., is a smooth manifold~$K^8$.
The equation~$d\bfeta = -\bigl(\bftheta - 2\ub\,dx_3\bigr)\w\bfeta$
shows that on~$\bbR\times K^8$, with coordinate~$x_3$ on the first
factor, there is a $\{1\}\times\Spin(7)$-structure, which
can be thought of as a 1-parameter family of torsion-free
$\Spin(7)$-structures on~$K^8$ (the parameter is~$x_3$, of course).
This 1-parameter family is not arbitrary because the matrix~$\ub$ is
symmetric. This condition is equivalent to saying that if~$\Phi$
is the canonical $\Spin(7)$-invariant $4$-form (depending on~$x_3$, of
course) then
$$
{{\partial\Phi}\over{\partial x_3}} = \lambda\,\Phi + \Upsilon
$$
for some function~$\lambda$ on~$\bbR\times K^8$ and~$\Upsilon$ is
an anti-self dual 4-form (via the $x_3$-dependent metric on the
fibers of~$\bbR\times K\to \bbR$, of course). It is not hard to see
that this is 7 equations on the variation of torsion-free
$\Spin(7)$-structures and that, moreover, given any 1-parameter variation of
torsion-free $\Spin(7)$-structures, one can (locally) gauge this family by
diffeomorphisms preserving the fibers of~$\bbR\times K\to \bbR$ so that
it satisfies these equations. (In fact, if $K$ is compact and the
cohomology class of~$\Phi$ in~$H^4(K,\bbR)$ is independent of~$x_3$ then
this can be done globally.) Call such a variation {\it conformally
anti-self dual}.
Now from the above calculations, this process can be reversed:
One starts with any conformally anti-self dual variation
of $\Spin(7)$-structures on~$K^8$, then on $\bbR^3\times K$ one forms the
Lorentzian metric
$$
ds^2 = -4\,dx_1\,dx_3 + {dx_2}^2 - 4g\,{dx_3}^2 + \bfeta\cdot\bfeta
$$
where~$g$ is any function on~$\bbR^3\times K$ that satisfies ${\partial g}/
{\partial x_1} = 0$ and~$\bfeta\cdot\bfeta$ is the $x_3$-dependent metric
associated to the variation of~$\Spin(7)$-structures. Then this Lorentzian
metric has a parallel null spinor. For generic choice of the variation
of~$\Spin(7)$-structures and the function~$g$,
this will yield a Lorentzian metric whose holonomy is the desired
stabilizer group of dimension~30. This can be seen by combining
the standard generality result for~$\Spin(7)$-metrics on $8$-manifolds,
which shows that for generic choices as above the curvature tensor has
range equal to the full~$\rho'(\euh)$ at the generic point,
with the Ambrose-Singer holonomy Theorem, which implies that such
a metric will have its holonomy equal to the full group of dimension~$30$.
In particular, it follows that, up to diffeomorphism, the local solutions to
this problem depend on one arbitrary function of 10 variables. It has to
be remarked, though, that such a solution is not, in general, Ricci flat,
in contrast to the case where a $(10,1)$ metric has a non-null parallel
spinor field.
\smallskip
Note, by the way, that the 4-form~$\Phi$ will not generally be closed,
let alone parallel. However, the 5-form~$dx_3\w\Phi$ will be closed
and parallel. The other non-trivial parallel forms are the 1-form~$dx_3$,
the 2-form~$dx_2\w dx_3$, and the 6-, 9-, and 10-forms that are the
duals of these.
\bye