# Riemannian geometry without indices.

V.V.Fock  and  P.Goussard  V.F. : IRMA, Université de Strasbourg, 7 rue René
Descartes 67084 Strasbourg France<EMAIL_ADDRESS>P.G. : ??
<EMAIL_ADDRESS>
###### Abstract.

We suggest an index-free formalism allowing to simplify many computations in
Riemann geometry. The main ingredients are forms with values in a Clifford
algebra and an action of the group $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$
on such forms.

##  1\. Introduction

Working with Levi-Civita connection, curvature, Weyl and Ricci tensors and in
particular deriving Einstein equation out of the Hilbert action is a painful
struggle with indices (see for example [7]). In the present note we suggest a
version of Cartan-Palatini formalism [1][2][3] allowing to make most of the
computations without indices at all. The main objects for this approach are
Clifford forms: differential forms with values in the Clifford algebra
$\mathit{Cl}(\mathrm{V})$ generated by a vector space $\mathrm{V}$ provided
with a nondegenerate quadratic form $\eta$. Since Clifford algebra is graded
as a vector space, such differential forms are bigraded. The corresponding
graded components are denoted by
$\Omega^{pq}=\Omega^{q}(M,\mathit{Cl}^{p}(\mathrm{V}))$. The elements of such
components are called $(p,q)$-forms.

Clifford forms are known to be useful to work with spinors, but here we would
like to emphasize that they simplify computations even if no spinors are under
consideration.

The Riemann metric on a manifold is replaced by a vielbein $\theta$ which is
an (1,1)-form and and a spin connection $\omega$, which is a (2,1)-form. The
torsion can be written as a (1,2)-form $t=d\theta+\omega\theta+\theta\omega$
and the curvature as a (2,2)-form $R=d\omega+\omega^{2}$. The Hilbert action
takes in these variables the form

$S=\operatorname{str}\int\theta^{n-2}(d\omega+\omega^{2}),$

where $\operatorname{str}$ denotes the super-trace on the Clifford algebra.
The Hilbert action is invariant under the gauge transformation $\theta\mapsto
g^{-1}\theta g$, $\omega\mapsto g^{-1}dg+g^{-1}dg$ with $g$ a function on the
manifold with values in the spin group considered as a subgroup of the
Clifford algebra.

In presence of the vielbein $\theta$ one can define an action of the Lie
algebra $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$ on the space of Clifford
forms analogous to the Lefschetz algebra $\mathfrak{sl}_{2}$ in topology of
Kähler manifolds. One can define two Hodge star operator on Clifford forms
acting in the fiber and in the base, respectively. The algebra
$\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$ is generated by external
multiplication by the form $\theta$ as well as by its conjugates by the Hodge
stars. This algebra simplifies working with the suggested formalism. In
particular Einstein equation together with the first Bianchi identity in four
dimensions amount to the invariance of the curvature tensor under this
algebra. Which in its turn implies self-duality and symmetry of the curvature.

In the end of this paper the construction is generalized to complex manifolds
where the space of Clifford forms is $\mathbb{Z}^{4}$ graded and the algebra
$\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$ is replaced by the affine Lie
algebra $\widehat{\mathfrak{sl}}_{4}$.

##  2\. Grassmann and Clifford algebras.

In this section we recall basic notions about Grassmann (exterior) and
Clifford algebras and relations between them. Then we define the algebra of
Clifford forms and the action of the algebra
$\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$ thereon.

A Clifford algebra $\mathit{Cl}(\mathrm{V})$ of a vector space $\mathrm{V}$
provided with a nondegenerate quadratic form
$\eta\in\mathrm{V}^{*}\otimes\mathrm{V}^{*}$ is an algebra generated by
$\mathrm{V}$ with relations $vv^{\prime}+v^{\prime}v=\eta(v,v^{\prime})$ for
any $v,v^{\prime}\in\mathrm{V}$. A Grassmann algebra $\Lambda(\mathrm{V})$ of
a vector space $\mathrm{V}$ is an algebra generated by $\mathrm{V}$ with
relations $vv^{\prime}+v^{\prime}v=0$ for any $v,v^{\prime}\in\mathrm{V}$. The
Grassmann algebra is graded with $\deg v=1$ for any $v\in\mathrm{V}$. The
corresponding graded components will be denoted by $\Lambda^{i}(\mathrm{V})$.

The Grassmann algebra $\Lambda(\mathrm{V})$ faithfully acts on the Clifford
algebra $\mathit{Cl}(\mathrm{V})$ by $v\wedge
a\mapsto\frac{1}{2}(va+(-1)^{\deg a}av)$ for any
$v\in\mathrm{V}\subset\Lambda(\mathrm{V})$ and any homogeneous
$a\in\mathit{Cl}(\mathrm{V})$. This action allows to define an isomorphism
between Grassmann and Clifford algebras as modules over the Grassmann algebra.
The isomorphism is fixed by the requirement that $1\mapsto 1$. Denote by
$\mathit{Cl}^{i}(\mathrm{V})$ the subspaces of the Clifford algebra
corresponding to the homogeneous components $\Lambda^{i}(\mathrm{V})$ of the
Grassmann algebra with respect to this isomorphism and
$\Pi^{i}:\mathit{Cl}(\mathrm{V})\to\mathit{Cl}^{i}(\mathrm{V})$ the projection
on the corresponding component.

The component $\mathit{Cl}^{n}(\mathrm{V})$ is of dimension one and contains
two elements $v$ such that $v^{2}=1$. Choosing one of these two elements fixes
the orientation of the space $V$. There is a second action of $\Lambda(V)$
given by $v\vdash a\mapsto\frac{1}{2}(va-(-1)^{\deg a}av)$ and the second
isomorphism corresponding of Grassmann and Clifford algebras as modules
sending $1$ to $v$. Define the Hodge star operator as the automorphism $*$ of
the Clifford algebra as a vector space intertwining these two isomorphisms,
namely such that $v\wedge(*a)=*(v\vdash a)$. Obviously
$*:\mathit{Cl}^{i}(\mathrm{V})\to\mathit{Cl}^{n-i}(\mathrm{V})$. The
identification of $\mathit{Cl}(\mathrm{V})$ and $\Lambda(\mathrm{V})$ allows
to consider $*$ also as an automorphism of the Grassmann algebra (as a vector
space). The actions $v\wedge$ and $v\vdash$ on the Clifford algebra
transferred to the Grassmann algebra are just the usual external and internal
multiplication by $v$.

Observe that though the isomorphism of $\mathit{Cl}(\mathrm{V})$ and
$\Lambda(\mathrm{V})$ is not an algebra isomorphism one can express the
external and the internal product in terms of Clifford product and the
projections:

$\displaystyle x\wedge y=\Pi^{\deg x+\deg y}xy,$ $\displaystyle x\vdash
y=\Pi^{\deg y-\deg x}xy.$

Given a basis in the space $V$ the Grassmann algebra can be identified with
the algebra of polynomials of $n$ Grassmann variables $\xi_{1},\ldots,\xi_{n}$
and the Hodge $*$ operator to a Grassmann Fourier transformation, see e.g.
[5], (here and below we assume the Einstein summation convention over repeated
indices):

$*f(\xi_{1},\ldots,\xi_{n})=\int
e^{\displaystyle\eta^{ij}\xi_{i}\zeta_{j}}f(\zeta_{1},\ldots,\zeta_{n})d\zeta_{n}\cdots
d\zeta_{1}.$

One can easily verify the properties analogous to the usual Fourier
transformation:

$\frac{\partial}{\partial\xi_{i}}*f=*(\eta^{ij}\xi_{j}f),\
\xi_{i}*f=*(\eta_{ij}\frac{\partial}{\partial\xi_{j}}f)$

Here $\eta_{ij}=\eta(\xi_{i},\xi_{j})$ is the scalar product matrix and
$\eta^{ij}$ is its inverse.

###  2.1. Action of the Lie algebra $\mathfrak{sl_{2}}\times\mathfrak{sl_{2}}$
on Clifford forms.

Consider now the algebra
$\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}}=\mathit{Cl}(V)\otimes\Omega(M)$
of forms on an $n$-dimensional manifold $M$ with values in the Clifford
algebra. Let the vielbein $\theta\in\Omega^{11}$ defines an isomorphism
between the tangent bundle to $M$ and a trivial bundle with fiber
$\mathrm{V}$. One can define two Hodge $*$ operators
$*_{1}:\Omega^{pq}\to\Omega^{(n-p)q}$ and
$*_{2}:\Omega^{pq}\to\Omega^{p(n-q)}$ acting on the Clifford algebra and on
forms, respectively. Define an operator $E:\Omega^{pq}\to\Omega^{p+1,q+1}$ by
$2Ex=\theta x+(-1)^{p+q}x\theta$ and the operators
$E^{\prime}=*_{1}^{-1}E*_{1}$, $F^{\prime}=*_{2}^{-1}E*_{2}$,
$F=*_{1}^{-1}*_{2}^{-1}E*_{2}*_{1}$, $H=p+q-n$ and $H^{\prime}=p-q$.

These operators satisfy the following properties:

######  Proposition 1.

1\. The operators $E,F,H,E^{\prime},F^{\prime},H^{\prime}$ satisfy the
relations of the Lie algebra $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$,
namely

$[H,E]=2E,[H,F]=-2F,[H^{\prime},E^{\prime}]=2E^{\prime},[H^{\prime},F^{\prime}]=-2F^{\prime}$

and all other pairs of generators commute.

2\. $\deg E=(1,1)$, $\deg F=(-1,-1)$, $\deg E^{\prime}=(-1,1)$, $\deg
F^{\prime}=(1,-1)$, , $\deg H=\deg H^{\prime}=(0,0)$.

3\. $2E^{\prime}x=\theta x-(-1)^{p+q}x\theta.$

4\.
$\exp(\frac{\pi}{2}(E^{\prime}-F^{\prime}))f(\xi_{1},\ldots,\xi_{n},\xi^{1},\ldots,\xi^{n})=\\\
~{}~{}~{}~{}~{}~{}~{}=f(\eta_{1j}\xi^{j}\,\ldots,\eta_{nj}\xi^{j},\eta^{1j}\xi_{j}\,\ldots,\eta^{nj}\xi_{j}).$

5\. $\exp(\frac{\pi}{2}(E-F))=*_{1}*_{2}$

###### Proof.

The generators act on Clifford forms without any differentiation so we need to
verify the proposition at one point of the manifold. At one point $m\in M$ a
Clifford form takes value in $\mathit{Cl}(\mathrm{V})\otimes\Lambda(T_{m}M)$
which, using the isomorphism defined by $\theta$, can be identified with
$\mathit{Cl}(\mathrm{V})\otimes\Lambda(\mathrm{V}^{*})$.

To make explicit computation one can further identify this algebra with the
algebra $\Lambda(\mathrm{V})\otimes\Lambda(\mathrm{V}^{*})$. Given a basis in
$\mathrm{V}$ the algebras $\Lambda(\mathrm{V})$ and $\Lambda(\mathrm{V}^{*})$
can be identified with the space of functions of $n$ Grassmann variables
$\xi_{i}$ and $\xi^{i}$, respectively, and the vielbein writes down as
$\theta=\xi_{i}\otimes\xi^{i}$. The following computation in
$\mathit{Cl}(\mathrm{V})\otimes\Lambda(\mathrm{V}^{*})$

(1) $\begin{split}2Ex=&\theta
x+(-1)^{p+q}x\theta=\xi_{i}a\otimes\xi^{i}\alpha+(-1)^{p+q}a\xi_{i}\otimes\alpha\xi^{i}=\\\
=&\xi_{i}a\otimes\xi^{i}\alpha+(-1)^{p}a\xi_{i}\otimes\xi^{i}\alpha=\xi_{i}\wedge
a\otimes\xi^{i}\alpha,\end{split}$

where $x=a\otimes\alpha$ with $a\in\mathit{Cl}^{p}(\mathrm{V})$ and
$\alpha\in\Omega^{q}(M)$, shows that the operators $E$ and $E^{\prime}$ can be
written in $\Lambda(\mathrm{V})\otimes\Lambda(\mathrm{V}^{*})$ just as
multiplication operator by $\xi_{i}\otimes\xi^{i}$. Therefore all operators
can be re written as

$\displaystyle E$ $\displaystyle=\xi_{i}\otimes\xi^{i},$ $\displaystyle F$
$\displaystyle=\frac{\partial}{\partial\xi_{i}}\otimes\frac{\partial}{\partial\xi^{i}}$
$\displaystyle H$
$\displaystyle=\,\xi_{i}\frac{\partial}{\partial\xi_{i}}\otimes
1+1\otimes\frac{\partial}{\partial\xi^{i}}\xi^{i}$ $\displaystyle E^{\prime}$
$\displaystyle=\eta_{ij}\frac{\partial}{\partial\xi_{i}}\otimes\xi^{j},$
$\displaystyle F^{\prime}$
$\displaystyle=\xi_{i}\otimes\eta^{ij}\frac{\partial}{\partial\xi^{j}},$
$\displaystyle H^{\prime}$
$\displaystyle=\,1\otimes\xi^{i}\frac{\partial}{\partial\xi^{i}}-\xi_{i}\frac{\partial}{\partial\xi_{i}}\otimes
1.$

The property 1 can be verified by a simple straightforward computation. The
property 2 is obvious. The property 3 follows from the computation similar to
(1).

$\displaystyle 2E^{\prime}x$ $\displaystyle=\theta
x-(-1)^{p+q}x\theta=\xi_{i}a\otimes\xi^{i}\alpha-(-1)^{p+q}a\xi_{i}\otimes\alpha\xi^{i}=$
$\displaystyle=\xi_{i}a\otimes\xi^{i}\alpha-(-1)^{p}a\xi_{i}\otimes\xi^{i}\alpha=\xi_{i}\vdash
a\otimes\xi^{i}\alpha=\eta_{ij}\frac{\partial}{\partial\xi_{j}}a\otimes\xi^{i}\alpha.$

The properties 5 and 6 follow from the fact that the operators in the left
hand side conjugate the generators exactly as the operation in the left hand
side does.

The properties 2,3,5 and 6 follow from these expressions immediately. The
property 1 also can be verified by a direct computation. The property 4 is a
standard statement about finite-dimensional representations of the algebra
$\mathfrak{sl}_{2}$. ∎

Other obvious but useful relations are:

7\. $\theta x=(E+E^{\prime})x,\ x\theta=(-1)^{p+q}(E-E^{\prime})x$,

8\. $E^{k}x=\Pi^{p+k}\theta^{k}x=(-1)^{k(p+q)}\Pi^{p+k}x\theta^{k}$,

9\. $E^{k}$ acting on the weight space with weight (eigenvalue of $H$) equal
to $l$ is an injection if $k+l\leqslant 0$ and a surjection if $k+l\geqslant
0$.

###  2.2. Supertrace.

A supertrace of a Clifford algebra is a linear function on
$\mathit{Cl}(\mathrm{V})$ satisfying the identity
$\operatorname{str}(ab)=(-1)^{\deg a\deg b}\operatorname{str}(ba)$ and
normalized by the condition that
$\operatorname{str}\operatorname{\boldmath{v}}=1$. If $n$ is even then the
Clifford algebra has a unique representation and the supertrace is given by
$\operatorname{str}a=\operatorname{tr}{a\operatorname{\boldmath{v}}}$, where
$\operatorname{tr}$ is the usual trace in this representation. For $n$ odd the
space $\mathrm{V}$ can be embedded into a space $\mathrm{V}^{+}$ of dimension
$n+1$ with one extra generator $\xi^{0}$ orthogonal to $\mathrm{V}$. The
supertrace is given by
$\operatorname{str}a=\operatorname{tr}{a\xi^{0}\operatorname{\boldmath{v}}}$
with the trace taken in the representation of $\mathit{Cl}(\mathrm{V}^{+})$.
The supertrace vanishes on $\mathit{Cl}^{i}(\mathrm{V})$ for $i\neq n$.

In particular for two forms $\alpha\in\Omega^{pq}$ and
$\beta\in\Omega^{p^{\prime}q^{\prime}}$ we have

$\operatorname{str}\alpha\beta=(-1)^{pp^{\prime}+qq^{\prime}}\operatorname{str}\beta\alpha.$

###  2.3. The action of the group $Pin$.

The invertible elements of $g\in\mathit{Cl}(\mathrm{V})$ such that
$g^{-1}\mathit{Cl}^{1}(\mathrm{V})g\subset\mathit{Cl}^{1}(\mathrm{V})$ form a
Lie group denoted by $Pin(\mathrm{V})$. The Lie algebra of this group is
$\mathit{Cl}^{2}(\mathrm{V})$ with respect to the commutator. It is isomorphic
to the Lie algebra $\mathfrak{so}(\mathrm{V})$ and acts on the whole Clifford
algebra preserving degree. The action of the group $Pin(\mathrm{V})$ obviously
commutes with the action of the algebra
$\mathfrak{sl_{2}}\times\mathfrak{sl_{2}}$.

##  3\. Use of the formalism

Let $M$ be an $n$-dimensional manifold and $\mathrm{V}$ an $n$-dimensional
vector space with a nondegenerate quadratic form $\eta$. Let we are given two
Clifford algebra valued 1-forms $\theta\in\Omega^{11}$ and
$\omega\in\Omega^{21}$ such that the form $\theta$ is nondegenerate in the
sense that it induces an isomorphism at every point of $M$ between the tangent
and a trivial bundle with fiber $\mathrm{V}$ by the rule $X\mapsto
i_{X}\theta$. These data define a Riemannian or pseudo-Riemannian metric on
$M$ and a metric preserving connection in the tangent bundle in it. Indeed,
the form $\omega$ defines a connection in the trivial bundle with fiber
$\mathrm{V}$ by the formula $\nabla e=de+\omega e-e\omega$, where
$e\in\Omega^{10}$. The isomorphism $\theta$ induces the metric and the
connection on $TM$.

###  3.1. Gauge group action.

A pair $\theta,\omega$ defines the same metric and connection as a pair
$\theta^{\prime},\omega^{\prime}$ if and only if there exists a gauge
transformation relating them, namely if there exists a function $g$ on $M$
with values in $Pin(\mathrm{V})\subset\mathit{Cl}(\mathrm{V})$ such that
$\theta^{\prime}=g^{-1}\theta g,\ \omega^{\prime}=g^{-1}\omega g+g^{-1}dg$.

Conversely, any metric and a connection in a tangent bundle preserving it can
be represented in this way if and only if the tangent bundle is trivializable.
In particular it always can be done locally and glued together by gauge
transformations.

###  3.2. Curvature.

The connection $\nabla$ can be extended in a standard way to $\Omega^{pq}$ by
the formula $\nabla e\alpha=(\nabla e)\alpha+ed\alpha$ for $e\in\Omega^{p0}$
and $\alpha\in\Omega^{0q}$. It can be therefore written as $\nabla x=dx+\omega
x-(-1)^{q}x\omega$ for any $x\in\Omega^{pq}$. This allows to compute
$\nabla^{2}x=(d\omega+\omega^{2})x-x(d\omega+\omega^{2})$ and thus the
curvature takes the form

$R=d\omega+\omega^{2}\in\Omega^{22}.$

###  3.3. Torsion.

The torsion can be defined as an element $t\in\Omega^{12}$ such that for any
two vector fields $X$ and $Y$ on $M$ we have $i_{Y}i_{X}t=i_{X}\nabla
i_{Y}\theta-i_{Y}\nabla i_{X}\theta-i_{[X,Y]}\theta$. Taking into account the
identity $i_{X}i_{Y}d\alpha=i_{Y}di_{X}\alpha-
i_{X}di_{Y}\alpha+i_{[X,Y]}\alpha$ valid for any 1-form $\alpha$ one can
easily verify that

$t=d\theta+\omega\theta+\theta\omega=d\theta+2E^{\prime}\omega.$

The operator $E^{\prime}:\Omega^{21}\to\Omega^{12}$ is an isomorphism since
$H^{\prime}|_{\Omega^{21}}=-1$ (property 9). It implies that given $\theta$
the torsion determines the form $\omega$ unambiguously. In particular for a
given $\theta$ there exists a unique form $\omega$ such that the torsion
vanishes. A pair $\theta,\omega$ for which the torsion vanishes is called
torsionless.

###  3.4. Bianchi identities.

Computing the covariant derivative of the torsion one gets

$\nabla t=dt+\omega t-t\omega=R\theta-\theta R=-2E^{\prime}R.$

In particular this identity implies that if the torsion vanishes we have
$E^{\prime}R=0$. Since $H^{\prime}R=0$ we have $F^{\prime}R=0$ and therefore
the curvature form is invariant with respect to the second $\mathfrak{sl_{2}}$
algebra and therefore by the property 5 the form $R$ considered as a function
with values in $\Lambda^{2}(\mathrm{V})\otimes\Lambda^{2}(\mathrm{V})$ is
symmetric. The observation that the Bianchi identity is a consequence of an
$\mathfrak{sl_{2}}$ invariance belongs to P.Ševera [6].

The second Bianchi identity $\nabla R=0$ can be verified by a short direct
calculation.

###  3.5. Conformal transformations and the Weyl tensor.

Let $(\theta,\omega)$ be a pair of Clifford forms with vanishing torsion. Let
$\tilde{\theta}=e^{\phi}\theta$ be a conformal transformation of the form
$\theta$. This transformation corresponds to a conformal transformation of the
metric. Compute now the induced transformation of the form $\omega$ and of the
curvature $R$.

######  Proposition 2.

Let $\theta,\omega$ be a torsionless pair of Clifford forms, $R$ be its
curvature and $\phi$ a function on $M$. Than the pair

$\tilde{\theta}=e^{\phi}\theta,\
\tilde{\omega}=\omega+\theta\varepsilon-\varepsilon\,\theta,$

is also torsionless with curvature

$\tilde{R}=R-\theta\rho-\rho\theta=R-2E\rho$

where

$\varepsilon=\frac{1}{4}F^{\prime}d\phi\in\Omega^{10}$

and

$\rho=d\varepsilon+\omega\varepsilon-\varepsilon\omega+\varepsilon\theta\varepsilon\in\Omega^{11}.$

It implies that the image of the curvature $R$ in the cokernel of $E$ in
$\Omega^{22}$ is invariant under conformal transformations. In dimension four
or higher this cokernel is nontrivial. Taking into account that for any $k$ we
have
$\Lambda^{\boldsymbol{\cdot}\boldsymbol{\cdot}}=\operatorname{im}E\oplus\ker
F$. The projection of $R$ onto $\ker F\cap\Lambda^{22}$ along $E(\Omega^{11})$
is called the Weyl tensor and is denoted by $W$.

###### Proof.

Observe first that $E^{\prime}F^{\prime}\alpha=\alpha$ for
$\alpha\in\Omega^{01}$. Therefore
$E^{\prime}F^{\prime}d\phi=\frac{1}{2}\theta(F^{\prime}d\phi)+\frac{1}{2}(F^{\prime}d\phi)\theta=d\phi$.
The torsion and the curvature can be computed directly:

$\displaystyle
e^{-\phi}(d\tilde{\theta}+\tilde{\theta}\tilde{\omega}+\tilde{\omega}\tilde{\theta})=e^{-\phi}d(e^{\phi}\theta)+\theta\omega+\omega\theta+$
$\displaystyle+\theta(\theta\varepsilon-\varepsilon\theta)+(\theta\varepsilon-\varepsilon\theta)\theta=d\theta+\theta\omega+\omega\theta+d\phi\theta+$
$\displaystyle+\theta(\theta\varepsilon+\varepsilon\theta)-(\theta\varepsilon+\varepsilon\theta)\theta=d\phi\,\theta+\frac{1}{2}\theta
d\phi-\frac{1}{2}d\phi\,\theta=0.$

and

$\displaystyle\tilde{R}=$ $\displaystyle
d\tilde{\omega}+\tilde{\omega}^{2}=R+d(\theta\varepsilon-\varepsilon\theta)+\omega(\theta\varepsilon-\varepsilon\theta)+(\theta\varepsilon-\varepsilon\theta)\omega+(\theta\varepsilon-\varepsilon\theta)^{2}=$
$\displaystyle=$
$\displaystyle-(\omega\theta+\theta\omega)\varepsilon+\varepsilon(\omega\theta+\theta\omega)-\theta
d\varepsilon-d\varepsilon\theta+\omega(\theta\varepsilon-\varepsilon\theta)+(\theta\varepsilon-\varepsilon\theta)\omega+$
$\displaystyle+$
$\displaystyle(\theta\varepsilon-\varepsilon\theta)^{2}=R-\theta(d\varepsilon+\omega\varepsilon-\varepsilon\omega+\varepsilon\theta\varepsilon)-(d\varepsilon+\omega\varepsilon-\varepsilon\omega+\varepsilon\theta\varepsilon)\theta.$

∎

###  3.6. Hilbert action.

On the space of pairs $\theta,\omega$ define the Hilbert functional:

$S(\theta,\omega)=\operatorname{str}\int_{M}\theta^{n-2}(d\omega+\omega^{2})=\operatorname{str}\int_{M}E^{n-2}R.$

######  Proposition 3.

The Hilbert functional is gauge invariant. The variation of this functional is

$\displaystyle\frac{\delta S}{\delta\theta}=$
$\displaystyle(-1)^{(n^{2}+n)/2}(n-2)E^{n-3}(d\omega+\omega^{2})$
$\displaystyle\frac{\delta S}{\delta\omega}=$
$\displaystyle(-1)^{(n^{2}-n)/2}E^{n-3}(d\theta+\theta\omega+\omega\theta).$

The induced closed 2-form on the space $\Omega^{11}(\partial
M)\oplus\Omega^{n-1,n-2}(\partial M)$ is

$w=\pm(n-2)\operatorname{str}\int_{\partial M}\delta\theta
E^{n-3}\delta\omega$

Since $E^{n-3}:\Omega^{12}\to\Omega^{n-2,n-1}$ is an isomorphism it implies
that the vanishing of $\frac{\delta S}{\delta\omega}$ is equivalent to the
vanishing of the torsion $t=d\theta+\omega\theta+\theta\omega$. Similarly
since $\ker E^{n-3}:\Omega^{22}\to\Omega^{n-1,n-1}=\ker
F:\Omega^{22}\to\Omega^{11}$, the vanishing of $\frac{\delta S}{\delta\theta}$
is equivalent to the vanishing of the Ricci tensor $r=FR$.

###### Proof.

Computing the variation of the Hilbert functional

$\delta S=\operatorname{str}\int_{\partial
M}\theta^{n-2}\delta\omega+\operatorname{str}\int_{M}\frac{\delta
S}{\delta\theta}\delta\theta+\operatorname{str}\int_{M}\frac{\delta
S}{\delta\omega}\delta\omega$

one gets

$\frac{\delta
S}{\delta\theta}=\Pi^{n-1}\left(\sum_{i=0}^{n-3-i}\theta^{n-3-i}(d\omega+\omega^{2})\theta^{i}\right)=(n-2)\Pi^{n-1}\left((d\omega+\omega^{2})\theta^{n-3}\right).$

On the other hand

$\displaystyle\frac{\delta S}{\delta\omega}=$
$\displaystyle\Pi^{n-2}\left((-1)^{n-1}d\theta^{n-2}+\theta^{n-2}\omega+(-1)^{n-1}\omega\theta^{n-2}\right)=$
$\displaystyle=$
$\displaystyle\Pi^{n-2}(\theta^{n-3}(d\theta+\omega\theta+\theta\omega)).$

Using the property 8 one gets the desired answer.

The expression for the 2-form is given by the variation of the boundary term.

$\displaystyle w=$ $\displaystyle\delta\operatorname{str}\int_{\partial
M}\theta^{n-2}\delta\omega=\operatorname{str}\int_{\partial
M}\sum_{i+j=n-3}\theta^{i}\delta\theta\theta^{j}\delta\omega=$
$\displaystyle=$ $\displaystyle\operatorname{str}\int_{\partial
M}\sum_{i+j=n-3}(-1)^{i}\delta\theta\Pi^{n-1}(\theta^{j}\delta\omega\theta^{i})=$
$\displaystyle=$ $\displaystyle(n-2)(-1)^{n-1}\operatorname{str}\int_{\partial
M}\delta\theta\Pi^{n-1}(\delta\omega\theta^{n-3})=$ $\displaystyle=$
$\displaystyle(n-2)(-1)^{n-1}\operatorname{str}\int_{\partial M}\delta\theta
E^{n-3}\delta\omega.$

∎

$R,W$$\theta$,$r$$\omega$$t$$E^{\prime}$$E$$F^{\prime}$$F$$q$$p$Zoo of
Clifford forms and the action of $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$
$(n=4)$

###  3.7. Special case $n=4$.

The dimension 4 of the manifold $M$ is special in particular since in this
case $HR=0$ and the equation of motion (Einstein equation) amounts to $ER=0$.
Together with the Bianchi identity $E^{\prime}R=0$ it implies that the
curvature $R$ is invariant under the action of both $\mathfrak{sl}_{2}$
algebras. In this dimension the action of the operators $*_{1}$ and $*_{2}$
are involutions preserving the subspace $\Omega^{22}$ and therefore the
curvature can be decomposed as $R=R_{++}+R_{+-}+R_{-+}+R_{--}$ according to
the eigenvalues of the operators $*_{1}$ and $*_{2}$. By the property 6 the
curvature $R$ is invariant under the product $*_{1}*_{2}$ and therefore it is
self-dual, i.e., $R_{+-}=R_{-+}=0$, see [4]. The components $R_{++}$ and
$R_{--}$ are symmetric and traceless, since $E^{2}R=E^{\prime 2}R=0$.
Therefore the curvature $R$ takes value in a 10-dimensional space, which is
possible to deduce also from the analysis of the decomposition of
$\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}}$ into irreducible components
w.r.t to the $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$ action. Namely the
decomposition into irreducibles is given by

$\displaystyle\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}}=W_{40}\oplus
W_{04}\oplus\mathbb{C}^{4}\otimes W_{13}\oplus\mathbb{C}^{4}\otimes
W_{31}\oplus\mathbb{C}^{6}\otimes W_{22}\oplus$
$\displaystyle\oplus\mathbb{C}^{9}\otimes W_{20}\oplus\mathbb{C}^{9}\otimes
W_{02}\oplus\mathbb{C}^{16}\otimes W_{11}\oplus\mathbb{C}^{10}\otimes W_{00},$

where $W_{ij}$ is an irreducible representation with the highest weight
$(i,j)$ of dimension $(i+1)(j+1)$.

##  4\. Complex structure

In this section we generalize the construction for the case where the space
$\mathrm{V}$ is provided with a complex structure $J$ compatible with the
metric. In order to simplify notation we rename this space, vielbein,
connection, torsion, curvature etc. into $\mathrm{V}^{\mathbb{R}}$,
$\theta^{\mathbb{R}}$ $t^{\mathbb{R}}$, $R^{\mathbb{R}}$ etc.

Recall that a complex structure of on $\mathrm{V}^{\mathbb{R}}$ is an operator
$J:\mathrm{V}^{\mathbb{R}}\to\mathrm{V}^{\mathbb{R}}$ with $J^{2}=-1$ and
preserving the metric $\eta$, namely, such that
$\eta(v_{1},v_{2})=\eta(Jv_{1},Jv_{2})$. Such operator gives a decomposition
of the complexification $\mathrm{V}^{\mathbb{C}}$ of the space
$\mathrm{V}^{\mathbb{R}}$ into a sum of isotropic eigenspaces of this operator
$\mathrm{V}^{\mathbb{C}}=\mathrm{V}\oplus\bar{\mathrm{V}}$ and therefore the
algebra
$\Lambda(\mathrm{V}^{\mathbb{R}})\otimes\Lambda((\mathrm{V}^{\mathbb{R}})^{*})=\Lambda(\mathrm{V}+\bar{\mathrm{V}})\otimes\Lambda(\mathrm{V}^{*}+\bar{\mathrm{V}}^{*})$
acquires a $\mathbb{Z}^{4}$ grading. The corresponding graded components are
denoted by $\Lambda^{p\bar{p}q\bar{q}}$.

Given a complex structure in $\mathrm{V}^{\mathbb{R}}$ the vielbein
$\theta^{\mathbb{R}}$ induces a complex structure on every tangent space to
the manifold $M$ and thus an almost complex structure on $M$. The space of
Clifford forms on $M$ is therefore graded by $\mathbb{Z}_{4}$. We denote by
$\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}}$
the space of $\mathit{Cl}(\mathrm{V}^{\mathbb{C}})$-valued forms on $M$. The
corresponding graded components are denoted by $\Omega^{p\bar{p}q\bar{q}}$ and
the whole graded algebra by
$\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}}$.
With respect to this complex structure we have $\theta\in\Omega^{1010}$ and
$\bar{\theta}\in\Omega^{0101}$. The complex structure is integrable if
$d\theta\in\Omega^{1020}\oplus\Omega^{1011}$ (and the component in
$\Omega^{1002}$ vanishes). In this case one can define a decomposition of the
differential $d=\partial+\bar{\partial}$ with the property
$\partial:\Omega^{p\bar{p}q\bar{q}}\to\Omega^{p\bar{p}(q+1)\bar{q}}$ and
$\bar{\partial}:\Omega^{p\bar{p}q\bar{q}}\to\Omega^{p\bar{p}q(\bar{q}+1)}$.

###  4.1. Action of the algebra $\widehat{\mathfrak{sl}}_{4}$ on complex
Clifford forms.

Define the operators

$\displaystyle 2E_{0}x$ $\displaystyle=\theta
x+(-1)^{(p+\bar{p}+q+\bar{q})}x\theta,$ $\displaystyle 2F_{1}x$
$\displaystyle=\theta x-(-1)^{(p+\bar{p}+q+\bar{q})}x\theta$ $\displaystyle
2F_{2}x$
$\displaystyle=\bar{\theta}x+(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta},$
$\displaystyle 2E_{3}$
$\displaystyle=\bar{\theta}x-(-1)^{(p+\bar{p}+q+\bar{q})}x\bar{\theta}$

Observe that $E_{3}=*_{1}^{-1}F_{2}*_{1}$ and $F_{1}=*_{1}^{-1}E_{0}*_{1}$.
Define also $E_{1}=*_{2}^{-1}F_{2}*_{2}$,
$E_{2}=*_{1}^{-1}*_{2}^{-1}E_{0}*_{2}*_{1}$,
$F_{0}=*_{1}^{-1}*_{2}^{-1}F_{2}*_{2}*_{1}$, $H_{0}=n-q^{\prime\prime}-p$,
$H_{1}=q^{\prime\prime}-p^{\prime\prime}$, $H_{3}=p^{\prime}-q^{\prime}$ and
$H_{2}=n-q^{\prime}-p^{\prime\prime}$. These operators have degrees

$\displaystyle\deg{E_{0}}$ $\displaystyle=(1,0,1,0),$
$\displaystyle\deg{E_{1}}$ $\displaystyle=(0,1,-1,0),$
$\displaystyle\deg{E_{2}}$ $\displaystyle=(0,-1,0,-1),$
$\displaystyle\deg{E_{3}}$ $\displaystyle=(-1,0,0,1),$
$\displaystyle\deg{F_{0}}$ $\displaystyle=(-1,0,-1,0),$
$\displaystyle\deg{F_{1}}$ $\displaystyle=(0,-1,1,0),$
$\displaystyle\deg{F_{2}}$ $\displaystyle=(0,1,0,1),$
$\displaystyle\deg{F_{3}}$ $\displaystyle=(1,0,0,-1),$ $\displaystyle\deg
H_{i}$ $\displaystyle=(0,0,0,0).$

######  Proposition 4.

The operators $\\{E_{i},F_{i},H_{i}\\}$ generate the action of the affine Lie
algebra $\widehat{\mathfrak{sl}_{4}}$ on
$\Omega^{\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}\boldsymbol{\cdot}}$

###### Proof.

We need to check the following commutation relations:

(2)
$\displaystyle[E_{i},F_{i}]=H_{i},\quad[H_{i},E_{i}]=2E_{i},\quad[H_{i},F_{i}]=-2F_{i},$
(3) $\displaystyle[E_{i},[E_{i},E_{i\pm 1}]]=0,\ [F_{i},[F_{i},F_{i\pm
1}]]=0,$ (4) $\displaystyle[E_{i},F_{j}]=0\mbox{ for }i\neq j,$ (5)
$\displaystyle[E_{i},E_{i+2}]=0$

Since the operators are not differential it is sufficient to check the
relation in every point of the manifold $M$ in the algebra
$\Lambda(\mathrm{V}^{\mathbb{R}})\otimes\Lambda((\mathrm{V}^{\mathbb{R}})^{*})=\Lambda(\mathrm{V}+\bar{\mathrm{V}})\otimes\Lambda(\mathrm{V}^{*}+\bar{\mathrm{V}}^{*})$.

Introducing a base $\xi_{i}$ of the space $\mathrm{V}^{\prime}$ one can
identify the algebra $\Lambda(\mathrm{V}+\bar{\mathrm{V}})$ with polynomials
of $2n$ Grassmann variables $\xi_{k}$, $\bar{\xi}_{k}$ and the algebra
$\Lambda(\mathrm{V}^{*}+\bar{\mathrm{V}}^{*})$with polynomials of $2n$
Grassmann variables $\xi^{k}$, $\bar{\xi}^{k}$. Taking into account that

$\displaystyle*^{-1}\frac{\partial}{\partial\xi_{i}}*=\eta^{i\bar{j}}\bar{\xi}_{j},$
$\displaystyle*^{-1}\frac{\partial}{\partial\bar{\xi}_{i}}*=\eta^{\bar{i}j}\xi_{j},$
$\displaystyle*^{-1}\xi_{i}*=\eta_{\bar{j}i}\frac{\partial}{\partial\bar{\xi}_{j}},$
$\displaystyle*^{-1}\bar{\xi}_{i}*=\eta_{j\bar{i}}\frac{\partial}{\partial\xi_{j}},$

and analogously for the dual variables one can write the expressions for the
generators $\\{E_{i},F_{i},H_{i}\\}$ in the form.

$\displaystyle E_{0}$ $\displaystyle=\xi_{i}\otimes\xi^{i},$ $\displaystyle
E_{1}$
$\displaystyle=\bar{\xi_{i}}\otimes\eta^{j\bar{i}}\frac{\partial}{\partial\xi^{j}}$
$\displaystyle E_{2}$
$\displaystyle=\frac{\partial}{\partial\bar{\xi_{i}}}\otimes\frac{\partial}{\partial\bar{\xi^{i}}}$
$\displaystyle E_{3}$
$\displaystyle=\eta_{j\bar{i}}\frac{\partial}{\partial\xi_{j}}\otimes\bar{\xi}^{i},$
$\displaystyle F_{0}$
$\displaystyle=\frac{\partial}{\partial\xi_{i}}\otimes\frac{\partial}{\partial\xi_{i}},$
$\displaystyle F_{1}$
$\displaystyle=\eta_{i\bar{j}}\frac{\partial}{\partial\bar{\xi}_{j}}\otimes\xi^{i},$
$\displaystyle F_{2}$ $\displaystyle=\bar{\xi}_{i}\otimes\bar{\xi}^{i},$
$\displaystyle F_{3}$
$\displaystyle=\xi_{i}\otimes\eta^{i\bar{j}}\frac{\partial}{\partial\bar{\xi}^{j}}.$

The relations (2), (4) and (5) are obvious from the expressions of the
generators in coordinates. To prove (3) compute the commutators:

$\displaystyle[E_{0},E_{1}]$
$\displaystyle=\eta^{i\bar{j}}\,\xi_{i}\bar{\xi}_{j}\otimes 1,$
$\displaystyle[E_{1},E_{2}]$
$\displaystyle=1\otimes\eta^{i\bar{j}}\,\frac{\partial^{2}}{\partial\xi^{i}\partial\bar{\xi}^{j}},$
$\displaystyle[E_{2},E_{3}]$
$\displaystyle=-\eta_{i\bar{j}}\,\frac{\partial^{2}}{\partial\xi_{i}\partial\bar{\xi}_{j}}\otimes
1,$ $\displaystyle[E_{3},E_{0}]$
$\displaystyle=-1\otimes\eta_{i\bar{j}}\,\xi^{i}\bar{\xi}^{j},$
$\displaystyle[F_{0},F_{1}]$
$\displaystyle=\eta_{i\bar{j}}\frac{\partial^{2}}{\partial\xi_{i}\partial\bar{\xi}_{j}}\otimes
1,$ $\displaystyle[F_{1},F_{2}]$
$\displaystyle=1\otimes\eta_{i\bar{j}}\,\xi^{i}\bar{\xi}^{j},$
$\displaystyle[F_{2},F_{3}]$
$\displaystyle=-\eta^{i\bar{j}}\,\xi_{i}\bar{\xi}_{j}\otimes 1,$
$\displaystyle[F_{3},F_{0}]$
$\displaystyle=-1\otimes\eta^{i\bar{j}}\,\frac{\partial^{2}}{\partial\xi^{i}\partial\bar{\xi}^{j}},$
$\displaystyle[E_{0},F_{0}]$
$\displaystyle=\xi_{i}\frac{\partial}{\partial\xi_{i}}\otimes
1+1\otimes\xi_{i}\frac{\partial}{\partial\xi_{i}}-n=p^{\prime}+q^{\prime}-n$
$\displaystyle[E_{1},F_{1}]$
$\displaystyle=\bar{\xi}_{i}\frac{\partial}{\partial\bar{\xi}_{i}}\otimes
1-1\otimes\xi_{i}\frac{\partial}{\partial\xi_{i}}=p^{\prime\prime}-q^{\prime}$
$\displaystyle[E_{2},F_{2}]$
$\displaystyle=n-\bar{\xi}_{i}\frac{\partial}{\partial\bar{\xi}_{i}}\otimes
1+1\otimes\bar{\xi}_{i}\frac{\partial}{\partial\bar{\xi}_{i}}=n-p^{\prime\prime}-q^{\prime\prime}$
$\displaystyle[E_{3},F_{3}]$
$\displaystyle=1\otimes\bar{\xi}_{i}\frac{\partial}{\partial\bar{\xi}_{i}}-\xi_{i}\frac{\partial}{\partial\xi_{i}}\otimes
1=q^{\prime\prime}-p^{\prime}$

The relations (3) follow. ∎

Observe that the embedding
$\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\to\widehat{\mathfrak{sl}}_{4}$ is
given by $E=E_{0}+F_{2}$, $E^{\prime}=E_{1}+F_{3}$, $F=F_{0}+E_{2}$,
$F^{\prime}=F_{1}+E_{3}$.

###  4.2. Kähler condition.

The Lie algebra of the unitary group (the group preserving both the complex
structure $J$ and the scalar product $\eta$) is embedded into the Clifford
algebra as $\mathit{Cl}^{11}(\mathrm{V}^{\mathbb{C}})$. The connection
preserving unitary structure is therefore a form
$\omega^{\mathbb{R}}=\omega+\bar{\omega}$ with $\omega\in\Omega^{1110}$ and
$\bar{\omega}\in\Omega^{1101}$.

Observe that $w=\theta\bar{\theta}-\bar{\theta}\theta\in\Omega^{0011}$ is just
an ordinary (1,1)-form.

######  Proposition 5.

A torsion-free connection preserving the metric and the complex structure
exists if and only if the form $w$ is closed. In this case such connection is
unique.

###### Proof.

The torsion
$t^{\mathbb{R}}=(\partial+\bar{\partial})(\omega+\bar{\omega})+(\theta+\bar{\theta})(\omega+\bar{\omega})+(\omega+\bar{\omega})(\theta+\bar{\theta})$
decomposes into four components

$\displaystyle t^{1020}$
$\displaystyle=\partial\theta+\omega\theta+\theta\omega=\partial\theta+2F_{1}\omega,$
$\displaystyle t^{0111}$
$\displaystyle=\partial\bar{\theta}+\omega\bar{\theta}+\bar{\theta}\omega=\partial\bar{\theta}+2E_{3}\omega,$
$\displaystyle t^{0102}$
$\displaystyle=\bar{\partial}\bar{\theta}+\bar{\omega}\bar{\theta}+\bar{\theta}\bar{\omega}=\bar{\partial}\bar{\theta}+2E_{3}\bar{\omega},$
$\displaystyle t^{1011}$
$\displaystyle=\bar{\partial}\theta+\bar{\omega}\theta+\theta\bar{\omega}=\bar{\partial}\theta+2F_{1}\bar{\omega},$

and vanishing of $t^{\mathbb{R}}$ is equivalent to vanishing of each of these
components. It implies that

$0=E_{3}t^{1020}-F_{1}t^{0111}=E_{3}\partial\theta-2E_{3}F_{1}\omega-
F_{1}\partial\bar{\theta}-2F_{1}E_{3}\omega=\partial w/2.$

The last equality is a consequence of

$\partial
w=(\partial\theta)\bar{\theta}+\bar{\theta}(\partial\theta)-\theta(\partial\bar{\theta})-(\partial\bar{\theta})\theta=2E_{3}\partial\theta-2F_{1}\partial\bar{\theta}$

and of the relation $[F_{1}E_{3}]=0$. The proof that $\bar{\partial}w=0$ is
analogous and thus the form $w$ is necessarily closed.

Conversely, the condition $dw=0$, the relation $t^{0111}=0$ and the
invertibility of $E_{3}:\Omega^{1110}\to\Omega^{0111}$ imply that the
condition $t^{1020}=0$ is also satisfied. Indeed the condition $t^{0111}=0$
implies $\omega=-\frac{1}{2}E_{3}^{-1}\partial\bar{\theta}$. Substituting it
into the expression for $t^{1020}$ one gets

$t^{1020}=\partial\theta-
F_{1}E_{3}^{-1}\partial\bar{\theta}=E_{3}^{-1}(E_{3}\partial\theta-
F_{1}\partial\bar{\theta})=\frac{1}{2}E_{3}^{-1}\partial w=0$

. ∎

###  4.3. Bianchi identity in the Kähler case.

In the Kahler case the first Bianchi identity implies that only the component
$R\in\Omega^{1111}$ of the curvature $R^{\mathbb{R}}$ is nonzero and that it
is invariant under a subalgebra
$\widehat{\mathfrak{sl}}_{2}\times\widehat{\mathfrak{sl}}_{2}$ of
$\widehat{\mathfrak{sl}}_{4}$.

######  Proposition 6.

The curvature is given by
$R^{\mathbb{R}}=R=\partial\bar{\omega}+\bar{\partial}\omega+\omega\bar{\omega}+\omega\bar{\omega}\in\Omega^{1111}$
It is invariant under the subalgebra or $\hat{\mathfrak{sl}}_{4}$ generated by
$E_{3},F_{3},H_{3}$ and $E_{1},F_{1},H_{1}$.

###### Proof.

The curvature
$R^{\mathbb{R}}=(\partial+\bar{\partial})(\omega+\bar{\omega})+(\omega+\bar{\omega})^{2}$
decomposes into three components
$(\partial\omega+\omega^{2})\in\Omega^{1120}$,
$\bar{\partial}\bar{\omega}+\bar{\omega}^{2}\in\Omega^{1102}$ and
$\partial\bar{\omega}+\bar{\partial}\omega+\omega\bar{\omega}+\omega\bar{\omega}\in\Omega^{1111}$.
Prove first that $\partial\omega+\omega^{2}=0$. Indeed

$\begin{split}0=&\partial
t^{0111}=\partial\omega\bar{\theta}-\bar{\theta}\partial\omega-\omega\partial\bar{\theta}+\partial\bar{\theta}\omega=\\\
=&(\partial\omega+\omega^{2})\bar{\theta}-\bar{\theta}(\partial\omega+\omega^{2})=-2E_{3}(\partial\omega+\omega^{2})\end{split}$

On the other hand since $H_{3}|_{\Omega^{1120}}=-1$ the operator $E_{3}$ is
bijective in this degree and therefore $\partial\omega+\omega^{2}=0$. The
identity $\bar{\partial}\bar{\omega}+\bar{\omega}^{2}=0$ can be proven
analogously.

Similarly $0=\bar{\partial}t^{1020}+\partial t^{1011}=R\theta-\theta
R=-2E_{3}R$. On the other hand $H_{3}R=0$ since $H_{3}|_{\Omega^{1111}}=0$ and
therefore $F_{3}R=0$. Invariance under $E_{1},F_{1},H_{1}$ can be proven
analogously. ∎

###  4.4. Einstein equation in the Kähler case.

In the Kähler case the Einstein equation is equivalent to the equations
$F_{0}R=0$ and $E_{2}R=0$. In four real dimension it is equivalent to the full
$\widehat{\mathfrak{sl}}_{4}$ invariance of the curvature. Indeed, the
Einstein equation reads as $FR^{\mathbb{R}}=0$. In the complex case it amounts
to $(F_{0}+E_{2})R=0$. Since the two terms have different gradings this
equation is equivalent to $F_{0}R=0$ and $E_{2}R=0$. Since in dimension four
$H_{0}R=H_{2}R=0$ it implies together with the Bianchi identities that all
generators of $\widehat{\mathfrak{sl}}_{4}$ annihilate the curvature.

##  5\. Conclusion.

First of all we hope that the developed formalism allows to simplify learning
Riemann differential geometry for students as well as to better understand the
logic of the theory. Emergence of the affine group
$\widehat{\mathfrak{sl}}_{4}$ seems mysterious for us and requires better
understanding of its consequences. In the forthcoming version of the paper we
plan to apply the construction to more elaborated structures such as
hyperkähler, $G_{2}$ and to supergravity. The main idea of the paper is
presented in the master thesis of the second author defended at the University
of Strasbourg 19/6/2017.

## References

  * [1] A.Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend.Circ.Mat. Palermo 43, 203 (1919).
  * [2] E.Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Acad. Sci. 174, 593-595 (1922).
  * [3] E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B311 (1988) 46.
  * [4] M.F.Atiyah, N.J.Hitchin, I.M.Singer: Self-duality in four-dimensional Riemannian geometry, Proc.Roy.Soc.Lond.A 362 (1978), 425–461.
  * [5] A.Losev, From Berezin integral to Batalin-Vilkovisky formalism. A mathematical physicist’s point of view. in M.Shifman (ed.) Felix Berezin. Life and Death of the Mastermind of Supermathematics. World Scientific 2007.
  * [6] P.Ševera, A remark on the symmetries of the Riemann curvature tensor.,  arxiv:math/0204191.
  * [7] B.A.Dubrovin, A.T.Fomenko, and S.P.Novikov, Modern Geometry - Methods and Applications : Part I, 2nd Edition, Springer-Verlag, N.Y., 1992.