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 M. Verbitsky}}


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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\LARGE\bf
Derived brackets\\[4mm]
and generalized complex manifolds
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[4mm]

Misha Verbitsky
 

\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small \hspace{0.10\linewidth}
\begin{minipage}[t]{0.85\linewidth}
{\bf Abstract} \\
This is a short note purporting to explain the
generalized complex geometry through superalgebra and
Clifford multiplication.
\end{minipage}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hfill

There is nothing essentially new in this note.
It is a write-up of a talk given June 27 at ``Geometric structures''
seminar in Moscow, HSE. The idea was to explain 
the definition of Courant bracket in terms of
spinors, and prove some basic results using this 
description. The basic reference
is \cite{_GK:Gualtieri_}, 
\cite{_Hitchin:GCY_} and \cite{_Deri:YKS_}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Clifford algebras}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\definition
{\bf A Clifford algebra} $\Cl(V,q)$ of a vector space $V$ with
a scalar product $q$ is an algebra generated by $V$ with 
a relation $xy+yx = q(x,y) 1$.

\newcommand{\Mat}{\operatorname{Mat}}

\hfill

\example
Suppose that $q=0$. Then $xy=-yx$, hence 
the Clifford algebra $\Cl(V,q)$ is isomorphic 
to the Grassmann algebra: $\Cl(V,q)=\Lambda^* V$.

\hfill

\example 
Denote the $k$-dimensional space $\R^k$
with a scalar product of signature $(q,p)$
by $(\R^n, \underbrace{+, ..., +}_q, \underbrace{-, ..., -}_p)$.
Clearly \[ \Cl(\R,-)=\R[t]/(t^2=-1)=\C,\]
and \[ \Cl(\R,+)=\R[t]/(t^2=1)=\R\oplus \R.\]

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that $\Cl(\R^2, -,-)$ is isomorphic to the quaternion algebra,
and $\Cl(\R^2, +,+)$, $\Cl(\R^2, +,-)$ are isomorphic to the 
algebra of 2x2-matrices, 
$\Cl(\R^2, +,+)\cong \Cl(\R^2, +,-)\cong \Mat(2,\R)$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\proposition\label{_Clifford_dimension_Proposition_}
$\dim \Cl(V)=2^{\dim V}$.

\hfill

Before I give a proof of this result, let me introduce the
filtered algebras.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $A_0\subset A_1\subset A_2 \subset ...$
be a sequence of subspaces of an algebra $A=\bigcup A_i$.
We say that $\{A_i\}$ is {\bf a multiplicative filtration}
if $A_i \cdot A_j \subset A_{i+j}$. In this case $A$
is called {\bf a filtered algebra}. 

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that the direct sum $\bigoplus_i A_{i}/A_{i-1}$
is equipped with an algebra structure: $a\in A_{i}\mod A_{i-1}$
multiplied by $a'\in A_{j}\mod A_{j-1}$ gives
$aa' \in A_{ij}\mod A_{ij-1}$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $A_0\subset A_1\subset A_2 \subset ... \subset A$ be a
filtered algebra. {\bf Associated graded algebra} of this
filtration is $\bigoplus_i A_{i}/A_{i-1}$ with the algebra structure
defined above.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\claim
Let $\Cl(V,q)$ be a Clifford algebra, and $\Cl_0(V,q)=k\cdot 1$ 
the field of constants, $\Cl_1(V)= \Cl_0(V,q)\oplus V$,
and $\Cl_i(V,q):= \underbrace{\Cl_1(V,q), ..., Cl_1(V,q)}_{\text{$i$ times}}$.
This gives a filtration on $\Cl(V,q)$. Then the associated
graded algebra is the Grassmann algebra $\Lambda^*V$.

\hfill

{\bf Proof:} Modulo lower terms of the filtration,
the Clifford relations give $xy+yx=0$. \endproof

\hfill

Now the proof of \ref{_Clifford_dimension_Proposition_}
is apparent; indeed, taking associated graded algebra does
not change the dimension, hence $\dim \Cl(V,q)=\dim \Lambda^* V= 2^{\dim V}$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theorem\label{_Spinors_Lambda^*_Theorem_}
Let $V:= W\oplus W^*$, with the usual pairing
$\langle(x+\xi), (x' + \xi')\rangle= \xi(x')+\xi'(x)$.
Then $\Cl(V)$ is naturally isomorphic to $\Mat(\Lambda^* V^*)$.

\hfill

{\bf Proof:}
Consider the convolution map 
$W \otimes \Lambda^i W^* \arrow \Lambda^{i-1} W^*$, with
$v\otimes \xi \arrow \xi(v, \cdot, \cdot, \dots, \cdot )$
denoted by $v, \xi \arrow i_v(\xi)$ and the extertior
multiplication map $W^* \otimes \Lambda^i W^* \arrow \Lambda^{i-1} W^*$,
with $\nu\otimes \xi \arrow \nu\wedge \xi$, denoted by
$\nu, \xi \arrow e_\nu(\xi)$.
Let $V\otimes \Lambda^* W^* \stackrel \Gamma \arrow \Lambda^* W^*$
map $(v,\nu)\otimes \xi$ to $i_v(\xi) + e_\nu(\xi)$.
It is easy to check that all $i_v$ pairwise anticommute,
all $e_\nu$ pairwise anticommute, and the anticommutator
$\{i_v, e_\nu\}$ is a scalar operator of multiplication
by a number $\nu(v)$.

To prove the last assertion without any calculations,
we notice that $i_v$ is an odd derivation of the Grassmann algebra,
$e_\nu$ is a linear operator, and a commutator of a derivation
and a linear operator is linear, hence one has
\[ 
\{i_v, e_\nu\}(a)= \{i_v, e_\nu\}(1) \wedge a= \nu(v)\cdot a.
\]

These anticommutator relations immediately imply that the map
$V\otimes \Lambda^* W^* \stackrel \Gamma \arrow \Lambda^* W^*$,
called {\bf the Clifford multiplication map}, is extended
to a homomorphism $\Cl(V) \arrow \Mat(\Lambda^* W^*)$.
An elementary calculation (left as an exercise) proves that
this map is injective. Since $\dim \Cl(V) = 2^{\dim V}= 2^{2\dim W}=
\dim \Mat(\Lambda^* W^*)$, this also implies that
$\Cl(V) \cong \Mat(\Lambda^* W^*)$.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $(V, q)$ be a vector space equipped with a scalar product,
and $\Cl(V,q) \cong \Mat (S)$ an isomorphism (such an isomorphism
is possible only when $V$ is even-dimensional, because otherwise
$2^{\dim V}$ is not a square of anything). Then $S$ is called
{\bf the space of spinors over $(V, q)$}.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark The Lie group $SO(V,q)$ acts on $\Cl(V,q)$
by automorphisms. However, $\Aut(\Mat(S))=PSL(S)$
(this is left as an exercise). This gives a group
homomorphism $SO(V,q) \arrow PSL(S)$. Lifting this homomorphism
to the universal covering $\Spin(V) \arrow SL(S)$,
we obtain {\bf the spinorial representation}
of the spin group $\Spin(V)$; it is a smallest faithfull
representation of the spin group.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition\label{_i_e_Definition_}
Let $M$ be a smooth manifold.
Consider the Clifford multiplication $V \otimes S \arrow S$.
Apply this construction 
to the bundle $\Lambda^* M$ taken, fiberwise, as spinors 
over $V:=TM\oplus T^*M$. We obtain the Clifford
multiplication map $\Gamma:\; V\otimes \Lambda^* M \arrow \Lambda^*M$
written as $(v,\nu)\otimes \xi\stackrel \Gamma \mapsto i_v(\xi) + e_\nu(\xi)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Derived brackets}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Graded algebras}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
A {\bf graded vector space} is a space $V^* =\bigoplus_{i\in \Z} V^i$.

\hfill


\remark If $V^*$ is graded, the endomorphisms space
\[ \End(V^*)=\bigoplus_{i\in \Z} \End^i(V^*)\] is also graded, with
\[ \End^i(V^*)= \bigoplus_{j\in \Z} \Hom(V^j, V^{i+j}).\]

\hfill


\definition
A {\bf graded algebra} (or ``graded associative algebra'')
is an associative algebra $A^*=\bigoplus_{i\in \Z} A^i$, with the product 
compatible with the grading: $A^i \cdot A^j \subset A^{i+j}$.

\hfill


\definition A bilinear map of graded spaces which satisfies
$A^i \cdot A^j \subset A^{i+j}$ is called {\bf graded},
or {\bf compatible with grading}.

\hfill

\remark 
The category of graded spaces can be defined as a  category
of vector spaces with $U(1)$-action, with the weight decomposition
providing the grading. Then  a graded algebra is an 
associative algebra in the category of spaces with $U(1)$-action.

\hfill


\definition
An operator on a graded vector space is called {\bf 
even} ({\bf  odd}) if it shifts the grading by even 
(odd) number. The {\bf parity} $\tilde a$ of an operator
$a$ is 0 if it is even, 1 if it is odd. We say that
an operator is {\bf  pure} if it is even or odd.

\hfill


\definition
A {\bf supercommutator}  of pure operators
on a graded vector space is defined by a formula
$\{a,b\}= ab - (-1)^{\tilde a \tilde b}ba$.

\hfill

\definition
A graded associative algebra is called {\bf 
graded commutative} (or ``supercommutative'')
if its supercommutator vanishes.

\hfill


\example The Grassmann algebra is supercommutative.


\hfill

\definition
{\bf A graded Lie algebra} (Lie superalgebra)
is a graded vector space $\g^*$
equipped with a bilinear graded map 
$\{\cdot,\cdot\}:\; \g^*\times \g^* \arrow \g^*$
which is graded anticommutative:
$\{a,b\} = - (-1)^{\tilde a \tilde b}\{b,a\}$
and satisfies {\bf the super Jacobi identity}
$\{c, \{a,b\}\} = \{\{c, a\},b\}+ (-1)^{\tilde a \tilde c}\{a,\{c, b\}\}$

\hfill


\example
Consider the algebra $\End(A^*)$ of operators on a 
graded vector space, with supercommutator as above.
Then $\End(A^*), \{\cdot,\cdot\}$ is a graded Lie algebra.


\hfill


{\bf Lemma 1:}
Let $d$ be an odd element of a Lie superalgebra, satisfying
$\{d,d\}=0$, and $L$ an even or odd element. Then $\{\{L, d\}, d\}=0$.

\hfill

{\bf Proof:} 
$0=\{L,\{d,d\}\}= \{\{L, d\}, d\}+(-1)^{\tilde L}\{d,\{L,
d\}\}=2\{\{L, d\}, d\}.$ 
\endproof

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Loday bracket}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the history and different versions of 
the definition of derived brackets please see \cite{_Deri:YKS_}.
For the present purposes, we need only one of them,
namely the Loday bracket.

From now on, we use the notation $[\cdot, \cdot]$
for the supercommutator.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $A =\bigoplus A^i$ be a Lie superalgebra, and
$d:\; A \arrow A$ an odd endomorphism satisfying $d^2=0$.
Define {\bf the Loday bracket} $[a,b]_d:= (-1)^{\tilde a}
  [d(a),b]$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that the Loday bracket satisfies the 
graded Jacobi identity: 
\[
[a,[b,c]_d]_d= [[a,b]_d,c]_d+(-1)^{\tilde a\tilde b}[b,[a,c]_d]_d.
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\example
Now let $A$ be the superalgebra $\End(\Lambda^* M)$,
where $M$ is a smooth manifold, and $d$ the de Rham
differential. Cartan formulas can be written in terms
of Loday bracket as follows (we use the notation
$e_\nu$, $i_v$ introduced in \ref{_i_e_Definition_}).
\begin{align*}
[i_x,i_y]_d=& i_{[x,y]}\\
[e_\eta, i_x]_d =& [ e_{d\eta}, i_x]=e_{d\eta\cntrct x}\\
[i_x, e_\eta]_d=\Lie_x e_\eta= e_{\Lie_x\eta}
\end{align*}
for all $x, y\in TM, \eta\in \Lambda^1M$.

\hfill

{\bf Proof:} Cartan's formula gives $[d,i_v]= \Lie_v$
(note that  $[\cdot, \cdot]$ here denotes a
supercommutator).
Then $[i_x,i_y]_d= \Lie_xi_y=i_{\Lie_x y}= i_{[x,y]}$.

Since $i_x$ is a derivation of the de Rham algebra
(prove this), the commutator $[i_x, e_\eta]$ is linear,
and this gives 
$[i_x, e_\xi](a)=i_xe_\xi(1)\cdot a=\xi\cntrct x$
where $\xi\cntrct x$ is contraction, $\xi\cntrct
x=i_x(\xi)$. This takes care about the formula
$[e_\eta, i_x]_d=e_{d\eta\cntrct x}$.

Finally, the last formula is self-evident.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\corollary\label{_Cou_bra_Loday_Corollary_}
Let $V:=TM\oplus T^*M$. Consider the Clifford
multiplication map 
$\Gamma:\; V\otimes \Lambda^* M \arrow \Lambda^*M$,
and let $x, x'\in V$, with $x=(x,\nu), x'=(v', \nu')$. Then 
$[\Gamma_x, \Gamma_{x'}]_d= \Gamma_y$,
where $y= ([v, v'], (d\nu)\cntrct v-\Lie_{v'}\nu)$.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
We define {\bf the Courant bracket}
$[(v,\nu), (v', \nu')]_d:= ([v, v'], (d\nu)\cntrct v-\Lie_{v'}\nu)$

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark
From \ref{_Cou_bra_Loday_Corollary_}
it is apparent that the Courant bracket is
the Loday bracket applied to the Clifford multiplication
operators.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise\label{_Dorfman_bracket_Exercise_}
Prove that $[u, v]_d -[v,u]_d  = d\langle u,
v\rangle$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark
The skew-symmetric bracket $[u, v]_d -[u, v]_d$
is called {\bf the Dorfman bracket}, after
I. Ya. Dorfman.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Complex structures and generalized complex structures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $V$ be a real vector space. {\bf A complex structure operator}
on $V$ is $I\in \Hom(V,V)$ satisfying $I^2=-\Id_V$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\claim
The eigenvalues $\alpha_i$ of $I$ are $\pm \1$.
Moreover, 
$I$ diagonalizable over $\C$.

\hfill

{\bf Proof:} 
The operator $I$ is unitary with respect to the Hermitian
form $g_I(x,y):= g(x,y) + g(Ix,Iy)$, where
$g$ is an arbitrary Hermitian form. 
Any unitary matrix is diagonalizable.
Finally,  $\alpha_i^2=-1$, because $I^2=-\Id$.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $V$ be a vector space, and $I\in \End(V)$ a complex structure
operator. The eigenvalue decomposition $V\otimes_\R \C = V^{1,0}\oplus V^{0,1}$
is called {\bf the Hodge decomposition}; here $I\restrict{V^{1,0}}=\1\Id$,
and $I\restrict{V^{0,1}}=-\1\Id$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark
One can reconstruct $I$ from the space $V^{1,0}\subset V\otimes_\R \C$.
Indeed, take $V^{0,1}= \overline{V^{1,0}}$, and let $I$ act
on $V^{0,1}$ as $\1\Id$, and on $V^{0,1}$ as $-\1\Id$.
Since thus defined operator $I\in \End(V\otimes_\R \C)$ commutes with the 
complex conjugation, it is {\bf real}, that is, preserves
$V\subset V\otimes_\R \C$. This gives an identification
between the set of complex structures on $V, \dim_\R V=2n$, and
an open part of the Grassmann space $Gr_n(V\otimes_\R \C)$
consisting of all subspaces $W\subset V\otimes_\R \C$
satisfying $W \cap \bar W=0$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
{\bf An almost complex structure} on a real $2n$-manifold
$M$ is an operator $I\in \End(TM)$ satisfying $I^2=-\Id_{TM}$,
or, equivalently, an $n$-dimensional sub-bundle 
$T^{1,0}M\subset TM\otimes_\R \C$ such that
$T^{1,0}M \cap \overline{T^{1,0}M}=0$. 
The almost complex structure called {\bf integrable}
(and $M$ {\bf a complex manifold}) if $T^{1,0}M$ is
{\em involutive}, that is, satisfies $[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $M$ be a real $2n$-manifold, and $V=TM \oplus T^*M$.
Consider the standard symmetric pairing on  $V$ 
of signature $(2n, 2n)$, 
\[ \langle(v,\nu), (v', \nu')\rangle:=
   \nu(v')+ \nu'(v).
\] Let
$I\in \End V$ an orthogonal operator satisfying 
$I^2=-\Id_V$. Then $I$ is called {\bf a
generalized almost complex structure}.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $V$ be an even-dimensional vector space equipped with
a non-degenerate scalar product $h$, and $W\subset V$
a subspace. Then $W$ is called {\bf isotropic}
if $h\restrict W=0$, and {\bf maximal isotropic}
if $\dim W = \frac 1 2 \dim V$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove the dimension
of an isotropic subspace is always $\leq \frac 1 2 \dim V$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark\label{_maxi_iso_from_almost_gc_Remark_}
Let $V=TM \oplus T^*M$,
$I\in \End(V)$ a generalized almost complex structure,
and $V^{1,0}\subset V \otimes_\R \C$ be the $\1$-eigenspace.
Then $V^{1,0}$ is maximal isotropic. Indeed, 
$\langle v, v'\rangle= \langle Iv, Iv'\rangle = - \langle v, v'\rangle$
for all $v, v' \in V^{1,0}$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\claim\label{_maxi_iso_bije_Claim_}
Let $M$ be a smooth manifold, $V=TM \oplus T^*M$. The generalized almost
complex structures $I\in \End(V)$ are in bijective corresponidence
with maximal isotropic subbundles $V^{1,0}\subset V \otimes_\R \C$ 
satisfying $V^{1,0}\cap \overline{V^{1,0}}=0$.

\hfill

{\bf Proof:} 
 $I\in \End(V)$ a generalized almost complex structure,
and $V^{1,0}\subset V \otimes_\R \C$ its $\1$-eigenspace.
As shown in \ref{_maxi_iso_from_almost_gc_Remark_},
$V^{1,0}$ is maximal isotropic. It remains to 
show that this correspondence is bijective.

Let $V^{1,0}\subset V \otimes_\R \C$ be a maximal isotropic
subbundle, satisfying $V^{1,0}\cap \overline{V^{1,0}}=0$.
Then $V\otimes_\R \C = V^{1,0}\oplus \overline{V^{1,0}}$.
Define $I\in \End(V)$ using $I\restrict{V^{1,0}}=\1\Id$,
and $I\restrict{\overline{V^{1,0}}}=-\1\Id$.
Then $I^2=-\Id_V$; to prove \ref{_maxi_iso_bije_Claim_}
it remains only to show that $I$ is orthogonal.

However, 
$\langle\cdot,\cdot\rangle\restrict{V^{1,0}}=
\langle\cdot,\cdot\rangle\restrict{\overline{V^{1,0}}}=0$,
because $V^{1,0}$ is isotropic, and for any $v\in V^{1,0}$,
$v'\in \overline{V^{1,0}}$, one has 
$\langle Iv, Iv'\rangle = \langle \1v, -\1v'\rangle=\langle v, v'\rangle$.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
A generalized almost complex structure $I$ on $M$ is {\bf integrable}
if $[V^{1,0}, V^{1,0}]_d \subset V^{1,0}$. Then $I$ is called
{\bf a generalized complex structure}, and $M$ {\bf a generalized
complex manifold}.

\hfill

The following examples explain the utility of generalized
complex structures, which unite into one usable definition
the notions of complex and symplectic structures. Their
integrability remains to be proven after the pure spinors
are introduced and used to prove integrability.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\example\label{_symplectic_Example_}
Let $(M,\omega)$ be a symplectic manifold. Consider
an almost complex structure $I\in \End(TM \oplus T^*M)$
written as
\[
I:= \begin{pmatrix} 0 & -\omega\\ \omega^{-1} & 0 
\end{pmatrix}.
\]
Then $I$ is integrable.

\hfill

\remark
Integrability of the generalized almost complex structure
is in fact equivalent to $d\omega=0$, which is an easy exercise
using the pure spinor approach.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\example\label{_complex_Example_}
Let $(M,J)$ be a complex manifold. Consider
an almost complex structure $I\in \End(TM \oplus T^*M)$
written as
\[
I:= \begin{pmatrix} J & 0\\ 0 & -J^* 
\end{pmatrix}.
\]
Then $I$ is integrable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Pure spinors and generalized complex structures}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark
Let $(V,h)$ be a vector space equipped with a scalar product,
$S$ the corresponding spinors, and $V\otimes S \arrow S$
the Clifford multiplication. Given a non-zero spinor $\Psi\in S$,
consider the space
\[
\ker \Psi:\; \{v\in V \ \ |\ \ v \cdot Psi =0\}.
\]
Then $\ker \Psi$ is isotropic. Indeed,
for each $u, v\in \ker \Psi$, one has
$0=uv\Psi+uv\Psi = h(u,v) \Psi$.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition 
(Cartan, Chevalley) $\Psi \in S$ is {\bf a pure spinor} if $\ker \Psi$
is maximal isotropic.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark
The following theorem gives a Pl\"ucker-type
embedding for the maximally isotropic Grassmannian.

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theorem
(Chevalley) 
Let $(V,h)$ be a vector space equipped with a scalar product,
and $S$ its spinor space. Then 
for each maximally isotropic subspace $W\subset V$,
$W=\ker \Psi$ for some pure spinor $\Psi\in S$, which is 
unique up to a scalar multiplier.

\hfill

{\bf Proof:} Identifying $V$ with $W\oplus W^*$,
we obtain an identification $S = \Lambda^* W$
(\ref{_Spinors_Lambda^*_Theorem_}).
Let $w_1, ..., w_n$ be a basis in $W$.
Then  $\ker w_1 \wedge w_2 \wedge ... \wedge w_n= W$.

Converse is also obvious: if
$\Psi\in \Lambda^*W$ satisfies $W \wedge \Psi=0$,
one has $\Psi = C w_1 \wedge w_2 \wedge ... \wedge w_n$.
\endproof

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\example
Let $(M,J)$ be a complex $n$-manifold, and 
$I$ the generalized complex structure constructed as in
\ref{_complex_Example_}. The corresponding pure spinor
is any non-degenerate section of $\Lambda^{n,0}(M,J)$
(check this).

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\example
Let $(M,\omega)$ be a symplectic $n$-manifold, and 
$I$ the generalized complex structure constructed as in
\ref{_symplectic_Example_}. The corresponding pure spinor
is $\Psi=e^{\1\omega}$. Indeed, $V^{1,0}$ is spanned by 
$i_x-\1 e_{\omega\cntrct x}$, where $x\in TM$. 
Since $i_x$ is a derivation, one has 
$i_x(e^{\1\omega})= \1 i_x(\omega)\wedge e^{\1\omega}$, giving
\[
i_x-\1 e_{\omega\cntrct x}(e^{\1\omega})=\
1 i_x(\omega) e^{\1\omega}-\1 e_{\omega\cntrct x}(e^{\1\omega})
= 0.
\]

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theorem
Let $L$ be a maximal isotropic subbundle in $V=TM \oplus T^*M$, and
$\Psi\in \Lambda^* M$ the corresponding spinor. Then $L$ satisfies
$[L,L]_d\subset L$ if and only if $d\Psi= t\cdot \Psi$,
for some $t\in V$.

\hfill

{\bf Proof:} $u, v\in \ker \Psi$, then
\[ [u, v]_d \Psi= [du+ud,v]\Psi =duv\Psi +udv\Psi - vdu\Psi-vud\Psi = -vud\Psi.
\]
If $d\Psi=0$, one has $[u, v]_d \Psi= -vud\Psi=0$.
 
To prove the converse, consider the filtration
% on the 
%Clifford algebra, 
%$\Cl_0(V)= \C$, $\Cl_1(V)=V$, $\Cl_d(V)=V \cdot \Cl_{d-1}(V)$,
%and the corresponding filtration
 on the spinor bundle,
$S_0=\Psi, S_1 = V\cdot \Psi, ...,  S_d= V \cdot S_{d-1}$.
Denote $\ker \Psi$ by $V^{1,0}$.
Let $\Lambda^dV^{1,0}\subset \Cl(V)$ be the
subspace in the Clifford algebra generated by the 
monomials of degree $d$ on $V^{1,0}$.
Clearly, $S_d = \{s\in S\ \ |\ \ \Lambda^dV^{1,0}s=0\}$.

Let now $v, u, [u,v]_d\in \ker\Psi$. The
same calculation as above gives $-vud\Psi=0$.
This implies that $d\Psi\in S_1$ for all pure spinors $\Psi$
inducing integrable generalized complex structure.
However, $S_1=V\cdot \Psi$.
\endproof

\hfill

{\small
\begin{thebibliography}{AV1}


\bibitem[G]{_GK:Gualtieri_}
Marco Gualtieri, 
{\em Generalized complex geometry}, 
arXiv:math/0401221

\bibitem[H]{_Hitchin:GCY_}
Nigel Hitchin
{\em Generalized Calabi-Yau manifolds},
arXiv:math/0209099, Quart. J. Math. Oxford Ser. 54:281-308, 2003.

\bibitem[KS]{_Deri:YKS_}
Yvette Kosmann-Schwarzbach, {\em 
Derived brackets},  arXiv:math/0312524.




\end{thebibliography}

\noindent {\sc Misha Verbitsky\\
{\sc Laboratory of Algebraic Geometry,\\
National Research University HSE,\\
Faculty of Mathematics, 7 Vavilova Str. Moscow, Russia,}\\
\tt  verbit@mccme.ru}, also: \\
{\sc Kavli IPMU (WPI), the University of Tokyo}
}


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