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\lhead{\tiny Hyperk\"ahler manifolds, HSE, Fall 2019} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  Class assignment \ \ \sc\version}
\rhead{{\tiny  Hyperk\"ahler manifolds, Misha Verbitsky}}


\begin{document}

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\listok{4}{Class assignment 4: spinors and holonomy}

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\definition
Let $U$ be a space equpped with scalar product.
Consider {\bf the exterior multiplication
operator} $e_u:\; \Lambda^*(U)\arrow \Lambda^{*+1}(U)$
with $e_u(\alpha) = u\wedge \alpha$ and {\bf the convolution
operator} $i_v:\; \Lambda^*(U)\arrow \Lambda^{*-1}(U)$, with\\
$i_v(\alpha)(v_1, ..., v_k)=\alpha(v,v_1, ..., v_k)$.
\ed


\exercise
Fix a basis $u_i$ in $U$, and let $v_j$ be the dual basis
in $U^*$. For any pair of monomials $A, B$ in
$\Lambda^*(U)$, find a sequence $z_1, ..., z_r$, with each
$z_k$ equal to $i_{v_i}$ or $e_{u_j}$, such that
$z_1z_2...z_r$ maps $A$ to $B$ and puts all
other monomials to 0.
\ez

\exercise
Let $(M, \nabla)$ be a manifold with holonomy $\Sp(n)$.
Prove that  all parallel $(p,0)$-forms on $M$ are powers of the
holomorphic symplectic form.
\ez


\exercise
Let $(M, \nabla)$ be a manifold with holonomy $SU(n)$.
Prove that  any holomorphic $(p,0)$-form on $M$ is a parallel
section of the canonical bundle.
\ez

\definition
Let $V$ be a vector space with non-degenerate scalar product $g$,
and $S$ a non-trivial, irreducible
module over $\Cl(V)$ (that is, a representation
of this algebra). Then $S$ is called {\bf the space of spinors}
over $V$.
\ed

\exercise
Let $V$ be a vector space with a non-degenerate scalar product $g$,
$v\in V$ a vector with $g(v,v)\neq 0$, and $S$ the space of spinors.
 Prove that the Clifford multiplication $S\stackrel{\sigma(v)}\arrow S$ 
is invertible. Is the condition $g(v,v)\neq 0$ necessary?
\ez

\exercise
Let $S$ be the space of spinors over $V$, and $\psi\in S$ a spinor.
Consider the Clifford multiplication map $r_\psi:\; V \arrow S$
mapping $v$ to $\sigma(v) \psi$. Prove that
$\dim \ker r_\psi \leq 1/2 \dim V$ when $\psi\neq 0$.
\ez

\exercise
Consider the space $\Lambda^2 V \subset V\otimes V$, 
identified with the Lie algebra $\goth {so}(V)$, and let
$\Lambda^2 V \subset V\otimes V\stackrel \sigma \arrow \Cl(V)$ be the 
Clifford multiplication map. Prove that it gives a Lie algebra
embedding $\goth {so}(V)\arrow \Cl(V)$.
\ez




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