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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex geometry\\[15mm]
\small lecture 11: Supersymmetry for K\"ahler manifolds: Hodge $*$ operator}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 27, 2020
}
\end{center}

\newpage

{\bf \blue Hodge $*$ operator}

Let $V$ be a vector space. {\bf \blue A metric $g$ on $V$ induces
a natural metric on each of its tensor spaces:}
$g(x_1\otimes x_2 \otimes ... \otimes x_k, x_1'\otimes x_2' \otimes ... \otimes x_k') = g(x_1, x'_1)g(x_2, x'_2) ... g(x_k, x'_k)$.

{\bf \purple This gives a natural positive definite scalar product
on differential $k$-forms over a Riemannian manifold $(M,g)$:}
$g(\alpha, \beta) := {\sqrt{k!}}\int_M g(\alpha, \beta) \Vol_M$.
{\bf \purple We rescale it in such a way that the standard monomials
become orthonormal.}

Another non-degenerate 
form is provided by the {\bf \blue Poincare pairing:}\\
$\alpha, \beta \arrow \int_M \alpha \wedge \beta$.

\definition
Let $M$ be a Riemannian $n$-manifold.
Define {\bf \blue the Hodge $*$ operator}
$*:\; \Lambda^k M \arrow \Lambda^{n-k} M$ by the following relation:
{\bf \red $g(\alpha, \beta) = \int_M \alpha \wedge *\beta.$}

\remark {\bf \red The Hodge $*$ operator always exists.} It is defined
explicitly in an orthonormal basis $\xi_1, ..., \xi_n \in \Lambda^1 M$:
\[ * (\xi_{i_1}\wedge\xi_{i_2} \wedge ... \wedge\xi_{i_k}) =
(-1)^s \xi_{j_1}\wedge\xi_{j_2} \wedge ... \wedge\xi_{j_{n-k}},
\]
where $\xi_{j_1},\xi_{j_2}, ...,  \xi_{j_{n-k}}$ is a complementary
set of vectors to $\xi_{i_1}, \xi_{i_2}, ..., \xi_{i_k}$, and
$s$ the signature of a permutation $(i_1, ..., i_k, j_1, ..., j_{n-k})$.

\remark $*^2\restrict{\Lambda^k(M)}=(-1)^{k(n-k)}\Id_{\Lambda^k(M)}$



\newpage

{\bf \blue $d^*=(-1)^{nk}*d*$}

\claim
On a compact Riemannian $n$-manifold, one has
$d^*\restrict{\Lambda^k M} = (-1)^{nk} *d*$,
where $d^*$ denotes {\bf \blue the adjoint operator}, which
is defined by the equation $(d\alpha, \gamma) = (\alpha, d^*\gamma)$.

{\bf \green Proof:} Since 
\[ 
  0=\int_M d(\alpha\wedge \beta) = \int_M d(\alpha)\wedge \beta +
  (-1)^{\tilde \alpha} \alpha \wedge d(\beta),
\]
one has $(d\alpha, *\beta) = (-1)^{\tilde \alpha} (\alpha, *d\beta)$.
Setting $\gamma:= *\beta$, we obtain
\[
  (d\alpha, \gamma) = (-1)^{\tilde \alpha} (\alpha, *d(*)^{-1}\gamma)=
  (-1)^{\tilde \alpha} (-1)^{\tilde \alpha(\tilde n-\tilde\alpha)}
  (\alpha, *d*\gamma)= (-1)^{\tilde \alpha\tilde n}(\alpha, *d*\gamma).
\]
\endproof

\remark Since in all applications which we consider,
$n$ is even, {\bf \purple 
I would from now on ignore the sign $(-1)^{nk}$.}


\newpage

{\bf \blue Graded vector spaces and algebras (reminder)}

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

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

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

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

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


\definition
An operator on a graded vector space is called {\bf \blue
even} ({\bf \blue odd}) if it shifts the grading by even 
(odd) number. The {\bf\blue 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 \blue pure} if it is even or odd.

\newpage


{\bf \blue Supercommutator}

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

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

\example {\bf \purple The Grassmann algebra is supercommutative.}

\definition
{\bf \blue 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 \blue the super Jacobi identity}
$\{c, \{a,b\}\} = \{\{c, a\},b\}+ (-1)^{\tilde a \tilde c}\{a,\{c, b\}\}$


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

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

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

\newpage

{\bf \blue Supersymmetry in K\"ahler geometry}

Let $(M, I, g)$ be a Kaehler manifold, $\omega$ its Kaehler form.
{\bf \blue On $\Lambda^*(M)$, the following operators are defined.}

0. $d$, $d^*$, $\Delta$, because it is Riemannian.

1. $L(\alpha):= \omega\wedge \alpha$

2. $\Lambda(\alpha) := * L * \alpha$. 
It is easily seen that $\Lambda= L^*$.

3. The Weil operator ${\cal W}\restrict{\Lambda^{p,q}(M)}=\1(p-q)$

\theorem
{\bf \red These operators generate a Lie superalgebra
$\goth a$ of dimension $(5|4)$,} 
acting on $\Lambda^*(M)$. Moreover, the Laplacian $\Delta$ is
central in $\goth a$, hence {\bf \purple $\goth a$ also acts on the
cohomology of $M$. }

\remark This is a convenient way to summarize 
the K\"ahler relations and the Lefschetz' $\goth{sl}(2)$-action.

\newpage

{\bf \blue The coordinate operators }

Let $V$ be an even-dimensional real vector space equipped with
a scalar product, and $v_1, ..., v_{2n}$ an orthonormal basis.
Denote by $e_{v_i}:\; \Lambda^k V  \arrow \Lambda^{k+1} V$
an operator of multiplication, $e_{v_i}(\eta) = v_i \wedge \eta$.
Let $i_{v_i}:\; \Lambda^k V  \arrow \Lambda^{k-1} V$
be an adjoint operator, $i_{v_i}= * e_{v_i} *$.

\claim
The operators $e_{v_i}$, $i_{v_i}$, $\Id$ are a basis of an {\blue \bf 
odd Heisenberg Lie superalgebra $\goth H$}, 
with {\bf \purple the only non-trivial 
supercommutator given by the formula $\{ e_{v_i}, i_{v_j}\} = \delta_{i,j}\Id$.}

Now, consider the tensor $\omega= \sum_{i=1}^n v_{2i-1} \wedge v_{2i}$,
and let $L(\alpha) = \omega \wedge \alpha$, and $\Lambda:= L^*$
be the corresponding {\bf \blue Hodge operators}.

\claim {\bf \blue (Lefschetz triples)}
From the commutator relations in $\goth H$,
one obtains immediately that
\[
H:=[L, \Lambda] = 
\left [\sum e_{v_{2i-1}} e_{v_{2i}}, \sum i_{v_{2i-1}} i_{v_{2i}}\right]
= \sum_{i=1}^{2n} e_{v_i} i_{v_i} - \sum_{i=1}^{2n} i_{v_i} e_{v_i},
\]
{\bf \red is a scalar operator acting as $k-n$ on $k$-forms.}

\corollary 
The triple $L, \Lambda, H$ satisfies the relations for the $\goth{sl}(2)$
Lie algebra: $[L,\Lambda]=H$, $[H,L]=2L$, $[H,\Lambda]=2\Lambda$.



\end{document}

