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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 13: symplectic reduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 16, 2021
}
\end{center}

\newpage


{\bf \blue Frobenius theorem and foliations (reminder)}


{\bf \green Frobenius Theorem:}
Let $B\subset TM$ be a sub-bundle. Then $[B,B]\subset B$
if and only if each point $x\in M$ has a neighbourhood
$U\ni x$ and {\bf \red a smooth submersion $U\stackrel \pi \arrow V$ 
such that $B$ is its vertical tangent space: $B= T_\pi M$.}

\definition
The fibers of $\pi$ are called {\bf \blue leaves},
or {\bf \blue integral submanifolds} of the distribution $B$.
Globally on $M$, {\bf \blue a leaf of $B$} is a maximal
connected manifold $Z\hookrightarrow M$ which is immersed to $M$ 
and tangent to $B$ at each point.
A distribution for which Frobenius theorem holds is called
{\bf \blue integrable}. If $B$ is integrable, the set of 
its leaves is called {\bf \blue a foliation}. The leaves
are manifolds which are immersed to $M$, but not necessarily closed.

\newpage

{\bf \blue Holonomy of a foliation with compact leaves (reminder)}

\definition
Let ${\cal F}$ be a smooth foliation on $M$.
Suppose that all leaves of ${\cal F}$ are compact
(in this case ${\cal F}$ is called {\bf\blue a foliation
with compact leaves}). 
 

Let $F\subset M$ be a compact leaf of ${\cal F}$, and
$U$ its tubular neighbourhood. Denote by $\pi$ a
smooth retraction of $U$ to $F$. For $U$ sufficiently
small, we may assume that $\pi$ is locally a diffeomorphism
on each leaf of ${\cal F}$. Then $\pi$ restricted
to a compact leaf $F_1 \subset F$ is a covering.
In particular, every path $\gamma \subset F$
can be lifted to a covering $\gamma_1 \in F_1$.
Let $S:= \pi^{-1}(x)$, where $x$ is the 
starting point of a loop $\gamma:\; [0,1] \arrow F$.
Then $\gamma_1$ is uniquely determined by
$\gamma_1(0)$ and gives a $H_\gamma:\; S \arrow S$
mapping $\gamma_1(0)$ to $\gamma_1(1)$.
{\bf \red This construction defines a group homomorphism
$\pi_1(F) \arrow \Diff_x(S_x)$,} where $S_x$
is a germ of $S$ in $x$, and $\Diff_x(S_x)$
denotes the group of diffeomorphisms of this 
germ.

\definition {\bf \blue (Ehresmann)}\\
The homomorphism $\pi_1(F) \arrow \Diff_x(S_x)$
is called {\bf \blue the holonomy of the foliation
${\cal F}$ in $F$.}

\remark Holonomy is well defined for any 
leaf of a foliation, {\bf \red compactness of its leaves
is not necessary. }
Moreover, a germ of a foliation
in a neighbourhood of a closed leaf {\bf \red is uniquely
(up to a diffeo) determined by its holonomy.}

\newpage

{\bf \blue Holonomy of a foliation with compact leaves (reminder)}

\remark
Holonomy of a foliation with compact leaves 
is finite in dimension 3, by a theorem of
D. B. A. Epstein. However, in dimension 5
D. Sullivan {\bf \red produced an $S^1$-foliation
on $S^5$ with infinite holonomy.}

\exercise
Let $G$ be a compact Lie group acting on
a manifold $M$. Prove that
{\bf \purple all orbits have dimension $\dim G$
if and only if for some basis
$g_1, ... g_n \in \Lie G$
the corresponding vector fields 
on $M$ are linearly independent everywhere.}

\definition
In this case, the action of $G$ on $M$
is called {\bf\blue locally free.}

\theorem
Let $G$ be a compact Lie group which 
locally freely acts on a manifold $M$,
and ${\cal F}$ the corresponding foliation,
with its leaves being the orbits of 
$G$. {\bf \red Then the holonomy of ${\cal F}$
is finite.} Moreover, the leaf space
$M/G$ locally in a neighbourhood of 
$[F]$ {\bf \red is homeomorphic to 
$\R^n/\Gamma$, where $\Gamma$ is the 
holonomy of ${\cal F}$ in $F$. }

\newpage

{\bf \blue Holonomy of a foliation with transitive group
action on its leaves (reminder)}


\theorem
Let $G$ be a compact Lie group with 
locally freely acts on a manifold $M$,
and ${\cal F}$ the corresponding foliation,
with its leaves being the orbits of 
$G$. {\bf \red Then the holonomy of ${\cal F}$
is finite.} Moreover, the leaf space
$M/G$ locally in a neighbourhood of 
$[F]$ {\bf \red is homeomorphic to 
$\R^n/\Gamma$, where $\Gamma$ is the 
holonomy of ${\cal F}$ in $F$. }

\pstep
Choose a $G$-invariant Riemannian metric on $M$
by taking any Riemannian metric and averaging it with $G$.
Then for any $r\in \R^{>0}$,  an $r$-neighbourhood
$U$ of a leaf $F$ is $G$-invariant. Therefore,
{\bf \purple $U$ contains the whole leaf $F_1$ of ${\cal F}$
if it contains a point of $F_1$.}

Let $\pi:\; U \arrow F$ a smooth retraction,
$S:= \pi^{-1}(x)$, and $\Gamma$ the holonomy of 
${\cal F}$ in $F$. Consider a leaf $F_1$ of ${\cal F}$
passing through $x_1 \in S$. {\bf \purple Then 
$F_1 \cap S= \Gamma\cdot x_1$, hence 
$U/G= S/\Gamma$.} We proved the second
claim of the theorem.


{\bf \green Step 2:} 
To see that $\Gamma$ is positive, we choose
$\sigma$ inverse to the Riemannian geodesic (exponential)
map in the direction orthogonal to $F$. 
Consider the map $\mu:\; S\times G\arrow U$ mapping $(s, g)$ 
to $g(s)$. This map is by construction surjective
and each point $x_1\in S\cap F_1$ has precisely
$\Gamma_{F_1}$ preimages. Let $\Gamma_F$ be the
subgroup of $G$ fixing $x\in F$. Then
$\mu(\Gamma_F)$ maps $S\subset U$ to itself, and
this action coincides with $\Gamma$.
Since $G$ is compact, $\Gamma_F$ is 
finite.
\endproof

\newpage

{\bf \blue Orbifolds}

\definition
We say that a topological space {\bf\blue has quotient
  singularities} if it is locally homeomorphic to
$\R^n/\Gamma$, where $\Gamma$ is a finite group acting
on $\R^n$ by diffeomorphisms.

It turns out that all ``analysis on manifolds''
can be performed on such varieties, if we define {\bf
  \green the orbifolds} (but it would take time). In
particular,

\theorem
Let $M$ be a manifold equipped with a locally free
action of a Lie group. {\bf \red Then $M/G$ is an orbifold.}

\newpage

{\bf \blue Basic forms}

\definition
Let $B \subset M$ be an involutive sub-bundle,
tangent to a foliation ${\cal F}$ on $M$, and
$U \stackrel {\pi_U} \arrow U/{\cal F}$ its leaf space, defined for a
sufficiently small $U\subset M$. {\bf \blue 
A basic form} on $(M, {\cal F})$
is a form $\alpha \in \Lambda^k(M)$ such that
for each local leaf space 
$U \stackrel {\pi_U} \arrow U/{\cal F}$,
one can represent $\alpha$ as a pullback,
$\alpha = \pi_U^*(\alpha_0)$ for a form
$\alpha_0$ on $ U/{\cal F}$.

\proposition
A form $\alpha \in \Lambda^k(M)$ 
{\bf \red is basic with respect to ${\cal F}$
if and only if for any vector field $X$ tangent to
${\cal F}$, one has $i_X(\alpha)=0$ and 
$\Lie_X(\alpha)=0$.}

\proof
Let $x_1, ..., x_n, y_1, ..., y_m$ be a coordinate
system on $U$ such that $\pi_U$ maps
$(x_1, ..., x_n, y_1, ..., y_m)$ to
$(x_1, ..., x_n)$. Then a form $\alpha$ is expressed
through a sum of coordinate monomials and
functions as $\alpha= \sum f_I \alpha_I$
is basic if and only if the functions $f_I$ are
independent from $y_i$ and the coordinate
monomials $\alpha_I$ do 
not contain $dy_i$. \endproof

\corollary
Let $\alpha$ be a closed differential form on a manifold $M$,
equipped with a foliation ${\cal F}$. 
{\bf \red Then $\alpha$ is basic if and only if
$i_X(\alpha)=0$ for any vector field $X$ tangent to
${\cal F}$.}

\proof
Cartan's formula gives $\Lie_X \alpha = i_X(d\alpha) + d(i_X\alpha)$;
when $\alpha$ is closed, this is equivalent to
$\Lie_X \alpha =d(i_X\alpha)$.
\endproof


\newpage

{\bf \blue Characteristic foliation}

\definition
Let $(M, \omega)$ be a symplectic manifold.
A submanifold $Z \subset M$ is called {\bf \blue
coisotropic} if $\dim_\R Z \geq \dim_\R M$ and
$\omega\restrict Z$ has rank $\dim_\R M- \dim_\R Z$
(minimal possible), or, equivalently, 
$(TZ)^{\bot_\omega} \subset TZ$, where
$(TZ)^{\bot_\omega}:= \{ x\in TZ\ \ |\ \ i_x\omega=0\}$

\definition
Let $Z\subset (M, \omega)$ be a coisotropic submanifold.
The bundle $(TZ)^{\bot_\omega}$ is called {\bf \blue
the characteristic bundle of $Z$}.

\theorem
Let  $Z\subset (M, \omega)$ be a coisotropic submanifold,
and $K\subset TM$ its characteristic bundle. {\bf \red Then
$[K, K]\subset K$, hence $K$ is tangent to a foliation} ${\cal F}$
which is called {\bf \blue the characteristic foliation of $Z$.} 
Moreover, {\bf \red the restriction $\omega\restrict Z$ is basic, and symplectic
on the leaf space of the characteristic foliation.}

\newpage

{\bf \blue Characteristic foliation (2)}

\theorem
Let  $Z\subset (M, \omega)$ be a coisotropic submanifold,
and $K\subset TM$ its characteristic bundle. {\bf \red Then
$[K, K]\subset K$, hence $K$ is tangent to a foliation} ${\cal F}$
which is called {\bf \blue the characteristic foliation of $Z$. }
Moreover, {\bf \red the restriction $\omega\restrict Z$ is basic, and symplectic
on the leaf space of the characteristic foliation.}

\proof
To see that $[K,K]\subset K$, we use Cartan's formula for
de Rham differential
\begin{multline*}
0 =d\omega(X,Y,Z) = \omega ([X,Y], Z) + \omega ([Y,Z], X)+
\omega ([Z,X], Y)+ \\+\Lie_X\omega(Y,Z)+ \Lie_Y\omega(Z,X)+ \Lie_Z\omega(X,Y).
\end{multline*}
If $X,Y\in K$, all terms in this sum vanish, except, possibly,
\[
0 =d\omega(X,Y,Z) = \omega ([X,Y], Z),
\]
hence $[X,Y]\in K$. 

Now, {\bf \purple $\omega$ is $K$-basic because it is closed and vanishes on $K$,}
and symplectic on the leaf space $M/{\cal F}$ because
it has rank $\dim_\R M- \dim_\R Z$, and 
$\dim M/{\cal F}=\dim_\R M- \dim_\R Z$. \endproof

\newpage

{\bf \blue Cartan's formula and symplectomorphisms (reminder)}

We denote the Lie derivative along a vector field
as $\Lie_x:\; \Lambda^i M \arrow \Lambda^i M$,
and contraction with a vector field by 
$i_x:\; \Lambda^i M \arrow \Lambda^{i-1} M$.

{\bf \blue Cartan's formula:} $d\circ i_x + i_x \circ d =\Lie_x$.

\remark
Let $(M,\omega)$ be a symplectic manifold, $G$ a 
Lie group acting on $M$ by symplectomorphisms, and $\goth g$
its Lie algebra. For any $g\in {\goth g}$,
denote by $\rho_g$ the corresponding vector field.
Then $\Lie_{\rho_g}\omega=0$, giving 
$d(i_{\rho_g}(\omega))=0$. {\bf \purple We obtain that 
$i_{\rho_g}(\omega)$ is closed, for any $g\in {\goth g}$.}

\definition
{\bf\blue A Hamiltonian} of $g\in {\goth g}$ is a function
$h$ on $M$ such that $dh=i_{\rho_g}(\omega)$.

\newpage

{\bf \blue Moment maps}

\definition
$(M,\omega)$ be a symplectic manifold, and $G$ a 
Lie group acting on $M$ by symplectomorphisms.
{\bf \blue A moment map} $\mu$ of this action is a linear map
${\goth g}\arrow C^\infty M$
associating to each $g\in G$ its Hamiltonian.

\remark
It is more convenient to consider $\mu$
as an element of ${\goth g}^* \otimes_\R C^\infty M$,
or (and this is most standard) {\bf \red as a function
with values in ${\goth g}^*$}. 

\remark
Moment map {\bf \purple always exists} if $M$ 
is simply connected.

\definition 
A moment map $M \arrow  {\goth g}^*$
is called {\bf \blue equivariant}
if it is equivariant with respect to the 
coadjoint action of $G$ on ${\goth g}^*$.

\remark
$M\stackrel\mu \arrow  {\goth g}^*$ is a moment map
iff for all $g\in {\goth g}$, 
$\langle d\mu,g\rangle= i_{\rho_g}(\omega)$.
Therefore, {\bf \purple a moment map is defined up to 
a constant ${\goth g}^*$-valued function.}
An equivariant moment map is is defined up to 
{\bf \purple a constant ${\goth g}^*$-valued function
which is $G$-invariant}, that is, up to addition of a {\bf \blue central}
(that is, $G$-invariant) vector $c\in \g^*$. 

\claim
{\bf \red
An equivariant moment map exists whenever $H^1(G, {\goth g}^*)=0$.}
In particular, when $G$ is reductive and $M$ is simply connected,
an equivariant moment map exists.
Further on, all moment maps will be tacitly considered equivariant.



\newpage

{\bf \blue Weinstein-Marsden theorem}

%\definition A $G$-invariant $c\in \goth g^*$ is called
%{\bf\blue central}.

\definition
(Weinstein-Marsden)
$(M,\omega)$ be a symplectic manifold, $G$ a 
compact Lie group freely acting on $M$ by symplectomorphisms,
$M\stackrel\mu \arrow  {\goth g}^*$ an equivariant
moment map, and $c\in {\goth g}^*$ a central element.
The quotient $\mu^{-1}(c)/G$ is called {\bf \blue
symplectic reduction} of $M$, denoted by $M\2 G$.

\claim
{\bf \red The symplectic quotient $M\2 G$ is a symplectic
manifold of dimension $\dim M - 2 \dim G$.}\\
\pstep
$T_x (\mu^{-1}(c))= d\mu^{-1}(0)$.
However, the space 
$\langle d\mu, g\rangle \subset \Lambda^1 M$ is $\omega$-dual to the space 
$\tau(\goth g)$ of vector fields
tangent to the $G$-action, {\bf \purple hence 
$d\mu^{-1}(c)= \tau(\goth g)^\bot$.}

{\bf \green Step 2:}
Since $\mu$ is $G$-equivariant, $G$ preserves
$\mu^{-1}(c)$, hence $\tau(\goth g)\subset d\mu^{-1}(0)$.
This implies that {\bf \purple $\tau(\goth g)\subset TM$ is isotropic}
(that is, $\omega\restrict{\tau(\goth g)}=0$).
Its $\omega$-orthogonal complement in $T_x M$ 
is $T_x (\mu^{-1}(c))$ (Step 1).

{\bf \green Step 3:}
Consider the {\bf \blue characteristic foliation}
${\cal F}$ on $\mu^{-1}(c)$. It is 
a bundle because $\mu^{-1}(c)\subset M$
is coisotropic. From Step 2 {\bf \purple we obtain that ${\cal F}=\tau(\goth g)$.}

{\bf \green Step 4:}
Since $\omega\restrict {\mu^{-1}(c)}$ is closed, it
satisfies  $\Lie_v (\omega)=0$ for all $v\in {\cal F}$.
This implies that it is {\em \green basic}, that is, 
lifted from the leaf space of characteristic foliation, 
identified with $M\2 G$.
\endproof


\end{document}
