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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hyperk\"ahler reduction \\[2mm] and the moduli of
flat bundles \\[7mm] on complex curves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf 
February 4, Thursday, 2016, \\[4mm] 
seminar on geometric structures on manifolds,\\ LAG HSE.
}
\end{center}


\newcommand{\4}{{/\!\!/\!\!/\!\!/}} 
\newcommand{\3}{{/\!\!/\!\!/}} 
\newcommand{\2}{{/\!\!/}}


\newpage

{\bf \blue Plan}

1. Symplectic reduction and GIT

2. Hyperk\"ahler reduction

3. Moment map for the gauge group action.

{\bf \green Conventions:} Further on, $G$ is a compact,
connected Lie group, $G_\C$ its complexification,
$\g$ and $\g_\C$ the corresponding Lie algebras.
{\bf \blue Central element} of $\g^*$ is one which
is fixed by the adjoint action of $G$.

\newpage

{\bf \blue Cartan's formula and symplectomorphisms}

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, $G$ 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 central 
vector $c\in \g^*$. 

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

\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
(Weinstein-Marsden)
$(M,\omega)$ be a symplectic manifold, $G$ a 
compact Lie group 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, $d\mu$ is $\omega$-dual to the space 
$\tau(\goth g)$ of vector fields
tangent to the $G$-action, {\bf \purple hence 
$d\mu^{-1}(0)= \tau(\goth g)^\bot$.}

{\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 $\mu^{-1}(c)$, that is, the set of all $v \in T_x (\mu^{-1}(c))$
such that $\omega(v, w)=0$ for all $w\in T_x (\mu^{-1}(c))$
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 lifted from the leaf space
of characteristic foliation, identified with $M\2 G$.
\endproof


\newpage

{\bf \blue Symplectic reduction and GIT}

\theorem
Let $(M,I, \omega)$ be a K\"ahler manifold,
$G_\C$ a complex reductive Lie group acting on $M$ by
holomorphic automorphisms, and $G$ its compact form
acting isometrically. {\bf \red
Then $M\2 G$ is a K\"ahler manifold.}

\proof
Since the orbits of the $G_\C$-action are complex
subvarieties, they are symplectic. Since
the orbits of $G\subset G_\C$ are isotropic,
and their dimension is half of dimension of 
orbits of $G_\C$, they are actially Lagrangian
subvarieties in orbits of $G_\C$. Therefore, $\mu^{-1}(c)$
intersects each orbit of $G_\C$ in a 
$G$-orbit. {\bf \purple We have identified 
$M\2 G$ with a space of $G_\C$-orbits
which intersect $\mu^{-1}(c)$.}
\endproof

\remark In such a situation, $M\2 G$
is called {\bf \blue the K\"ahler quotient},
or {\bf \blue GIT quotient}. The choice
of a central element  $c\in {\goth g}^*$
is known as a choice of {\bf \blue stability data}.

\remark
{\bf \purple The
points of $M\2 G$ are in bijective correspondence
with the orbits of $G_\C$ which intersect
$\mu^{-1}(c)$.} Such orbits are called
{\bf \blue polystable}, and the intersection
of a $G_\C$-orbit with $\mu^{-1}(c)$ is a $G$-orbit.

\newpage

{\bf \blue K\"ahler reduction and a K\"ahler potential}

\definition
{\bf \blue K\"ahler potential} on a K\"ahler manifold $(M, \omega)$
is a function $\psi$ such that $dd^c \psi=\omega$.

\proposition
Let $G$ be a real Lie group acting
on a K\"ahler manifold $M$ by holomorphic isometries,
and $\psi$ be a $G$-invariant K\"ahler potential.
{\bf \red Then the moment map $\g \times M 
\stackrel {\mu_g} \arrow \R$
can be written as $g, m \arrow - \Lie_{Iv}\psi$,}
where $v=\tau(g)\in TM$ is the tangent vector field
associated with $g\in \goth g$.

\proof
Since $\psi$ is $G$-invariant, and $I$ is $G$-invariant,
we  have $0 = \Lie_v d^c \psi = (dd^c\psi)\cntrct v +
d(\langle d^c \psi, v\rangle)$. Using $\omega=dd^c \psi$,
we rewrite this equation as
$\omega\cntrct v = -d(\langle d^c \psi, v\rangle)$,
giving an equation for the moment map 
$\mu_g= -\langle d^c \psi, v\rangle.$
Acting by $I$ on both sides, we obtain
$\mu_g= -\langle d \psi, Iv\rangle = -\Lie_{Iv}\psi$.
\endproof

\corollary Let $V$ be a Hermitian representation of
a compact Lie group $G$. {\bf \purple Then the corresponding moment
map can be written as $\mu_g(v)= -\Lie_{Ig}|v|^2=
-\frac 1 2 \langle v, Ig(v)\rangle.$} \endproof


\newpage

{\bf \blue Hyperk\"ahler manifolds}

\definition
A {\bf \blue hyperk\"ahler structure} on a manifold $M$
is a Riemannian structure $g$ and a triple of complex
structures $I,J,K$, satisfying quaternionic relations
$I\circ J = - J \circ I =K$, such that $g$ is K\"ahler
for $I,J,K$.

\remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
$\omega_K:=  g(K\cdot, \cdot)$.}


{\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$:
the parallel transport along the connection preserves $I, J,K$.

\remark \\
The form $\Omega:= \omega_J + \1 \omega_K$
is {\bf \red holomorphic and symplectic} on $(M,I)$.

\newpage

{\bf\blue Hyperk\"ahler reduction}

\definition
Let $G$ be a compact Lie group, $\rho$ its action 
on a hyperk\"ahler manifold $M$ by hyperk\"ahler isometries, and
$\g^*$ a dual space to its Lie algebra. {\bf \blue A 
hyperk\"ahler moment map} is a $G$-equivariant 
smooth map $\mu: M\to\g^*\otimes\R^3$ such that
 $\langle \mu_i(v),g \rangle = \omega_i(v,d\rho(g))$, 
for every $v\in TM$, $g\in\g$ and $i=1,2,3$,
where $\omega_i$ is one three K\"ahler forms associated 
with the hyperk\"ahler structure.

\definition
Let $\xi_1,\xi_2,\xi_3$ be three $G$-invariant vectors
in $\g^*$.
The quotient manifold $M\3 G := \mu^{-1}(\xi_1,\xi_2,\xi_3)/G$ 
is called {\bf\blue  the hyperk\"ahler quotient} of $M$.

\theorem (Hitchin, Karlhede, 
 Lindstr\"om, Ro\v cek)\\ {\bf \red  The quotient $M\3 G$
is hyperkaehler}.

\newpage

{\bf \blue Holomorphic moment map}

Let $\Omega:=\omega_J+ \1\omega_K$. This is a holomorphic
symplectic (2,0)-form on $(M,I)$.

{\bf \green The proof of HKLR theorem. Step 1:}
Let $\mu_J, \mu_K$ be the moment map associated with 
$\omega_J, \omega_K$, and $\mu_\C:=  \mu_J+ \1\mu_K$.
Then $\langle d\mu_\C,g\rangle= i_{\rho_g}(\Omega)$.
Therefore, $d\mu_\C\in \Lambda^{1,0}(M,I)\otimes {\goth g}^*$.

{\bf \green Step 2:} This implies that the map $\mu_\C$ is
holomorphic. It is called {\bf \blue the holomorphic moment map.}

{\bf \green Step 3:} 
By definition, $M\3 G=\mu_{\C}^{-1}(c)\2 G$, 
where $c\in {\goth g}^*\otimes_\R \C$
is a central element.
{\bf \red This is a K\"ahler manifold,} because it is a K\"ahler
quotient of a K\"ahler manifold.

{\bf \green Step 4:} We obtain 3 complex structures
$I,J,K$ on the hyperk\"ahler quotient $M\3 G$.
{\bf \purple They are compatible in the usual way} 
(an easy exercise).
\endproof

\newpage

{\bf \blue Gauge group}

\definition
Let $G$ be a Lie group, and $P$ a principal
$G$-bundle on $M$.
{\bf \blue The gauge group} of $P$ is the
group of $G$-invariant automorphisms of $P$.

\remark
Let $G_\C$ be a complex Lie group,  $P$ a principal
$G_\C$-bundle,  ${\cal A}$ the space of all 
$G_\C$-invariant connections on $P$, and $\pgoth_\C$ the bundle
of $G$-invariant vector fields tangent to the fibers of $P$.
{\bf \purple Then ${\cal A}$
is a complex affine vector space with linearization 
$\Lambda^1M \otimes \pgoth_\C$.} It is equipped with the
gauge group action.

\definition
Let $P_G$ be a reduction of the principal $G_\C$-bundle
$P$ to $G$, and $g \arrow g^t$ be the corresponding real structure
operator on $\pgoth_\C$. Then  $\Aut(P_G)$ is called
{\bf \blue the real gauge group}, and $\Aut(P)$
{\bf \blue the complex gauge group}. 
The Killing form on $\pgoth_\C$ is denoted as
$a, b \arrow \Tr(ab)\in C^\infty M$.

\newpage

{\bf \blue Hyperk\"ahler structure on the space of connections}

\claim Let $P_\C$ be a complexification of a principal $G$-bundle 
$P$ on  a compact Riemann surface $M$, and $\nabla\in {\cal A}$
a connecion in $P_\C$. Denote by $g \arrow g^t$ the
real involution on $\pgoth_\C$ fixing $\pgoth$. {\bf \purple Then
the tangent space $T_\nabla{\cal A}$ is equipped with a real gauge invariant
Hermitian form $g(a, b)= \int_M \Re \Tr(a\wedge b^t)$ and
a complex linear 2-form $\Omega(u, v) := \int_M \Tr(u\wedge v)$.} \endproof

\definition
Define quaternionic structure on $T_\nabla{\cal A}$ as
follows. The complex structure $I$ comes from
the complex structure on $\pgoth_\C$, and $J$ comes from
$\Re \Omega(x, Jy) = g(a,b)$. 

\remark 
For $\lambda \otimes a\in T_\nabla{\cal A}=\Lambda^1(M)\otimes_\C\g_C$,
we can write $J(\lambda \otimes a)= I_M(\lambda) \otimes a^t$,
where $I_M$ is a complex structure operator on $M$
acting on $\Lambda^1(M)$.

\corollary The manifold ${\cal A}$ {\bf \purple is 
equipped with a natural real gauge invariant flat
hyperk\"ahler structure.}
\endproof




\newpage


{\bf \blue Hyperk\"ahler moment map}

\remark The tangent space to the gauge group $\Aut(P)$ can be 
identified with $\pgoth_\C$. Therefore, {\bf \purple the gauge moment map
takes values in $\pgoth_\C^*=\pgoth_\C$.}

\theorem
Let $(M, I, \omega)$ be a Riemannian surface equipped
with a Hermitian form $\omega$, $P$ a principal $G_\C$-bundle,
${\cal A}$ the space of connections on $P$, and $G\subset G_\C$. 
a compact real form. Consider the hyperk\"ahler structure
on ${\cal A}$ defined above. {\bf \red Then the holomorphic moment map
associated with the gauge action can be written
as $\nabla \arrow -\frac {\Theta_\nabla}{\omega}$,
where $\Theta_\nabla \in \Lambda^2 M \otimes \pgoth_\C$
is the curvature of $\nabla$.}

\pstep
If $\nabla_1 = \nabla +A$, we have 
$\theta_{\nabla_1}=\Theta_\nabla+\nabla(A)+A\wedge A$. 
Therefore, differential of $\Theta_\nabla$ takes 
$A\in \Lambda^1M\otimes \pgoth_\C$ to 
$\int_M \nabla(A)$. If we pair this to $b\in \pgoth_\C$,
we obtain $A \arrow \int_M \Tr(b \nabla(A))$.
{\bf \purple This is the differential of a moment map evaluated in $\nabla, b$.}

{\bf \green Step 2:}
The holomorphic symplectic form on ${\cal A}$
is expressed as $\Omega(A, B)= \int_M \Tr(A \wedge B)$.
For any $b\in \pgoth_\C$,  
the corresponding vector field on ${\cal A}$
is written as $\nabla(b)$, and the corresponding 1-form as
$\Omega(\nabla(b), A) = \int_M \Tr(\nabla(b) \wedge A)$.
Here $A\in T{\cal A}$ is a tangent vector, and  $\Omega(\nabla(b), \cdot)$
is considered as a 1-form on ${\cal A}$.

{\bf \green Step 3:} It remains to compare the 1-forms obtained
in Step 1 and Step 2: $\int_M \Tr(\nabla(b) \wedge A) = - \int_M \Tr(b \nabla(A)).$
\endproof

\newpage


{\bf \blue Space of flat connections}

\theorem
The space of stable flat connections on a compact
Riemann surface up to (complex) gauge equivalence
{\bf \red is obtained as ${\cal M}={\cal A}\3 {\cal G}$,} where ${\cal G}$
is the real gauge group. {\bf \purple 
In particular, ${\cal M}$ is hyperk\"ahler.}

\proof 
Denote the space of flat connections by ${\cal A}_{\fl}\subset {\cal A}$.
Flatness is vanishing of the holomorphic moment map,
hence ${\cal A}_{\fl}\2 G = {\cal A}\3 {\cal G}$.
However, ${\cal A}_{\fl}\2 G$ is precisely the space of
stable flat connections up to complex gauge equivalence.
\endproof


\newpage


{\bf \blue Addendum: Real moment map}


\proposition
Since the Hermitian form on ${\cal A}$ is 
fixed by affine transforms, {\bf \purple it can be written
as $dd^c$ of a quadratic function 
which is its K\"ahler potential.} 
{\bf \red The latter can be written as
$\psi(\nabla) = \int_M\Tr[\nabla, \nabla^t]$}
(here $\nabla^t$ is a connection operator which
is real conjugate to $\nabla$).

\proof
Second derivative of this quadratic map
is $\Hess(x, y) = \int_M \Re(x\wedge y^t)$.
\endproof

\proposition
The {\bf \purple real moment map for gauge action
on the space of connections is $\frac{[\nabla,\nabla^t]}{\omega}$.}

\proof
The gauge Lie algebra action is written
as $g(\nabla) = \nabla + \nabla(g)$, for all $g\in \pgoth_\C$.
This gives an expression for the real moment
map: 
\[ \mu_g(\nabla) 
= \frac{d}{dg}\int_M\Tr [\nabla+\1\nabla(g), \nabla^t-\1 \nabla^ t(g)]
= \int_M \Tr ([\nabla,\nabla^t](g))
\]
We obtain that the real moment map $\mu:\; {\cal A} \arrow \pgoth_\C$
takes $\nabla$ and puts it to $\frac{[\nabla,\nabla^t]}{\omega}$.
\endproof



\end{document}

