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Step 1: }} \newcommand{\proof}{{\bf \green Proof: }} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Hyperk\"ahler reduction\hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}