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Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}