\documentclass[10pt]{article} \usepackage{url} %version 1.0,\ \ 22.09.2021 %version 1.1,\ \ 29.09.2021, опечатки в 3.2, 3.3 %version 1.2,\ \ 02.10.2021, опечатки в 3.8 %version 1.3,\ \ 09.10.2021, опечатки в 3.10 %version 2.0,\ \ 13.10.2021, миллион опечаток %version 2.1,\ \ 20.10.2021, some more %version 2.2,\ \ 23.10.2021, linear vector field in 3.7b \newcommand{\version}{version 2.2,\ \ 23.10.2021} \newcommand{\firstdate}{22.09.2021} \addtolength{\topmargin}{-15mm} \addtolength{\textheight}{30mm} \addtolength{\oddsidemargin}{-10mm} \addtolength{\textwidth}{20mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Symplectic geometry, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{3}{Symplectic handout 3: Symplectic reduction} \definition Let $(M, \omega)$ be a symplectic manifold. A vector field $v\in TM$ is called {\bf symplectomorphic} if $\Lie_v \omega=0$. \ed \exercise Let $\rho$ be a closed $k$-form on $M$, and $v$ a vector field. Prove that $\Lie_v\rho=0$ if and only if $d(i_v\rho)=0$, where $i_v(\rho) = \rho(v, \cdot, ..., \cdot)$ denotes the contraction operator. \ez \definition A symplectomorphic vector field $v$ on $(M, \omega)$ is called {\bf Hamiltonian} if the 1-form $i_v(\omega)$ is exact. {\bf Hamiltonian symplectomorphism} is $\Psi_1$, where $\Psi_t$ is a flow of symplectomorphisms associated with a smooth family $v_t$, $t\in [0,1]$ of Hamiltonian vector fields, with $\Psi_0=\Id$, and $\Psi_t^{-1}\frac{d\Psi_t}{dt}=v_t$. \ed \definition Let $G$ be a Lie group acting on a symplectic manifold $(M, \omega)$ by symplectomorphisms, $\g=T_e G$ its Lie algebra, and $\rho:\; \g \arrow TM$ the corresponding Lie algebra action, that is, the differential of the action of $G$ in $e\in G$. Suppose that all vector fields in $\rho(\g)$ are Hamiltonian. Consider a linear map $\mu_1:\; \g\arrow C^\infty M$ which takes each vector field $x\in \g$ to the Hamiltonian of $\rho(x)$, and let $\mu:\; M \arrow \g^*$ take $X \in \g$ to $\mu_1(X)$. The map $\mu:\; M \arrow \g^*$ is called {\bf the moment map} of the Lie group action. It is called {\bf an equivariant moment map} if $\mu(g(x)) = \coad(g)(\mu(x))$, where $g\in G$, and $\coad$ denotes the coadjoint action of $G$ on $\g^*$. \ed \exercise Let $G$ be a Lie group acting on a simply connected symplectic manifold by symplectomorphisms. Prove that the moment map $\mu:\; M \arrow \g^*$ always exists.\footnote{Here we don't assume that $\mu$ is equivariant.} Prove that the set of moment maps is a finite-dimensional affine space with free, transitive action of the space $\g^*$ considered as an abelian Lie group. \ez \exercise Let $G$ be a Lie group which acts on $M$ by Hamiltonian symplectomorphisms. We say that this action is {\bf locally free} if each orbit is diffeomorphic to a quotient of $G$ by a discrete subgroup. Prove that $G$ acts locally freely if and only if the moment map $\mu$ is a submersion. \ez \exercise\label{_Poisson_via_moment_Exercise_} Let $\mu:\; M \arrow \g^*$ be an equivariant moment map, $\rho:\; \g \arrow TM$ the Lie algebra action, and $x, y\in \g$. Prove that $\omega(\rho(x), \rho(y))= \mu([x, y])$. \ez \exercise\label{_central_Exercise_} An covector $t\in \g^*$ is called {\bf central} if it is $\coad$-invariant. Prove that any central element vanishes on $[\g,\g]$. \ez \exercise Prove that the space of equivariant moment maps is a finite-dimensional affine space with free, transitive action of the space $(\g^*)^G$ of central covectors. \ez \exercise Let $\omega=\sum_i dx_i\wedge dy_i$ be the standard symplectic form on $\C^n$, and $X\in \goth u(n)$ a complex linear vector field acting by isometries. \enum \ite Prove that the symplectic gradient $\omega^{-1}(dH)$ of a function $H\in C^\infty(\C^n)$ can be written as $I(\grad H)$, where $\grad H= (dH)^\sharp$ is the usual gradient. \ite Let $X$ be a vector field acting on $\C^n$ by linear holomorphic isometries. Prove that $4 \omega= d I d (r)$, where $r(z) =|z|^2$. Prove that $\Lie_X (I dr) = 0 = d(\langle I dr, X\rangle) + 2 i_X \omega$. Deduce from this $\Lie_{IX}(dr)=d(\langle dr, IX\rangle) = 2 i_X \omega$. \ite Prove that the moment map for the action of the Lie group $\R= e^{tX}$ can be written as $\mu(v) = \frac 1 4 \Lie_{IX} (r)$. \ee \ez \exercise Let $\mu:\; M \arrow \g^*$ be an equivariant moment map, and $t\in \g^*$ a central covector. Prove that for any and any $x, y \in \rho(g)\restrict {T(\mu^{-1}(t))}$, one has $\omega(x, y) =0$. \ez \hint Use Exercise \ref{_Poisson_via_moment_Exercise_} and Exercise \ref{_central_Exercise_}. \eh \exercise\label{_fiber_of_mu_coisotro_Exercise_} Let $G$ be a connected Lie group which acts on $(M,\omega)$ locally freely by Hamiltonian symplectomorphisms, $\mu:\; M \arrow \g^*$ an equivariant moment map, $t\in \g^*$ a central element and $Z:= \mu^{-1}(t)$. \enum \ite Prove that $Z$ is coisotropic (that is, for each $z\in Z$ one has $(T_z Z)^\bot \subset T_zZ$). \ite Prove that $(T_z Z)^\bot=\rho(\g)$, where $\rho:\; \g \arrow TM$ is the corresponding action of the Lie algebra $\Lie(G)$. \ee \ez \hint Use the previous exercise. \eh \exercise Let $Z\subset M$ be a coisotropic submanifold, and $B\subset TZ$ the sub-bundle defined as $B:= (TZ)^\bot$. Prove that $[B, B]\subset B$. \ez \remark By Frobenius theorem, the condition $[B, B]\subset B$ means that $B$ is is a tangent bundle to a foliation on $M$. This foliation is called {\bf the characteristic foliation} of $Z\subset M$. Further on, you are allowed to apply Frobenius theorem, if you can state it correctly. \er \exercise Let $Z\subset M$ be a coisotropic subvariety in $(M, \omega)$, and $\pi:\; Z \arrow Z_0$ the projection to the leaf space of the characteristic foliation. \enum \ite Let $\pi:\; X \arrow Y$ be a smooth submersion, and $\alpha\in \Lambda^k (X)$ a closed form. Prove that $\alpha = \pi^* \alpha_0$ if and only if $i_v \alpha=0$ for any vector field $v$ tangent to the fibers of $\pi$. \ite Prove that $Z_0$ is equipped with a symplectic form $\omega_0$ such that $\pi^* \omega_0 = \omega\restrict Z$. \ee \ez \exercise In assumptions of Exercise \ref{_fiber_of_mu_coisotro_Exercise_}, let $Z:= \mu^{-1}(t)$, where $t\in \g^*$ is a central covector. \enum \ite Prove that $Z$ is $G$-invariant. \ite Prove that the leaves of the characteristic foliation on $Z$ coincide with the orbits of $G$. \ite Assume that $G$ is compact and its action is free. Prove that the quotient space $\frac{\mu^{-1}(t)}{G}$ is a smooth manifold which is equipped with a natural symplectic structure. \ee \ez \hint Use the previous exercise. \eh \definition The manifold $M\2 G:=\frac{\mu^{-1}(t)}{G}$ is called {\bf the symplectic reduction} of $M$. \ed \exercise Consider the natural action of the group $U(n+1)$ on $\C P^n$. \enum \ite Prove that there exists an $U(n+1)$-invariant non-degenerate 2-form on $\C P^n$. \ite Prove that such a form is always closed. \ite Prove that it is unique up to a constant multiplier.\footnote{This form is called {\bf the Fubini-Study symplectic form}.} \ee \ez \exercise Consider the action of $U(1)$ on $\C^{n+1}$ by complex rotations. Prove that the symplectic quotient $\C^{n+1}\2 U(1)$ is $\emptyset$, a point, or $\C P^n$ with the Fubini-Study symplectic form, depending on the choice of the moment map. \ez \exercise[*] Let $\gamma\subset S^1 \times S^1$ be a circle $S^1\times \{x\}$, and $g$ a Hamiltonian diffeomorphism of $S^1\times S^1$. Prove that $g(\gamma)\cap \gamma$ is non-empty. \ez \end{document} \enum \ite Prove that the central covector on the Lie algebra ${\goth u}(n+1)$ is unique up to a constant multiplier. \ite Prove that the symplectic reduction