LECTURE 9: SYMPLECTIC REDUCTION Contents 1. Reduction 1 2. Reduction at other levels 5 3. Reduction in stages 7 1. Reduction ¶ The tangent map of the moment map. Let G be a compact Lie group action on M. Recall • The orbit G · m = {g · m | g ∈ G} is an embedded submanifold with tangent space at m equals Tm(G · m) = {XM(m) | X ∈ g}. • The stabilizer subgroup of each m ∈ M, Gm = {g ∈ G | g · m = m}, is a Lie subgroup of G whose Lie algebra gm is gm = {X ∈ g | XM(m) = 0} ⊂ g. Now suppose (M, ω, G, µ) is a Hamiltonian G-manifold. As before, we let (Tm(G · m))ωm = {v ∈ TmM | ωm(v, w) = 0 for any w ∈ Tm(G · m)} be the symplectic ortho-complement of Tm(G · m) in (TmM, ωp), and let g 0 m = {ξ ∈ g ∗ | hξ, Xi = 0 for any X ∈ gm} be the annihilator of gm in g ∗ . Lemma 1.1. For any m ∈ M, (1) ker(dµm) = (Tm(G · m))ωm (2) Im(dµm) = g 0 m 1
2 LECTURE 9: SYMPLECTIC REDUCTION Proof. For any v ∈ TmM and any X ∈ g one has ωm(XM(m), v) = hdµm(v), Xi. From this one immediately get (1): ker(dµm) = (Tm(G · m))ωm, and half of (2): Im(dµm) ⊂ g 0 m. The other half of (2) follows from dimension counting: dim Im(dµm) = dim TmM − dim ker(dµm) = dim Tm(G · m) = dim Im(dτm), where dτm is the linear map dτm : g → TmM, X 7→ XM(m), and thus dim Im(dτm) = dim g − dim ker(dτm) = dim g − dim gm = dim g 0 m. ¶ Consequences of freeness. Recall that an Lie group action is • locally free at m if gm = {0}. • free if Gm = {e} for all m. Also recall that a regular value of a smooth map f : M → N is a point n ∈ N so that either f −1 (n) = ∅ or dµ is surjective at each point in f −1 (n). According to Sard’s theorem, the set of points which are not regular values is a measure zero set. A standard result in manifold theory is: if n is a regular value of f : M → N, then f −1 (n) is a smooth submanifold of M. Back to the case of Hamiltonian G-manifold (M, ω, G, µ). According to the equivariance of µ, µ(g · m) = Ad∗ g ◦ µ(m), and use the fact that the co-adjoint action is linear, we conclude Lemma 1.2. If µ(m) = 0. then for any g ∈ G, µ(g · m) = 0. In other words, the G-actions on M induces a G-action on µ −1 (0). To proceed we want µ −1 (0) to be a smooth manifold. Proposition 1.3. The G-action is locally free at each m ∈ µ −1 (0) if and only if dµm is surjective, i.e. m is a regular point of µ. Proof. The action is locally free if and only if gm = {0}, if and only if Im(dµm) = g ∗ , i.e. dµm is surjective. As a consequence,
LECTURE 9: SYMPLECTIC REDUCTION 3 Corollary 1.4. If the G-action on µ −1 (0) is free, then 0 is a regular value of µ, and thus µ −1 (0) is a closed submanifold of M whose codimension is dim G. Since G is compact, any G-action is proper. So according to Lie theory, Proposition 1.5. If the G-action on µ −1 (0) is free, then µ −1 (0)/G is a smooth manifold of dimension 2 dim M − 2 dim G and the projection map π : µ −1 (0) → µ −1 (0)/G is a principal G-bundle. Remark. If the G-action on µ −1 (0) is not free but locally free (i.e. only assume 0 is a regular value of µ), then by the slice theorem, locally µ −1 (0)/G is just (G ×Gm D)/G ' D/Gm. In other words, the quotient µ −1/G is an orbifold (a topological space that locally looks like R n divided by a finite group action, slightly generalizes the conception of manifolds). ¶ The linear reduction. We would like to construct a symplectic structure on the quotient space µ −1 (0)/G. The following theorem appeared in lecture 1: Theorem 1.6. Let W be a coisotropic subspace of a symplectic vector space (V, Ω), then the induced 2-form on W/WΩ, (1) Ω0 ([v1], [v2]) := Ω(v1, v2), is a symplectic 2-form. To construct a symplectic structure on µ −1 (0)/G, we need to take a closer look at µ −1 (0): Lemma 1.7. Suppose G acts freely on µ −1 (0). Then for any m ∈ µ −1 (0), Tm(µ −1 (0)) is a co-isotropic subspace of TmM whose symplectic ortho-complement is Tm(G· m). Proof. Since 0 is a regular value, for each m ∈ µ −1 (0) we have Tmµ −1 (0) = ker(dµm). In other words, Tmµ −1 (0) is the symplectic ortho-complement of Tm(G · m). Since G acts on µ −1 (0), m ∈ µ −1 (0) implies G · m ∈ µ −1 (0). So Tm(G · m) ⊂ Tmµ −1 (0). The conclusion follows. ¶ The Marsden-Weinstein-Meyer theorem. Now we are ready to prove Theorem 1.8 (Marsden-Weinstein-Meyer). Suppose G is a compact Lie group and (M, ω, G, µ) a Hamiltonian G-space. Let ι : µ −1 (0) ,→ be the inclusion map and assume that G acts freely on µ −1 (0). Then (1) The orbit space Mred := µ −1 (0)/G is a smooth manifold. (2) The projection π : µ −1 (0) → Mred is a principal G-bundle (and in particular π is a submersion).
4 LECTURE 9: SYMPLECTIC REDUCTION (3) There is a symplectic form ωred on Mred such that ι ∗ω = π ∗ωred. We will call (Mred, ωred) the symplectic quotient of (M, ω). Sometimes we will use the notion Mred = M//G. Proof. We have already proved (1) and (2). To prove (3) we shall apply theorem 1.6 and lemma 1.7, to conclude that for each m ∈ µ −1 (0), one can construct canonically a symplectic structure on Tmµ −1 (0)/Tm(G · m). On the other hand, at each [m] ∈ Mred = µ −1 (0)/G, T[m]Mred is canonically isomorphic to Tmµ −1 (0)/Tm(G · m). So we get a non-degenerate 2-form on Mred. Moreover, by construction (1) one gets ι ∗ω = π ∗ωred. As a consequence, π ∗ dωred = dπ∗ωred = dι∗ω = ι ∗ dω = 0. Since π ∗ is injective, we conclude dωred = 0, i.e. ωred is a symplectic form on Mred. ¶ Several examples. Example. Consider the S 1 action on S 2 by rotations with respect to z axis. The action is Hamiltonian with moment map µ(x, y, z) = z. Obviously S 1 acts freely on the equator µ −1 (0), and the quotient is a single point which is a 0 dimensional manifold. Example. Consider the diagonal S 1 action on C n by e iθ · (z1, · · · , zn) := (e iθz1, · · · , eiθzn). This is a Hamiltonian action with moment maps µ(z) = 1 2 (|z1| 2 + · · · + |zn| 2 ) + c, where c ∈ R is arbitrary and we identified Lie(S 1 ) ∗ = R. In particular if we take c = − 1 2 , then µ −1 (0) = S 2n−1 is the unit sphere in C n , and the symplectic quotient is µ −1 (0)/S1 = S 2n−1/S1 ' CPn−1 . The quotient symplectic form is the same (up to a constant factor) as the Fubini-Study symplectic form on CPn−1 that we constructed in problem set 2.
LECTURE 9: SYMPLECTIC REDUCTION 5 2. Reduction at other levels ¶ Reduction at other level sets - the case of torus. The Marsden-Weinstein-Meyer theorem tells us how to reduce at the 0 level set of the moment map µ. A natural question is whether we can perform reduction at other level sets of µ. The the case of torus, this is trivial, since if G = T n , and µ : M → g ∗ ' R n is a moment map for some Hamiltonian G-action, then for any ξ ∈ g ∗ , the map µξ = µ − ξ is a moment map for the same action. So to reduce at the ξ level set of µ is the same as to reduce at the 0 level set of this new moment map µξ. However, for more general Lie groups the argument above does not work. In fact, for a given moment map µ and an arbitrary ξ ∈ g ∗ , if m ∈ M is a point with µ(m) = ξ, in general µ(g · m) 6= ξ. In other words, G does not acts on µ −1 (ξ). Remark. Of course one can modify the torus argument above to general compact Lie groups to get reduction at special level sets: If ξ ∈ H1 (g, R) = [g, g] 0 , then one can reduce at the level set µ −1 (ξ), since in this case µξ = µ − ξ is a moment map. ¶ G-action on the pre-image of coadjoint orbits. Suppose G is a compact Lie group and (M, ω, G, µ) a Hamiltonian G-space. According to the equivariance of µ, µ(g · m) = Ad∗ g (µ(m)). So for any m ∈ M and any g ∈ G, g · m ∈ µ −1 (Oµ(m)), where Oµ(m) = G · µ(m) = Ad∗ G(µ(m)) is the orbit of the coadjoint action Ad∗ : G → Aut(g ∗ ) containing µ(m). As a consequence, Lemma 2.1. For each coadjoint orbit O in g ∗ , one gets an induced G-action on the subset µ −1 (O) ⊂ M. The three interesting special cases we studied are: (1) Since the coadjoint action is linear, the coadjoint orbit through 0 ∈ g ∗ is {0}. So we get a G-action on µ −1 (0). (2) If G is abelian, i.e. G = T n is a torus, then the coadjoint action is the trivial action Ad∗ g (ξ) = ξ. So each ξ ∈ g ∗ is a coadjoint orbit. As a consequence, G acts on µ −1 (ξ). (3) For more general compact connected Lie group G, if ξ ∈ [g, g] 0 , again we have Ad∗ g ξ = ξ for any g ∈ G. To prove this we write g = exp(Y ) (which
6 LECTURE 9: SYMPLECTIC REDUCTION is possible since G is compact, in general we can write g as a product of elements of this form and the same proof below works), then for any X ∈ g, hAd∗ exp Y ξ, Xi = hξ, Adexp(−Y )Xi = hξ, exp(−adY )Xi = hξ, Xi. ¶ The shifting trick. Now let O be a coadjoint orbit of g. According to Yuguo’s presentation on coadjoint orbits, on O one can define a G-invariant symplectic form ωO (called the Kostant-Kirillov symplectic structure) via (ωO)ξ(X, Y ) := hξ, [X, Y ]i. One can prove that the induced G-action on O is Hamiltonian with moment map µ(ξ) = ξ. Now consider the product manifold M × O, equipped with a twisted product symplectic form ω × (−ωO). Then the natural product action of G on (M × O, ω × (−ωO) is Hamiltonian with moment map µO(m, ξ) = µ(m) − ξ. Note that if µO(m, ξ) = 0 if and only if µ(m) ∈ O. So the natural projection F : µ −1 O (0) → µ −1 (O), (m, ξ) 7→ m is bijective, and thus one can identify µ −1 (0) with µ −1 O (0). As a consequence, Theorem 2.2. If the G-action on µ −1 (O) is free, then the quotient µ −1 (O)/G admits a quotient symplectic structure ω O red. The quotient (µ −1 (O)/G, ωO red) is called the symplectic quotient of (M, ω) with respect to the coadjoint orbit O. ¶ An alternative approach. Here is an alternative approach to perform symplectic reduction at other levels. Consider the coadjoint action of G on g ∗ . Then the stabilizer of ξ ∈ g ∗ is Gξ = {g ∈ G | Ad∗ g ξ = ξ}. This is a closed subgroup of G, which acts on the level set µ −1 (ξ). So if this Gα- action on µ −1 (ξ) is free, we again get a quotient symplectic manifold (M ξ red, ω ξ red). This is in fact the same as what we get by the shifting trick above. In fact, Proposition 2.3. Suppose (M, ω, G, µ) is a Hamiltonian G space, and ξ ∈ g ∗ . (1) G acts freely on µ −1 (Oξ) if and only if Gξ acts freely on µ −1 (α). (2) The map ψ : µ −1 (ξ)/Gξ → µ −1 (Oξ)/G induced by the inclusion µ −1 (ξ) ,→ µ −1 (Oα) is a diffeomorphism. Proof.
LECTURE 9: SYMPLECTIC REDUCTION 7 3. Reduction in stages student presentation