Symplectic Geometry and Moment Maps Flashcards
What one definition and 4 theorems are the core of the course?
Def. A Hamiltonian T-Space is (M manifold with T action, w T-invariant 2-form, mu: M –> Lie(T)*) s.t.
1. (M, mu) is symplectic
2. mu is T-invariant
3. fundamental v.f. exact form…
Thm 1 (Symplectic reduction)
Thm 2 (Convexity)
Thm 3 (Localization)
Thm 4 ([Q, R] = 0)
Define: symplectic vector space, w flat, pi sharp
f.d vector aspace over R with 2-form w s.t. ker(w) = 0
w flat (x) = w(x, _)
pi sharp = pi(z, _)
pg 4, 5
Prove: (V,w) symplectic <=> w flat an iso
pg 4
Prove: (V, w) symplectic <=> exist unique pi s.t. (w flat)^(-1) = pi sharp
pg 5
Define: standard symplectic form, show symplectic
pg 6
Discuss how to define a symplectic form on V + V*
pg 7
Discuss how to define a symplectic form on a Hermitian vector space over C. Show symplectic
pg 8
Define U^(w) and compare to U perp
pg 9
What is (U^w)^w? Proof?
= U
pg 10
Define: isotropic, co-isotropic, Lagrangian, symplectic
relationships?
iso: U < U^w
co-iso: U^w < U
Lagrangian: U^w = U
symplectic: U int U^w = {0}
pg 10
Lagrangian subspaces in L(+)L*?
pg 11
What is Linear Reduction Lemma? Proof?
(V,w) symplectic, U < V isotropic => W = U^w/U is symplectic.
pg 12
Prove there always exist Lagrangian subspaces in a symplectic vector space. Dimension?
dim V even
L = max isotropic
pg 13-14
Prove: L < V Lagrangian => there exists M < V Lagrangian s.t. V = L (+) M
pg 15
Prove any symplectic vector space is iso to standard symplectic vector space
pg 15
Relationship between Liouville volume form and symplectic form?
pg 16-17
Define: symplectic manifold
Smooth manifold with non-degenerate closed 2-form pg 18
Discuss orientation, dimension, and volume of symplectic manifold. Cohomology?
Even dimensional
Louiville volume form gives canonical orientation and symplectic volume (if M compact)
Pg 18-19
Define: symplectomorphism, Sp(M, w), Chi(M, w)
symplectomorphism if diffeo and f*w2 = w1
Sp(M,w) = group of symplectomorphisms inside Diff+(M)
Chi(M, w) = Lie algebra of symplectic vector fields
Discuss relationship between symplectomorphisms and symplectic vector fields
If M compact, then have bijective correspondence. Only obstruction to this is integrability of vector field
pg 20
Define: Hamiltonian vector field of f, Chi_Ham(M, w)
Equivalent defs?
pi#(df) := v_f
Just turn differential of f, a 1-form, into a vector field by the uniquely defined map pi# : T*M –> TM
Chi_Ham(M,w) is Lie algebra of all Hamiltonian vector fields
pg 21
Prove Chi(M,w) is a Lie subalgebra of Chi(M) and Chi_Ham(M,w) a Lie subalgebra of Chi(M,w)
Define: Poisson bracket, prove properties
Show map from C^inf(M) –> Chi_Ham(M, w) is a Lie algebra hom
That Chi(M,w) a Lie subalgebra is easy from Lie derivative commutes with bracket.
- Check v_f symplectic (Cartan magic)
- Prove Lie subalgebra
- Show different expressions for Poisson bracket are equivalent
- Show {,} is Lie bracket (Skew automatic, Jacobi just compute from defs)
- Show C^inf(M) –> Chi_Ham(M, w) is a Lie algebra hom (already proved above)
pg 22-24
Discuss symplectic structure on R^2n
2-form, volume form, Hamiltonian VFs, flows, Poisson structure
pg 25
Discuss symplectic structure on orientable 2-manifolds
No constraint on w - choose any non-degenerate 2-form and it is automatically closed by dimension
pg 26
Discuss symplectic structure on Cotangent bundles.
What does it look like in local coordinates?
Prop. Exists unique one form on cotangent bundle s.t. if we pull it back to base manifold with 1-form alpha we get alpha.
pg 26 - 30
Discuss/define Kahler manifolds
N = mfld/C with Hermitian structure
M = N_R <– viewed as real manifold
w(u,v) = Im h(u,v)
(N,h) is Kahler if dw = 0 => (M,w) symplectic
pg 31 - 32
Show N = C^n is Kahler. What is w in basis zj = qj + ipj?
pg 32
Show N = CP^1 is Kahler.
Have SU(2) action on CP^1 by mobius transforms. Define hermitian structure on CP^1 by SU(2) invariance. Unlike defining structures on Lie groups, we don’t have a free action here - need to make sure this is well-defined. (Working with symmetric space not Lie group)
Check stabilizer of point preserves structure.
Get SU(2) invariant symplectic structure on CP^1
pg 33 - 35
Are complex submanifolds of Kahler manifold Kahler? Proof?
Hermitian structure restricts nicely
Differential commutes with pullback –> 2-form still closed.
pg 35
Discuss time dependent vector fields
Lee
What is Moser Lemma/Trick? Proof?
M = mfld, w_t smooth family of symplectic forms on M.
Assume: dw_t/dt is exact 2-form => [w_t] constant.
Define: Moser vector field
Assume v_t integrates to theta_t
Then (M, w0) and (M, wt) symplectomorphic.
PF. Again Cartan magic.
pg 36 - 37
State and Prove Darboux Thm.
There exists a cover of any symplectic manifold by Darboux charts –> symplectic manifolds have no local invariants, in sharp contrast with curvature in Riemannian geometry. All patches look the same.
pg 38-41
Discuss when two Kahler structures on N yield symplectomorphic manifolds.
Banyaga. 2 Kahler structures, only need to compare cohomology class of symplectic form.
pg 41
Define: normal bundle
Example of Lagrangian submanifold?
Lee
pg 45
Discuss Tubular Neighborhood Thm using Riemannian metric
Lee
pg 46
State and Prove a normal form theory for Lagrangian submanifolds
Weinstein Lagrangian Embedding Thm. (M,w) symplectic, L < M cpt Lagrangian submfd => exists tubular neighborhood symplectomorphic to a neighborhood of zero-section in T*L.
pg 47 - 51
State and discuss (without proof) Co-isotropic embedding thm
Again gives symplectomorphism of tubular neighborhoods, but not a standard model - in absence of more structure there is not a convincing normal form for co-isotropic submanifolds
Discuss torus representations over R
pg 53-55
Discuss stabilizer subgroups for different S^1 actions on R^2.
pg 59
Discuss orbits of S^1 on R^2 and S^2. Oribits of U(n) on Hermitian matrices
pg 60-61
Define: fundamental vector fields compare to treatment in ch 20 of Lie on infinitesimal generator of group action. Relation to orbits?
action yields vector field in M associated to each element of g <– fund vector field.
Remember g = left-invariant vector fields on G. VF <=> flows…
Orbits are submanifolds - tangent space at a point of orbit spanned by fundamental VFs
pg 63-64
Discuss G-action on Normal bundle to an orbit. Gm-action?
Example of S^1 acting on S^2?
pg 64-65
Discuss tubular neighborhoods of orbit
We can always choose tubular neighborhood using geodesic rays - diffeo between subset of M and NOrbit. If G compact, can average Riemannian metric over group action –> get G-invaraint metric –> geodesics now G invaraint so get our diffeo G-invaraint
pg 66
Show normal bundle of orbit equivalent to kind of product. Proof?
pg 67-68
Prove: Connected components of M^G are submanifolds of M
pg 69-70
In what sense are group actions rigid?
pg 71
Discuss the relationship between zero-set of fundamental vector field and fixed points of 1-param subgroups.
Structure of zero-set? What about for arbitrary vector fields?
zeros of vector field = fixed points of 1-param subgroup –> a smooth manifold (assuming group compact)
Zero set need not be a subman if vector field doesn’t come from group action
pg 72-73
What is the kernel of the map from Lie(G) = g –> TmOm (the tangent space of the orbit of m at m)?
What about the orbit map G –> Om?
Relationship?
g_m ={elements of Lie algebra s.t. associated fundamental vector field vanishes at m}
g_m = {0} <=> G_m finite
dim O_m = dim g - dim g_m
pg 74
Define: equivariant map, maps between stabilizer subgroups?
f(gm) = gf(m)
G_m includes into G_n
pg 75
Define: Adjoint action, examples?
pg 76
Define: coadjoint action, examples?
pg 77
Define: Hamiltonian G-Space
Triple: (Manifold, 2-form, moment map) s.t.
- (M, w) symplectic
- mu: M –> g* equivariant w.r.t. coadjoint action
- Contracting w w.r.t fundamental v.f. = differential of natural pairing of moment map and element of g yielding fund v.f. <– compare to def of Hamiltonian vector field
pg 78
Discuss the kernel and image of differential of moment map
- kernel at m = TmOn^w = subspace orthogonal to tangent space of orbit
- Image at m = g_m^perp = elements of g* vanishing on g_m
pg 79-81
Discuss properties of Hamiltonian G-spaces were G is abelian
pg 81
Discuss with proof: regular values of moment map and symplectic structures on regular level sets
x regular value <=> every point mapping to x has finite stabilizer
pg 83
