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Master Class - Geometry, Topology, Physics > Symplectic Geometry and Moment Maps > Flashcards

Symplectic Geometry and Moment Maps Flashcards

1
Q

What one definition and 4 theorems are the core of the course?

A

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)

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2
Q

Define: symplectic vector space, w flat, pi sharp

A

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

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3
Q

Prove: (V,w) symplectic <=> w flat an iso

A

pg 4

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4
Q

Prove: (V, w) symplectic <=> exist unique pi s.t. (w flat)^(-1) = pi sharp

A

pg 5

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5
Q

Define: standard symplectic form, show symplectic

A

pg 6

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6
Q

Discuss how to define a symplectic form on V + V*

A

pg 7

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7
Q

Discuss how to define a symplectic form on a Hermitian vector space over C. Show symplectic

A

pg 8

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8
Q

Define U^(w) and compare to U perp

A

pg 9

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9
Q

What is (U^w)^w? Proof?

A

= U
pg 10

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10
Q

Define: isotropic, co-isotropic, Lagrangian, symplectic

relationships?

A

iso: U < U^w
co-iso: U^w < U
Lagrangian: U^w = U
symplectic: U int U^w = {0}

pg 10

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11
Q

Lagrangian subspaces in L(+)L*?

A

pg 11

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12
Q

What is Linear Reduction Lemma? Proof?

A

(V,w) symplectic, U < V isotropic => W = U^w/U is symplectic.

pg 12

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13
Q

Prove there always exist Lagrangian subspaces in a symplectic vector space. Dimension?

A

dim V even

L = max isotropic

pg 13-14

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14
Q

Prove: L < V Lagrangian => there exists M < V Lagrangian s.t. V = L (+) M

A

pg 15

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15
Q

Prove any symplectic vector space is iso to standard symplectic vector space

A

pg 15

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16
Q

Relationship between Liouville volume form and symplectic form?

A

pg 16-17

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17
Q

Define: symplectic manifold

A

Smooth manifold with non-degenerate closed 2-form pg 18

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18
Q

Discuss orientation, dimension, and volume of symplectic manifold. Cohomology?

A

Even dimensional

Louiville volume form gives canonical orientation and symplectic volume (if M compact)

Pg 18-19

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19
Q

Define: symplectomorphism, Sp(M, w), Chi(M, w)

A

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

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20
Q

Discuss relationship between symplectomorphisms and symplectic vector fields

A

If M compact, then have bijective correspondence. Only obstruction to this is integrability of vector field

pg 20

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21
Q

Define: Hamiltonian vector field of f, Chi_Ham(M, w)

Equivalent defs?

A

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

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22
Q

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

A

That Chi(M,w) a Lie subalgebra is easy from Lie derivative commutes with bracket.

  1. Check v_f symplectic (Cartan magic)
  2. Prove Lie subalgebra
  3. Show different expressions for Poisson bracket are equivalent
  4. Show {,} is Lie bracket (Skew automatic, Jacobi just compute from defs)
  5. Show C^inf(M) –> Chi_Ham(M, w) is a Lie algebra hom (already proved above)

pg 22-24

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23
Q

Discuss symplectic structure on R^2n
2-form, volume form, Hamiltonian VFs, flows, Poisson structure

A

pg 25

24
Q

Discuss symplectic structure on orientable 2-manifolds

A

No constraint on w - choose any non-degenerate 2-form and it is automatically closed by dimension

pg 26

25
Q

Discuss symplectic structure on Cotangent bundles.

What does it look like in local coordinates?

A

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

26
Q

Discuss/define Kahler manifolds

A

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

27
Q

Show N = C^n is Kahler. What is w in basis zj = qj + ipj?

A

pg 32

28
Q

Show N = CP^1 is Kahler.

A

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

29
Q

Are complex submanifolds of Kahler manifold Kahler? Proof?

A

Hermitian structure restricts nicely

Differential commutes with pullback –> 2-form still closed.

pg 35

30
Q

Discuss time dependent vector fields

A

Lee

31
Q

What is Moser Lemma/Trick? Proof?

A

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

32
Q

State and Prove Darboux Thm.

A

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

33
Q

Discuss when two Kahler structures on N yield symplectomorphic manifolds.

A

Banyaga. 2 Kahler structures, only need to compare cohomology class of symplectic form.

pg 41

34
Q

Define: normal bundle
Example of Lagrangian submanifold?

A

Lee
pg 45

35
Q

Discuss Tubular Neighborhood Thm using Riemannian metric

A

Lee
pg 46

36
Q

State and Prove a normal form theory for Lagrangian submanifolds

A

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

37
Q

State and discuss (without proof) Co-isotropic embedding thm

A

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

38
Q

Discuss torus representations over R

A

pg 53-55

39
Q

Discuss stabilizer subgroups for different S^1 actions on R^2.

A

pg 59

40
Q

Discuss orbits of S^1 on R^2 and S^2. Oribits of U(n) on Hermitian matrices

A

pg 60-61

41
Q

Define: fundamental vector fields compare to treatment in ch 20 of Lie on infinitesimal generator of group action. Relation to orbits?

A

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

42
Q

Discuss G-action on Normal bundle to an orbit. Gm-action?

Example of S^1 acting on S^2?

A

pg 64-65

43
Q

Discuss tubular neighborhoods of orbit

A

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

44
Q

Show normal bundle of orbit equivalent to kind of product. Proof?

A

pg 67-68

45
Q

Prove: Connected components of M^G are submanifolds of M

A

pg 69-70

46
Q

In what sense are group actions rigid?

A

pg 71

47
Q

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?

A

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

48
Q

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?

A

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

49
Q

Define: equivariant map, maps between stabilizer subgroups?

A

f(gm) = gf(m)

G_m includes into G_n

pg 75

50
Q

Define: Adjoint action, examples?

A

pg 76

51
Q

Define: coadjoint action, examples?

A

pg 77

52
Q

Define: Hamiltonian G-Space

A

Triple: (Manifold, 2-form, moment map) s.t.

  1. (M, w) symplectic
  2. mu: M –> g* equivariant w.r.t. coadjoint action
  3. 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

53
Q

Discuss the kernel and image of differential of moment map

A
  1. kernel at m = TmOn^w = subspace orthogonal to tangent space of orbit
  2. Image at m = g_m^perp = elements of g* vanishing on g_m

pg 79-81

54
Q

Discuss properties of Hamiltonian G-spaces were G is abelian

A

pg 81

55
Q

Discuss with proof: regular values of moment map and symplectic structures on regular level sets

A

x regular value <=> every point mapping to x has finite stabilizer

pg 83

Master Class - Geometry, Topology, Physics (5 decks)

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