Symplectic geometry of deformation spaces
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- Everett Clarke
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1 July 15, 2010
2 Outline What is a symplectic structure?
3 What is a symplectic structure? Denition A symplectic structure on a (smooth) manifold M is a closed nondegenerate 2-form ω.
4 Examples Darboux coordinates If (x 1,..., x n, y 1,..., y n ) are global coordinates on M, ω = n i=1 dx i dy i is a symplectic structure. (x i, y i ) are called Darboux coordinates for the symplectic structure ω.
5 Examples Darboux coordinates If (x 1,..., x n, y 1,..., y n ) are global coordinates on M, ω = n i=1 dx i dy i is a symplectic structure. (x i, y i ) are called Darboux coordinates for the symplectic structure ω. Darboux theorem : If (M, ω) is a symplectic manifold, there are Darboux coordinates in the neighborhood of any point. Any two symplectic manifolds of the same dimension are locally symplectomorphic.
6 Examples The total space of a cotangent bundle Let π : T M M be a cotangent bundle. Dene the canonical one-form ξ on the total space T M by ξ α = π α [i.e. ξ α (v) = α(t α π(v)) for v T α T M].
7 Examples The total space of a cotangent bundle Let π : T M M be a cotangent bundle. Dene the canonical one-form ξ on the total space T M by ξ α = π α [i.e. ξ α (v) = α(t α π(v)) for v T α T M]. ω = dξ is the canonical symplectic structure of T M.
8 Examples The total space of a cotangent bundle Let π : T M M be a cotangent bundle. Dene the canonical one-form ξ on the total space T M by ξ α = π α [i.e. ξ α (v) = α(t α π(v)) for v T α T M]. ω = dξ is the canonical symplectic structure of T M. In Hamiltonian mechanics, T M represents the phase space of a mechanical system.
9 Examples Kähler manifolds Let M be a complex manifold. A hermitian metric h on M is a hermitian inner product in each tangent space.
10 Examples Kähler manifolds Let M be a complex manifold. A hermitian metric h on M is a hermitian inner product in each tangent space. g = Re(h) is a Riemannian metric on M and ω = Im(h) is a nondegenerate 2-form.
11 Examples Kähler manifolds Let M be a complex manifold. A hermitian metric h on M is a hermitian inner product in each tangent space. g = Re(h) is a Riemannian metric on M and ω = Im(h) is a nondegenerate 2-form. When ω is closed, it is a symplectic structure. h is called a Kähler metric and ω the Kähler form of h.
12 Examples Complex symplectic structures Denition Let M be a complex manifold. A complex symplectic structure on M is a complex-valued nondegenerate closed (2, 0)-form ω.
13 Examples Complex symplectic structures Denition Let M be a complex manifold. A complex symplectic structure on M is a complex-valued nondegenerate closed (2, 0)-form ω. Complex symplectic structures are holomorphic.
14 Examples Complex symplectic structures Denition Let M be a complex manifold. A complex symplectic structure on M is a complex-valued nondegenerate closed (2, 0)-form ω. Complex symplectic structures are holomorphic. If ω is a complex symplectic structure on M, then Re(ω) and Im(ω) are real symplectic structures on M.
15 The character variety Let S be a closed oriented surface of genus g 2, π = π 1 (S). Let G be a reductive Lie group, g the Lie algebra of G. Let X = X (S, G) = Hom(π, G)//G be the character variety of S. Note : I will ignore issues involving singularities.
16 The tangent space At a regular point [ρ] of X, the tangent space is given by T [ρ] X = H 1 (π, g Ad ρ ). Explanation : let ρ t : π G be a path in Hom(π, G) such that ρ 0 = ρ. Dene ρ : α π d dt t=0 ρ t(α)ρ 1 (α). ρ is a map π g. ρ t (αβ) = ρ t (α)ρ t (β) ρ(αβ) = ρ(α) + Ad ρ(α) ρ(β), i.e. ρ Z 1 (π, g Ad ρ ). If ρ t = g t 1 ρg t for some path g t G, then ρ(α) = Ad ρ(α) ġ ġ, i.e. ρ B 1 (π, g Ad ρ ).
17 The symplectic product in the tangent space Let B be a nondegenerate symmetric bilinear form on g invariant under the adjoint representation. If G is semisimple, take the Killing form (on sl(n, R), B(A, B) = 2ntr(AB)). One can take the cup product with B as coecient pairing : H 1 (π, g) H 1 (π, g) H 2 (π, g g) H 2 (π, R) (u, v) u v B(u v) Note : the cup-product is the exterior (wedge) product in de Rham cohomology.
18 Note that H 2 (π, R) H 2 (S, R) R under integration. We have dened a skew-symmetric nondegenerate bilinear form ω [ρ] : (u, v) B(u v) on the tangent space T [ρ] X.
19 Note that H 2 (π, R) H 2 (S, R) R under integration. We have dened a skew-symmetric nondegenerate bilinear form ω [ρ] : (u, v) B(u v) on the tangent space T [ρ] X. Theorem (Goldman, following Atiyah-Bott) The 2-form ω G on X dened by (ω G ) T[ρ] X = ω [ρ] is closed. I will call this symplectic form Goldman's symplectic form.
20 Symplectic geometry of Teichmüller space Goldman's symplectic form on Teichmüller space Let T hyp (S) be the hyperbolic Teichmüller space of S (a point in T hyp (S) is a marked hyperbolic surface). It embeds as a connected component of X (S, G), where G = PSL(2, R). As a consequence of, we get a symplectic structure ω G on Teichmüller space.
21 Symplectic geometry of Teichmüller space Teichmüller space as a complex manifold Let T C (S) be the conformal Teichmüller space of S (a point X in T C (S) is a marked Riemann surface). T C (S) is a complex manifold. The cotangent space is given by T X T C(S) = Q(X ), where Q(X ) is the space of holomorphic quadratic dierentials on X.
22 Symplectic geometry of Teichmüller space The Weil-Petersson metric There is a Hermitian product on Q(X ) given by (φ, ψ) = X ϕσ 1 ψ, where σ is the area form of the Poincaré metric on X. By duality, we get a Hermitian product on T X T C (S). In other words, we have dened a Hermitian metric on T C (S), it is called the Weil-Petersson metric.
23 Symplectic geometry of Teichmüller space The Weil-Petersson metric There is a Hermitian product on Q(X ) given by (φ, ψ) = X ϕσ 1 ψ, where σ is the area form of the Poincaré metric on X. By duality, we get a Hermitian product on T X T C (S). In other words, we have dened a Hermitian metric on T C (S), it is called the Weil-Petersson metric. Theorem (Ahlfors, Weil) The Weil-Petersson metric is Kähler. In other words, ω WP = Im(h) is a (real) symplectic structure on T C (S).
24 A rst identication Goldman showed the two symplectic structures agree under the identication T hyp (S) T C (S) (given by uniformization) : Theorem (Goldman) ω G = ( 2)ω WP
25 Symplectic geometry of Teichmüller space The Fenchel-Nielsen symplectic structure Let (α 1,..., α N ) be a pants decomposition of S and FN = (l αi, τ αi ) be associated Fenchel-Nielsen coordinates on T (S). FN : T (S) R 2N are global coordinates on Teichmüller space. ω FN = dl i dτ i is a symplectic structure on Teichmüller space. It depends a priori on the choice of the pants decomposition.
26 Symplectic geometry of Teichmüller space Lemma (Wolpert) If α, β are disjoint simple closed curves on S, then l α and l β Poisson-commute (for the Weil-Petersson symplectic structure) : {l α, l β } = 0. In particular, the functions l αi for the pants decomposition dene a completely integrable Hamiltonian system.
27 Symplectic geometry of Teichmüller space Lemma (Wolpert) If α, β are disjoint simple closed curves on S, then l α and l β Poisson-commute (for the Weil-Petersson symplectic structure) : {l α, l β } = 0. In particular, the functions l αi for the pants decomposition dene a completely integrable Hamiltonian system. Theorem (Wolpert) Fenchel-Nielsen coordinates are Darboux coordinates for the Weil-Petersson symplectic structure.
28 Conclusion To summarize, we have constructed three examples of symplectic structures on Teichmüller space using apparently very dierent techniques, but they turn out to be the same : Theorem ω G = ω WP = ω FN (maybe up to simple constants). This both shows how natural this symplectic structure is and provides dierent angles to comprehend it.
29 Goldman's symplectic form on CP(S) Let CP(S) be the deformation space of marked projective structures on S. The holonomy map gives a local (biholomorphic) identication hol : CP(S) X = X (S, G), where G = PSL(2, C).
30 Goldman's symplectic form on CP(S) Let CP(S) be the deformation space of marked projective structures on S. The holonomy map gives a local (biholomorphic) identication hol : CP(S) X = X (S, G), where G = PSL(2, C). As a consequence of, we get a symplectic structure ω G on X. Because G is a complex Lie group, ω G is a complex symplectic form (take the complex Killing form for B). Pulling back by the holonomy map, we get a complex symplectic structure ω G on CP(S).
31 Let σ F : T (S) CP(S) be the (standard) Fuchsian section (given by uniformization) and F(S) = σ F (T (S)) be the standard Fuchsian slice in CP(S). It is a copy of Teichmüller space in CP(S).
32 Let σ F : T (S) CP(S) be the (standard) Fuchsian section (given by uniformization) and F(S) = σ F (T (S)) be the standard Fuchsian slice in CP(S). It is a copy of Teichmüller space in CP(S). The following diagram commutes : CP(S) σ F T hyp (S) hol CP hol T X (S, PSL(2, C)) ι X (S, PSL(2, R))
33 Let σ F : T (S) CP(S) be the (standard) Fuchsian section (given by uniformization) and F(S) = σ F (T (S)) be the standard Fuchsian slice in CP(S). It is a copy of Teichmüller space in CP(S). The following diagram commutes : CP(S) σ F T hyp (S) hol CP hol T X (S, PSL(2, C)) ι X (S, PSL(2, R)) As a consequence, Goldman's symplectic structure is the Weil-Petersson Kähler form in restriction to the Fuchsian slice : σ F ω G = ω WP.
34 Complex Fenchel-Nielsen coordinates on QF(S) Let QF(S) be the (standard) quasifuchsian space of S. It is an open set in T (S) (which contains the Fuchsian slice).
35 Complex Fenchel-Nielsen coordinates on QF(S) Let QF(S) be the (standard) quasifuchsian space of S. It is an open set in T (S) (which contains the Fuchsian slice). Kourouniotis and Tan introduced a system of global holomorphic coordinates on QF(S), called complex Fenchel-Nielsen coordinates. These coordinates (l C 1,..., l C N, τ C 1,... τ C N) depend on the choice of a pants decomposition (α 1,..., α N ) of S, and can be thought of as a complexication of the real Fenchel-Nielsen coordinates on the Fuchsian slice.
36 Complex Darboux coordinates on QF(S) Theorem Complex Fenchel-Nielsen coordinates are Darboux coordinates for the complex symplectic structure ω G on QF(S)
37 Complex Darboux coordinates on QF(S) Theorem Complex Fenchel-Nielsen coordinates are Darboux coordinates for the complex symplectic structure ω G on QF(S) Functions l C i dene a holomorphic Hamiltonian ow called complex twisting (corresponds to -or is a generalization of- what authors might have called complex bending, complex earthquakes, quakebending...)
38 Another identication Let T T (S) be the total space of the holomorphic cotangent bundle of Teichmüller space. It is equipped with a canonical complex symplectic structure ω C.
39 Another identication Let T T (S) be the total space of the holomorphic cotangent bundle of Teichmüller space. It is equipped with a canonical complex symplectic structure ω C. Recall that the Schwarzian parametrization gives an identication CP(S) T T (S). However, this identication depends on the choice of a zero section σ. Name τ σ : CP(S) T T (S) the identication using the section σ.
40 Another identication Let T T (S) be the total space of the holomorphic cotangent bundle of Teichmüller space. It is equipped with a canonical complex symplectic structure ω C. Recall that the Schwarzian parametrization gives an identication CP(S) T T (S). However, this identication depends on the choice of a zero section σ. Name τ σ : CP(S) T T (S) the identication using the section σ. How do the symplectic structures ω G and (τ σ ) ω C on CP(S) compare?
41 Another identication, continued Theorem (Kawai) If σ is any Bers slice, then (τ σ ) ω C = iω G.
42 Another identication, continued Theorem (Kawai) If σ is any Bers slice, then (τ σ ) ω C = iω G. This result relates to the following facts : Bers slices are Lagrangian in CP(S). Quasifuchsian reciprocity (McMullen) If M is any 3-manifold, the restriction map : X (M, G) X ( M, G) is a Lagrangian immersion (at regular points)
43 Conclusion and questions Conclusion As in the case of Teichmüller space, we have seen in various ways that CP(S) carries a natural (complex) symplectic structure.
44 Conclusion and questions Conclusion As in the case of Teichmüller space, we have seen in various ways that CP(S) carries a natural (complex) symplectic structure. In fact, everything seems to be happening in CP(S) as a complexication of the situation in Teichmüller space.
45 Conclusion and questions Conclusion As in the case of Teichmüller space, we have seen in various ways that CP(S) carries a natural (complex) symplectic structure. In fact, everything seems to be happening in CP(S) as a complexication of the situation in Teichmüller space. The one situation that did not occur in the case of CP(S) was obtaining the symplectic structure as the imaginary part of some natural metric.
46 Conclusion and questions Questions Does the complex symplectic structure on CP(S) come as the imaginary part of some hyperkähler metric on CP(S)?
47 Conclusion and questions Questions Does the complex symplectic structure on CP(S) come as the imaginary part of some hyperkähler metric on CP(S)? How would such a hyperkähler metric compare to the canonical hyperkhäler metric on T T (S)?
48 Conclusion and questions Questions Does the complex symplectic structure on CP(S) come as the imaginary part of some hyperkähler metric on CP(S)? How would such a hyperkähler metric compare to the canonical hyperkhäler metric on T T (S)? What about the hyperkähler structure given by the description in terms of Higgs bundles?
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