Symplectic Construction of Moduli Spaces Eugene Z. Xia

Symplectic Construction of Moduli Spaces

Eugene Z. Xia

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X is a compact K¨ahler manifold of (complex) dimen- sion k with form ω.

G a (complex) reductive group with algebra g.

K ⊆ G a real (compact) form with algebra k.

B a non-degenerate Ad(G)-invariant (resp. Ad(K)- invariant) bi-linear form on g (resp. k).

g = k ⊕ p. (Example: sl(n,C) = su(n) ⊕ p.)

Example: If g is semi-simple, then B is the Killing form.

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The Betti moduli M_B:

Assume π = π₁(X) is finitely generated with n gen- erators and finitely many relations.

Hom(π₁, G) is an affine algebraic variety over C, in fact a subvariety of Gⁿ.

There is an algebraic adjoint G-action on Hom(π₁, G), G × Hom(π₁, G)−→ Hom(π₁, G)

(g, ρ) 7→ Ad(g)(ρ)

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Categorical quotient:

Hom(π₁, G) is an affine variety, so is defined by a ring R.

Hence the Ad(G)-action on Hom(π₁, G) induces a G- action on R. Denote by R^G the invariant subring.

The Betti moduli

M_B(G) = Hom(π₁, G)/G is the variety defined by R^G.

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Geometric explanation:

The orbits Ad(G)-action may not be closed in Hom(π₁, G).

This implies that the geometric quotient of the Ad(G)- action may not be Hausdorff.

Must identify each orbit with the orbits in its closure.

M_B(G) parameterizes the reductive representations.

Remark: M_B(G) has a variety structure coming from the variety structure on G.

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Example: X a compact Riemann surface of genus g.

G = C^×.

Set [A, B] = ABA⁻¹B⁻¹ π₁ = hA_i, B_i, 1 ≤ i ≤ g | Qg

i=1[A_i, B_i]i Hom(π₁, C^×) ∼= (C^×)^2g.

M_B(C^×) ∼= (C^×)^2g.

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Example: G = SL(2, C).

Hom(π₁, G) ∼=

{a_i, b_i : 1 ≤ i ≤ g,Qg

i=1[a_i, b_i] = e ∈ G}.

Consider the representation ρ corresponding to a_i, b_i =

1 1 0 1

, 1 ≤ i ≤ g.

The G-orbit of ρ is not closed because it does not contain the trivial representation.

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Symplectic structures

ρ ∈ Hom(π, G) together with the Ad(G)-action on g defines a π-module structure on g.

The tangent space at [ρ] ∈ M_B(G) is H¹(π,Ad_ρg).

Suppose X is a compact Riemann surface.

Then there is a symplectic structure Ω_J on M_B(G):

H¹(π,Ad_ρg) × H¹(π,Ad_ρg)−→H²(π,C) ∼= C.

M_B(K) is symplectic while M_B(G) is complex sym- plectic, i.e. M_B(G) has two symplectic structures.

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The de Rham construction We begin with bilinear forms Ω(η₁, η₂) = R

X B(η₁, η₂)ωⁱ.

Notice that multiplication on the form is the wedge.

For each η ∈ Ω¹(g), the curvature is F(η) = dη + ¹₂[η, η].

Ω²(g) is canonically duel to Ω⁰(g):

Ω⁰(g) × Ω²(g)−→C, (η, w) = R

X B(η, w)ω^k−1.

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Complex symplectic quotient There is a (gauge) action

Ω⁰(G)×Ω¹(g)−→Ω¹(g), (g, η) 7→ (dg)g⁻¹+Ad(g)(η).

Define the (moment) map by curvature

µ : Ω¹(g)−→Ω²(g) ∼= Ω⁰(g), µ(η) = F(η).

The de Rham moduli M_dR ∼= µ⁻¹(0)//Ω⁰(G).

M_dR acquires a natural symplectic form Ω_J from the reduction.

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Suppose that ρ ∈ Hom(π, G) is a representation.

Let ˜d : Ω⁰( ˜X, g)−→Ω¹( ˜X,g),

be the exterior derivative operator.

Form the ρ-twisted bundle ˜X ×_Ad(ρ) g.

d˜ descents to a flat connection d + η on X, where η ∈ Ω¹(g).

This defines M_B −→ M^w _dR.

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Example: G = C. (a not so great example) This is simply the de Rham cohomology:

Gauge group is Ω⁰(C) and µ(A) = dA 0−→Ω⁰(C) −→^d Ω¹(C) −→^d Ω²(C) · · ·. M_dR ∼= H¹_dR(Ω^•(C)).

Suppose X is a Riemann surface of genus g.

Then M_dR ∼= (C)^2g.

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Example: G = C^× and G = SL(2, C).

Gauge group is Ω⁰(C^×) and µ(A) = dA.

Ω⁰(C^×) −→^dlog Ω¹(C) −→^d Ω²(C)−→ · · · 0−→Z−→C −→^exp C^×−→0

· · · → H¹(Z) → H¹(C) → H¹(C^×) → H²(Z) → · · · M_dR ∼= H¹(C^×)⁰ ∼= H¹(C)/im(H¹(Z)).

Suppose X is a Riemann surface of genus g.

Then M_dR ∼= (C^×)^2g.

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The Dolbeault construction

Let h be the Hermitian metric h(X, Y ) = B(X,Y¯ ).

Let K ⊂ G be the compact form preserving h.

The K¨ahler metric on X together with h defines a Hermitian metric h_j on Ω^j(g):

h_j(η₁, η₂) = R

X h(η₁, η₂)ω^k. For A ∈ Ω¹(k), let

D_A : Ω⁰(g)−→Ω¹(g), D_A(f) = df + [A, f].

Let D_A^∗ : Ω¹(g)−→Ω⁰(g), h₀(D_A^∗η, f) = h₁(η, D_Af).

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HyperK¨ahler structure on Ω¹(g) ∼= g ⊗ Ω¹(X) I = i ⊗ 1, J = τ ⊗ j, K = IJ, where

j is the complex structure on X,

τ : g−→g is the adjoint with respect to B.

The Riemannian metric from h is compatible with I, J, K. Together, they form a HyperK¨ahler struc- ture on Ω¹(g).

The following action is symplectic with respect to Ω_I, Ω_J, Ω_K:

Ω⁰(K)×Ω¹(g)−→Ω¹(g), (g, η) 7→ (dg)g⁻¹+Ad(g)(η).

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HyperK¨ahler reduction η ∈ Ω¹(g), η = A + Ψ,

where A ∈ Ω¹(k), Ψ ∈ Ω¹(p)

The action is also Hamiltonian with respect to the three symplectic structures, resulting in the HyperK¨ahler quotient M with moment maps







µ_I(η) = D_A^∗Ψ

µ_J(η) = Im(F(η)) µ_K(η) = Re(F(η))

The smooth part of M_Dol ∼= Ω¹(g)///Ω⁰(K) has a HyperK¨ahler structure.

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The twister space.

Let a ∈ S² ⊂ R³. Then

a₁I + a₂J + a₃K is a complex structure on M com- patible with g.

This gives a family of K¨ahler structures on M, pa- rameterized by S².

All these structures are isomorphic to each other, ex- cept J.

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The Abelian case (Hodge Theory):

X a compact Riemann surface, G = C (Bad example)

Fix the trivial Hermitian metric h on X × C. 0−→Ω⁰(C) −→^d Ω¹(C) −→^d Ω²(C) · · ·.

η ∈ Ω¹(C), η = 0 + Ψ, Ψ ∈ Ω¹(C) dA = dΨ = 0, d^∗Ψ = 0.

Ψ ∈ H¹(C) is harmonic.

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X a compact Riemann surface, G = C^×:

η ∈ Ω¹(C), η = A + Ψ, A ∈ Ω¹(iR), Ψ ∈ Ω¹(R) dA = dΨ = 0, d^∗Ψ = 0.

dA = 0 ⇒ A is closed.

dΨ = 0, d^∗Ψ = 0 ⇒ Ψ is harmonic.

0−→Ω⁰(U(1)) −→^dlog Z¹(iR) × H¹(R) −→^d Ω²(C)· · ·. The complex structure J gives the complex structure on M_Dol = T^∗J ac(X).

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Real forms:

If K ⊂ G is a real form, then I and Ω_I disappear, but J and Ω_J still exist. Hence M_Dol(K) is a K¨ahler quotient with respect to J.

Example: K = U(1), M_Dol(K) ∼= J ac(X).

Example: K = R^×, M_Dol(K) = J²(X) × H⁰(Ω¹), where Ω¹ is the sheaf of holomorphic 1-forms on X.

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The complex structure J depends on the complex structure of X.

The symplectic structure Ω_J depends on the symplec- tic structure ω on X only.

In general, there are more symplectomorphisms than there are complex automorphisms.

Example: If X is a compact Riemann surface, then the space of complex automorphism is finite when g > 1. By the M_B construction, Ω_J is topological.

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Dynamics of the mapping class group: X a Riemann surface of genus g > 0.

M CG = Homeo(X)/Homeo(X)₀. There are actions

Homeo(X) × π−→π.

M CG × π−→π.

M CG × Hom(π, G)−→Hom(π, G) M CG × M_B−→M_B.

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Hamiltonian actions on M_B

Let X be of genus g > 1 and γ ⊂ X a based simple closed curve.

Tr_γ : M_B−→C, Tr_γ([ρ]) = Tr(ρ(γ)).

There are 3g − 3 mutually non-intersecting simple closed curves.

There is a (C^×)^3g−3-action on M_B(G).

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Suppose K ⊂ G is a compact form.

Then M_B(K) is compact.

There is a U(1)^3g−3-action on M_B(K).

If K = SU(2), then M_B(K) is integrable.

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X is a (punctured) torus; K = SU(2).

π = hA, B|[A, B]i

M_B is defined by x² + y² + z² − xyz − 2.

k = x² + y² + z² − xyz − 2 is the boundary trace function.

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