The moduli spaces of symplectic vortices on an orbifold Riemann - - PowerPoint PPT Presentation

The moduli spaces of symplectic vortices

• n an orbifold Riemann surface

Hironori Sakai

Higher Structures in Algebraic Analysis Feb 13, 2014 Mathematisches Institut, WWU M¨ unster http://sakai.blueskyproject.net/

Plan of this talk 1) What are Symplectic Vortex Equations (SVE)? Hamiltonian G-space → SVE → moduli → invariants 2) Motivation of my research 3) The reason for differentiable stacks 4) Moduli spaces for special cases

Next: Notation 1 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Notation

[§1 What are SVE?]

⋄ G compact and connected Lie group. ⋄ g = g∨ thru an , on g. g := Lie(G) ⋄ A Hamiltonian G-space is a triple (G-manifold M, symp form ω, moment map µ). ⋄ A moment map is µ ∈ C∞

G (M, g) satisfying

dµ, ξ = −ι(ξM)ω (∀ξ ∈ g)

Next: Examples of Hamiltonian G-spaces 2 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Examples of Hamiltonian G-spaces

[§1 What are SVE?]

1) Ham Ud-space

• Mat(d×n, C), ω, µ0
• (Grassmannian)

⋄ Mat(d×n, C) Ud; A · g := g−1A ⋄ µ0 : Mat(d×n, C) → ud; µ0(A) = − i 2(AA† −τ1 l) (τ ∈ R) 2) Ham U1-space

• C, ω, µa
• (a ∈ Z>0)

(WP pt) ⋄ C U1; z · t = t−az ⋄ µa : C → iR; µa(z) = i 2(a|z|2 − τ) (τ ∈ R)

Next: Symplectic Vortex Equations 3 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Symplectic Vortex Equations

[§1 What are SVE?]

Fixed Data      (M, ω, µ) Hamiltonian G-space (Σ, j, dvolΣ) closed Riemann surface P → Σ principal G-bundle

✓ ✏

SVE

• ∂Au = 0

∗FA + µ ◦ u = 0 for

• A ∈ A(P) ⊂ Ω1(P, g)G

u ∈ C∞

G (P, M)

✒ ✑

⋄ ∂Au = 0 ⇐ ⇒ u : Σ → P ×G M is pseudo-hol

Next: “Hitchin–Kobayashi correspondence” 4 /14 Hironori Sakai <h.sakai@uni-muenster.de>

“Hitchin–Kobayashi correspondence”

[§1 What are SVE?]

⋄ Hamiltonian Ud-space

• Mat(d × n, C), ω, µ0
• ⋄ SVE: ∂Au = 0 and ∗FA − i

2(uu† − τ1 l) = 0 (τ ∈ R) [H–K corresp. (Bertram–Daskalopoulos–Wentworth)]

✓ ✏

{SV (A, u)}

• gauge ↔ {τ-stable n-pairs (∂E, s)}
• isom

✒ ✑

⋄ E = P ×Ud Mat(d × n, C), s ∈ H0(Σ, E⊕n). ⋄ τ-stable

def

⇐ ⇒ deg(E′) rk(E′) < τVol(Σ) 4π (∀E′⊂E) and something

Next: Moduli space and invariants 5 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Moduli space and invariants

[§1 What are SVE?]

Assumptions: µ−1(0)G is free and more. [Thm (Cieliebak–Gaio–Mundet–Salamon)]

✓ ✏

M(P) = {(A, u) | SVE}

• (gauge) is an oriented closed mfd.

✒ ✑

⋄ SVI: Hdim M(P)

G

(M) → R; α →

• M(P)

ev∗α (Intuitive def’ n!) [Thm (Gaio–Salamon)]

✓ ✏

Under several topological conditions, SVI for M = GWI of M with fixed marked points Here M := µ−1(0)/G.

✒ ✑ Next: Q. GW theory is enough, isn’t it? 6 /14 Hironori Sakai <h.sakai@uni-muenster.de>

• Q. GW theory is enough, isn’t it?

[§1 What are SVE?]

• A. No!!

⋄ Applications! – Periodic orbits, SW inv, GW inv, QH∗(M) ⋄ Exciting Topics! – Geometry and topology of moduli spaces – H–K correspondence – Hamiltonian invariants – Differentiable stacks (today)!

Next: Motivation: SVI=GWI for orbifold 7 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Motivation: SVI=GWI for orbifold

[§2 Motivation]

⋄ Unnatural assumption: µ−1(0) G is free. ⋄ M (= µ−1(0)/G) is usually an orbifold. ⋄ Orbifold GWI ⋄ (SVI of M) = (GWI for orbifold M) ⋄ ∵ SVE do not care about singularities. [Conjecture]

✓ ✏

“SVI=GWI” holds for orbifolds after modifying SVE.

✒ ✑ Next: Idea: a “variation” of the eqn of J-hol curve 8 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Idea: a “variation” of the eqn of J-hol curve

[§2 Motivation]

Don’t Look at solutions! [Idea for “SVI=GWI”]

✓ ✏

• ∂Au = 0

∗FA + µ ◦ u = 0

dvolΣ→+∞

• ∂Au = 0

µ ◦ u = 0

SVE Eqn of J-hol Σ → M

✒ ✑

⋄ Orbi GW: pseudo-hol maps from orbifold Riemann surf ⋄ Everything is an orbifold → Terrible! ⋄ Strategies: 1) “holonomy data” on smooth Σ (majority) 2) differentiable stacks! (today)

Next: Q. Why do we need diff stacks? 9 /14 Hironori Sakai <h.sakai@uni-muenster.de>

• Q. Why do we need diff stacks?

[§2 Motivation]

• A. SVE are PDEs on differentiable stacks!

⋄ SVE

• ∂Au = 0

∗FA + µ ◦ u = τ for

• A ∈ A(P)

u ∈ C∞

G (M, P)

⋄ P

u

• π

M

• Σ

φ

[M/G]

(π, u) ← → map of stacks φ : Σ → [M/G] (smooth Σ) ⋄ Idea: An orbifold Riemann surf Σ as a stacks.

Next: Q. What should we do for P? 10 /14 Hironori Sakai <h.sakai@uni-muenster.de>

• Q. What should we do for P?

[§2 Motivation]

• A. Use the cat PG(Σ).

(orbifold Σ) ⋄ PG(Σ) = cat of prin G-bdl over Σ with smooth total space. ⋄ (P → Σ) ∈ PG(Σ) = ⇒ A(P) = ∅ [Theorem]

✓ ✏

Take P → Σ in PG(Σ). Then

• ∂Au = 0

∗FA + µ(u) = 0

dvolΣ→+∞

• ∂Au = 0

µ(u) = 0

SVE (nothing to change!) Eqn of J-hol orbicurve Σ → M

✒ ✑ Next: Moduli spaces (special cases) 11 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Moduli spaces (special cases)

[§4 Moduli space]

⋄ Recall: Ham U1-space (C, ωstd, µa) (a ∈ Z>0) z · t = t−az (z ∈C, t ∈U1), µa(z) = i 2

• a|z|2 − τ
• (τ ∈ R)

[Theorem]

✓ ✏

⋄ π1(Σ) = 1 for Σ = (Σ;

sing pts

• z1, . . . , zk;
• rder
• m1, . . . , mk)

⋄ a ∈ lcm(m1, . . . , mk)Z (⇐ ∃ of J-hol orbicurve) = ⇒ M(P) ∼ = CPad if d < τVol(Σ) 4π

• d := i

2π

• Σ

FA

• ✒

✑ Next: What will come next? 12 /14 Hironori Sakai <h.sakai@uni-muenster.de>

What will come next?

[§4 Moduli space]

1) Moduli spaces ⋄ π1(Σ) = 1

• M(P) ∼

= covering sp of Symad(Σ)?

• ⋄ Linear Hamiltonian Tr-space (Cn, ωstd, µ)

[WANTED] Specialist of geometric analysis of G-mfd! 2) Construction of SVI 3) SVI=GWI for orbifold M

Next: Summary 13 /14 Hironori Sakai <h.sakai@uni-muenster.de>

Summary

[§4 Moduli space]

⋄ Hamiltonian G-space → SVE → moduli → invariants ⋄ Exciting topics and appl: H–K corresp, GW theory, etc. ⋄ SVE as PDEs on differentiable stacks work! Thank you for your attention!

Next: http://sakai.blueskyproject.net/ 14 /14 Hironori Sakai <h.sakai@uni-muenster.de>