The moduli spaces of symplectic vortices on 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 d � µ, ξ � = − ι ( ξ M ) 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 U d -space Mat( d×n, C ) , ω, µ 0 (Grassmannian) A · g := g − 1 A ⋄ Mat( d×n, C ) � U d ; ⋄ µ 0 : Mat( d×n, C ) → u d ; µ 0 ( A ) = − i 2( AA † − τ 1 l) ( τ ∈ R ) � � 2) Ham U 1 -space C , ω, µ a ( a ∈ Z > 0 ) (WP pt) z · t = t − a z ⋄ C � U 1 ; ⋄ µ a : C → i R ; µ a ( 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?] ( M, ω, µ ) Hamiltonian G -space Fixed Data (Σ , j, dvol Σ ) closed Riemann surface P → Σ principal G -bundle ✓ ✏ � � A ∈ A ( P ) ⊂ Ω 1 ( P, g ) G ∂ A u = 0 SVE for u ∈ C ∞ ∗ F A + µ ◦ u = 0 G ( P, M ) ✒ ✑ ⋄ ∂ A u = 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 U d -space Mat( d × n, C ) , ω, µ 0 ⋄ SVE: ∂ A u = 0 and ∗F A − i 2( uu † − τ 1 l) = 0 ( τ ∈ R ) ✓ [H–K corresp. (Bertram–Daskalopoulos–Wentworth)] ✏ � � { SV ( A, u ) } gauge ↔ { τ -stable n -pairs ( ∂ E , s ) } isom ✒ ✑ s ∈ H 0 (Σ , E ⊕ n ). ⋄ E = P × U d Mat( d × n, C ), ⇒ deg( E ′ ) rk( E ′ ) < τ Vol(Σ) def ( ∀ E ′ ⊂ E ) and something ⋄ τ -stable ⇐ 4 π 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: H dim M ( P ) ev ∗ α ( M ) → R ; α �→ (Intuitive def’ n!) G M ( P ) ✓ [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”] ✏ � � ∂ A u = 0 ∂ A u = 0 dvol Σ → + ∞ ∗F A + µ ◦ u = 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! � � ∂ A u = 0 A ∈ A ( P ) ⋄ SVE for u ∈ C ∞ ∗F A + µ ◦ u = τ G ( M, P ) u ⋄ P M ( π, u ) ← → map of stacks φ : Σ → [ M/G ] π � (smooth Σ) Σ � [ M/G ] φ ⋄ 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 P G (Σ) . (orbifold Σ) ⋄ P G (Σ) = cat of prin G -bdl over Σ with smooth total space. ⋄ ( P → Σ) ∈ P G (Σ) = ⇒ A ( P ) � = ∅ ✓ [Theorem] ✏ Take P → Σ in P G (Σ). Then � � ∂ A u = 0 ∂ A u = 0 dvol Σ → + ∞ ∗ F A + µ ( u ) = 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 U 1 -space ( C , ω std , µ a ) ( a ∈ Z > 0 ) z · t = t −a z ( z ∈ C , t ∈ U 1 ) , µ a ( z ) = i � a | z | 2 − τ � ( τ ∈ R ) 2 ✓ [Theorem] ✏ sing pts order � �� � � �� � ⋄ π 1 (Σ) = 1 for Σ = (Σ; z 1 , . . . , z k ; m 1 , . . . , m k ) ⋄ a ∈ lcm( m 1 , . . . , m k ) Z ( ⇐ ∃ of J -hol orbicurve) � = C P ad if d < τ Vol(Σ) � d := i � M ( P ) ∼ = ⇒ F A 4 π 2 π Σ ✒ ✑ Next: What will come next? 12 / 14 Hironori Sakai <h.sakai@uni-muenster.de>
What will come next? [ § 4 Moduli space] 1) Moduli spaces M ( P ) ∼ � � = covering sp of Sym ad (Σ)? ⋄ π 1 (Σ) � = 1 ⋄ Linear Hamiltonian T r -space ( C n , ω std , 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>
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