The moduli spaces of symplectic vortices on an orbifold Riemann surface Hironori Sakai

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1 The moduli spaces of symplectic vortices on an orbifold Riemann surface Hironori Sakai Higher Structures in Algebraic Analysis Feb 13, 2014 Mathematisches Institut, WWU Münster

2 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>

3 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>

4 Examples of Hamiltonian G-spaces 1) Ham U d -space ( ) Mat(d n, C), ω, µ 0 [ 1 What are SVE?] (Grassmannian) Mat(d n, C) U d ; A g := g 1 A µ 0 : Mat(d n, C) u d ; µ 0 (A) = i 2 (AA τ1l) (τ R) 2) Ham U 1 -space ( C, ω, µ a ) (a Z >0 ) (WP pt) C U 1 ; z t = t a z µ a : C ir; µ a (z) = i 2 (a z 2 τ) (τ R) Next: Symplectic Vortex Equations 3 /14 Hironori Sakai <h.sakai@uni-muenster.de>

5 Symplectic Vortex Equations (M, ω, µ) Hamiltonian G-space Fixed Data (Σ, j, dvol Σ ) closed Riemann surface P Σ principal G-bundle [ 1 What are SVE?] SVE A u = 0 F A + µ u = 0 for A A(P) Ω 1 (P, g) G u C 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>

6 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 τ1l) = 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 H 0 (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>

7 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. dim M(P) SVI: H (M) R; α ev 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>

8 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>

9 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>

10 Idea: a variation of the eqn of J-hol curve [ 2 Motivation] Don t Look at solutions! A u = 0 F A + µ u = 0 dvol Σ + [Idea for SVI=GWI ] A 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>

11 Q. Why do we need diff stacks? [ 2 Motivation] A. SVE are PDEs on differentiable stacks! A u = 0 A A(P) SVE for F A + µ u = τ u CG (M, P) P π u M (π, u) map of stacks φ : Σ [M/G] Σ φ [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>

12 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) Take P Σ in P G (Σ). Then A u = 0 F A + µ(u) = 0 dvol Σ + A u = 0 µ(u) = 0 [Theorem] 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>

13 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 2( a z 2 τ ) (τ R) π 1 (Σ) = 1 for Σ = (Σ; a lcm(m 1,..., m k )Z sing pts }} z 1,..., z k ; = M(P) = CP ad if d < τvol(σ) 4π [Theorem] order }} m 1,..., m k ) ( of J-hol orbicurve) ( d := i 2π Σ F A ) Next: What will come next? 12 /14 Hironori Sakai <h.sakai@uni-muenster.de>

14 What will come next? [ 4 Moduli space] 1) Moduli spaces π 1 (Σ) 1 ( M(P) = covering sp of Sym ad (Σ)? ) 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>

15 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: 14 /14 Hironori Sakai <h.sakai@uni-muenster.de>

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