A Quantum Kirwan Map: Bubbling and Fredholm Theory for ... - KIAS

CHAPTER 2Bubbling for vortices over the planeIn this chapter, stable maps consisting of vortex classes over the plane C, holomorphicspheres in the symplectic quotient, and marked points, are defined, andthe first main result of this memoir, Theorem 3, is proven. This result states thatgiven a sequence of vortex classes over C, with uniformly bounded energies, andsequences of marked points, there exists a subsequence that converges to some stablemap. We also describe stable maps and convergence in the simplest interestingexample, the Ginzburg-Landau setting.2.1. Stable mapsLet M,ω,G,g, 〈·, ·〉 g ,µ,J be as in Chapter 1. Our standing hypothesis (H)implies that the symplectic quotient(M = µ −1 (0)/G,ω )is well-defined and closed. The structure J induces an ω-compatible almost complexstructure on M as follows. For every x ∈ M we denote by L x : g → T x M theinfinitesimal action at x. We define the horizontal distribution H ⊆ T(µ −1 (0)) byH x := kerdµ(x) ∩ imL ⊥ x , ∀x ∈ µ −1 (0).Here ⊥ denotes the orthogonal complement with respect to the metric ω(·,J·) onM. We denote by π : µ −1 (0) → M := µ −1 (0)/G the canonical projection. Wedefine ¯J to be the unique isomorphism of TM such that(2.1) ¯J dπ = dπJ on H.1We identify C ∪ {∞} with S 2 . The (Connectedness) condition in the definitionof a stable map below will involve the evaluation of a vortex class at the point∞ ∈ S 2 . In order to make sense of this, we need the following. We denote by Gxthe orbit of a point x ∈ M. Let P be a smooth (principal) G-bundle over C 2 andu ∈ CG ∞ (P,M) a map. We denote by M/G the orbit space of the action of G onM, and defineū : C → M/G, ū(z) := Gu(p),where p ∈ P is an arbitrary point in the fiber over z. For W ∈ B we define(2.2) ū W := ū,where w = (P,A,u) is any representative of W. This is well-defined, i.e., does notdepend on the choice of w. Recall the definition (1.15) of M, and that by the image1 Such a ¯J exists and is unique, since the map dπ is an isomorphism from H to TM, and Jpreserves H.2 Such a bundle is trivializable, but we do not fix a trivialization here.17