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

. .2.1. STABLE MAPS 19.z 0α 0. ... .. .Figure 1. Stable map consisting of vortex classes over C andpseudo-holomorphic spheres in M.(iii) (Connectedness) Let α,β ∈ T 1 ∪ T ∞ be such that αEβ. Thenev zαβ (W α ) = ev zβα (W β ).Here ev is defined as in (2.4) and (2.5) and we set W α := ū α if α ∈ T ∞ .(iv) (Stability) Let α ∈ T.• If α ∈ T 1 and E(W α ) = 0 then the set{ } {(2.8)β ∈ T |αEβ ∪ i ∈ {0,...,k} |αi = α }contains at least two points.• If either α ∈ T ∞ and E(ū α ) = 0, or α ∈ T 0 , then the set (2.8) consists ofat least three points.This definition is modeled on the notion of a genus 0 pseudo-holomorphic stablemap, as introduced by Kontsevich in [Ko]. 4 Roughly speaking, a stable map in thesense of Definition 15 can be thought of as a collection of vortex classes over C,pseudo-holomorphic spheres in the symplectic quotient M, “ghost spheres of type0” corresponding to the vertices of T 0 , and marked and nodal points. A vortex classmay be connected to a sphere in M at the nodal point ∞, and to “ghost spheres oftype 0” at points in C. Furthermore, spheres of the same type may be connectedat nodal points. The “ghost spheres of type 0” should be thought of as constant4 For an exhaustive exposition of those stable maps see the book by D. McDuff and D. A. Salamon[MS2].