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

2.4. THE ACTION OF THE REPARAMETRIZATION GROUP 29Then ( W ν ,z0 ν := ∞ ) converges to the stable map consisting of the sets T 0 := ∅,T 1 := {1,2}, T ∞ := {0}, the tree relation E := {(1,0),(0,1),(2,0),(0,2)}, thevortices W 1 and W 2 , the unique constant ¯J-holomorphic sphere ū 0 in M, the nodalpoints z 1 0 := z 2 0 := ∞, z 0 1 := 1, z 0 2 := 2 ∈ S 2 ∼ = C ∪ {∞}, and the marked point(α 0 ,z 0 ) := (0, ∞) ∈ T × S 2 . This follows from Proposition 27.It follows from this example that in general, the additional marked pointsz0 ν := ∞ are needed in the bubbling result, Theorem 3. Without these points, nosubsequence of W ν as above converges to a stable map. 14 This follows from anelementary argument. ✷2.4. The action of the reparametrization groupThis section covers an additional topic, which will not be used in this memoir,but will be relevant in a future definition of the quantum Kirwan map. Namely, weintroduce a natural group of reparametrizations and show that this group acts freelyon the set of simple stable maps consisting of vortex classes over C and pseudoholomorphicspheres in the symplectic quotient. The relevance of this result is thefollowing. For the definition of the quantum Kirwan map it will be necessary toshow that a certain natural evaluation map on the set of vortex classes over C (see[Zi1, Proposition 6.1] and [Zi4]) is a pseudo-cycle. This will rely on the fact thatits omega limit set has codimension at least two. In order to show this, one needsto cut down the dimension of each “boundary stratum” by dividing by the action ofthe reparametrization group. Heuristically, the freeness of the action of this groupimplies that the quotient is a smooth manifold, hence providing a meaning to thisprocedure.∐We∐fix finite sets T 0 ,T 1 ,T ∞ and a tree relation E on the disjoint union T :=T 0 T1 T∞ . We define the reparametrization group G T as follows. We defineAut(T) := Aut ( T 0 ,T 1 ,T ∞ ,E ) to be the subgroup of all automorphisms f of thetree (T,E), satisfying f(T i ) = T i , for i = 0,1, ∞. We denote by PSL(2, C) thegroup of Möbius transformations and by T C the group of translations of the planeC. We define{TC , if α ∈ TAut α :=1 ,PSL(2, C), if α ∈ T 0 ∪ T ∞ .We denote by Aut T the set of collections (ϕ α ) α∈T , such that ϕ α ∈ Aut α , for everyα ∈ T. The group Aut(T) acts on Aut T byf · (ϕ α ) α∈T := (ϕ f −1 (α)) α∈T .29. Definition. We define G T := G T0,T 1,T ∞,E to be the semi-direct productof Aut(T) and Aut T induced by this action.The group PSL(2, C) acts on the set of ¯J-holomorphic maps S 2 → M byϕ ∗ f := f ◦ ϕ.Furthermore, the group T C acts on the set M