By Fabian Ziltener
Think of a Hamiltonian motion of a compact attached Lie workforce on a symplectic manifold M ,w. Conjecturally, below compatible assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten idea of M , w to the Gromov-Witten thought of the symplectic quotient. The morphism may be a deformation of the Kirwan map. the belief, because of D. A. Salamon, is to outline any such deformation through counting gauge equivalence sessions of symplectic vortices over the complicated airplane C. the current memoir is a part of a undertaking whose aim is to make this definition rigorous. Its major effects care for the symplectically aspherical case
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Extra resources for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane
COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 37 embedding theorem and the Arzel`a-Ascoli theorem, shrinking I 1 , we may assume that A1ν converges (strongly) in C 0 on X 1 . Iterating this argument, for every i ≥ 2 there exist an inﬁnite subset I i ⊆ I i−1 and gauge transformations gνi ∈ W 2,p (X 1 , G), for ν ∈ I i , such that X i ⊆ Ων , for every ν ∈ I i , and the sequence Aiν := (gνi )∗ (Aν |X i ) converges to some W 1,p connection Ai over X i , weakly in W 1,p and in C 0 on X i .
46. Remark. Let M, ·, · M be a Riemannian manifold, G a compact Lie group that acts on M by isometries, P a G-bundle over [0, 1] 28 , A ∈ A(P ) a ∞ (P, M ) a map. We deﬁne connection, and u ∈ CG 1 |dA u|dt, (A, u) := 0 where dA u = du + Lu A, and the norm is taken with respect to the standard metric on [0, 1] and ·, · M . Furthermore, we deﬁne u ¯ : [0, 1] → M/G, u ¯(t) := Gu(p), where p ∈ P is any point over t. 59). Then for every pair of points x ¯0 , x ¯1 ∈ M/G, we have ¯ x0 , x ¯1 ) ≤ inf (A, u) (P, A, u) as above: u ¯(i) = x ¯i , i = 0, 1 .
21) π deg(W ) min1 |v(z)|2 ≤ z∈SR 1 SR v ∗ α ≤ π deg(W ) max1 |v(z)|2 . 1 implies that |μ ◦ v(Rz)| = 12 (1 − |v(Rz)|2 ) converges to 0, uniformly in z ∈ S 1 , as R → ∞. 19) follows. This proves Proposition 24. This result has the following consequence. 25. Corollary. Let w := (P, A, u) be a smooth vortex over C with positive and ﬁnite energy. Then the image of u contains the open unit ball B1 ⊆ C. Proof of Corollary 25. Consider the set X := |u(p)| p ∈ P . This set is connected, and hence an interval.