By Fabian Ziltener
Give some thought to a Hamiltonian motion of a compact hooked up Lie crew on a symplectic manifold M ,w. Conjecturally, less than appropriate assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten conception of M , w to the Gromov-Witten idea of the symplectic quotient. The morphism can be a deformation of the Kirwan map. the belief, as a result of D. A. Salamon, is to outline one of these deformation through counting gauge equivalence sessions of symplectic vortices over the advanced aircraft C. the current memoir is a part of a venture whose aim is to make this definition rigorous. Its major effects take care of the symplectically aspherical case
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Additional resources for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane
For every (f, (ϕα )) ∈ GT and (W, z) ∈ M(T ) we deﬁne Wα := ϕ∗f (α) Wf (α) , ∀α ∈ T1 , u ¯α := u ¯α ◦ ϕf (α) , ∀α ∈ T∞ , zαβ := ϕ−1 f (α) (zf (α)f (β) ), αi := f (αi ), zi := ϕ−1 αi (zαi ), ∀αEβ, ∀i = 0, . . , k. (Here we set Wα := u ¯α if α ∈ T∞ . ) 14 For this statement to make sense, here we adjust the notion of “stable map” by discarding the marked point (α0 , z0 ). 30 FABIAN ZILTENER 30. Definition. ,k . This deﬁnes an action of GT on M(T ). Let now (M, J) be an almost complex manifold. Recall that a J-holomorphic map u : S 2 → M is called multiply covered iﬀ there exists a holomorphic map ϕ : S 2 → S 2 of degree at least two, and a J-holomorphic map v : S 2 → M , such that u = v ◦ ϕ.
29) eR w = eR w ωΣ ∈ [0, ∞]. 30) X Remark. Consider the case (Σ, j) = C, equipped with the standard area form ω0 . Assume that 0 < R < ∞, and consider the map ϕ : C → C deﬁned by ϕ(z) := Rz. 26) implies that 2 ω0 ,i eR ◦ ϕ. 31) ∂¯J,A (u) = 0, FA + R2 (μ ◦ u)ωΣ = 0. 32) as μ ◦ u = 0. 32) an R-vortex over Σ. Remarks. Consider the case (Σ, j) = C, equipped with ω0 , and let 0 < R < ∞. We deﬁne the map ϕ : C → C by ϕ(z) := Rz. Then w ∈ BC is a vortex if and only if ϕ∗ w is an R-vortex. The rescaled energy density has the following important property.
We need the following two lemmata. 41. Lemma (Hofer). Let (X, d) be a metric space, f : X → [0, ∞) a continuous ¯2δ (x) is complete. Then function, x ∈ X, and δ > 0. Assume that the closed ball B there exists ξ ∈ X and a number 0 < ε ≤ δ such that d(x, ξ) < 2δ, sup f ≤ 2f (ξ), Bε (ξ) εf (ξ) ≥ δf (x). 5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 39 Proof. 4]. The next lemma ensures that for a suitably convergent sequence of rescaled vortices in the limit ν → ∞ no energy gets lost on any compact set.