A quantum Kirwan map: bubbling and Fredholm theory for by Fabian Ziltener

By Fabian Ziltener

Think about a Hamiltonian motion of a compact hooked up Lie crew on a symplectic manifold M ,w. Conjecturally, below appropriate 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 will be a deformation of the Kirwan map. the assumption, as a result of D. A. Salamon, is to outline this kind of deformation through counting gauge equivalence sessions of symplectic vortices over the advanced aircraft C. the current memoir is a part of a venture whose target is to make this definition rigorous. Its major effects care for the symplectically aspherical case

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Additional info for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane

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2 π|z|2 We define a := |z|/(2r). Then a ≥ 2 and B|z|/2 (z) is contained in A(ar, a−1 R). 61) we have E w, B |z| (z) ≤ 16r 2−ε |z|−2+ε E(w). 73) and the fact |dA u|(z) ≤ 1− ε2 −2+ ε2 |dA u(z)v| ≤ Cr |z| E(w)|v|, √ where C := 32/ π. 75) |dA u(z)v| ≤ CR−1+ 2 |z|− 2 ε ε E(w)|v|, 2ew (z), it follows that √ ∀z ∈ A(4r, rR), v ∈ C. √ ∀z ∈ A( rR, R/4). Let now a ≥ 4 and z, z ∈ A(ar, a−1 R). Assume that ε ≤ 1. ) We define γ : [0, 1] → C to be the radial path of constant speed, such that γ(0) = z and |γ(1)| = |z |.

This density has the following transformation property: Let Σ be another surface, and ϕ : Σ → Σ a smooth immersion. Consider the pullback ϕ∗ w := ϕ∗ P, Φ∗ A, u ◦ Φ , where the bundle isomorphism Φ : ϕ∗ P → P is defined by Φ(z, p) := p. 26) ϕ∗ (ω ,j) eϕ∗ w Σ ωΣ ,j = ew ◦ ϕ. Note also that w is a vortex with respect to (ωΣ , j) if and only if ϕ∗ w is a vortex with respect to ϕ∗ (ωΣ , j). ✷ 36. Remark. 9). ✷ Let R ∈ [0, ∞] and w ∈ BΣ . Consider first the case 0 < R < ∞. 28) 15 The 2 R eR w := R ew 2 ωΣ ,j .

It also uses the fact that at each bubbling point at least the energy Emin > 0 is lost, which is the minimal energy of a vortex over C or pseudo-holomorphic sphere in M . This is the content of Proposition 40 below, which is proved by a hard rescaling argument, using Proposition 38 and Hofer’s lemma. Another ingredient of the proof of Proposition 37 is Lemma 42 below, which says that the energy densities of a convergent sequence of rescaled vortices converge to the density of the limit. In order to explain the main result of this section, let M, ω, G, g, ·, · g , μ, J, Σ, j, and ωΣ be as in Chapter 1.

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