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

By Fabian Ziltener

Think of a Hamiltonian motion of a compact hooked up Lie team on a symplectic manifold M ,w. Conjecturally, below appropriate assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten thought of M , w to the Gromov-Witten thought of the symplectic quotient. The morphism might be a deformation of the Kirwan map. the belief, as a result of D. A. Salamon, is to outline this sort of deformation through counting gauge equivalence periods of symplectic vortices over the advanced aircraft C. the current memoir is a part of a undertaking whose aim is to make this definition rigorous. Its major effects take care of the symplectically aspherical case

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Let z ∈ A(4r, rR). 73) ew (z) ≤ 32 E w, B |z| (z) . 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 |.

Furthermore, if α, β ∈ T1 α ∈ T∞ the J¯-holomorphic map u are such that α = β and E(Wα ) = 0, and ϕ ∈ TC , then ϕ∗ Wα = Wβ . Moreover, if ¯α is nonconstant, and if ϕ ∈ PSL(2, C), then α, β ∈ T∞ are such that α = β and u ¯β . We denote by u ¯α ◦ ϕ = u M∗ (T ) := M∗ T0 , T1 , T∞ , E ⊆ M(T ) the subset of all simple stable maps. The action of GT on M(T ) leaves M∗ (T ) invariant. 31. Proposition. The action of GT on M∗ (T ) is free. The proof of this result uses the following lemma. 32. Lemma. The action of TC on M<∞ is free.

Such a map is needed for the definition of the quantum Kirwan map. Note here that we cannot define evaluation of a vortex class W at some point z ∈ Σ by choosing a representative (P, A, u) of W and evaluating u at some point in the fiber over z, thus obtaining a point in M , since this point depends on the choices. 12 Another reason for allowing only translations as reparametrizations (for α ∈ T1 ) is that the action of Isom+ (Σ) on the set of vortex classes with positive energy is not always free.

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