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

By Fabian Ziltener

Think of a Hamiltonian motion of a compact attached Lie workforce on a symplectic manifold M ,w. Conjecturally, lower than compatible assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten conception of M , w to the Gromov-Witten concept of the symplectic quotient. The morphism may be a deformation of the Kirwan map. the assumption, because of D. A. Salamon, is to outline this kind of deformation via counting gauge equivalence periods 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 care for the symplectically aspherical case

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C \ Z open subsets such that ν Ων = C \ Z, and for ν ∈ N let wν = (uν , Aν ) ∈ WΩp ν be an Rν -vortex. Assume that there exists a compact subset K ⊆ M such that for ν large enough uν (Ων ) ⊆ K. 36) sup ν eR wν L∞ (Q) ν ∈ N : Q ⊆ Ων < ∞. Then there exists an R0 -vortex w0 := (A0 , u0 ) ∈ WC\Z , and passing to some 2,p (C \ Z, G), such that the subsequence, there exist gauge transformations gν ∈ Wloc following conditions are satisfied. (i) If R0 < ∞ then gν∗ wν converges to w0 in C ∞ on every compact subset of C \ Z.

46). We prove statement (ii). Assume that there exists a compact subset Q ⊆ Ω ν such that supν ||eR wν ||C 0 (Q) = ∞. Let zν ∈ Q be such that fν (zν ) → ∞. We choose a pair (r0 , w0 ) as in the claim. 47) and Remark 43 24 Here 25 see the norm is taken with respect to the metric ω(·, J·) on M . g. 2] 42 FABIAN ZILTENER (in the case r0 = ∞), we have E r0 (w0 ) ≥ Emin . 44) follows. This proves (ii) and concludes the proof of Proposition 40. We are now ready to prove Proposition 37 (p. 33). ν Proof of Proposition 37.

62) supz,z E w, A(ar, a−1 R) ≤ 4a−2+ε E(w), ∀a ≥ 2, −1+ε ¯ E(w), ∀a ≥ 4. 1]. 2). 2). 62) also uses the following remark. 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.

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