By Fabian Ziltener

ISBN-10: 0821894722

ISBN-13: 9780821894729

Think about a Hamiltonian motion of a compact attached Lie staff on a symplectic manifold M ,w. Conjecturally, lower than compatible assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten idea of M , w to the Gromov-Witten idea of the symplectic quotient. The morphism can be a deformation of the Kirwan map. the assumption, because of D. A. Salamon, is to outline this kind of deformation by way of counting gauge equivalence periods of symplectic vortices over the complicated 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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**Example text**

Zσ−1 (d) . 22) z∈C by assigning to z := [z1 , . . , zd ] ∈ Symd (C) the multiplicity map mz : C → N0 , given by mz (z) := # i ∈ {1, . . , d} | zi = z . We can now characterize vortex classes with energy dπ as follows. 26. Proposition. 23) Md := W ∈ M | E(W ) = dπ d W → degW ∈ Sym (C) = Symd (C) is a bijection. Proof of Proposition 26. This follows from [JT, Chap. 1]. 28 FABIAN ZILTENER As a consequence of Proposition 26, we obtain a classiﬁcation of stable maps in the sense of Deﬁnition 15 in the present setting: Here the symplectic quotient M = μ−1 (0)/S 1 consists of a single point, the orbit S 1 ⊆ M = C.

Remark. Convergence in conditions (i,ii) should be understood as convergence of the subsequence labelled by those indices ν for which Brν contains the given compact set. ✷ This proposition will be proved on page 42. The strategy of the proof is the ν following. Assume that the energy densities eR wν are uniformly bounded on every compact subset of C. Then the statement of Proposition 37 with Z = ∅ follows from an argument involving Uhlenbeck compactness, an estimate for ∂¯J , elliptic bootstrapping (for statement (i)), and a patching argument.

Assume that (M, ω) is aspherical. Let r > 0, z0 ∈ C, Rν > 0 be a sequence that converges to ∞, p > 2, and for every ν ∈ N let p be an Rν -vortex, such that the following conditions are wν := (Aν , uν ) ∈ WB r (z0 ) satisﬁed. (a) There exists a compact subset K ⊆ M such that uν (Br (z0 )) ⊆ K for every ν. 56) E(ε) := lim E Rν (wν , Bε (z0 )) ν→∞ exists and Emin ≤ E(ε) < ∞. 57) (0, r] ε → E(ε) ∈ R is continuous. Then there exist R0 ∈ {1, ∞}, a ﬁnite subset Z ⊆ C, an R0 -vortex w0 := (A0 , u0 ) ∈ WC\Z , and, passing to some subsequence, there exist sequences εν > 0, zν ∈ C, 2,p (C \ Z, G), such that, deﬁning and gν ∈ Wloc ϕν : C → C, ϕν (z) := εν z + zν , the following conditions hold.

### A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane by Fabian Ziltener

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