New PDF release: A quantum Kirwan map: bubbling and Fredholm theory for

By Fabian Ziltener

Examine a Hamiltonian motion of a compact hooked up Lie team on a symplectic manifold M ,w. Conjecturally, less than appropriate assumptions there exists a morphism of cohomological box theories from the equivariant Gromov-Witten concept of M , w to the Gromov-Witten conception of the symplectic quotient. The morphism will be a deformation of the Kirwan map. the belief, because of D. A. Salamon, is to outline this sort of deformation by means of counting gauge equivalence sessions of symplectic vortices over the advanced aircraft C. the current memoir is a part of a venture whose objective is to make this definition rigorous. Its major effects care for the symplectically aspherical case

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44) holds. Proof of the claim. For = 0 the assertion holds with Z0 := ∅. We fix ≥ 1 and assume by induction that there exists a finite subset Z −1 ⊆ C such that the assertion with replaced by − 1 holds. 54) is satisfied for every compact subset Q ⊆ C \ Z −1 , then the statement for holds with Z := Z −1 . 54) does not hold. 55) −1 | ≥ − 1. 44) holds, for every ε > 0. We set Z := Z −1 ∪ {z0 }. 54) does not hold and statement (i) of Proposition 40 that R0 = limν→∞ Rν = ∞. 55) implies that |Z | ≥ . It follows that the statement of the claim for is satisfied.

It follows that the statement of the claim for is satisfied. By induction, the claim follows. We fix an integer supν E Rν (wν , Brν ) Emin and a finite subset Z := Z ⊆ C that satisfies the conditions of the claim. 44) that > |Z|. 36) of Proposition 38 is satisfied with Ων := Brν \ Z. Applying that result and passing to some subsequence, there exist an R0 -vortex w0 ∈ WC\Z and 2,p gauge transformations gν ∈ Wloc (C \ Z, G), such that the statements (i,ii) of Proposition 37 are satisfied. ) We prove statement (iii).

Furthermore, if R0 = ∞ then on every compact subset of C\Z, the sequence gν∗ ϕ∗ν Aν converges to A0 in C 0 , and the sequence gν−1 (uν ◦ ϕν ) converges to u0 in C 1 . (iv) Fix z ∈ Z and a number ε0 > 0 such that Bε0 (z) ∩ Z = {z}. Then for every 0 < ε < ε0 the limit Ez (ε) := lim E εν Rν ϕ∗ν wν , Bε (z) ν→∞ exists and Emin ≤ Ez (ε) < ∞. Furthermore, the function (0, ε0 ) Ez (ε) ∈ R is continuous. 58) lim lim sup E Rν wν , BR−1 (z0 ) \ BRεν (zν ) = 0. R→∞ ν→∞ ε → 44 FABIAN ZILTENER Roughly speaking, this result states that given a sequence of “zoomed out” vortices for which a positive amount of energy is concentrated around some point z0 , after “zooming back in”, some subsequence converges either to a vortex over C or a holomorphic sphere in the symplectic quotient, up to bubbling at finitely many points.

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