By Fabian Ziltener
Give some thought to 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 thought of M , w to the Gromov-Witten concept of the symplectic quotient. The morphism might be a deformation of the Kirwan map. the assumption, because of D. A. Salamon, is to outline the sort of deformation via counting gauge equivalence sessions of symplectic vortices over the complicated airplane C. the current memoir is a part of a undertaking whose target is to make this definition rigorous. Its major effects take care of the symplectically aspherical case
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Extra info for A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane
Hence condition (iii) makes sense. ✷ The proof of Proposition 44 is given on page 48. It is based on the following result, which states that the energy of a vortex over an annulus is concentrated near the ends, provided that it is small enough. For 0 ≤ r, R ≤ ∞ we denote the open annulus around 0 with radii r, R by ¯r . A(r, R) := BR \ B ¯r , and A(r, R) = ∅ in the case r ≥ R. 59) d¯ : M/G × M/G → [0, ∞], ¯ x, y¯) := d(¯ min d(x, y). 7 this is a distance function on M/G which induces the quotient topology.
Consider the action of G := S 1 ⊆ C on M := C by multiplication. Let d ∈ N0 be an integer. By Proposition 26 there exists a unique ﬁnite energy vortex class W over C such that degW (z) = d, 0, if z = 0, otherwise. 5. COMPACTNESS MODULO BUBBLING AND GAUGE FOR RESCALED VORTICES 31 For every rotation R ∈ SO(2) we have degR∗ W = R∗ degW = degW , where SO(2) acts in a natural way on Symd (C). Thus the action of Isom+ (C) on the set of vortex classes of positive energy is not free. 5. Compactness modulo bubbling and gauge for rescaled vortices In this section we formulate and prove a crucial ingredient (Proposition 37) of the proof of Theorem 3, which states the following.
T1 Furthermore, there exist M¨obius transformations ϕνα : S 2 → S 2 for α ∈ T and ν ∈ N such that conditions (i,ii,iv) of Deﬁnition 20 are satisﬁed, and for every α ∈ T1 the point in the symmetric product degWν ◦ϕνα ∈ Symd (C) ⊆ Symd (S 2 ) converges to ιd (degWα ) ∈ Symd (S 2 ), as ν → ∞. 8. 28. Example. Let k := 0, and for ν ∈ N let W ν ∈ M7 be the unique vortex satisfying degW ν (−2 − i) = 1, degW ν (3 + 4i) = 2, degW ν (νeiν ) = 4. Let W1 ∈ M3 be the unique vortex satisfying degW1 (−2 − i) = 1, degW1 (3 + 4i) = 2, and W2 ∈ M4 be the unique vortex satisfying degW2 (0) = 4.
A quantum Kirwan map: bubbling and Fredholm theory for symplectic vortices over the plane by Fabian Ziltener