2012/09/26 by Fabian Ziltener, Ziltener, Fabian
Mathematics · #53D20 #53D45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AP #math.SG #msc:53D20 #msc:53D45
paper · pdf · doi:10.48550/arxiv.1209.5866
This is an improved version of a combination of arXiv:0905.4047 and arXiv:1106.1729, 129 pages, 5 figures
arxiv created 2012/09/27 · arxiv updated 2012/09/28
Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold (M,ω). Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of (M,ω) to the Gromov-Witten theory of the symplectic quotient. The morphism should be a deformation of the Kirwan map. The idea, due to D. A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane C. The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case. The first one states that every sequence of equivalence classes of vortices over the plane has a subsequence that converges to a new type of genus zero stable map, provided that the energies of the vortices are uniformly bounded. Such a stable map consists of equivalence classes of vortices over the plane and holomorphic spheres in the symplectic quotient. The second main result is that the vertical differential of the vortex equations over the plane (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially the quantum Kirwan map can be used to compute the quantum cohomology of symplectic quotients.