A Quantum Kirwan Map: Bubbling and Fredholm Theory for ... - KIAS

ContentsChapter 1. Motivation and main results 11.1. Quantum deformations of the Kirwan map 11.2. Symplectic vortices, idea of the proof of existence of a quantumKirwan map 31.3. Bubbling for vortices over the plane 61.4. Fredholm theory for vortices over the plane 71.5. Remarks, related work, organization 11Chapter 2. Bubbling for vortices over the plane 172.1. Stable maps 172.2. Convergence to a stable map 212.3. An example: the Ginzburg-Landau setting 252.4. The action of the reparametrization group 292.5. Compactness modulo bubbling and gauge for rescaled vortices 312.6. Soft rescaling 432.7. Proof of the bubbling result 522.8. Proof of the result in Section 2.3 characterizing convergence 62Chapter 3. Fredholm theory for vortices over the plane 693.1. Equivariant homology, the Chern number, and the Maslov index 693.2. Proof of the Fredholm result 73Appendix A. Auxiliary results about vortices, weighted spaces, and othertopics 91A.1. Auxiliary results about vortices 91A.2. The invariant symplectic action 96A.3. Proofs of the results of Section 3.1 97A.4. Weighted Sobolev spaces and a Hardy-type inequality 103A.5. Smoothening a principal bundle 110A.6. Proof of the existence of a right inverse for d ∗ A 112A.7. Further auxiliary results 120Bibliography 127iii