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

Format:

• Book

Author/Creator:

• Ziltener, Fabian, 1977- author.

Series:

• Memoirs of the American Mathematical Society ; Volume 230, Number 1082.
• Memoirs of the American Mathematical Society, 1947-6221 ; Volume 230, Number 1082

Language:

English

Subjects (All):

• Geometry, Differential.
• Gromov-Witten invariants.

Physical Description:

1 online resource (142 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2013.

Language Note:

English

Summary:

Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold $(M\omega)$. Conjecturally under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of $(M\omega)$ 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 $\mathbb{C}$. 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.

Contents:

• ""Contents""; ""Chapter 1. Motivation and main results""; ""1.1. Quantum deformations of the Kirwan map""; ""1.2. Symplectic vortices, idea of the proof of existence of a quantum Kirwan map""; ""1.3. Bubbling for vortices over the plane""; ""1.4. Fredholm theory for vortices over the plane""; ""1.5. Remarks, related work, organization, and acknowledgments""; ""Chapter 2. Bubbling for vortices over the plane""; ""2.1. Stable maps""; ""2.2. Convergence to a stable map""; ""2.3. An example: the Ginzburg-Landau setting""; ""2.4. The action of the reparametrization group""
• ""2.5. Compactness modulo bubbling and gauge for rescaled vortices""""2.6. Soft rescaling""; ""2.7. Proof of the bubbling result""; ""2.8. Proof of the result in Section 2.3 characterizing convergence""; ""Chapter 3. Fredholm theory for vortices over the plane""; ""3.1. Equivariant homology, the Chern number, and the Maslov index""; ""3.2. Proof of the Fredholm result""; ""Appendix A. Auxiliary results about vortices, weighted spaces, and other topics""; ""A.1. Auxiliary results about vortices""; ""A.2. The invariant symplectic action""; ""A.3. Proofs of the results of Section 3.1""
• ""A.4. Weighted Sobolev spaces and a Hardy-type inequality""""A.5. Smoothening a principal bundle""; ""A.6. Proof of the existence of a right inverse for _{ }*""; ""A.7. Further auxiliary results""; ""Bibliography""

Notes:

• "Volume 230, Number 1082 (fourth of 5 numbers)."
• Includes bibliographical references.
• Description based on print version record.

ISBN:

1-4704-1672-7