**Memoirs of the American Mathematical Society**

2014;
129 pp;
Softcover

MSC: Primary 58; 53;

Print ISBN: 978-0-8218-9472-9

Product Code: MEMO/230/1082

List Price: $76.00

AMS Member Price: $45.60

MAA Member Price: $68.40

**Electronic ISBN: 978-1-4704-1672-0
Product Code: MEMO/230/1082.E**

List Price: $76.00

AMS Member Price: $45.60

MAA Member Price: $68.40

# A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane

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*Fabian Ziltener*

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.

#### Table of Contents

# Table of Contents

## A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane

- Chapter 1. Motivation and main results 18 free
- Chapter 2. Bubbling for vortices over the plane 1724
- 2.1. Stable maps 1724
- 2.2. Convergence to a stable map 2128
- 2.3. An example: the Ginzburg-Landau setting 2532
- 2.4. The action of the reparametrization group 2936
- 2.5. Compactness modulo bubbling and gauge for rescaled vortices 3138
- 2.6. Soft rescaling 4350
- 2.7. Proof of the bubbling result 5259
- 2.8. Proof of the result in Section 2.3 characterizing convergence 6269

- Chapter 3. Fredholm theory for vortices over the plane 6976
- Appendix A. Auxiliary results about vortices, weighted spaces, and other topics 9198
- A.1. Auxiliary results about vortices 9198
- A.2. The invariant symplectic action 95102
- A.3. Proofs of the results of Section 3.1 97104
- A.4. Weighted Sobolev spaces and a Hardy-type inequality 102109
- A.5. Smoothening a principal bundle 110117
- A.6. Proof of the existence of a right inverse for 𝑑_{𝐴}* 112119
- A.7. Further auxiliary results 120127

- Bibliography 127134