Floer homology groups in Yang-Mills theory / S.K. Donaldson with the assistance of M. Furuta and D. Kotschick.

Format:

Book

Author/Creator:

Donaldson, S. K., author.

Furuta, M., author.

Kotschick, D., author.

Series:

Cambridge tracts in mathematics ; 147.

Cambridge tracts in mathematics ; 147

Language:

English

Subjects (All):

Yang-Mills theory.

Floer homology.

Geometry, Differential.

Physical Description:

1 online resource (vii, 236 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2002.

Language Note:

English

Summary:

The concept of Floer homology was one of the most striking developments in differential geometry. It yields rigorously defined invariants which can be viewed as homology groups of infinite-dimensional cycles. The ideas led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory. The first half of this book gives a thorough account of Floer's construction in the context of gauge theory over 3 and 4-dimensional manifolds. The second half works out some further technical developments of the theory, and the final chapter outlines some research developments for the future - including a discussion of the appearance of modular forms in the theory. The scope of the material in this book means that it will appeal to graduate students as well as those on the frontiers of the subject.

Contents:

Yang-Mills theory over compact manifolds

The case of a compact 4-manifold

Technical results

Manifolds with tubular ends

Yang-Mills theory and 3-manifolds

Initial discussion

The Chern-Simons functional

The instanton equation

Linear operators

Appendix A: local models

Appendix B: pseudo-holomorphic maps

Appendix C: relations with mechanics

Linear analysis

Separation of variables

Sobolev spaces on tubes

Remarks on other operators

The addition property

Weighted spaces

Floer's grading function; relation with the Atiyah, Patodi, Singer theory

Refinement of weighted theory

L[superscript p] theory

Gauge theory and tubular ends

Exponential decay

Moduli theory

Moduli theory and weighted spaces

Gluing instantons

Gluing in the reducible case

Appendix A: further analytical results

Convergence in the general case

Gluing in the Morse

Bott case

The Floer homology groups

Compactness properties

Floer's instanton homology groups

Independence of metric

Orientations

Deforming the equations

Transversality arguments

U(2) and SO(3) connections

Floer homology and 4-manifold invariants

The conceptual picture

The straightforward case

Review of invariants for closed 4-manifolds

Invariants for manifolds with boundary and b[superscript +]] 1

Reducible connections and cup products

The maps D[subscript 1], D[subscript 2]

Manifolds with b[superscript +] = 0, 1

The case b[superscript +] = 1.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 231-233) and index.

ISBN:

1-107-12463-8

1-280-43046-X

9786610430468

0-511-17547-7

0-511-15583-2

0-511-30404-8

0-511-54309-3

0-511-04453-4

OCLC:

475916909