From Quantum Cohomology to Integrable Systems

by Martin A. Guest

Overview

Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry. Certain kinds of differential equations (or D-modules) provide the key links between quantum cohomology and traditional mathematics; these links are the main focus of the book, and quantum cohomology and other integrable PDEs such as the KdV equation and the harmonic map equation are discussed within this unified framework. Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.

Product Details

ISBN-13:	9780191606960
Publisher:	OUP Oxford
Publication date:	03/13/2008
Series:	Oxford Graduate Texts in Mathematics , #15
Sold by:	Barnes & Noble
Format:	eBook
File size:	31 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface v

Acknowledgements xi

Introduction xvii

1 Cohomology and quantum cohomology xvii

2 Differential equations and D-modules xx

3 Integrable systems xxii

1 The many faces of cohomology 1

1.1 Simplicial homology 2

1.2 Simplicial cohomology 3

1.3 Other versions of homology and cohomology 4

1.4 How to think about homology and cohomology 6

1.5 Notation 7

1.6 The symplectic volume function 10

2 Quantum cohomology 12

2.1 3-point Gromov-Witten invariants 12

2.2 The quantum product 16

2.3 Examples of the quantum cohomology algebra 19

2.4 Homological geometry 29

3 Quantum differential equations 33

3.1 The quantum differential equations 33

3.2 Examples of quantum differential equations 39

3.3 Intermission 43

4 Linear differential equations in general 46

4.1 Ordinary differential equations 46

4.2 Partial differential equations 53

4.3 Differential equations with spectral parameter 62

4.4 Flat connections from extensions of D-modules 67

4.5 Appendix: connections in differential geometry 71

4.6 Appendix: self-adjointness 89

5 The quantum D-module 100

5.1 The quantum D-module 100

5.2 The cyclic structure and the J-function 102

5.3 Other properties 106

5.4 Appendix: explicit formula for the J-function 112

6 Abstract quantum cohomology 116

6.1 The Birkhoff factorization 116

6.2 Quantization of an algebra 124

6.3 Digression on D[superscript h]-modules 125

6.4 Abstract quantum cohomology 130

6.5 Properties of abstract quantum cohomology 135

6.6 Computations for Fano type examples 138

6.7 Beyond Fano type examples 144

6.8 Towards integrable systems 152

7 Integrable systems 154

7.1 The KdV equation 155

7.2The mKdV equation 160

7.3 Harmonic maps into Lie groups 164

7.4 Harmonic maps into symmetric spaces 171

7.5 Pluriharmonic maps (and quantum cohomology) 176

7.6 Summary: zero curvature equations 178

8 Solving integrable systems 182

8.1 The Grassmannian model 183

8.2 The fundamental construction 186

8.3 Solving the KdV equation: the Guiding Principle 191

8.4 Solving the KdV equation 197

8.5 Solving the KdV equation: summary 202

8.6 Solving the harmonic map equation 206

8.7 D-module aspects 218

8.8 Appendix: the Birkhoff and Iwasawa decompositions 219

9 Quantum cohomology as an integrable system 223

9.1 Large quantum cohomology 224

9.2 Frobenius manifolds 229

9.3 Homogeneity 236

9.4 Semisimple Frobenius manifolds 239

10 Integrable systems and quantum cohomology 243

10.1 Motivation: variations of Hodge structure (VHS) 244

10.2 Mirror symmetry: an example 255

10.3 h-version 265

10.4 Loop group version 270

10.5 Integrable systems of mirror symmetry type 276

10.6 Further developments 287

References 293

Index 303

Editorial Reviews

"This book is written in the style of lecture notes, which...is quite nice. It begins with basic materials, followed by separate discussions on the main topics, and eventually puts them all together. This makes the book quite accessible to readers with little or no experience in the subject."—Mathematical Reviews

From the Publisher