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.
Author(s): Martin A. Guest
Series: Oxford Graduate Texts in Mathematics
Publisher: Oxford University Press
Year: 2008
Language: English
Pages: 336
0198565992......Page 1
Contents......Page 14
Preface......Page 6
Acknowledgements......Page 12
1. Cohomology and quantum cohomology......Page 18
2. Differential equations and D-modules......Page 21
3. Integrable systems......Page 23
1 The many faces of cohomology......Page 32
1.1 Simplicial homology......Page 33
1.2 Simplicial cohomology......Page 34
1.3 Other versions of homology and cohomology......Page 35
1.4 How to think about homology and cohomology......Page 37
1.5 Notation......Page 38
1.6 The symplectic volume function......Page 41
2.1 3-point Gromov–Witten invariants......Page 43
2.2 The quantum product......Page 47
2.3 Examples of the quantum cohomology algebra......Page 50
2.4 Homological geometry......Page 60
3.1 The quantum differential equations......Page 64
3.2 Examples of quantum differential equations......Page 70
3.3 Intermission......Page 74
4.1 Ordinary differential equations......Page 77
4.2 Partial differential equations......Page 84
4.3 Differential equations with spectral parameter......Page 93
4.4 Flat connections from extensions of D-modules......Page 98
4.5 Appendix: connections in differential geometry......Page 102
4.6 Appendix: self-adjointness......Page 120
5.1 The quantum D-module......Page 131
5.2 The cyclic structure and the J-function......Page 133
5.3 Other properties......Page 137
5.4 Appendix: explicit formula for the J-function......Page 143
6.1 The Birkhoff factorization......Page 147
6.2 Quantization of an algebra......Page 155
6.3 Digression on D[sup(h)]-modules......Page 156
6.4 Abstract quantum cohomology......Page 161
6.5 Properties of abstract quantum cohomology......Page 166
6.6 Computations for Fano type examples......Page 169
6.7 Beyond Fano type examples......Page 175
6.8 Towards integrable systems......Page 183
7 Integrable systems......Page 185
7.1 The KdV equation......Page 186
7.2 The mKdV equation......Page 191
7.3 Harmonic maps into Lie groups......Page 195
7.4 Harmonic maps into symmetric spaces......Page 202
7.5 Pluriharmonic maps (and quantum cohomology)......Page 207
7.6 Summary: zero curvature equations......Page 209
8 Solving integrable systems......Page 213
8.1 The Grassmannian model......Page 214
8.2 The fundamental construction......Page 217
8.3 Solving the KdV equation: the Guiding Principle......Page 222
8.4 Solving the KdV equation......Page 228
8.5 Solving the KdV equation: summary......Page 233
8.6 Solving the harmonic map equation......Page 237
8.7 D-module aspects......Page 249
8.8 Appendix: the Birkhoff and Iwasawa decompositions......Page 250
9 Quantum cohomology as an integrable system......Page 254
9.1 Large quantum cohomology......Page 255
9.2 Frobenius manifolds......Page 260
9.3 Homogeneity......Page 267
9.4 Semisimple Frobenius manifolds......Page 270
10 Integrable systems and quantum cohomology......Page 274
10.1 Motivation: variations of Hodge structure (VHS)......Page 275
10.2 Mirror symmetry: an example......Page 286
10.3 h-version......Page 296
10.4 Loop group version......Page 301
10.5 Integrable systems of mirror symmetry type......Page 307
10.6 Further developments......Page 318
References......Page 324
G......Page 334
P......Page 335
Z......Page 336
