From Quantum Cohomology to Integrable Systems (Oxford Graduate Texts in Mathematics)/ Martin A.Guest

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
Contents
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 45
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 -IOq
5.1 The quantum D-module 100
5.2 The cyclic structure and the^-function 102
5.3 Other properties 106
5.4 Appendix: explicit formula for the>function 112
6 Abstract quantum cohomology 116
6.1 The Birkhoff factorization 116
6.2 Quantization of algebra 124
6.3 Digression on -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 I55
7.2 The mKdV equation I60
7.3 Harmonic maps into Lie groups 164
7.4 Harmonic maps into symmetric spaces 17i
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
8.2 The fundamental construction
186 Solving the KdV equation: the Guiding Principle
8.4 Solving.the KdV equation
8.5 Solving the KdV equation: summary 202
8.6 Solving the harmonic map equation
8.7 D-module aspects
Appendix: the Birkhoff and Iwasawa decompositions
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 /?-version 265
10.4 Loop group version 270
10.5 Integrable systems of mirror symmetry type 276
10.6 Further developments 287