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E-grāmata: From Quantum Cohomology to Integrable Systems

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(Tokyo Metropolitan University)
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From Quantum Cohomology to Integrable SystemsPalielināt
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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.

Recenzijas

Well-written and reasonably paced, From Quantum Cohomology to Integrable Systems is a good introduction to a rich and fascinating subject that is still in its early stages of evolution. The book is indispensable for mathematicians interested in pursuing these ideas and themes. * Mathematical Association of America * This book is written in the style of lecture notes, which, in the reviewers opinion, 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. * American Mathematical Society *

Preface v
Acknowledgements xi
Introduction xvii
Cohomology and quantum cohomology xvii
Differential equations and D-modules xx
Integrable systems xxii
The many faces of cohomology
1(11)
Simplicial homology
2(1)
Simplicial cohomology
3(1)
Other versions of homology and cohomology
4(2)
How to think about homology and cohomology
6(1)
Notation
7(3)
The symplectic volume function
10(2)
Quantum cohomology
12(21)
3-point Gromov-Witten invariants
12(4)
The quantum product
16(3)
Examples of the quantum cohomology algebra
19(10)
Homological geometry
29(4)
The quantum differential equations
33(13)
Examples of quantum differential equations
39(4)
Intermission
43(3)
Linear differential equations in general
46(54)
Ordinary differential equations
46(7)
Partial differential equations
53(9)
Differential equations with spectral parameter
62(5)
Flat connections from extensions of D-modules
67(4)
Appendix: connections in differential geometry
71(18)
Appendix: self-adjointness
89(11)
The quantum D-module
100(16)
The quantum D-module
100(2)
The cyclic structure and the J-function
102(4)
Other properties
106(6)
Appendix: explicit formula for the J-function
112(4)
Abstract quantum cohomology
116(38)
The Birkhoff factorization
116(8)
Quantization of an algebra
124(1)
Digression on Dh-modules
125(5)
Abstract quantum cohomology
130(5)
Properties of abstract quantum cohomology
135(3)
Computations for Fano type examples
138(6)
Beyond Fano type examples
144(8)
Towards integrable systems
152(2)
Integrable systems
154(28)
The KdV equation
155(5)
The mKdV equation
160(4)
Harmonic maps into Lie groups
164(7)
Harmonic maps into symmetric spaces
171(5)
Pluriharmonic maps (and quantum cohomology)
176(2)
Summary: zero curvature equations
178(4)
Solving integrable systems
182(41)
The Grassmannian model
183(3)
The fundamental construction
186(5)
Solving the KdV equation: the Guiding Principle
191(6)
Solving the KdV equation
197(5)
Solving the KdV equation: summary
202(4)
Solving the harmonic map equation
206(12)
D-module aspects
218(1)
Appendix: the Birkhoff and Iwasawa decompositions
219(4)
Quantum cohomology as an integrable system
223(20)
Large quantum cohomology
224(5)
Frobenius manifolds
229(7)
Homogeneity
236(3)
Semisimple Frobenius manifolds
239(4)
Integrable systems and quantum cohomology
243(50)
Motivation: variations of Hodge structure (VHS)
244(11)
Mirror symmetry: an example
255(10)
h-version
265(5)
Loop group version
270(6)
Integrable systems of mirror symmetry type
276(11)
Further developments
287(6)
References 293(10)
Index 303
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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