From quantum cohomology to integrable systems:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2008
|
| Edition: | 1. publ. |
| Series: | Oxford graduate texts in mathematics
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 305 S. |
| ISBN: | 9780198565994 0198565992 |
Staff View
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| 100 | 1 | |a Guest, Martin A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a From quantum cohomology to integrable systems |c Martin A. Guest |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2008 | |
| 300 | |a XXIX, 305 S. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Oxford graduate texts in mathematics |v 15 | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Homology theory | |
| 650 | 4 | |a Mappings (Mathematics) | |
| 650 | 4 | |a Quantum theory | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
Acknowledgements
Introduction
χι
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
Simplicia!
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
1
о
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
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
Démodules
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
7.2
The mKdV equation
160
Contents
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
/i-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
|
| adam_txt |
Contents
Preface
Acknowledgements
Introduction
χι
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
Simplicia!
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
1
о
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
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
Démodules
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
7.2
The mKdV equation
160
Contents
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
/i-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 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:54:34Z |
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| spelling | Guest, Martin A. Verfasser aut From quantum cohomology to integrable systems Martin A. Guest 1. publ. Oxford [u.a.] Oxford Univ. Press 2008 XXIX, 305 S. txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts in mathematics 15 Quantentheorie Differential equations Homology theory Mappings (Mathematics) Quantum theory Integrables System (DE-588)4114032-1 gnd rswk-swf Quantenkohomologie (DE-588)4684891-5 gnd rswk-swf Quantenkohomologie (DE-588)4684891-5 s Integrables System (DE-588)4114032-1 s DE-604 Oxford graduate texts in mathematics 15 (DE-604)BV011416591 15 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guest, Martin A. From quantum cohomology to integrable systems Oxford graduate texts in mathematics Quantentheorie Differential equations Homology theory Mappings (Mathematics) Quantum theory Integrables System (DE-588)4114032-1 gnd Quantenkohomologie (DE-588)4684891-5 gnd |
| subject_GND | (DE-588)4114032-1 (DE-588)4684891-5 |
| title | From quantum cohomology to integrable systems |
| title_auth | From quantum cohomology to integrable systems |
| title_exact_search | From quantum cohomology to integrable systems |
| title_exact_search_txtP | From quantum cohomology to integrable systems |
| title_full | From quantum cohomology to integrable systems Martin A. Guest |
| title_fullStr | From quantum cohomology to integrable systems Martin A. Guest |
| title_full_unstemmed | From quantum cohomology to integrable systems Martin A. Guest |
| title_short | From quantum cohomology to integrable systems |
| title_sort | from quantum cohomology to integrable systems |
| topic | Quantentheorie Differential equations Homology theory Mappings (Mathematics) Quantum theory Integrables System (DE-588)4114032-1 gnd Quantenkohomologie (DE-588)4684891-5 gnd |
| topic_facet | Quantentheorie Differential equations Homology theory Mappings (Mathematics) Quantum theory Integrables System Quantenkohomologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136181&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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