Sheinman Current Algebras on Riemann Surfaces

New Results and Applications
1. Auflage 2012
ISBN: 978-3-11-026452-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 163 Seiten

Reihe: ISSN

Zielgruppe

Researchers, Lecturers, PhD and Graduate Students in Mathematics and Mathematical Physics; Academic Libraries

Autoren/Hrsg.

Sheinman, Oleg K.

Fachgebiete

• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1;1 Krichever-Novikov algebras: basic definitions and structure theory;15
1.1;1.1 Current, vector field, and other Krichever-Novikov algebras;15
1.2;1.2 Meromorphic .-forms and Krichever-Novikov duality;16
1.3;1.3 Krichever-Novikov bases;18
1.4;1.4 Almost-graded structure, triangle decompositions;20
1.5;1.5 Central extensions and 2-cohomology; Virasoro-type algebras;23
1.6;1.6 Affine Krichever-Novikov, in particular Kac-Moody, algebras;27
1.7;1.7 Central extensions of the Lie algebra D1g;29
1.8;1.8 Local cocycles for sl(n) and gl(n);30
2;2 Fermion representations and Sugawara construction;33
2.1;2.1 Admissible representations and holomorphic bundles;33
2.2;2.2 Holomorphic bundles in the Tyurin parametrization;35
2.3;2.3 Krichever-Novikov bases for holomorphic vector bundles;37
2.4;2.4 Fermion representations of affine algebras;40
2.5;2.5 Verma modules for affine algebras;43
2.6;2.6 Fermion representations of Virasoro-type algebras;45
2.7;2.7 Sugawara representation;48
2.8;2.8 Proof of the main theorems for the Sugawara construction;53
2.8.1;2.8.1 Main theorems in the form of relations with structure constants;54
2.8.2;2.8.2 End of the proof of the main theorems;57
3;3 Projective flat connections on the moduli space of punctured Riemann surfaces and the Knizhnik-Zamolodchikov equation;69
3.1;3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces;70
3.2;3.2 Sheaf of conformal blocks and other sheaves on the moduli space M(1,0)g,N+1;76
3.3;3.3 Differentiation of the Krichever-Novikov objects in modular variables;77
3.4;3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation;81
3.5;3.5 Explicit form of the Knizhnik-Zamolodchikov equations for genus 0 and genus 1;86
3.5.1;3.5.1 Explicit form of the equations for g = 0;86
3.5.2;3.5.2 Explicit form of the equations for g = 1;90
3.6;3.6 Appendix: the Krichever-Novikov base in the elliptic case;95
4;4 Lax operator algebras;98
4.1;4.1 Lax operators and their Lie bracket;99
4.1.1;4.1.1 Lax operator algebras for gl(n) and sl(n);99
4.1.2;4.1.2 Lax operator algebras for sv(n);100
4.1.3;4.1.3 Lax operator algebras for sp(2n);102
4.2;4.2 Almost-graded structure;104
4.3;4.3 Central extensions of Lax operator algebras: the construction;106
4.4;4.4 Uniqueness theorem;112
5;5 Lax equations on Riemann surfaces, and their hierarchies;115
5.1;5.1 M-operators;117
5.2;5.2 L-operators and Lax operator algebras from M-operators;120
5.3;5.3 g-valued Lax equations;121
5.4;5.4 Hierarchies of commuting flows;125
5.5;5.5 Symplectic structure;127
5.6;5.6 Hamiltonian theory;131
5.7;5.7 Examples: Calogero-Moser systems;138
6;6 Lax integrable systems and conformal field theory;143
6.1;6.1 Conformal field theory related to a Lax integrable system;143
6.2;6.2 From Lax operator algebra to commutative Krichever-Novikov algebra;145
6.3;6.3 The representation of AL;146
6.4;6.4 Sugawara representation;148
6.5;6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection;149
6.6;6.6 The representation of the algebra of Hamiltonian vector fields and commuting Hamiltonians;149
6.7;6.7 Unitarity;150
6.8;6.8 Relation to geometric quantization and quantum integrable systems;152
6.9;6.9 Remark on the Seiberg-Witten theory;152
7;Bibliography;155
8;Notation;161
9;Index;163

Über Autor(innen)

Sheinman, Oleg K.
Oleg K. Sheinman, Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia; Independent University of Moscow, Russia.