Current Algebras on Riemann Surfaces (eBook)

New Results and Applications

Oleg K. Sheinman (Autor)

eBook Download: PDF

2012
163 Seiten
De Gruyter (Verlag)
978-3-11-026452-4 (ISBN)

Infos

This book is an introduction into a new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory, conformal field theory, and the theory of integrable systems. For beginners, it provides a short way to join in the investigations in these fields. For experts, it sums up the recent advances in the theory of almost graded infinite-dimensional Lie algebras and their applications. The majority of results is presented for the first time in the form of a monograph.

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

Autor

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

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 1.10.2012
• Reihe/Serie: De Gruyter Expositions in Mathematics
• Reihe/Serie: De Gruyter Expositions in Mathematics
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Zusatzinfo: 6 b/w ill.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: conformal field theory • Current Algebra • Integrable Systems • Krichever-Novikov algebras • Lax Integrable System • Lax operator algebras • Lie algebra • Riemann surface
• ISBN-10: 3-11-026452-8 / 3110264528
• ISBN-13: 978-3-11-026452-4 / 9783110264524