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1 Krichever-Novikov algebras：basic definitions and structure theory 1

1.1 Current,vector field,and other Krichever-Novikov algebras 1

1.2 Meromorphic λ-forms and Krichever-Novikov duality 2

1.3 Krichever-Novikov bases 4

1.4 Almost-graded structure,triangledecompositions 6

1.5 Central extensions and 2-cohomology;Virasoro-type algebras 9

1.6 Affine Krichever-Novikov,in particular Kac-Moody,algebras 13

1.7 Central extensions ofthe Lie algebra D？ 15

1.8 Local cocycles for？(n)and g？(n) 16

2 Fermion representations and Sugawara construction 19

2.1 Admissible representations and holomorphic bundles 19

2.2 Holomorphic bundles in the Tyurin parametrization 21

2.3 Krichever-Novikov bases for holomorphic vector bundles 23

2.4 Fermion representations of affine algebras 26

2.5 Verma modules for affine algebras 29

2.6 Fermion representations of Virasoro-type algebras 31

2.7 Sugawara representation 34

2.8 Proof of the main theorems for the Sugawara construction 39

2.8.1 Main theorems in the form of relations with structure constants 40

2.8.2 End ofthe proof ofthe main theorems 43

3 Projective fiat connections on the moduli space of punctured Riemann surfaces and the Knizhnik-Zamolodchikov equation 55

3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces 56

3.2 Sheaf of conformal blocks and other sheaves on the moduli space M？ 62

3.3 Differentiation of the Krichever-Novikov objects in modular variables 63

3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation 67

3.5 Explicit form of the Knizhnik-Zamolodchikov equations for genus 0 and genus 1 72

3.5.1 Explicit form of the equations for g=0 72

3.5.2 Explicit form of the equations for g=1 76

3.6 Appendix:the Krichever-Novikov base in the elliptic case 81

4 Lax operator algebras 84

4.1 Lax operators and their Lie bracket 85

4.1.1 Lax operator algebras for g？(n)and？(n) 85

4.1.2 Lax operator algebras for ？(n) 86

4.1.3 Lax operator algebras for ？(2n) 88

4.2 Almost-graded structure 90

4.3 Central extensions of Lax operator algebras:the construction 92

4.4 Uniqueness theorem 98

5 Lax equations on Riemann surfaces,and their hierarchies 101

5.1 M-operators 103

5.2 L-operators and Lax operator algebras from M-operators 106

5.3 g-valued Lax equations 107

5.4 Hierarchies of commuting flows 111

5.5 Symplectic structure 113

5.6 Hamiltonian theory 117

5.7 Examples:Calogero-Moser systems 124

6 Lax integrable systems and conformal field theory 129

6.1 Conformal field theory related to a Lax integrable system 129

6.2 From Lax operator algebra to commutative Krichever-Novikov algebra 131

6.3 The representation of AL 132

6.4 Sugawara representation 134

6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection 135

6.6 The representation of the algebra of Hamiltonian vector fields and commuting Hamiltonians 135

6.7 Unitarity 136

6.8 Relation to geometric quantization and quantum integrable systems 138

6.9 Remark on the Seiberg-Witten theory 138

Bibliography 141

Notation 147

Index 149