黎曼曲面上的流代数【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Krichever-Novikov algebras：basic definitions and structure theory1

1.1 Current,vector field,and other Krichever-Novikov algebras1

1.2 Meromorphic λ-forms and Krichever-Novikov duality2

1.3 Krichever-Novikov bases4

1.4 Almost-graded structure,triangledecompositions6

1.5 Central extensions and 2-cohomology;Virasoro-type algebras9

1.6 Affine Krichever-Novikov,in particular Kac-Moody,algebras13

1.7 Central extensions ofthe Lie algebra D？15

1.8 Local cocycles for？(n)and g？(n)16

2 Fermion representations and Sugawara construction19

2.1 Admissible representations and holomorphic bundles19

2.2 Holomorphic bundles in the Tyurin parametrization21

2.3 Krichever-Novikov bases for holomorphic vector bundles23

2.4 Fermion representations of affine algebras26

2.5 Verma modules for affine algebras29

2.6 Fermion representations of Virasoro-type algebras31

2.7 Sugawara representation34

2.8 Proof of the main theorems for the Sugawara construction39

2.8.1 Main theorems in the form of relations with structure constants40

2.8.2 End ofthe proof ofthe main theorems43

3 Projective fiat connections on the moduli space of punctured Riemann surfaces and the Knizhnik-Zamolodchikov equation55

3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces56

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 variables63

3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation67

3.5 Explicit form of the Knizhnik-Zamolodchikov equations for genus 0 and genus 172

3.5.1 Explicit form of the equations for g=072

3.5.2 Explicit form of the equations for g=176

3.6 Appendix:the Krichever-Novikov base in the elliptic case81

4 Lax operator algebras84

4.1 Lax operators and their Lie bracket85

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 structure90

4.3 Central extensions of Lax operator algebras:the construction92

4.4 Uniqueness theorem98

5 Lax equations on Riemann surfaces,and their hierarchies101

5.1 M-operators103

5.2 L-operators and Lax operator algebras from M-operators106

5.3 g-valued Lax equations107

5.4 Hierarchies of commuting flows111

5.5 Symplectic structure113

5.6 Hamiltonian theory117

5.7 Examples:Calogero-Moser systems124

6 Lax integrable systems and conformal field theory129

6.1 Conformal field theory related to a Lax integrable system129

6.2 From Lax operator algebra to commutative Krichever-Novikov algebra131

6.3 The representation of AL132

6.4 Sugawara representation134

6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection135

6.6 The representation of the algebra of Hamiltonian vector fields and commuting Hamiltonians135

6.7 Unitarity136

6.8 Relation to geometric quantization and quantum integrable systems138

6.9 Remark on the Seiberg-Witten theory138

Bibliography141

Notation147

Index149