1 Krichever-Novikov algebras：basic definitions and structure theory

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

1.2 Meromorphic λ-forms and Krichever-Novikov duality

1.3 Krichever-Novikov bases

1.4 Almost-graded structure,triangledecompositions

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

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

1.7 Central extensions ofthe Lie algebra D？

1.8 Local cocycles for？(n)and g？(n)

2 Fermion representations and Sugawara construction

2.1 Admissible representations and holomorphic bundles

2.2 Holomorphic bundles in the Tyurin parametrization

2.3 Krichever-Novikov bases for holomorphic vector bundles

2.4 Fermion representations of affine algebras

2.5 Verma modules for affine algebras

2.6 Fermion representations of Virasoro-type algebras

2.7 Sugawara representation

2.8 Proof of the main theorems for the Sugawara construction

2.8.1 Main theorems in the form of relations with structure constants

2.8.2 End ofthe proof ofthe main theorems

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

3.1 Virasoro-type algebras and moduli spaces of Riemann surfaces

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

3.3 Differentiation of the Krichever-Novikov objects in modular variables

3.4 Projective flat connection and generalized Knizhnik-Zamolodchikov equation

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

3.5.1 Explicit form of the equations for g=0

3.5.2 Explicit form of the equations for g=1

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

4 Lax operator algebras

4.1 Lax operators and their Lie bracket

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

4.1.2 Lax operator algebras for ？(n)

4.1.3 Lax operator algebras for ？(2n)

4.2 Almost-graded structure

4.3 Central extensions of Lax operator algebras:the construction

4.4 Uniqueness theorem

5 Lax equations on Riemann surfaces,and their hierarchies

5.1 M-operators

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

5.3 g-valued Lax equations

5.4 Hierarchies of commuting flows

5.5 Symplectic structure

5.6 Hamiltonian theory

5.7 Examples:Calogero-Moser systems

6 Lax integrable systems and conformal field theory

6.1 Conformal field theory related to a Lax integrable system

6.2 From Lax operator algebra to commutative Krichever-Novikov algebra

6.3 The representation of AL

6.4 Sugawara representation

6.5 Conformal blocks and the Knizhnik-Zamolodchikov connection

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

6.7 Unitarity

6.8 Relation to geometric quantization and quantum integrable systems

6.9 Remark on the Seiberg-Witten theory

Bibliography

Notation

Index