| 1 Krichever-Novikov algebras: basic definitions and structure theory | 15 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| Notation | 161 |
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| Index | 163 |
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