Quantization, Classical and Quantum Field Theory and Theta-Functions.

Arnaud Beauville’s survey ”Vector bundles on Curves and Generalized Theta functions: Recent Results and Open Problems” [Be] appeared 10 years ago. This elegant survey is short (16 pages) but provides a complete introduction to a specific part of algebraic geometry. To repeat his succes now we need more pages, even though we assume that the reader is already acquainted with the material presented there. Moreover, in Beauville’s survey the relation between generalized theta functions and conformal field theories (classical and quantum) was presented already. Following Beauville’s strategy we do not provide any proof or motivation. But we would like to propose all constructions of this large domain of mathematics in such a way that the proofs can be guessed from the geometric picture. Thus this text is not a mathematical monograph yet, but rather a digest of a field of mathematical investigations. In the abelian case ( the subject of several beautiful classical books ([B], [C], [Wi], [F1] and many others ) fixing some combinatorial structure (a so called theta structure of level k) one obtains special basis in the space of sections of powers of the canonical polarization powers on jacobians. These sections can be presented as holomorphic functions on the ”abelian Schottky” space (C). This fact provides various applications of these concrete