On the correlation functions of the vector bundlegeneralization of the bc - systemMatthias

It is shown that the determinants of the correlation functions of the generalized bc-system introduced recently are given as pullbacks of the non-abelian theta divisor. The usual bc-system appearing in bosonic string theory 1;2 is very well understood 1?4 and has also been considered in a rigorous algebro-geometric way by Raina 5;6. Assuming some natural physical axioms, Raina showed the existence and uniqueness of the correlation functions and was able to rederive the explicit expressions using the geometry of the theta divisor. It is important to note that one considers in this approach not the quantum elds b; c themselves (which should be \operator valued sections" of certain line bundles), but their correlation functions inheriting the symmetries of the operators. A closely related cousin of the bc-sytem based on a Hermitian vector bundle of rank r was introduced in Ref. 7 and the existence and uniqueness of correlation functions was established for a particular class of bundles. The hope was that the correlation functions of this bc r-system are determined completely by the geometry of the non-abelian theta divisor, in complete analogy to the usual rank one case. Since at the time of writing the necessary formulae were lacking, this remained a hope, but in the meantime Ref. 8 appeared, providing some useful results. Unfortunately, only the determinants of the correlation functions can be described with the help of the results of Ref. 8, so there is still much to be done to realize this hope and it is unclear whether one will be able to do so along the lines pursued here. In the second section we brieey recall the geometry of the system of rank one before we consider the higher rank case in the third section. Some of the diiculties concerning the current and the energy-momentum tensor are indicated. For the convenience of the reader we have stated the required result of Ref. 8 in an appendix. In the following g will be a Riemann surface of genus g 2 with canonical bundle K K g. The group of (isomorphism classes of) line bundles of degree d will be denoted by P ic d ((g) and there is the canonical theta divisor := fL 2 P ic g?1 ((g) j h 0 ((g ; L) 6 = 0g P ic g?1 ((g). We will denote by ?1 the inverse of the line bundle and by E _ …