A Canonical Decomposition of Generalized Theta Functions on the Moduli Stack of Gieseker Vector Bundles

In this paper we prove a decomposition formula for generalized theta functions which is motivated by what in conformal field theory is called the factorization rule. A rational conformal field theory associates a finite dimensional vector space (called the space of conformal blocks) to a pointed nodal projective algebraic curve over C whose marked points are labelled by elements of a certain finite set. If we choose a singular point p in such a labelled pointed curve X, then we get a new pointed curve X̃ by taking the partial normalization at p and marking all points which lie either over one of the marked points of X (old points) or over the singularity p (two new points). The factorization rule gives a canonical direct sum decomposition of the space of conformal blocks associated to X with its labelled marked points, such that the summands appearing in that decomposition are spaces of conformal blocks associated to the pointed curve X̃ whose old marked points are labelled by the same elements as the corresponding points of X. The direct sum runs over a certain finite set of labellings of the two new points. In the case of a Wess-Zumino-Witten conformal field theory associated to a simply connected semisimple algebraic group G and a natural number κ ≥ 1 Tsuchiya, Ueno and Yamada have given a mathematical definition of the spaces of conformal blocks in terms of the representation theory of the affine Lie algebra associated to G and they have shown that these spaces satisfy the factorization rule ([TUY], [U], cf. also [So] for an overview). It has been conjectured by physicists and later proved by various mathematicians that the spaces of conformal blocks of Tsuchiya, Ueno, Yamada have an algebro-geometric interpretation: In the case of smooth labelled pointed curves they can be identified with spaces of global