Conformal blocks attached to twisted groups

The aim of this paper is to generalize the notion of conformal blocks to the situation in which the Lie algebra they are attached to is replaced with a sheaf of Lie algebras depending on covering data of curves. The result is a vector bundle of finite rank on the stack Hur(,.)g,n parametrizing -coverings of curves. Many features of the classical sheaves of conformal blocks are proved to hold in this more general setting, in particular the factorization rules, the propagation of vacua and the WZW connection.