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 not defined over a field, but depends on covering data of curves. The result will be a sheaf of conformal blocks on the Hurwitz stack parametrizing Galois coverings of curves. Many features of the classical sheaves of conformal blocks are proved to hold in this more general setting, in particular the fusion rules, the propagation of vacua and the WZW connection.