Twisted conformal blocks and their dimension

A bstract . Let Γ be a finite group acting on a simple Lie algebra g and acting on a s -pointed projective curve ( Σ , (cid:126) p = { p 1 , . . . , p s } ) faithfully (for s ≥ 1). Also, let an integrable highest weight module H c ( λ i ) of an appropriate twisted a ffi ne Lie algebra determined by the ramification at p i with a fixed central charge c is attached to each p i . We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of Γ acting on g by diagram automorphisms and acting on a quotient of Σ . Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when Γ acts on g by diagram automorphisms and covers of P 1 with 3 marked points. Assuming a twisted analogue of Teleman’s vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general Γ -curves (with mild restrictions on ramification types). In particular, if the Lie algebra g is not of type D 4 , there are no restrictions on ramification types.