Twisted Modules over Vertex Algebras on Algebraic Curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.