Algebraic curves, twisted vertex operators, and Prym varieties

This dissertation consists of two parts. In the first, we extend the geometric vertex algebra formalism developed by Edward Frenkel to treat twisted vertex operators on algebraic curves. We develop the notion of conformal blocks for twisted vertex algebra modules, and show that in the Heisenberg case, these have a simple realization in terms of twisted coinvariants with respect to a twisted Heisenberg algebra. Furthermore, in the Heisenberg case, these conformal blocks naturally give rise to sheaves of twisted differential operators on Prym varieties.; In the second part, we apply the representation theory of vertex algebras and their twisted modules to construct Wakimoto modules for twisted affine algebras. Using the Wakimoto construction, we prove the Kac-Kazhdan conjecture on the characters of irreducible modules with generic critical highest weights in the twisted case. We provide explicit formulas for the twisted fields in the case of A22 .