Area dependence in gauged Gromov-Witten theory

Abstract. We study the variation of the moduli space of symplectic vortices on a fixed holomorphic curve with respect to the area form. For compact, convex varieties we define symplectic vortex invariants and prove a wall-crossing formula for them. As an application, we prove a vortex version of the abelianization conjecture of Bertram, Ciocan-Fontanine, and Kim [4], which related Gromov-Witten invariants of geometric invariant theory quotients by a group and its maximal torus, for vortices on non-trivial bundles.