  We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically
gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold
target spaces. For a subset of toric manifolds, these equivariant branes have a
mirror description as branes in gauged Landau-Ginzburg models with neutral
matter. We then study correlation functions in the topological A-twisted
version of the GNLSM, and identify their values with open Hamiltonian
Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted
GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST
quantize the mirror theory to analyze this phenomenon.
