I describe the notion of a categorified representation of a compact Lie group G, which is the mathematical counter-part to a topological boundary condition for (pure) 3-dimensional gauge theory. The main examples come from the Gromov-Witten theories of compact symplectic manifolds with Hamiltonian group action. The character theory of these representations is captured, in the spirit of quantum mechanics, by the holomorphic symplectic geometry of a certain manifold, now recognised as the `Coulomb brancha** of the pure 3D gauge theory. Equivariant quantum cohomology is shown to be Legendre dual to the Seidel action, and the combined information allows for a character calculus in the world of categories.