Branes on $G$-manifolds

Abstract: Let $X$ be Calabi-Yau manifold acted by a group $G$. We give a definition of $G$-equivariance for branes on $X$, and assign to each equivariant brane an element of the equivariant cohomology of $X$ that can be considered as a charge of the brane. We prove that the spaces of strings stretching between equivariant branes support representations of $G$. This fact allows us to give formulas for the dimension of some of such spaces, when $X$ is a flag manifold of $G$.