Brane Superpotential and Local Calabi–yau Manifolds

We briefly report on some recent progress in the computation of B-brane superpotentials for Type II strings compactified on Calabi–Yau manifolds, obtained by using a parametrization of tubular neighborhoods of complex submanifolds, also known as local spaces. In particular, we propose a closed expression for the superpotential of a brane on a genus-g curve in a Calabi–Yau threefold in the case in which there exists a holomorphic projection from the local space around the curve to the curve itself.