The simplest example of a D-brane is a smooth holomorphic submanifold of a smooth complex manifold. In the B-model, to compute the space of open string states between two such branes one needs to understand their intersection, understood in the sense of derived algebraic geometry. I will discuss recent results with Damien Calaque and Junwu Tu which express this derived intersection in terms of natural Lie algebroids associated to the subvarieties. Our work generalizes classic work of Kapranov and Kontsevich on a metric-free definition of Rozansky-Witten invariants.