The Ginzburg-Landau functional was originally proposed as a model for superconductivity in Euclidean domains. However, invariance with respect to gauge transformations - which is one of the most prominent features of the model - suggests that the functional can be naturally defined in the setting of complex line bundles, where it can be regarded as an Abelian Yang-Mills-Higgs theory. In this talk, we shall consider the Ginzburg-Landau functional on an Hermitian line bundle over a closed Riemannian manifold, in the so-called "non-self dual scaling" (which is closer to the original motivation from superconductivity theory). We shall focus on the variational aspects of the problem; more precisely, we will discuss a Gamma-convergence result for sequences whose energy grows at most logarithmically in the Ginzburg-Landau coupling parameter. As we shall see, the London equation for superconductivity plays a significant role in our analysis. The talk is based on a joint work with Federico Dipasquale (Università Federico II, Napoli) and Giandomenico Orlandi (Verona).