Exploring a simple sector of the Einstein-Maxwell landscape

We explore the four-dimensional Einstein-Maxwell landscape as a toy model in which we can formulate a sphere compactification stabilized by an electromagnetic field. Replacing the compactification sphere by J spheres, we obtain a simple sector of the (2J+2)-dimensional Einstein-Maxwell landscape. In this toy model, we analyze some properties which are very difficult to uncover in the string theory landscape, including complete moduli stabilization, stability conditions, and state counting. We also show how to construct anthropic states in this model. A detailed comparison between the main features of this landscape and the Bousso-Polchinski landscape is given. We finally speculate on the impact of these phenomena in the string theory landscape.