F-theory on K3 × K3 and instantons on 7-branes

We discuss N = 2 supersymmetric compactifications to four dimensions from the point of view of F-theory and heterotic theory. In a relatively simple setup, we illustrate the spectral theory for vector bundles on K3 × T2 and discuss the heterotic-F-theory map. The moduli space of instantons on the 7-brane wrapped around K3 is discussed from the point of view of the Higgs mechanism in the effective four-dimensional N = 2 theory. This allows us to elaborate on the transitions between various branches of the moduli space of the heterotic-F-theory. We also describe the F-theory compactification on smooth K3 × K3 without the 3-branes in which the anomaly is canceled by the gauge fields.