Explicit Non-Abelian Gerbes with Connections

We derive the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is a 2-group with adjustment datum. We depart from the common fake-flat connections and employ adjusted connections, which is important for physical applications, especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the principal bundle corresponding to an instanton-anti-instanton pair to a string 2-group bundle. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.