Multiplicative bundle gerbes with connection

Multiplicative bundle gerbes are gerbes over a Lie group which are compatible with the group structure. In this article connections on such bundle gerbes are introduced and studied. It is shown that multiplicative bundle gerbes with connection furnish geometrical constructions of the following objects: smooth central extensions of loop groups, Chern–Simons actions for arbitrary gauge groups, and symmetric bi-branes for WZW models with topological defect lines.