A Lie based 4–dimensional higher Chern–Simons theory

We present and study a model of 4–dimensional higher Chern-Simons theory, special Chern–Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2–algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2–connection coupled to a background closed 3–form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2–group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3–form. Finally, SCS theory is related to a 3–dimensional special gauge theory whose 2–connection space has a natural symplectic structure with respect to which the 1–gauge transformation action is Hamiltonian, the 2–curvature map acting as moment map.