Types of quantum field thories
A general gauge field is a cocycle in differential cohomology. The case of the electromagnetic field and the Yang-Mills fields with 2-form curvature is just a very special case of this, in general a gauge field has higher curvature forms. For instance the B-field has a curvature 3-form, the C3-field has a curvature 4-form and the RR-field has a curvature form in every even or every odd degree.
For emphasis, therefore, one sometimes speaks of higher gauge fields and higher gauge theory , where the “higher” is as in higher category theory . Indeed, where an ordinary Yang-Mills field is a connection on a bundle defined in ordinary category theory, a higher gauge field is modeled by a connection on a 2-bundle or generally by a connection on a principal infinity-bundle.
See gauge theory for more.
Introductory lecture notes include
For foundations of higher prequantum field theory see
For foundations of higher gauge theory in homotopy type theory see