Ordinary $U(1)$ gauge fields can naturally couple to classical fields such as spin-$1/2$ fields via the Dirac Lagrangian, or to complex spin-$0$ fields via the obvious covariant derivative coupling, or indeed to fields valued in any bundle $E$ carrying a representation of $U(1)$ on its fiber. They also couple naturally to the world-lines of particles, just by integration of the 1-form along the particle world-line. More generally, $U(1)$ $p$-form gauge fields naturally couple to $(p-1)$-branes, again simply via integration along the world-sheet of the corresponding brane.
My question is: are there known natural coupling terms between $p$-form gauge fields and other fields (not branes) that behave like a "covariant derivative" coupling between gauge fields and spinors/scalar bosons, like those that I mentioned above? More broadly, what are some good examples of coupling terms between $p$-form gauge fields and other fields (again, not branes)?
I am particularly interested in couplings that resemble the Dirac Lagrangian coupling between gauge field and the Diract spinor, and coupling terms that respect gauge symmetry with respect to the group of gauge transformations of abelian $p$-form gauges fields.