# What kind of fields can couple naturally to a $p$-form gauge fields in a Lagrangian?

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.

• I think you are mixing two distinct notions. One is the manifold on which you are integrating the Lagrangian density (it can be a world-line, a world-sheet, or a higher dimensional brane), and the other is the field which lives on it. In other words, all these objects are described by fields, the difference between particles, strings and branes is the number of spatial coordinates they depend on. To answer your question, what couples to $p$-form gauge fields are fields living on $(p-1)$-branes. May 9, 2016 at 12:35
• When I say "a $p$-form gauge field couples to a $p$-brane" I mean that if I have an embedding $\phi$ from a $p$-dimensional manifold $\Sigma$ into a $n$-dimensional manifold $X$, then I can write down: May 9, 2016 at 17:51
• Whoops, I pressed enter to early on the last comment! May 9, 2016 at 17:59
• @user40085 When I say "a $p$-form gauge field couples to a $p$-brane" I mean that if I have an embedding $\phi$ from a $p$-manifold $\Sigma$ into a $n$-manifold $X$ and a $p$-form gauge field $A$, then I can write down the total action $S(A,\phi) = \dots + \int_{\Sigma} \phi^*A$ and that the last part acts as a coupling term. So in that sense, sure, this is still a coupling of fields, since $\phi$ and $A$ are both fields (albeit on different manifolds). The convention of saying that something "couples to a $p$-brane" as opposed to a field on the $p$-brane is pretty common though. May 9, 2016 at 17:59
• @user40085 Anyway, I think that this answer that "what couples to $p$-form gauge fields are fields living on $(p-1)$-branes" is not the whole story. For instance, a regular $U(1)$ gauge field $A$ can couple to a Dirac-Fermion $\psi$ via the Dirac coupling term $\bar{\psi}D_A\psi$ (here $D_A$ is the covariant Dirac operator) in the Lagrangian density. But a spinor is not a field on a $0$-brane (which is a point), it's a field on the whatever $n$-manifold $X$ that $A$ and $\psi$ live on. So what gives? May 9, 2016 at 18:06