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I am looking for a few examples of field theories (classical or quantum) that can be formulated taking the fields to be differential forms at least of degree 1 (not counting 0-forms) excluiding electromagnetism.

In particular differential forms $ \omega \in \Omega^*(M)$ for a Riemannian manifold $(M,g)$ with a fixed metric $g$ or $(M\times \mathbb{R}, g)$ (the latter being time, space and time do not mix through $g$). (Not Lorentzian)

Theories of the Yang-Mills type are not in this list as they formed with fields valued on $\Omega^* (M) \otimes \mathfrak{g}$ with some Lie algebra $\mathfrak{g}$.

Such list would include $M$ and if possible the action of the field theory in terms of the differential forms, or the name by which it is known.

Spinor fields are exlcuded, so the theory (even if its dynamics be somewhat trivial) should be formulated in terms of differential forms only, with a non-dynamic metric.

Some examples have been given in the answers, are there more?

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closed as too broad by ACuriousMind, user81619, Sebastian Riese, John Rennie, DanielSank Oct 22 '15 at 7:00

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Every field theory can be cast in the language of forms and sections, physicists just tend to just the local coordinate expressions instead of the abstract coordinate-free notation. This question is too broad. $\endgroup$ – ACuriousMind Oct 21 '15 at 14:25
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    $\begingroup$ The question (v2) seems like a list question. $\endgroup$ – Qmechanic Oct 21 '15 at 14:37
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    $\begingroup$ @ACuriousMind: I know that already, but I am not asking about sections but specifically differential forms. You cannot write a spinor theory using differential forms I believe, as the spinors contain more information. Neither would I consider scalar fields as forms (they are 0-forms, just not interesting to me) What I am looking for are specific examples of field theories based on ``abelian'' differential forms, other than electromagnetism. I do not consider that as too broad a question. $\endgroup$ – Rogelio Molina Oct 21 '15 at 18:43
  • $\begingroup$ @Qmechanic: indeed it is a list question, but I tried to find a list tag and found none. I have edited the question to make it clear what I am looking for. I do not see why it should remain on hold now, if it is still deemed too broad, then it should be very easy to produce even a short list, which would suffice. Good answers need not be too long for this format, I am looking for a list, that should not take more than a few lines. $\endgroup$ – Rogelio Molina Oct 25 '15 at 5:44
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The "Kalb-Ramond B-field" is a 2-form field, locally given by $B \in \Omega^2(M)$.

The supergravity C-field is a 3-form field, locally given by $C \in \Omega^3(M)$.

The "Ramond-Ramond field" is locally and rationally a formal sum of differential forms in all even degrees $\Omega^{2\bullet}(M)$ or in all odd degrees $\Omega^{2\bullet+1}(M)$.

The next article (due in a few days) in the "higher prequantum theory" series on PhysicsForums will have more on this.

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To complement Urs Schreiber's answer, there are also some intersting field theories of forms for which it is not possible to write down a Lagragian with a local kinetic term. For instance, type II B supergravity in 10 dimension has a self-dual five form $F_5=*F_5$, for which a kinetic term like $F_5\wedge *F_5 = F_5\wedge F_5=0$ simply vanishes.

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