p-forms contracted with spinors

Question

Does one encounter terms in the action such as
\(\int \bar\psi B^{\mu\nu\theta}\gamma^\mu \gamma^\nu \gamma^\theta \psi d^nx\)

Where B is some antisymmetric 3 form field.

This can be generalized to p form fields.

Answer 1

Before you contract with a spinor, you should think what kind of a field $B$ is. It is a tensor, so there must be a gauge invariance to get rid of the timelike negative-norm degrees of freedom. But if there is this invariance, one may write the physical degrees of freedom in terms of $F=dB$ which is a four-form, and that's equivalent to a scalar. Moreover, the kinetic term is then $F^2$ which means that in terms of $F$, there are no derivatives, and the equations of motion are $F=0$. Locally, there are no dynamical degrees of freedom. In higher dimensions, $d\gt 4$, the term you mention does exist. For example, it is exactly the coupling of the 3-form with the fermions in the 11D supergravity (low-energy limit of M-theory).

Comments

@RonMaimon done.

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But what if $B^{\mu\nu\theta}(x)$ is an external field (a known function)?

What "property" of an electron (fermion) could describe such a term?