# Gauge symmetry for p-forms

It is well known that the Lorentz invariance of the S-matrix implies gauge redundancy for 1-forms or 'photons'. Does this argument go through to $p$-forms? That is, does Lorentz invariance of the S-matrix of these fields imply that $A_{a_1 \dots a_p} \to A_{a_1 \dots a_p} + \partial_{[a_1}\lambda_{a_2 \dots a_p]}$ is a symmetry of the action, where $\lambda$ is a $(p-1)$-form?

• In one word - Yes. (I had to expand on the answer because of the minimum letter limit in comments! :P) May 9, 2014 at 15:33
• See also this relevant question: physics.stackexchange.com/q/95911 May 9, 2014 at 23:13

We may easily generalize the Maxwell Lagrangian$^\star$ for any $p$-form connection. If we denote, $A^{(p)}$ a $p$-form gauge connection, then the field-strength is given simply by, $F=\mathrm{d}A^{(p)}$. To be gauge invariant, the Lagrangian must be invariant under,$^\dagger$

$$A^{(p)}\to A^{(p)}+\mathrm{d}\alpha^{(p-1)}$$

where $\alpha$ is an exact $(p-1)$-form. Obviously if the action only depends on $F$, then it is gauge invariant as the exterior derivative applied twice is nilpotent, i.e. $\mathrm{d}^2=0$. An example for the case $p=2$:

$$H_{\lambda\mu\nu}= \partial_\lambda B_{\mu\nu} + \partial_\nu B_{\lambda \mu} + \partial_\mu B_{\nu \lambda}$$

and the Lagrangian is given by $\mathcal{L}\sim H^2$, for potential $B$. In general, the field-strength is given by,

$$(\mathrm{d}A)_{a_1 a_2 ... a_n} = \frac{1}{p!} \partial_{[a_1}A_{a_2 ... a_n]}$$

$\star$ The definition of the field-strength as simply the exterior derivative of the gauge field only holds if the connection is a Lie algebra valued form for an abelian group $G$.

$\dagger$ In fact, the term added to the potential does not need to be exact. The only requirement is that the exterior derivative vanishes, i.e. it is closed. Being exact implies it is closed. For the case $p=1$, we often phrase the term as being a 'total derivative.'

• In what sense are $p$-form gauge fields connections? Do you have any references about the mathematical theory behind higher gauge fields?
– Danu
Oct 26, 2015 at 16:27

You can include p-forms in a theory in a way that doesn't imply gauge-invariance, just as you can write down a theory of a massive vector by including an $m^2 A^{\mu} A_{\mu}$ term in the Lagrangian. But in many theories with p-forms there is a gauge invariance like the one you wrote down. In that case the Lagrangian would be constructed from the field strength $F=dA$ (which is itself manifestly gauge invariance for Abelian fields), and possibly other terms depending on $p$, the dimension of spacetime, etc.

• @JamalS , My question was given a massless p-form field, does the Lorentz invariance of the S-matrix imply Gauge redundancy. This has been proven by Weinberg for vector fields and gravitons. May be I should find his paper. May 9, 2014 at 16:45
• There is a discussion of $p$-forms in Weinberg's first volume. I'm not too familiar with it myself so I'll just direct you to it. May 9, 2014 at 16:47