# Higher form symmetries and massive gauge fields

I have seen all kinds of questions and answers about how to identify a higher-form symmetries, but they all seem rather abstract. What I would like to do is investigate two very simple examples.

1. Consider ordinary E&M theory with Lagrangian $$\mathcal L = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}.$$ The equations of motion are $$\partial_\nu F^{\mu\nu}=0$$ and the Bianchi identity tells us that $$\partial_\nu\star F^{\mu\nu}=0$$. Thus, we have the two conserved two-form currents $$F^{\mu\nu}=0$$ and $$\star F^{\mu\nu}=0$$.

2. Consider massive E&M theory with Lagrangian $$\mathcal L = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}-\frac{1}{2} m^2 A_\mu A^\mu.$$ Now the equations of motion are $$\partial_\nu F^{\mu\nu} = m^2 A^\mu$$ and the Bianchi identity is as before $$\partial_\nu\star F^{\mu\nu}=0$$.

QUESTION: Evidently, in the massive case $$F^{\mu\nu}$$ is no longer a conserved two-form current. But it appears that $$\star F^{\mu\nu}$$ is a conserved current. Does this mean that massive E&M has a one-form symmetry?

I have never heard anyone claim that massive gauge fields have higher-form symmetries, leading me to suspect they do not. But I cannot figure out why this should be so.

• My naive answer is that, indeed, Proca has a $U(1)$ one-form symmetry. But it is the first time I think about this, so perhaps I am wrong. It's funny that this rather simple example is never mentioned when presenting one-form symmetries. The similar situation in 3d is much more studied; here you can also add a mass term (which is topological, i.e., Chern-Simons), and the model still has the one-form center symmetry. So I see no reason this should not be true in 4d. Commented Mar 2, 2021 at 23:59

There is a one-form symmetry in the Higgs phase of electromagnetism. And as the Higgs action can be rewritten into a Proca action, I would think the answer to your question is affirmative.

In Maxwell electromagnetism, there are indeed two 1-form symmetries, which we can call electric and magnetic. Furthermore, these are both spontaneously broken, in the sense that their Wilson loops obey a perimeter law. The photon is the associated Nambu-Goldstone boson.

Coupling the Maxwell field to any source, the electric 1-form symmetry is explicitly broken. The magnetic symmetry persists.

Moreover, in the Higgs phase of the Abelian-Higgs model where the matter field condenses, the Wilson loop now obeys an area law and the magnetic 1-form symmetry is unbroken.

In the Higgs phase, one can define a new gauge field $$B_\mu = A_\mu - \frac{1}{e} \partial_\mu \phi$$ with $$\phi$$ the phase of the Higgs field and $$e$$ the coupling constant; the action for $$B_\mu$$ is the Proca action, while its field strength is identical to the field strength of $$A_\mu$$. Therefore, it has the same unbroken magnetic 1-form symmetry.

References

1. https://physics.stackexchange.com/a/522942/288815
2. 1412.5148
3. 1802.07747
4. 1802.09512
5. 1102.0468 (for the redefinition of the gauge field)