Trivial vs nontrivial TQFT

Question

This question is inspired by Examples of "gauging a global symmetry" and answer to that question.

I list main statements from answer:

1) We start from free scalar field $\phi$ in d+1 spacetime

$$L_0 = d\phi \wedge\star d\phi.$$

This has a manifest global symmetry $\phi \mapsto \phi + \theta$.

2) If we perform a local variation where $\theta$ has a small first derivative, then the Lagrangian is not invariant, instead, up to boundary terms

$$\delta L_0 = 2 \theta d\star d\phi + \mathcal{O}(\theta^2) = \theta\ dj + \mathcal{O}(\theta^2),$$

where we identify the Noether current $d$-form $j = \star d \phi$. The conservation law

$$dj = 2 d\star d\phi = 0$$

is equivalent to the equations of motion. To gauge this symmetry, we couple to a $U(1)$ gauge field $A$.

3) Minimal coupling is

$$L_0 - A \wedge j = L_0 - 2 A \star d\phi.$$

This action is not yet gauge invariant, but we're allowed to add local terms possibly depending on $\phi$ and at least secord order in $A$. We're missing a term like $A \wedge \star A$. If we put it all together we get

$$(d\phi - A) \wedge \star (d\phi - A).$$

You can check that this is a trivial theory (!).

4) However, if instead the symmetry was $\phi \mapsto \phi + 2\theta$, we would end up with $j = 4 \star d\phi$ and a gauged Lagrangian

$$(d\phi - 2A) \wedge \star (d\phi - 2A),$$

which you can check is a nontrivial TQFT. It's the $\mathbb{Z}_2$ gauge theory. You can see this theory has a $\mathbb{Z}_2$ 1-form symmetry which if you gauge takes you back to the trivial theory above.

Questions:

1) What is trivial theory?

2) I wanna to understand why first theory is trivial, but second theory is not? I am confusing, because it is seem as trivial rescaling of fields.

3) To understand this example, I think it is necessary understand following question:

What will if I will consider $\phi \mapsto \phi + n\theta$?

Answer 1

The field $\phi$ is actually $2\pi$ periodic: $\phi\sim\phi+2\pi$, also $\oint A\sim \oint A +2\pi$. In particle physics notation, the Lagrangian in the first case (3rd point in your question) is (ignoring the Maxwell term) simply $$ L=\frac{1}{2}(\partial_\mu \phi-A_\mu)(\partial^\mu\phi-A^\mu). $$ The equation of motion gives $\partial_\mu \phi = A_\mu$. This implies that the theory does not have any non trivial Guage invariant ovservables. So, the theory is trivial in this sense. Even if you add a Maxwell term, the theory describes a massive photon, which is not observable at the low energy.

The second case (the 4th point in your question) gives the equation of motion $A_\mu =\frac{1}{2}\partial_\mu \phi$. In this case again there is no local degree of freedom. But now the guage invariant observable $\oint A$ can take values $0$ and $\pi$. These are $Z_2$ topological degree of freedom. So this theory is non trivial.