# Trivial vs nontrivial TQFT

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 $$\phi \mapsto \phi + n\theta$$?

• Does $\phi$ live on a circle? Commented Feb 20, 2020 at 10:34
• what is "trivial"? meaning why is the theory in your (3) a "trivial theory"
– user21299
Commented Feb 20, 2020 at 11:26
• I think that we can consider circle and line... And I think that we obtain different result. Commented Feb 20, 2020 at 11:28
• It is good question about trivial... I don't know Commented Feb 20, 2020 at 11:37

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.