Does topological mass imply preservation of global symmetry whose current is topological?

Question

This question is general but the motivation for it lies within the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics". On pages 20-23, they consider a system which has four phases depending on the VEV of \(\phi\) and \(\hat\phi\) fields. They then proceed to claim that these 4 phases are indeed different because of the gapping (or lack thereof) of the gauge fields \(b\) and \(\hat b\) associated with \(\phi\) and $\hat\phi$. In the phase where both fields are in their Coulomb regime, i.e. the VEV of both is zero, the system is gapped because $b$ and $\hat b$ acquire topological mass through the BF couplings present in the original Lagrangian.

I am, however, trying to understand those different phases through the underlying symmetries that they posses or that have been broken, since this view is more suited for the discussion that comes later on the paper.

We have here two global U(1) symmetries whose currents are the topological currents $db$ and $d\hat b$. Generally, these symmetries are broken when their associated field ($\phi$ or $\hat\phi$) is in its Coulomb phase and they are preserved when the field is in its Higgs phase. Thus, when both fields are in their Coulomb phase, in principle this should mean that both symmetries are broken. However, if we accept this, then when in the discussion they add a monopole operator that breaks the symmetry associated with $d\hat b$ for the system, the symmetries of the three phases which they intend to merge will not match, as shown here (green stands for symmetries that are preserved, red means broken symmetries).

They will match, however, if somehow initially the presence of the topological mass in that phase preserves both symmetries, as shown here.

Hence, I am trying to understand this. Did I perhaps get all of these symmetries wrong? If not, how exactly does topological mass in the upper left phase preserve both global U(1) symmetries? I am also trying to understand this under the light of "Generalized Global Symmetries" so feel welcome to refer to that paper if necessary. Thank you!

Comments

What do you mean by topological mass? Why does the BF coupling gives the gauge field mass?

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The conservation of topological currents $db$ and $d\hat{b}$ breaks by adding monopole operators. This has nothing to do with the Higgs mechanism. Higgs mechanism is the spontaneous breaking of local gauge symmetry, not the conservation of topological currents. The topological currents are conserved automatically for topological reasons, even without equation of motion, and they break for topological reasons (when monopoles present).

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Could you please be more specific about your definition of topological mass? Can you write down its expression?

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Hi New Student, thank you for your comments. What I mean by topological mass is when the original Lagrangian of the problem does not contain any mass terms for a specific field, but if you compute the equations of motion for it, it is possible to show that they reduce to a massive Klein-Gordon equation, thus it has a mass that arises as a result of some contact term. That is the topological mass, and it is very common in theories that contain Chern-Simons or BF terms. You can see a simple example of this here (section 2.2). In the case at hand, in order to see the topological mass more clearly, you have to add to the Larangian Maxwell terms for $b$ and $\hat b$, which are IR-irrelevant in 3d. Then computing the topological mass is a matter of calculating the equations of motion and subtituting.

Regarding your comment on the Higgs mechanism, I understand that it breaks local gauge symmetry, however, from what I understand it is also related in some cases with the topological global U(1) symmetry whose current is $db$ or $d\hat b$. This, if I understand correctly, is because this current can only be nonzero and topologically robust if we are in a phase whose vacuum is degenerate. Since the Higgs phase has a vacuum which is degenerate, it allows for the existence of this current, whereas the Coulomb vacuum doesn't. This is at least what I understood from this (Sections 3.2 to 3.4). This is also touched upon by Tong in his QHE lectures (section 5.3, in the Abelian Higgs model explanation), although he does not go into too much detail as to why the symmetries are broken or conserved. 

Anyway, I hope this is clearer now. Thanks again for your comment!

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1. The mass in your reference is derived from Yang-Mills Chern Simons theory, which is different from this case. In this theory, there is no Chern-Simons couplings, and the Maxwell term vanishes in deep IR limit. If you do the variation with respect to the gauge field, you don't have mass. To be more specific, let us follow your reference and see what you get.

$$\partial_{\mu}f^{\mu\nu}+k\epsilon^{\nu\alpha\beta}\hat{f}_{\alpha\beta}=0$$

where $f=db$ and $\hat{f}=d\hat{b}$, and $k$ is some unimportant constant. Clearly, this cannot be made a Proca equation. There is no mass of the gauge field propagating in this theory.

2.  The second reference you linked is about 2D soluble models, Sine-Gordon and Thirring model. They are not related with this theory. Let us now focus on the Lagrangian 
$$\mathcal{L}=|D_{b}\phi|^{2}+|D_{\hat{b}}\hat{\phi}|^{2}-V(|\phi|,|\hat{\phi}|)+\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}b_{\alpha}\partial_{\beta}\hat{b}_{\gamma}+\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}b_{\alpha}\partial_{\beta}B_{\gamma}$$
As I said from another the answer of the other question you posted, the conservation of the topological current has nothing to do with the Noether theorem. In Noether theorem, the current is conserved if and only if the field is on-shell, i.e. equation of motion holds. Here, in contrast, the conservation is trivially 
$$ddb=0$$
However, adding a monopole breaks this conservation because the vector potential for your electromagnetic field is no longer globally defined. This is explained in the classic papers about Dirac monopole by Wu and Yang.