Duality Web in 2+1D

Question

I've been studying the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics".
 https://arxiv.org/abs/1606.01989
On page 22, starting with the Lagrangian 

$$|D_{b}\phi|^{2}+|D_{\hat{b}}\hat{\phi}|^{2}-V(|\phi|,|\hat{\phi}|)+\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}b_{\alpha}\partial_{\beta}\hat{b}_{\gamma}+\frac{1}{2\pi}\epsilon^{\alpha\beta\gamma}b_{\alpha}\partial_{\beta}B_{\gamma},$$
in the phase $<\phi>=<\hat{\phi}>=0$, the two $U(1)$-gauge symmetries are not Higgsed. $\phi$ and $\hat{\phi}$ are massive and can be integrated out. The gauge fields coupled through $b\wedge d\hat{b}$ and $b\wedge dB$ makes the spectrum gapped and the low energy effective field theory is topological and is trivial. 

From the above statements, the exact expression of the potential $V$ is not given, and I cannot understand why $\phi$ and $\hat{\phi}$ are massive. How do I perform the path-integral over these two fields? Why do the last two BF terms make the spectrum gapped? Why is the low energy effective field theory topological?

I tried to perform the following path-integral of the complex scalars: 

$$\int(\mathcal{D}\phi^{\dagger}\mathcal{D}\phi)\exp \left\{i\int d^{3}x \phi^{\dagger}(-\partial_{\mu}\partial^{\mu}+ib_{\mu}\partial^{\mu}+i\partial_{\mu}b^{\mu}+b_{\mu}b^{\mu})\phi+V(|\phi|,|\hat{\phi}|)\right\}$$

Does the $b_{\mu}b^{\mu}|\phi|^{2}$ term give the complex scalar mass?  

Comments

The "coupling" you're trying to add is not gauge invariant and should not appear. Massive nature is hidden in potential term.

Answer 1

All of these 2+1D dualities are IR dualities, which hold after turning on all relevant operators which respect the symmetries and for lack of a better word, "phase constraints" like vanishing expectation value of some relevant charged operators like $\phi, \phi'$, which are insensitive to small enough perturbations. So you can imagine $V$ contains terms like $m^2 |\phi^2| + m'^2|\phi'|^2$ and the condition on the expectation value is saying $m^2 , m'^2 > 0$. $m^2 = 0$ for instance would be fine tuned, and one would need to locate in the generically dual theory the relevant operator corresponding to $|\phi^2|$ and tune it also. Sometimes this works and I don't think anyone knows a good argument.

Then, if we just want to see what the spectrum of the theory is like in the IR, we can replace these massive fields with their expectation values (which is just zero for both). What remains is a Chern-Simons-Maxwell theory which is known to be gapped.  See this paper about "topologically massive gauge theory".

Comments

Thank you. I edited my question. I tried to do the path-integral over the complex scalars. Could you please leave a comment?

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The mass will come from V. Try writing down all the relevant operators in $\phi$ and $\phi'$ (independent of derivatives).

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How about the $b_{\mu}b^{\mu}|\phi|^{2}$ term?

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That term is not gauge-invariant.

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Why is what remains a Maxwell theory after replacing the massive fields with their vacuum expectation values?

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Well if b is dynamical you need to add a kinetic term and Maxwell is the usual choice.