I've been studying the paper A Duality Web in 2+1 Dimensions and Condensed Matter Physics. 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?


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