In terms of the TQFTs in continuous differential form gauge fields, what would the Walker-Wang lattice model describe? Obviously, there is a $BF$ theory part:

$$\frac{N}{2 \pi}\int B dA$$

if it contains the discrete $\mathbb{Z}_N$ gauge fields. But what else does the model contain, in any dimensions, or in 3+1d?

They mention the non Abelian BF+BB theory in their paper, but is that TQFT really a part of their model?

$$\frac{N}{2 \pi}\int Tr[ B \wedge dA + (\Lambda/12) B \wedge B ]$$ $$\frac{N}{2 \pi}\int Tr[ B \wedge dA + F \wedge F] $$

There is no explicit argument why such a continuous TQFT shows up from the discrete Walker-Wang lattice model. Are threre complete or partial lists of such differential form TQFTs for Walker-Wang model? What are they?

p.s. Please DO correct my TQFT quantization factor if it is wrong. I copied down from their paper. Thank you.


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