In Weyl semimetal, there is an analog of ABJ anomaly, which is a $E \cdot B$ term. The ABJ anomaly can be viewed as winding number because of the homotopy group of sphere $\pi_3(S^3)= \mathbb{Z}$ for non-abelian $SU(2)$ gauge group. Usually for abelian $U(1)$ group, we don't have this anomaly because $\pi_i(S^1) = 0$ for $i>1$. I guess the reason we can write the anomaly term $E \cdot B$ is that the 3D Brillouin zone is not a sphere, then my question is what is the topology of 3D Brillouin zone? Is this some kind of high dimensional torus? What is the homotopy group for mapping this 3D Brillouin zone to $U(1)$ gauge group?
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