## Scattered Thoughts

Conformal field theories in even space-time dimensions(e.g. 1+1, 3+1,…) have trace anomaly and one can define a central charge for the theory which effectively counts how many degrees of freedom there are in the theory. This is related to the stability of  topologically-ordered phases in odd space-time dimensions, since their boundaries are described by CFT in even space-time dimensions. Therefore, we can understand the IQHE in 2+1 and 4+1 dimensions. Relation to chiral anomaly?

FQHE can be understood from parton(or slave particle) construction as the IQHE of partons. This approach has been developed by Wen to derive the effective field theories of various non-Abelian QHS. Recently discovered FQHE in flat-band lattice models can also be understood using this approach. An interesting question is what if the mean-field ansatz breaks the internal gauge symmetry(i.e. the mean-field ansatz has nontrivial internal gauge fluxes). The most extreme case would be that the gauge group is completely broken to its center, which is usually a discrete group(i.e. Z_n). Does the topological order remain the same? The Abelian case is studied by Lu & Ran. The non-Abelian case is still an open question.

A deep result due to Dijkgraaf and Witten states that the Chern-Simons actions for a compact gauge group G are in one–to–one correspondence with the elements of the cohomology group $H^4(BG, Z)$ of the classifying space BG with integer coeﬃcients Z. In particular, this classiﬁcation includes the case of ﬁnite groups H. The isomorphism $H^4(BH, Z)\simeq H^3(H, U(1))$ which is only valid for ﬁnite H then implies that the diﬀerent CS theories for a ﬁnite gauge group H correspond to the diﬀerent
elements $\omega \in H^3(H, U(1))$, i.e. algebraic 3-cocycles ω taking values in U(1).