**Group structure** in Chern-Simons theory?

A non-Abelian Chern-Simons(C-S) has the action $$S=\int L dt=\int \frac{k}{4\pi}Tr[\big( A \wedge d A + (2/3) A \wedge A \wedge A \big)]$$

We know that the common cases, $A=A^a T^a$ is the connection as a Lie algebra valued one form. $T^a$ is the generator of the Lie group.

The well-known case is the well-defined SU(2) C-S theory and SO(3) C-S theory.

SU(2) is a compact, simple, simply-connected Lie group.

SO(3) is a compact, simple, connected but not simply connected Lie group.

Question: what is the minimum requirement on the group structure of $A$ in Chern-Simons theory?

Can we have the group of $A$ of C-S theory:

(1) to be NOT a Lie group?

(2) to be NOT compact?

(3) to be NOT connected?

(4) to be a Lie group but NOT a simple-Lie group?

Please could you also explain why is it so, and better with some examples of (1),(2),(3),(4).

ps. Of course, I know C-S theory is required to be invariant under a gauge transformation $$A \to U^\dagger(A-id)U$$ with a boundary derives a Wess-Zumino-Witten term. Here I am questioning the constraint on the group. Many thanks!

• I need at least 10-ish reputation to add some more web links on those terms on semi-simple, etc. If someone can help, please...
– user36094
Dec 21, 2013 at 4:50
• There is a formulation of $2+1$ gravity as a C-S theory. The groups in this case are $ISO(2,1)$, $SO(3,1)$ and $SO(2,2)$ (depending on the cosmological constant choice). So, I guess that compactness and (semi)simplicity is optional. The reference is: Witten, E. (1988). 2+1 dimensional gravity as an exactly soluble system. Nuclear Physics B, 311(1), 46-78. doi. Dec 21, 2013 at 5:42
– user36094
Dec 21, 2013 at 5:44
• Another reference is CS theories with a finite gauge groups: Dijkgraaf, R., & Witten, E. (1990). Topological gauge theories and group cohomology. Communications in Mathematical Physics, 129(2), 393-429. doi preprint. So we do not really need the group to be a Lie group. Dec 21, 2013 at 9:03
• The formulation of D-W theory through C-S action is here: Freed, D. S., & Quinn, F. (1993). Chern-Simons theory with finite gauge group. Communications in Mathematical Physics, 156(3), 435-472. arXiv:hep-th/9111004. There are indeed only trivial gauge transformations for discrete groups. Dec 22, 2013 at 6:32

As to examples, I believe the simplest is the D($\mathbb Z_2$) Dijkgraaf-Witten TQFT, which has a Hamiltonian realization in Kitaev's Toric code model.