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!

This post imported from StackExchange Physics at 2014-03-06 16:49 (UCT), posted by SE-user cycles