Why is the semiclassical approximation of the abelian Chern-Simons theory exact?
**Group structure** in Chern-Simons theory?
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?
(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!
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.
Good to know. Thanks! More comments/Ref will be helpful.
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.
But whether those Dijkgraaf-Witten theory can always be written as a continuous Chern-Simons action? Or may not be? Whether there is always a continuous gauge transformation for those discrete gauge theory?
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.
@user23660, can you specify what is your trivial gauge transformations? Do you have both finite and infinitesimal forms of gauge transformations? Thanks.
The are no infinitesimal gauge transformations for the finite group. In this case the gauge transformation is just a (global) deck transformation of underlying covering space.
The nLab is a great reference for all of these things and seems to answer all of your questions. They do a better job of explaining why than I would probably do.
Their page on Chern-Simons Theory seems to answer questions (2)-(4). They give a method for constructing Chern-Simons theories from generic compact Lie groups in the page listed. They also have a section describing Witten's construction for 2+1 gravity, which should constitute a set of examples.
Dijkgraaf-Witten Theories can be thought of as Chern-Simons theories because they are constructed in the same way Chern-Simons theories are constructed. This can be seen by comparing the two definitions and constructions in the nLab pages linked here. In this sense the DW theories are CS theories constructed from discrete groups.
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.
Since I could only include two references above, the reference for the toric code is arxiv.org/abs/quant-ph/9707021.