# Degeneracy and the unitarity of a gauge theory with a non-compact gauge group

The topological ground state degeneracy(g.s.d.) provides useful information for a topological field theory(TQFT), such as this post shows some example.

To count g.s.d., it seems to be equivalent to count the volume of the symplectic phase space. As I have heard, it is known that a gauge theory with a non-compact gauge group the g.s.d. is infinity: $$\text{g.s.d.}=\infty$$ (specifically, my interest can be 2+1D Chern-Simons theory; or other cases).

Question 1: Are there some explicit ways to demonstrate this $\text{g.s.d.}=\infty$?

Question 2: Does non-compact gauge group necessarily(only if)/sufficiently(if) leads to a non-unitary theory?

Many thanks.

• For the question 2, non-compact groups have infinite unitary representations, so I don't think that this is contradictory with unitarity. – Trimok Jan 9 '14 at 21:54
• I strongly recommend reading Witten's Quantization of Chern-Simons gauge theory with complex gauge group. (This is a special case of a general princple: If you are working in a subject Witten has touched, you should read his papers on the subject.) – user1504 Jan 9 '14 at 22:23