Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).

However, it is not entirely clear for Yang-Mills theory with non-compact or non-semi-simple gauge groups.

One issue is that physical system governed by quantum theory and QFT, we hope to have

• unitarity (say the partition function and the probability will be always conserved)

• locality

However, it looks that Unitary for "Yang-Mills theory with non-compact or non-semi-simple gauge groups" may be an issue.

1. Opposition for Unitarity:

See discussion here: "a necessary condition for unitarity is that the Yang Mills (YM) gauge group $G$ with corresponding Lie algebra $g$ should be real and have a positive (semi)definite associative/invariant bilinear form " --- Why is the Yang-Mills gauge group assumed compact and semi-simple?

2. Neutral opinion for Unitary:

On gauge theories for non-semisimple groups A.A. Tseytlin The non-positivity of the metric implies that these theories are apparently non-unitary. However, the special structure of interaction terms (degenerate compared to non-compact YM theories) suggests that there may exist a unitary `truncation'. "Yang-Mills theories for non-semisimple real Lie algebras which admit invariant non-degenerate metrics. These 4-dimensional theories have many similarities with corresponding WZW models in 2 dimensions and Chern-Simons theories in 3 dimensions."

3. Supportive opinion for Unitary (Yang-Mills theory with non-compact gauge groups G):

or at least some attempts:

YANG-MILLS FIELD QUANTIZATION WITH NON-COMPACT GAUGE GROUP Article in Modern Physics Letters A 07(29) · November 2011

Unitary gauge theories of noncompact groups - Kevin Cahill and Sertaç Özenli - Phys. Rev. D 27, 1396 two pape papers] "It is noted that the use of an internal metric field allows one to gauge noncompact internal-symmetry groups without sacrificing unitarity. The possibility that such theories could be rendered renormalizable is discussed."

Question: So are their sharp mathematical statement to be made for "Yang-Mills theory with non-compact or non-semi-simple gauge groups" --- is the unitarity and locality an issue?