This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general dimensions NLSM.

For 2d NLSM, it can be a boundary theory's QFT of a 3d Chern-Simons theory.

Question 1: Are there any mathematical rigorous studies and mathematical/physics uses of 3d Chern-Simons theory with non-compact groups gauge G?

Question 2: What is the energy spectrum of this Chern-Simons theory with non-compact groups gauge G? Is it a TQFT with finite number of Hilbert space (for the zero energy states (zero modes, or flat connections))

Question 3: Is this Chern-Simons theory unitary or not-unitary? Does the unitarity depend on the Lie algebra ${\mathcal{G}}$ properties? (reductive, non-semi-simple, etc?)

(Namely, physically, does it preserve the norm in partition function $Z$ or so-called Feynman path integral and the probability in quantum mechanics.)

The standard 3d quantum Chern-Simons theory path integral is defined by, $$ \int [DA]\exp(i \frac{k}{4\pi} \int \mathrm{Tr}_{\mathcal{G}} (A \wedge dA + \frac{2}{3} A \wedge A \wedge A)) $$ see the details of the notations here.