This is related to [a previous question](https://mathoverflow.net/q/314104/27004), 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? does it preserve 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](https://ncatlab.org/nlab/show/Chern-Simons+theory#quantum_chernsimons_theory).