# Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?

Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals:

1. abelian Chern-Simons theory on non-spin manifolds --- $$\int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA ))$$

2. abelian Chern-Simons theory on spin manifolds --- $$\int [DA]\exp(i \frac{k}{4\pi} \int_X (A \wedge dA ))$$

3. non-abelian Chern-Simons theory --- $$\int [DA]\exp(i \frac{k}{4\pi} \int_X \mathrm{Tr}_{} (A \wedge dA + \frac{2}{3} A \wedge A \wedge A))$$ where $$A$$ takes values in the Lie algebra valued $$\mathcal{G}$$ 1-form. So does the Tr take the matrix representations in the Lie algebra $$\mathcal{G}$$.

What are the correct and rigorous ways to argue the quantization of values of $$k$$?

I think there are three possible helpful ideas:

• extend 3-manifolds $$X$$ to 4-manifolds $$Y$$?

• large gauge transformation.

• Use Wess Zumino Witten like terms.

Could any expert demonstrate these line by line?

• None of the actions you write are well-defined. Once you pick a proper definitions, the action will be valued in $\mathbb{R}/\mathbb{Z}$. The quantization follows accordingly. You can find one definition in the appendix to Freed’s arxiv.org/abs/0808.2507. – Aaron Bergman Aug 9 '20 at 3:01