Let $\mathbf G$ be a compact, simple, and simply-connected Lie group. Let $G: \Sigma \to \mathbf G$ be a smooth map, where $\Sigma$ is a Riemann surface. Also, let $\tilde G$ be an extension of $G$ to a 3-manifold $B$ with boundary $\partial B=\Sigma$.
A physics paper claims that, the integral $$\frac{ik}{24\pi} \int_{\tilde B} \langle \tilde G^{-1} d\tilde G, [\tilde G^{-1} d\tilde G, \tilde G^{-1} d\tilde G]\rangle$$ is $2\pi ik\mathbb Z$-valued, and furthermore it is related to the generalized degree of the map $\tilde G: \tilde B \to \mathbf G$. I am not sure what is $\tilde B$.
The following is the relevant part of the paper:
Question: Why the above integral is quantized, and related to the generalized degree?
The degree that I know is the "de Rham cohomology sense". Given a smooth map $f:M\to M$ on a compact connected orientable $m$-manifold $M$, the degree $\deg f \in \mathbb Z$ is characterized by the following condition. For all $\omega \in \Omega^m(M)$, we have $$\int_M f^*\omega = (\deg f) \int_M \omega.$$