I am reading Dan Freed's lectures on Quantum Groups on Path Integrals. I am picking up the required math as I go along and I am finding certain calculations hard to follow.
This is regarding the discussion on page 7. We have the following setup:
A two dimensional closed oriented manifold $X$, a Lie group $G=SU(2)$, a smooth map $\phi: X\to G$ and a $3$-form $\omega$ on $G$ with the property that $\int_X \omega = 1$.
Now the map $\phi$ is extended to a map $\Phi:W \to G$ where $W$ is a oriented three manifold whose boundary is $X$ and the restriction of $\Phi$ on the boundary is $\phi$. We define a functional $$S(\Phi) = \int_W \Phi^* \omega.$$
Apparently it is "straightforward" to see that if we choose a different pair $(\Phi, W)$, the functional differs by an integer. Why is this so?