I have questions about differential geometry calculations. If there is any misunderstanding of mine in the contents below, please let me know and help me to fix it.
Let's consider a 3-dimensional spacetime $\mathcal{M} = \mathbb{R} \times \Sigma^2$ ($\mathbb{R}$ for time) and three 1-forms $e^a$ ($a = 1, 2$) and $\omega$. Here, we assume $\omega_0 = 0$, where, for a given chart, $\omega = \omega_\mu dx^\mu$. (For those who are familiar with these, the 1-forms are known as the coframe fields and spin connection.)
From them, we introduce 2-forms as
$$T^a \equiv de^a + \Omega^a_{\ b} \wedge e^b.$$
where $\Omega^a_{\ b}$ is the full spin connection. Here, we assume $\Omega^1_{\ 2} = -\Omega^2_{\ 1} = \omega$ and vanishes otherwise. This assumption leaves
$$T^1 = (\partial_\mu e_\nu^{\ 1} + \omega_\mu e_\nu^{\ 2}) dx^\mu \wedge dx^\nu$$
$$T^2 = (\partial_\mu e_\nu^{\ 2} - \omega_\mu e_\nu^{\ 1}) dx^\mu \wedge dx^\nu$$
Now, we integrate those 2-forms over $D \subset \Sigma^2$:
$$\int_D T^1 = \int_D de^1 + \int_D \omega \wedge e^2 = \int_{\partial D} e^1 + \int_D \omega \wedge e^2$$
$$\int_D T^2 = \int_D de^2 - \int_D \omega \wedge e^1 = \int_{\partial D} e^2 - \int_D \omega \wedge e^1$$
What I want to do here is to promote those 1-forms to "gauge fields" just like the electromagnetic 1-form gauge field $A = A_\mu dx^\mu$. To do so, I defined a gauge transformations:
$$e_\mu^{\ a} \rightarrow e_\mu^{\ a} + \partial_\mu \epsilon^a$$
$$\omega_\mu \rightarrow \omega_\mu + \partial_\mu h$$
I expected that the 2-forms I introduced above should be gauge invariant, but they weren't. In particular, the $\omega \wedge e^a$ terms are not invariant:
$$\omega \wedge e^a \rightarrow [\omega_\mu e_\nu^{\ a} + e_\nu^{\ a} \partial_\mu h + \omega_\mu \partial_\nu \epsilon^a + (\partial_\mu h) (\partial_\nu \epsilon^a)] dx^\mu \wedge dx^\nu$$
After observing this, my questions are twofold:
Did I define the gauge transformations in a right way?
If I did it in a right way, then can I define these integrals (which may be considered as "fluxes") in a gauge invariant way?
