I am looking at an expression like
$$ \int_{\partial M}\langle D\psi, D\psi\rangle dx dy $$ where $\psi$ is some (well-behaved) function $\partial M \rightarrow \mathbb{C}$. $D=d+A$ is the connection one form. As this comes from a physical background, I assume $A$ is proportional to the magnetic vector potential.
Now I want to apply Stokes-Theorem to it:
$$ \int_{\partial M}\langle D\psi, D\psi\rangle dxdy=\int_M d \langle D\psi, D\psi\rangle dxdydz $$ I already know, there is a manifold $M$. First of all, this is probably: $$ \int_M \langle dD\psi, D\psi\rangle+\langle D\psi, dD\psi\rangle dx dydz $$ here then my confusion with dimensionality sets in. I am simply told $D$ is the connection one form, but if $\partial M$ is an e.g. 2d manifold, should, after applying $d$ the connection one form somehow change to reflect that it is know on a 3d manifold? Explicitly, if $$ D\psi=\left(\begin{array}{l} \partial_x+A_x\\ \partial_y+A_y \end{array}\right) \psi $$ do I then look at : $$ \partial_z\left(\begin{array}{l} \partial_x+A_x\\ \partial_y+A_y \end{array}\right) \psi $$ or $$ \partial_z\left(\begin{array}{l} \partial_x+A_x\\ \partial_y+A_y\\ \partial_z+A_z \end{array}\right) \psi $$ And what is the reasoning behind the specific choice? I am currently self studying differential geometry and would be very happy about any help.