Background: If we have a vector bundle $S^+(\tilde{P})$ over a Riemannian manifold $M$, then we have a metric $g$ and Levi-Civita connection $\nabla$ on $M$, and we can choose a metric $h$ and a compatible connection $\tilde{\nabla}$ on $S^+(\tilde{P})$.
With this, we can form covariant derivatives on all the bundles $T^*M^{\otimes k}\otimes S^+(\tilde{P})$. It would look like $$\underbrace{(\nabla\otimes 1\otimes\cdots\otimes 1)\otimes 1 + (1\otimes\nabla\otimes\cdots\otimes 1) \otimes 1+ \cdots + (1\otimes\cdots\otimes 1\otimes\nabla)\otimes 1}_{k} +(1\otimes 1\otimes\cdots\otimes 1)\otimes \tilde{\nabla}$$
And with this we can form $\nabla^k:C^\infty(S^+(\tilde{P}))\to C^\infty(T^*M^{\otimes k}\otimes S^+(\tilde{P}))$.
We also have metrics on these $T^*M^{\otimes k}\otimes S^+(\tilde{P})$ by $$\langle v^1\otimes\cdots\otimes v^k\otimes e, u^1\otimes \cdots \otimes u^k\otimes f\rangle = \langle v^1,u^1\rangle\cdots\langle v^k, u^k\rangle \langle e, f\rangle$$ and extending by linearity.
This allows us to form the Sobolev space $L^2_2(S^+(\tilde{P}))$ by completion of $C^\infty(S^+(\tilde{P}))$ with norm $$||u||_{L^2_2}=\left(\int_M|u|^2+|\nabla u|^2+|\nabla^2 u|^2dV_g\right)^{1/2}$$ that Morgan writes in his Seiberg-Witten book, which I copy below. All this was taken from Nicolaescu's answer on https://mathoverflow.net/questions/403546/sobolev-spaces-of-differential-forms-and-regular-atlases
Question: Now $\mathcal{L}=\Lambda^2S^+(\tilde{P})$. Shouldn't we be able to get an induced connection on $\mathcal{L}$ from one on $S^+(\tilde{P})$ (just like we get a metric on it too)? Probably no since it’d turn the space $\mathcal{A}_{L^2_2}(\mathcal{L})$ useless, but I feel like there should be, just like a connection on the cotangent bundle induces one on the space of 2-forms.
Question: Now assume we chose an unitary connection on $\mathcal{L}$. It is a section of the bundle $T^*M\otimes (i\mathbb{R}\times M)$. Then how are we defining its $L^2_2$ norm? To repeat the above construction for $L_2^2(S^+(\tilde{P}))$ we would need a metric and compatible connection on the trivial bundle $i\mathbb{R}\times M$. Are we just taking the Euclidean metric and flat connection $d$ on it for the construction? Is this implicitly assumed?