I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations. Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the projective plane", it is Lemma 2 therein. One step in the proof goes as follows: $$ \Delta \left| \Phi \right|^2 = 2 \langle \nabla_A^* \nabla_A \Phi , \Phi \rangle - 2 \langle \nabla_A \Phi, \nabla _A \Phi \rangle. $$ (This step with context is provided in the picture below) The proof that this is true comes from using the definition $\Delta = - \sum_i \nabla_i \nabla_i$ and using that $\nabla_A$ is metric. But I'm puzzled by the result.
Why is the right hand side not constant $0$? It should be, as $\langle \nabla_A^* \nabla_A \Phi , \Phi \rangle=\langle \nabla_A \Phi , \nabla_A \Phi \rangle$ by the definition of adjoint.
Without knowing the exact context, I would bet that $\nabla^*$ is not a pointwise adjoint on each fiber, but rather it is the $L^2$ adjoint, meaning $$\int \langle \nabla^* \phi , \psi \rangle = \int \langle \phi , \nabla\psi \rangle,$$ and since the theorem is about a $C^0$ bound you are working pointwise.
I don't know if it's as simple to define a pointwise adjoint for a differential operater, because $\nabla \phi$ depends on the local behavior of $\phi$.