Lemma: Infinite rank of the boundary energy

The form ${B_{g,A}}^{\Gamma }$ has infinite rank. Its zero-energy subspace has complex dimension at most two.

Proof. If $Q(f)=0$, then ${\nabla }_A{u}_f=0$. A parallel section on a connected manifold is determined by its value in one fibre, since it is transported along every path by a linear ordinary differential equation. The space of parallel sections has dimension at most two. Its boundary traces therefore have dimension at most two.

The patch-supported trace space ${\mathcal{H}}_{\Gamma }$ is infinite-dimensional. A nonnegative form on it whose kernel has dimension at most two cannot have finite rank.