A possible application of the the proposition 5.5 in physics occurs in the case of the Kapustin-Witten equation:

In this case $\phi$ is a tensorial 1-form of type $ad G$ and $d_{A} \phi$ is given by the proposition 5.5.

It is conjectured that the coefficients of the Jones polynomial of a knot can be computed by counting solutions of the KW equations on a half-space in $R^4$ with the generalized Nahm pole boundary conditions.

The Jones polynomial is a Laurent series $J \left( q \right) =\sum _ na_{{n}}{q}^{n} $, and the conjecture is that $a_ n$ is an algebraic count of the number of solutions of the KW equations with second Chern class equal to $n$.

Reference : https://arxiv.org/pdf/1712.00835.pdf