Neumann vs Dirichlet boundary conditions for vector fields From hep-th/9611230,

A vector field $A_\mu$ in $3 + 1$ dimensions may obey either Dirichlet boundary conditions,
in which the components of $F_{\mu\nu}$ with $\mu$ and $\nu$ tangent to the boundary vanish,
or Neumann boundary conditions, in which the components in which one index
is tangent to the boundary vanish.

I am struggling to see why this is the case. Consider for example a boundary in the $(x,y)$ plane (i.e. at $z=0$) and with normal coordinate $z$. Then, for the case of two indices being both along the boundary, e.g. $$F_{xy}=\partial_xA_y-\partial_yA_x+i[A_x,A_y]$$
If I impose this vanishes, I am not quite sure how this would recover Dirichlet boundary conditions, which I expect to be of the form $A_\mu(x,y,z)|_{z=0}=0$. Similarly, for the case of one index along the boundary, $$F_{xz}=\partial_xA_z-\partial_zA_x+i[A_x,A_z]$$ I am not sure how this recovers Neumann boundary conditions $\partial_\mu A_\nu(x,y,z)|_{z=0}=0$, and I fact I am struggling to see how this second equation wuld result in a different condition from the first one.
 A: Instead of $(w, x, y, z)$, I will use $(x^a, z)$ as the $\mathbb{R}^3$ and $\mathbb{R}_+$ directions.
For the Dirichlet part, the authors are simply saying that $F_{ab} |_{z = 0} = ( \partial_a A_b - \partial_b A_a ) |_{z = 0}$ becomes zero when you plug in $A_a |_{z = 0} = \mathrm{const}$. The reverse implication would be too strong since gauge transformations can change the value of $A_a$ but not $F_{ab}$.
The Neuman part is more interesting because this requires us to go back to the action
\begin{equation}
S = \int_{z \geq 0} \int d^3 x -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}.
\end{equation}
If we're careful when deriving the equations of motion, the step that involves integrating by parts produces the boundary term
\begin{align}
\delta S_\partial = \int d^3 x \delta A^a F_{za}
\end{align}
which has to vanish. If we're not using the Dirichlet condition, the only alternative is $F_{za} |_{z = 0} = 0$ which we define to be the Neuman condition.
For more about this, https://arxiv.org/abs/1902.09567 is a great modern paper.
