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.