When considering an AdS spacetime including a black hole, matter field and gauge field, the value of temporal component $A_t$ of the gauge potential $A_\mu$ on horizon always is set be zero, even the matter field is nonzero outsider black hole.
The explanation in paper Holographic phase transitions at finite baryon density (Page 6) is that the Killing vector on horizon $\partial_t$ vanishes, if the potential $A_\mu$ as a one-form is to be well-defined, then the temporal component $A_t(r_h)$ must vanish on horizon ($r_h$ is the horizon).
What the relation between Killing vector and gauge potential lead to the vanishing condition($A_t(r_h)=0$) ?
Is that vanishing value for $A_\mu$ on horizon dependent on a special gauge?
What about the SU(N) gauge theory in above case?