Scheme dependence and instability of double-trace deformations for gauge fields in AdS$_5$

In holography, double-trace deformations provide a general framework for deforming boundary field theories. In particular, they can be utilized to introduce dynamical gauge fields in the boundary theory through double-trace deformations of bulk gauge fields. In this work, we study this construction in the case where the bulk geometry is asymptotically AdS$_5$, and find that such a system involves tachyon and ghost modes. This instability originates from the logarithmic behavior of the gauge fields in the vicinity of the AdS boundary, which leads to a scheme-dependent ambiguity in the double-trace deformation. We investigate this instability by using both analytical and numerical methods in several holographic setups, including bottom-up models and the top-down D3-D7 construction.