Stability and boundedness in AdS/CFT with double trace deformations

Scalar fields on the bulk side of AdS/CFT correspondence can be assigned unconventional boundary conditions, related to the conventional one by Legendre transform. One can further perform double trace deformations which relate the two boundary conditions via renormalization group flow. Thinking of these operators as $S$ and $T$ transformations, respectively, we explore the $SL(2,{\bf R})$ family of models which naively emerges from repeatedly applying these operations. Depending on the parameters, the effective masses vary and can render the theory unstable. However, unlike in the $SL(2,{\bf Z})$ structure previously seen in the context of vector fields in $AdS_4$, some of the features arising from this exercise, such as the vacuum susceptibility, turns out to be scheme dependent. We explain how scheme independent physical content can be extracted in spite of some degree of scheme dependence in certain quantities.