  Higgs mechanism to massive higher-spin gauge fields is an outstanding open
problem. We investigate this issue in the context of Kaluza-Klein
compactification. Starting from a free massless higher-spin field in
$(d+2)$-dimensional anti-de Sitter space and compactifying over a finite
angular wedge, we obtain an infinite tower of heavy, light and massless
higher-spin fields in $(d+1)$-dimensional anti-de Sitter space. All massive
higher-spin fields are described gauge invariantly in terms of St\"ueckelberg
fields. The spectrum depends on the boundary conditions imposed at both ends of
the wedges. We obseved that higher-derivative boundary condition is inevitable
for spin greater than three. For some higher-derivative boundary conditions,
equivalently, spectrum-dependent boundary conditions, we get a non-unitary
representation of partially-massless higher-spin fields of varying depth. We
present intuitive picture which higher-derivative boundary conditions yield
non-unitary system in terms of boundary action. We argue that isotropic
Lifshitz interfaces in $O(N)$ Heisenberg magnet or $O(N)$ Gross-Neveu model
provides the holographic dual conformal field theory and propose experimental
test of (inverse) Higgs mechanism for massive and partially massless
higher-spin fields.
