In the self-dual higher-spin theory, a scalar vacuum that breaks AdS symmetry to 3D Poincaré makes all higher-spin gauge fields decouple except spin one, and makes higher-spin currents non-conserved except the spin-one current.
Toward higher-spin symmetry breaking in the bulk
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We present a new vacuum of the bosonic higher-spin gauge theory in $d+1$ dimensions, which has leftover symmetry of the Poincar\'{e} algebra in $d$ dimensions. Its structure is very simple: the space-time geometry is that of $AdS$, while the only nonzero field is a scalar. The scalar extends along the Poincar\'{e} radial coordinate $z$ and is shown to be linearly exact for an arbitrary mixture of its two $\Delta=2$ and $\Delta=d-2$ conformal branches. The obtained vacuum breaks the global higher-spin symmetry leading to a broken phase that lives in the Minkowski space-time.
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On symmetry breaking in the self-dual higher-spin theory
In the self-dual higher-spin theory, a scalar vacuum that breaks AdS symmetry to 3D Poincaré makes all higher-spin gauge fields decouple except spin one, and makes higher-spin currents non-conserved except the spin-one current.