On symmetry breaking in the self-dual higher-spin theory

We explore the symmetry-broken phase of the self-dual (chiral) sector of higher-spin theory in four dimensions. To that end, we construct a two-parameter vacuum that breaks the AdS symmetry but remains symmetric under the leftover Poincar\'{e} algebra in three dimensions. The vacuum non-zero fields include spin-two AdS frame fields and a scalar, which has a profile that extends along the AdS radial direction. The two free parameters correspond to two scalar branches of conformal dimensions $\Delta=1$ and $\Delta=2$. Focusing on the $\Delta=1$ branch, we analyze the dynamics of free fields around this vacuum and examine its holographic dual. We observe that certain higher spin states decouple in the broken phase. This is illustrated by a set of gauge fluctuations, which acquire no source from higher-spin currents, leading to their complete decoupling, except for the gauge field associated with spin one. The dual higher-spin currents appear to be disentangled from the gauge fields and generally do not conserve; however, their lower-spin components with helicities $s = -1, 0$, and $\pm 1/2$ remain unaffected by the symmetry breaking. Notably, the helicity $s=+1$ current, while deformed, remains conserved.