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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 anti–de Sitter (AdS) symmetry but remains symmetric under the leftover Poincaré algebra in three dimensions. The vacuum nonzero 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 Δ = 1 and Δ = 2 . Focusing on the Δ = 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 ± 1 / 2 remain unaffected by the symmetry breaking. Notably, the helicity s = + 1 current, while deformed, remains conserved.
Didenko et al. (Tue,) studied this question.