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 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Δ</a:mi> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi mathvariant="normal">Δ</d:mi> <d:mo>=</d:mo> <d:mn>2</d:mn> </d:math> . Focusing on the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi mathvariant="normal">Δ</g:mi> <g:mo>=</g:mo> <g:mn>1</g:mn> </g:math> 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>s</j:mi> <j:mo>=</j:mo> <j:mo>−</j:mo> <j:mn>1</j:mn> </j:math> , 0, and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mo>±</l:mo> <l:mn>1</l:mn> <l:mo>/</l:mo> <l:mn>2</l:mn> </l:math> remain unaffected by the symmetry breaking. Notably, the helicity <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>s</n:mi> <n:mo>=</n:mo> <n:mo>+</n:mo> <n:mn>1</n:mn> </n:math> current, while deformed, remains conserved.