Holography, Unfolding and Higher-Spin Theory

TL;DR

This work argues that holographic duality is a natural consequence of reformulating theories in the unfolded, first-order free differential algebra framework. By applying unfolded dynamics to higher-spin theories in AdS$_4$, Vasiliev shows a duality between bulk HS fields and a nonlinear 3d conformal HS theory of currents interacting with Chern-Simons-like boundary fields, with two special A- and B-model reductions yielding free 3d duals, in line with Klebanov-Polyakov and Maldacena-Zhiboedov. The analysis extends to AdS$_3$/CFT$_2$ and even a connection to nonrelativistic quantum mechanics via HS in matrix spaces, revealing a web of dualities that are made transparent by the twistor/Y-space formulation and the doubled bulk construction. The paper also outlines how boundary locality emerges at AdS infinity and sketches an off-shell, action-based generalization, highlighting open problems in fully nonlinear 3d conformal HS theory, boundary conditions, and potential implications for quantum mechanics and anomalies. Overall, the unfolded approach provides a unifying, highly geometric language for HS holography across dimensions and models.

Abstract

Holographic duality is argued to relate classes of models that have equivalent unfolded formulation, hence exhibiting different space-time visualizations for the same theory. This general phenomenon is illustrated by the $AdS_4$ higher-spin gauge theory shown to be dual to the theory of 3d conformal currents of all spins interacting with 3d conformal higher-spin fields of Chern-Simons type. Generally, the resulting 3d boundary conformal theory is nonlinear, providing an interacting version of the 3d boundary sigma model conjectured by Klebanov and Polyakov to be dual to the $AdS_4$ HS theory in the large $N$ limit. Being a gauge theory it escapes the conditions of the theorem of Maldacena and Zhiboedov, which force a 3d boundary conformal theory to be free. Two reductions of particular higher-spin gauge theories where boundary higher-spin gauge fields decouple from the currents and which have free boundary duals are identified. Higher-spin holographic duality is also discussed for the cases of $AdS_3/CFT_2$ and duality between higher-spin theories and nonrelativistic quantum mechanics. In the latter case it is shown in particular that ($dS$) $AdS$ geometry in the higher-spin setup is dual to the (inverted) harmonic potential in the quantum-mechanical setup.