Higher Spins and Matter Interacting in Dimension Three

TL;DR

This work analyzes Prokushkin–Vasiliev Theory in AdS$_3$ to address the Gaberdiel–Gopakumar duality by identifying a unique truncation at $oldsymbol{ u}=0$ (i.e., $oldsymbol{ ilde{C}}$ vacuum) and $oldsymbol{ u}=0$ (or $oldsymbol{ ilde{g}_0}=0$ frame) that removes the twisted sector at second order. The authors compute the explicit second-order backreaction of the scalar on the physical higher-spin sector, and determine the complete cubic action by fixing the spin-$s$ couplings with the admissibility condition, finding $g_s=1/(2s-2)!$. They develop a manifestly Lorentz-covariant perturbation theory in the Schwinger–Fock gauge, analyze the cohomologies that govern pseudo-local field redefinitions, and show that a consistent truncation to the physical sector is possible only at a unique parameter point; otherwise, twisted fields remain nontrivial at order two. The results illuminate the structure of backreactions and their locality properties, offering a controlled path toward a fully consistent cubic (and potentially quartic) theory and providing insights for higher-spin holography and AdS$_3$/CFT$_2$ correspondences, including connections to the D-dimensional Vasiliev theory at $oldsymbol{D}=3$ ($oldsymbol{ abla}$-frame with $oldsymbol{ u}=1$). Overall, the paper advances the understanding of matter–higher-spin interactions in 3D and clarifies the role of twisted sectors, truncations, and field redefinitions in establishing a consistent holographic framework.

Abstract

The spectrum of Prokushkin--Vasiliev Theory is puzzling in light of the Gaberdiel--Gopakumar conjecture because it generically contains an additional sector besides higher-spin gauge and scalar fields. We find the unique truncation of the theory avoiding this problem to order 2 in perturbations around AdS$_3$. The second-order backreaction on the physical gauge sector induced by the scalars is computed explicitly. The cubic action for the physical fields is determined completely. We comment on a different higher-spin theory without such additional fields at $λ=1$.