Differential Contracting Homotopy in the Linearized 3d Higher-Spin Theory

TL;DR

This paper applies a differential homotopy method to 3d higher-spin gauge theory, unifying known solutions and proposing new derivations for disentangling dynamical and topological fields at the linear level.

Contribution

It introduces a differential homotopy approach to systematically reproduce and extend solutions in 3d higher-spin gauge theory, aiding nonlinear analysis.

Findings

Unified form of known disentangling solutions

Reproduction of solutions from previous methods

New derivation approach for disentangled equations

Abstract

In this paper, the recently developed differential homotopy approach is applied to the problem of disentangling dynamical and topological fields of the $3d$ higher-spin gauge theory at the linear level. This formalism allows us to reproduce all known disentangling solutions in a unified form, including both the solutions obtained previously within the shifted homotopy approach in \cite{Korybut:2022kdx} and that derived by hand in \cite{Vasiliev:1992ix}, as well as other solutions including those associated with the cohomology of the background covariant derivative $D_0$. Also, within the differential homotopy framework an alternative way of derivation of disentangled equations through a non-conventional solution for the field $S_1$ is suggested. The obtained results are important for further analysis of nonlinear corrections to HS equations in $AdS_3$.