# Fronsdal fields from gauge functions in Vasiliev's higher-spin gravity

## TL;DR

This paper explores the relationship between gauge functions and Fronsdal fields in Vasiliev's higher-spin gravity, demonstrating how solutions can be parametrized and transformed to recover free-field limits and asymptotic behaviors.

## Contribution

It introduces a parametrization of linearized solutions via gauge functions, clarifies the embedding of Fronsdal fields, and proposes a Fefferman-Graham-like scheme for asymptotic analysis.

## Key findings

- Explicitly relates gauge functions to Fronsdal fields and Weyl tensors.
- Shows Vasiliev's equations admit correct free-field limits.
- Proposes a perturbative scheme for asymptotic master fields.

## Abstract

In this paper, we revisit a number of issues in Vasiliev's theory related to gauge functions, ordering schemes, and the embedding of Fronsdal fields into master fields. First, we parametrize a broad equivalence class of linearized solutions using gauge functions and integration constants, and show explicitly how Fronsdal fields and their Weyl tensors arise from these data in accordance with Vasiliev's central on mass shell theorem. We then gauge transform the linearized piece of exact solutions, obtained in a convenient gauge in Weyl order, to the aforementioned class, where we land in normal order. We spell out this map for massless particle and higher spin black hole modes. Our results show that Vasiliev's equations admit the correct free-field limit for master field configurations that relax the original regularity and gauge conditions in twistor space. Moreover, they support the off-shell Frobenius--Chern--Simons formulation of higher spin gravity for which Weyl order plays a crucial role. Finally, we propose a Fefferman-Graham-like scheme for computing asymptotically anti-de Sitter master field configurations, based on the assumption that gauge function and integration constant can be adjusted perturbatively so that the full master fields approach free master fields asymptotically.