The Generalized Curvature and Christoffel Symbols for a Higher Spin Potential in AdS_{d+1} Space

TL;DR

This work constructs the generalized curvature and Christoffel symbols for higher-spin gauge potentials in $AdS_{d+1}$ by extending the de Wit–Freedman framework with a controlled $L^{-2}$ expansion, enforcing gauge invariance to determine recursion relations for the expansion coefficients. It develops a differential-algebra built from covariant derivatives on symmetric tensors, derives level-by-level gauge variations, and solves the resulting linear systems up to second order, yielding explicit curvature and Riemann-tensor expressions. The results are explicit for spins up to $s\le 5$ and provide an algorithm for higher orders, laying groundwork for higher-spin geometry in AdS and potential links to unfolded formulations and AdS/CFT. Overall, the paper furnishes a gauge-invariant, geometrical description of HS fields in AdS via a constructive, recursive approach with clear pathways to compute higher-order corrections and associated invariants.

Abstract

The generalized curvature tensor and Christoffel symbols are determined in AdS_{d+1} background by a modified ansatz of the de Wit - Freedman type by imposing gauge invariance. The resulting set of recurrence relations and difference equations is solved. The Riemann curvature tensor is derived by antisymmetrization. All results are presented as finite power series in the inverse AdS radius and are unique. The fourth order, which is complete for fields up to spin five, is calculated explicitly. Higher orders can be obtained with the same method.