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

TL;DR

This paper derives explicit formulas for generalized curvature tensors and Christoffel symbols in AdS_{d+1} space for higher spin fields, providing a systematic method for calculations up to spin five and beyond.

Contribution

It introduces a modified ansatz and solves recurrence relations to explicitly compute curvature tensors for higher spin fields in AdS space, extending previous work.

Findings

Explicit formulas for curvature tensors up to spin five.

Finite power series in inverse AdS radius.

Method applicable to higher orders and spins.

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.