Continuous spin field in the $\mathbf{AdS_6}$ space

TL;DR

This paper develops a mathematical framework for continuous spin fields in six-dimensional anti-de Sitter space, deriving explicit algebraic expressions and constraints that define their properties.

Contribution

It introduces a new realization of the $ ext{so}(2,5)$ algebra for continuous spin fields in $ ext{AdS}_6$ using the Lie-Lorentz derivative and derives explicit Casimir operator expressions.

Findings

Explicit Casimir operators expressed in terms of covariant derivatives and spin invariants.

Complete set of operator constraints fixing Casimir eigenvalues.

Eigenvalues characterized by a real parameter and a positive (half-)integer.

Abstract

A representation of the $\mathfrak{so}(2,5)$ algebra corresponding to the continuous spin field in $\mathbf{AdS_6}$ is considered. The algebra is realized using the Lie-Lorentz derivative, which naturally incorporates $\mathbf{AdS_6}$ geometry and spin degrees of freedom. Within this framework, we derive explicit expressions for the Casimir operators in terms of both the covariant derivative and the spin invariants. The continuous spin representation under consideration is defined by a system of operator constraints that generalize those known for six-dimensional Minkowski space. We demonstrate that these constraints completely fix all Casimir operators of the $\mathfrak{so}(2,5)$ algebra, with the eigenvalues determined by a dimensional real parameter $\boldsymbol{\mu}$ and a positive (half-)integer $s$.