Wigner continuous-spin equations in $\mathbf{AdS_D}$: bosonic and fermionic cases

Abstract

We construct Wigner-like equations of motion for symmetric continuous-spin fields in anti-de Sitter space of arbitrary dimension, treating both the bosonic and fermionic cases. We generalise the classical flat-space Wigner constraints for bosonic continuous-spin fields, and for the fermionic case we adopt the equations proposed by Bekaert and Mourad as our starting point. This is achieved by covariantising the ordinary derivatives and deforming the resulting constraints so that they form a closed algebra. The construction is carried out in a metric-like formalism and yields a system of first-class constraints that define a representation of the $\mathfrak{so}(2,D-1)$ isometry algebra, realised via the Lie-Lorentz derivative. Using these constraints, we compute the eigenvalues of the quadratic and quartic Casimir operators and compare the obtained values with Metsaev's classification of continuous-spin representations for both the bosonic and fermionic cases. A crucial algebraic role is played by special operators of the (super)algebra Howe-dual to the Lorentz subalgebra $\mathfrak{so}(1,D-1)$: the $\mathfrak{sl}(2)$ Casimir operator in the bosonic case, and the Casimir's ghost of the $\mathfrak{osp}(1|2)$ superalgebra in the fermionic case. Both objects naturally organise the constraint algebra and ensure its consistency.