Constant Curvature Algebras and Higher Spin Action Generating Functions

TL;DR

The paper develops a comprehensive algebraic and generating-function framework for higher-spin fields on constant-curvature backgrounds. By enlarging the constant-curvature operator algebra to a Casimir-based presentation and introducing a robust normal-ordering/Hadamard-product calculus, it provides compact generating functions for massless, massive, and partially massless higher-spin actions, including both bosonic and fermionic sectors, with a minimal unconstrained field content. Radial dimensional reduction ties flat-space massless theories in $d+1$ dimensions to massive and partially massless theories in $d$ dimensions, and the approach naturally yields Stückelberg descriptions and explicit degeneracies that realize partial gauge invariances. The work also reformulates higher-spin dynamics as a scalar theory on the total space of the cotangent bundle, and extends the formalism to spinor-tensor fields via a corresponding superalgebra, offering a unifying, scalable toolkit potentially useful for interactions and connections to string theory.

Abstract

The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for the actions and gauge invariances of massive, partially massless and massless (for both bose and fermi statistics) higher spins on constant curvature backgrounds. These are formulated in terms of a minimal set of covariant, unconstrained, fields rather than towers of auxiliary fields. Partially massless gauge transformations are shown to arise as degeneracies of the flat, massless gauge transformation in one dimension higher. Moreover, our results and calculus offer a considerable simplification over existing techniques for handling higher spins. In particular, we show how theories of arbitrary spin in dimension d can be rewritten in terms of a single scalar field in dimension 2d where the d additional dimensions correspond to coordinate differentials. We also develop an analogous framework for spinor-tensor fields in terms of the corresponding superalgebra.

Figures (2)

• Figure 1: The algebra of operators on symmetric tensors living on $n+1$ dimensional constant curvature manifolds. The left-hand column enumerates the weight with respect to the index operator ${\space\bf N\space}$. The two rightmost columns are the results for non-trivial/vanishing commutators.
• Figure 2: The algebra of operators on symmetric spinor tensors living on $n+1$ dimensional constant curvature manifolds. The right-hand column enumerates the weight with respect to the index operator ${\space\bf N\space}$. Only non-vanishing or results different from the bosonic case in Figure \ref{['commutators']} are shown. Dashed lines denote anticommutators, while dashed loops are squares of operators (the diagram should be read from the middle column outwards).