Higher-spin Cotton tensors and massive gauge-invariant actions in AdS${}_3$

Abstract

In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor ${\mathfrak{C}}_{\alpha (n)}(h)$ is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential $h_{\alpha (n)}$. The explicit form of ${\mathfrak{C}}_{\alpha (n)}(h)$ is well known in Minkowski space. Here we solve the problem of extending the Minkowskian result to the case of anti-de Sitter (AdS) space and derive a closed-form expression for ${\mathfrak{C}}_{\alpha (n)}(h)$ in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action $S_{\text{CS}}^{(n)}[h]\propto \int {\text{d}}^{3}xe{h}^{\alpha (n)}{\mathfrak{C}}_{\alpha (n)}(h)$ factorises into a product of $(n-1)$ first-order operators that are associated with the spin-$n/2$ partially massless AdS values. Our findings greatly facilitate the on-shell analysis of massive higher-spin gauge-invariant actions in AdS${}_3$. The main results are extended to the case of $\mathcal{N}=1$ AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS${}_3$.

44 pages; V2: comments added and typos corrected; V3: published version

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Cited by in corpus (4)

• On higher spin analogues of linearized Topologically Massive Gravity and linearized "New Massive Gravity"
• Superfield approach to interacting N=2 massive and massless supermultiplets in 3d flat space
• On equivalence of gauge-invariant models for massive integer-spin fields
• AdS (super)projectors in three dimensions and partial masslessness