Perturbing AdS$_6 \times_w S^4$: linearised equations and spin-2 spectrum

TL;DR

This work develops the linearised framework for studying the Kaluza-Klein spectrum of massive IIA supergravity on a warped AdS$_6 \times_w S^4$ background. By deriving the bosonic and fermionic fluctuations and recasting the spin-2 sector as a warped Laplace eigenproblem, the authors reduce the internal problem to a hypergeometric equation, revealing two mass-branch structures bounded from below. The branch masses are $M^2 = ( \frac{3}{2} \ell + 3j )( \frac{3}{2} \ell + 3j + 5 )$ (branch A) and $M^2 = ( \frac{3}{2} \ell + 3j + 1 )( \frac{3}{2} \ell + 3j + 6 )$ (branch B), with dual operator dimensions $\Delta_A = \frac{3}{2}\ell + 3j + 5$ and $\Delta_B = \frac{3}{2}\ell + 3j + 6$. The bound-saturating states assemble into short multiplets of the exceptional $\mathfrak{f}(4)$ symmetry, aligning with known multiplet classifications, and the analysis provides a concrete step toward mapping the bulk spectrum to operators in the dual 5d SCFT. The paper also lays out the challenges and directions for extending the spectrum to all fluctuations in this warped setting.

Abstract

We initiate the analysis of the Kaluza--Klein mass spectrum of massive IIA supergravity on the warped AdS$_6 \times_w S^4$ background, by deriving the linearised equations of motion of bosonic and fermionic fluctuations, and determining the mass spectrum of those of spin-2. The spin-2 modes are given in terms of hypergeometric functions and a careful analysis of their boundary conditions uncovers the existence of two branches of mass spectra, bounded from below. The modes that saturate the bounds belong to short multiplets which we identify in the representation theory of the $\mathfrak{f}(4)$ symmetry superalgebra of the AdS$_6 \times_w S^4$ solution.