The structure of N=3 multiplets in AdS_4 and the complete Osp(3|4) X SU(3) spectrum of M-theory on AdS_4 X N^{010}

TL;DR

The paper computes the complete $Osp(3|4)\times SU(3)$ Kaluza-Klein spectrum for M-theory on $AdS_4\times N^{0,1,0}$, exploiting harmonic analysis and the known KK masses to classify all unitary irreducible representations with maximal spin $s_{\max}\le 2$. It provides a detailed decomposition into $Osp(2|4)$ multiplets and specifies the $SU(3)$ representation content $(M_1,M_2)$ and the energy $E_0$, including long, short, and massless sectors, as well as the unique Betti multiplet associated with $H^2(N^{0,1,0})$. The results delineate the full spectrum structure, clarifying how the KK tower arranges into $Osp(3|4)$ multiplets and how they map to boundary operators in the anticipated $\mathcal{N}=3$ SCFT, while identifying the geometric origin of Betti-charged states. This furnishes a concrete algebraic framework to guide the construction of the dual three-dimensional theory and illuminates the bulk gauged $\mathcal{N}=3$ supergravity description with scalar manifold $SU(3,9)/SU(3)\times SU(9)\times U(1)$ and gauge group $SO(3)_R\times SU(3)$.

Abstract

In this paper, relying on previous results of one of us on harmonic analysis, we derive the complete spectrum of Osp(3|4) X SU(3) multiplets that one obtains compactifying D=11 supergravity on the unique homogeneous space N^{0,1,0} that has a tri-sasakian structure, namely leads to N=3 supersymmetry both in the four-dimensional bulk and on the three-dimensional boundary. As in previously analyzed cases the knowledge of the Kaluza Klein spectrum, together with general information on the geometric structure of the compact manifold is an essential ingredient to guess and construct the corresponding superconformal field theory. This is work in progress. As a bonus of our analysis we derive and present the explicit structure of all unitary irreducible representations of the superalgebra Osp(3|4) with maximal spin content s_{max}>=2.