Abstract
New techniques based on Exceptional Field Theory have recently allowed for the calculation of the Kaluza-Klein spectra of certain AdS4 solutions of D = 11 and massive IIA supergravity. These are the solutions that consistently uplift on S7 and S6 from vacua of maximal four-dimensional supergravity with SO(8) and ISO(7) gaugings. In this paper, we provide an algorithmic procedure to compute the complete Kaluza-Klein spectrum of five such AdS4 solutions, all of them N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, and give the first few Kaluza-Klein levels. These solutions preserve SO(3) and U(1) × U(1) internal symmetry in D = 11, and U(1) (two of them) and no continuous symmetry in type IIA. Together with previously discussed cases, our results exhaust the Kaluza-Klein spectra of known supersymmetric AdS4 solutions in D = 11 and type IIA in the relevant class.
Highlights
Extremely hard and, for that reason, progress on finding the KK spectra of such solutions has been virtually inexistent over the last twenty years or so
In this paper we will focus on D = 11 [1] and massive type IIA [17] supersymmetric solutions of the form AdS4 ×w S7 and AdS4 ×w S6, that uplift consistently [36,37,38] from critical points of D = 4 N = 8 supergravity [39] with SO(8) [40] and dyonic ISO(7) [41] gaugings, respectively
In this work we present the KK spectrum of the AdS4 ×w S7 solutions of D = 11 supergravity that uplift from the N = 1 critical points of D = 4 N = 8 SO(8) supergravity [40] with SO(3) [42, 46] and U(1) × U(1) [47, 48] residual symmetry
Summary
The mass matrices for the bosonic and fermionic KK perturbations above the AdS4 solutions of interest have been derived from E7(7) ExFT [28,29,30,31] in [27, 44, 45, 50]. These vevs can be brought to the mass matrices (2.1)–(2.4), along with the expressions for the SO(8) or SO(7) generators given in appendix A of [44], in order to evaluate these mass matrices at each vacuum Diagonalising these matrices KK level by KK level, and removing the spurious spin-1 and spin-1/2 modes discussed at the end of section 2, we are able to find the KK towers of physical graviton, gravitino, vector and spin-1/2 fermion mass states, L2M22, LM 3 , L2M12, LM 1 , above each of these vacua. This is not surprising, because these larger groups are not symmetries of these solutions In these cases, one can only form OSp(4|N ) multiplets KK level by KK level when the SO(8) in D = 11 [4] or SO(7) in type IIA [44] state content at each level has already been broken down to the actual residual symmetry group SU(3).
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