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M-theory on AdS_4xM^{111}: the complete Osp(2|4)xSU(3)xSU(2) spectrum from harmonic analysis

Davide Fabbri, Pietro Fre`, Leonardo Gualtieri, Piet Termonia

TL;DR

This work delivers the complete Kaluza–Klein spectrum for M-theory on AdS_4 × M^{111}, an N=2 background with isometry SU(3) × SU(2) × U(1), by performing a systematic harmonic analysis on the seven-manifold M^{111}. The authors organize the spectrum into long, short, and massless Osp(2|4) multiplets, and provide the full SU(3) × SU(2) flavor representation content along with the exact Clifford-vacuum energies E_0 and hypercharges y_0 for each multiplet. They derive the eigenvalues of the Laplace–Beltrami operators on zero-, one-, two-, and three-forms, as well as spinors, and show how these feed into the multiplet structure, including the Betti (massless) multiplet and the universal long-multiplet framework applicable to other N=2 AdS_4 compactifications. The results supply a rigorous KK-side foundation for testing the AdS_4/CFT_3 correspondence in this class of models and pave the way for analogous analyses on other G/H cosets, with potential links to the dual 3D SCFTs and their operator spectra. The work thus advances nontrivial checks of holography beyond the well-studied S^7 case by exploiting the detailed geometric and harmonic-analysis structure of M^{111}.

Abstract

We reconsider the Kaluza Klein compactifications of D=11 supergravity on AdS_4x(G/H)_7 manifolds that were classified in the eighties, in the modern perspective of AdS_4/CFT_3 correspondence. We focus on one of the three N=2 cases: (G/H)_7=M^{111}=SU(3)xSU(2)xU(1)/SU(2)xU(1)'xU(1)''. Relying on the systematic use of the harmonic analysis techniques developed in the eighties by one of us (P. Fre') with R. D'Auria, we derive the complete spectrum of long, short and massless Osp(2|4)xSU(3)xSU(2) unitary irreducible representations obtained in this compactification. Our result also provides a general scheme for the other N=2 compactifications. Furthermore, it is a necessary comparison term in the AdS_4/CFT_3 correspondence: the complete AdS/CFT match of the spectra that we obtain will provide a much more stringent proof of the AdS/CFT correspondence than in the S^7 case, since the structure of the superconformal field theory on the M2-brane world volume must be such as to reproduce, at the level of composite operators, the flavor group representations, the conformal dimensions and the hypercharges that we obtain in the present article. The investigation of the match is left to future publications. Here we provide an exhaustive construction of the Kaluza Klein side of our spectroscopy.

M-theory on AdS_4xM^{111}: the complete Osp(2|4)xSU(3)xSU(2) spectrum from harmonic analysis

TL;DR

This work delivers the complete Kaluza–Klein spectrum for M-theory on AdS_4 × M^{111}, an N=2 background with isometry SU(3) × SU(2) × U(1), by performing a systematic harmonic analysis on the seven-manifold M^{111}. The authors organize the spectrum into long, short, and massless Osp(2|4) multiplets, and provide the full SU(3) × SU(2) flavor representation content along with the exact Clifford-vacuum energies E_0 and hypercharges y_0 for each multiplet. They derive the eigenvalues of the Laplace–Beltrami operators on zero-, one-, two-, and three-forms, as well as spinors, and show how these feed into the multiplet structure, including the Betti (massless) multiplet and the universal long-multiplet framework applicable to other N=2 AdS_4 compactifications. The results supply a rigorous KK-side foundation for testing the AdS_4/CFT_3 correspondence in this class of models and pave the way for analogous analyses on other G/H cosets, with potential links to the dual 3D SCFTs and their operator spectra. The work thus advances nontrivial checks of holography beyond the well-studied S^7 case by exploiting the detailed geometric and harmonic-analysis structure of M^{111}.

Abstract

We reconsider the Kaluza Klein compactifications of D=11 supergravity on AdS_4x(G/H)_7 manifolds that were classified in the eighties, in the modern perspective of AdS_4/CFT_3 correspondence. We focus on one of the three N=2 cases: (G/H)_7=M^{111}=SU(3)xSU(2)xU(1)/SU(2)xU(1)'xU(1)''. Relying on the systematic use of the harmonic analysis techniques developed in the eighties by one of us (P. Fre') with R. D'Auria, we derive the complete spectrum of long, short and massless Osp(2|4)xSU(3)xSU(2) unitary irreducible representations obtained in this compactification. Our result also provides a general scheme for the other N=2 compactifications. Furthermore, it is a necessary comparison term in the AdS_4/CFT_3 correspondence: the complete AdS/CFT match of the spectra that we obtain will provide a much more stringent proof of the AdS/CFT correspondence than in the S^7 case, since the structure of the superconformal field theory on the M2-brane world volume must be such as to reproduce, at the level of composite operators, the flavor group representations, the conformal dimensions and the hypercharges that we obtain in the present article. The investigation of the match is left to future publications. Here we provide an exhaustive construction of the Kaluza Klein side of our spectroscopy.

Paper Structure

This paper contains 23 sections, 187 equations, 8 figures, 15 tables.

Table of Contents

  1. Introduction to $M^{111}$: old and new viewpoints
  2. Old Kaluza Klein supergravity: the search for a realistic model based on $M^{pqr}$ spaces and harmonic analysis
  3. The results of this paper in the modern perspective of $AdS/CFT$ correspondence
  4. Structure of the paper
  5. Structure of the $Osp\left(2|4\right)$ multiplets
  6. The complete spectrum of $Osp(2\vert 4) \times SU\left(3\right)\times SU\left(2\right)$ supermultiplets
  7. The differential geometry of $M^{111}$
  8. Harmonic analysis on $M^{111}$
  9. The constraints on the irreducible representations
  10. Differential calculus via harmonic analysis
  11. The zero-form
  12. The one-form
  13. The two-form
  14. The three-form
  15. The spinor
  16. ...and 8 more sections

Figures (8)

  • Figure 1: ${\cal N}=2\rightarrow {\cal N}=1$ decomposition of the long graviton multiplet
  • Figure 2: ${\cal N}=2\rightarrow {\cal N}=1$ decomposition of the long gravitino multiplet
  • Figure 3: ${\cal N}=2\rightarrow{\cal N}=1$ decomposition of the long vector multiplet
  • Figure 4: ${\cal N}=2\rightarrow{\cal N}=1$ decomposition of the short graviton multiplet
  • Figure 5: ${\cal N}=2\rightarrow{\cal N}=1$ decomposition of the short gravitino multiplet
  • ...and 3 more figures