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Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities

José Figueroa-O'Farrill, George Papadopoulos

TL;DR

The paper delivers a complete local classification of maximally supersymmetric backgrounds in eleven- and ten-dimensional supergravity, showing that all such vacua arise from either AdS×S geometries, flat space, or Hpp-waves. The authors derive this by enforcing the vanishing of the supercovariant curvature $\\mathcal{R}_{MN}$, decomposing $F$ via gamma-matrix expansions, and proving that the four-form $F$ must be parallel and decomposable; this leads to a dichotomy between Freund–Rubin product geometries and Cahen–Wallach plane waves. The results yield explicit locally isometric backgrounds: in M-theory the AdS$_7\times$S$^4$, AdS$_4\times$S$^7$ vacua with corresponding $F$-flux, the flat vacuum, and the maximally supersymmetric Hpp-wave; in IIB the AdS$_5\times$S$^5$ and the maximally supersymmetric Hpp-wave; and in the remaining theories only flat space vacua or none in the massive IIA case. These findings have implications for AdS/CFT and the role of plane-wave limits in string theory, providing a geometric and algebraic framework for understanding all maximal SUSY backgrounds across ten and eleven dimensions.

Abstract

We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.

Maximally supersymmetric solutions of ten- and eleven-dimensional supergravities

TL;DR

The paper delivers a complete local classification of maximally supersymmetric backgrounds in eleven- and ten-dimensional supergravity, showing that all such vacua arise from either AdS×S geometries, flat space, or Hpp-waves. The authors derive this by enforcing the vanishing of the supercovariant curvature , decomposing via gamma-matrix expansions, and proving that the four-form must be parallel and decomposable; this leads to a dichotomy between Freund–Rubin product geometries and Cahen–Wallach plane waves. The results yield explicit locally isometric backgrounds: in M-theory the AdSS, AdSS vacua with corresponding -flux, the flat vacuum, and the maximally supersymmetric Hpp-wave; in IIB the AdSS and the maximally supersymmetric Hpp-wave; and in the remaining theories only flat space vacua or none in the massive IIA case. These findings have implications for AdS/CFT and the role of plane-wave limits in string theory, providing a geometric and algebraic framework for understanding all maximal SUSY backgrounds across ten and eleven dimensions.

Abstract

We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.

Paper Structure

This paper contains 25 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction and main results
  2. Maximal supersymmetry in eleven-dimensional supergravity
  3. Eleven-dimensional supergravity
  4. The zero curvature equations
  5. Terms linear in $\Gamma$
  6. Terms quadratic in $\Gamma$
  7. Terms cubic in $\Gamma$
  8. Terms quartic in $\Gamma$
  9. Terms quintic in $\Gamma$
  10. $F$ is decomposable
  11. The local geometries
  12. The Freund--Rubin Ansatz revisited
  13. The case of a null four-form
  14. Maximal supersymmetry in IIA supergravity
  15. Maximal supersymmetry in IIB supergravity
  16. ...and 10 more sections

Key Result

Theorem 1

Let $(M,g,F_4)$ be a maximally supersymmetric solution of eleven-dimensional supergravity. Then it is locally isometric to one of the following:

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6