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Some rigidity results for supergravity backgrounds in 11 dimensions

Emanuele Di Bella, Willem A. de Graaf, Andrea Santi

Abstract

This paper is a contribution to the supersymmetry gap problem for supergravity backgrounds $(M,g,F)$ in $11$ dimensions. We study restrictions on the curvature of $(M,g,F)$ and, using the bijective correspondence between the space of certain filtered deformations of Lie superalgebras and the space of highly supersymmetric supergravity backgrounds, we establish the following general rigidity result: if the $4$-form $F$ has rank $\operatorname{rk}(F)\leq 6$, Euclidean support, and the space $\mathfrak{k}_{\bar 1}$ of Killing spinors has dimension $\dim\mathfrak{k}_{\bar 1}> 26$ then $(M,g,F)$ is locally isometric to the maximally supersymmetric Minkowski spacetime or Freund Rubin background $\mathrm{AdS}_7\times\mathrm{S}^4$. The same rigidity result but with finer estimates on $\dim\mathfrak{k}_{\bar 1}$ is provided for certain types of $\mathfrak k_{\bar 1}$ and specific orbits of the $4$-form under the action of the Lorentz group.

Some rigidity results for supergravity backgrounds in 11 dimensions

Abstract

This paper is a contribution to the supersymmetry gap problem for supergravity backgrounds in dimensions. We study restrictions on the curvature of and, using the bijective correspondence between the space of certain filtered deformations of Lie superalgebras and the space of highly supersymmetric supergravity backgrounds, we establish the following general rigidity result: if the -form has rank , Euclidean support, and the space of Killing spinors has dimension then is locally isometric to the maximally supersymmetric Minkowski spacetime or Freund Rubin background . The same rigidity result but with finer estimates on is provided for certain types of and specific orbits of the -form under the action of the Lorentz group.
Paper Structure (14 sections, 24 theorems, 54 equations)

This paper contains 14 sections, 24 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. The Poincaré superalgebra, its filtered subdeformations, and realizability
  4. The Reconstruction Theorem and the Dirac kernel
  5. General strategy and first results
  6. Brief overview on general strategy
  7. The $\mathrm{SO}(V)$-orbits in $\Lambda^4 V$ of small rank and with Euclidean support
  8. Reduction to $\mathrm{SO}(V)$-orbits in $\Lambda^4 V$ of rank $6$
  9. The interplay of spinors and fourvectors
  10. Clifford algebras and admissible bilinear forms
  11. The Dirac current
  12. Curvature in superspace
  13. The main theorem: case of fourvectors of rank $6$ and length $3$
  14. The main theorem: case of fourvectors of rank $6$ and length $2$

Key Result

Theorem 1.1

GGPRGGP If a background of $D=11$ supergravity has at least $30$ Killing spinors, then it is locally isometric to a maximally supersymmetric background.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Reconstruction Theorem
  • Definition 2.6
  • ...and 32 more