Math 2.0

Relativity and gravitational theory Definition

Eleven-dimensional supergravity

Builds on SupergravityDifferential formExterior derivativeHodge star operator

Definition

Let MM be an eleven-dimensional spin manifold. Eleven-dimensional supergravity is the supergravity theory on MM whose fields are:

  1. a vielbein eaμe^a{}_\mu, giving a Lorentzian metric gg;
  2. one Majorana gravitino ψμ\psi_\mu, whose spinor index takes 3232 values;
  3. a three-form gauge potential AA, a differential form with field strength F=dAF = dA, where dd is the exterior derivative.

The bosonic part of its action is S=12κ2∫M(R ⋆1−12F∧⋆F−16A∧F∧F).S = \frac{1}{2\kappa^2} \int_M \left( R \, {\star 1} - \tfrac12 F \wedge {\star F} - \tfrac16 A \wedge F \wedge F \right). Here RR is the scalar curvature of gg, ⋆\star is the Hodge star of gg and κ\kappa is the gravitational coupling. The fermionic terms are fixed by local supersymmetry. The signs and normalizations depend on conventions.

Remarks

The gauge transformation A↦A+dΛA \mapsto A + d\Lambda, for a two-form Λ\Lambda, leaves FF unchanged. It changes the last term only by an exact form, so the action is gauge invariant on a closed MM. Counting physical degrees of freedom: the metric has 4444, the three-form has (93)=84\binom{9}{3} = 84, and the gravitino has 128128, so there are 128128 bosonic and 128128 fermionic ones. Example: AdS4×S7\mathrm{AdS}_4 \times S^7 and AdS7×S4\mathrm{AdS}_7 \times S^4, with FF proportional to the volume form of the four-dimensional factor, are solutions. Non-example: dropping the term A∧F∧FA \wedge F \wedge F gives a bosonic theory that is no longer invariant under local supersymmetry.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

Show the path as a graph
Differential form, concept, no evidence levelDifferential formconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHodge star operator, concept, no evidence levelHodge star operatorconceptSupergravity, concept, no evidence levelSupergravityconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementDifferential form, definition, no evidence levelDifferential formdefinitionParametrised curve, definition, no evidence levelParametrised curvedefinitionVector field, definition, no evidence levelVector fielddefinitionWedge product of forms, definition, no evidence levelWedge product of formsdefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionRectifiable curve, definition, no evidence levelRectifiable curvedefinitionCurl, definition, no evidence levelCurldefinitionDivergence, definition, no evidence levelDivergencedefinitionLine integral of a vector field, definition, no evidence levelLine integral of a ve…definitionOne-form, definition, no evidence levelOne-formdefinitionExterior derivative, definition, no evidence levelExterior derivativedefinitionEleven-dimensional supergravity, definition, no evidence levelEleven-dimensional su…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential form concept
  2. Euclidean space object
  3. Function concept
  4. Hodge star operator concept
  5. Supergravity concept
  6. Fréchet derivative definition
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Limit of a function of several variables concept
  10. Partial derivative concept
  11. Continuity in R^n definition
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Differentiable on an open set definition
  15. Gradient definition
  16. Jacobian matrix definition
  17. Higher partial derivatives definition
  18. Iterated derivative definition
  19. C^k function characterised by continuous derivatives statement
  20. Differential form definition
  21. Parametrised curve definition
  22. Vector field definition
  23. Wedge product of forms definition
  24. Conservative vector field definition
  25. Rectifiable curve definition
  26. Curl definition
  27. Divergence definition
  28. Line integral of a vector field definition
  29. One-form definition
  30. Exterior derivative definition
  31. Eleven-dimensional supergravity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Eleven-dimensional_supergravity
About this page
TagJBBFXDCNXV
URIm2://JBBFXDCNXV
Kinddefinition
WritingW0 stub
MSC83E50, 81T60, 83E15
DefinesEleven-dimensional supergravity
UsesSupergravity, Differential form, Exterior derivative, Hodge star operator
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.