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Flat Eleven-Dimensional CJS Bosonic Vacuum

Physical scope

Exact example of a source-free classical bosonic solution in the adopted two-derivative CJS truncation. It asserts neither supersymmetry nor quantum stability.

Statement

On $M = R^{1, 10}$ with coordinates $x^{0}, \dots, x^{10}$ , orientation $d x^{0} \land \dots \land d x^{10}$ , and metric

$g = η = - (d x^{0})^{2} + \sum_{i = 1}^{10} (d x^{i})^{2},$

take $C_{3} = 0$ , so $F_{4} = d C_{3} = 0$ . This is an exact source-free bosonic solution of the field equations in Source-Free Eleven-Dimensional CJS Bosonic Field Equations for the action of Eleven-Dimensional CJS Supergravity Bosonic Action.

Facts & Assumptions

Given: The adopted source-free CJS bosonic action and its field equations, and the stated smooth flat Lorentzian fields on $R^{1, 10}$ .

[F1]

Eleven-Dimensional Oriented Lorentzian Geometry fixes the signature and Hodge-star convention; Levi-Civita Connection on an Eleven-Dimensional Lorentzian Manifold gives the Levi-Civita coefficients from the metric.

[F2]

Eleven-Dimensional CJS Supergravity Bosonic Action and Source-Free Eleven-Dimensional CJS Bosonic Field Equations specify the model and the conditional classical equations being checked.

Proof

technique · direct substitution

1.1F1algebra

The metric components are constant, so every partial derivative $\partial_{M} g_{N Q}$ vanishes. The Christoffel formula in Levi-Civita Connection on an Eleven-Dimensional Lorentzian Manifold therefore gives $Γ^{P}_{M N} = 0$ . Substitution in the stated curvature convention gives $R^{P}_{Q M N} = 0$ , hence $R_{M N} = 0$ and $R = 0$ .

2.1F1F2step 1.1algebra

Since $C_{3} = 0$ , $F_{4} = d C_{3} = 0$ ; linearity of $d$ and $*_{g}$ gives $d F_{4} = 0$ , $*_{g} F_{4} = 0$ , $d *_{g} F_{4} = 0$ , and $F_{4} \land F_{4} = 0$ . Thus the four-form equation is zero equals zero. Every component $F_{M A B C} F_{N}^{A B C}$ and $F_{A B C D} F^{A B C D}$ is zero, while $R_{M N} = 0$ by step 1.1, so the Einstein equation also holds.

3.1F2step 1.1step 2.1∎

Each term of the action density vanishes for these fields. The configuration is therefore an exact solution of the stated source-free classical bosonic equations. No Killing-spinor condition or supersymmetry conclusion is asserted.

Flat Eleven-Dimensional CJS Bosonic Vacuum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels