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Eleven-Dimensional CJS Supergravity Bosonic Action

Physical scope

Adopted source-free, two-derivative, classical eleven-dimensional CJS bosonic action and Planck normalization. It is the fermion-free truncation of the low-energy supergravity framework; no brane sources, higher-derivative corrections, quantum effects, or field equations are asserted here.

Postulate

Adopt the following classical bosonic truncation of eleven-dimensional Cremmer–Julia–Scherk (CJS) supergravity. Let $M$ be an oriented Lorentzian eleven-manifold without boundary with metric $g$ of signature $(- + \dots +)$ , Levi-Civita connection from Levi-Civita Connection on an Eleven-Dimensional Lorentzian Manifold, scalar curvature $R$ , and Hodge star from Eleven-Dimensional Oriented Lorentzian Geometry. Let $C_{3} \in Ω^{3} (M)$ be a globally defined smooth real three-form potential and set $F_{4} : = d C_{3}$ . In natural units $ℏ = c = 1$ , take the action

$S [g, C_{3}] = \frac{1}{2 κ_{11}^{2}} \int_{M} (R {vol}_{g} - \frac{1}{2} F_{4} \land *_{g} F_{4} - \frac{1}{6} C_{3} \land F_{4} \land F_{4}) .$

Here $ℓ_{p} > 0$ is the eleven-dimensional Planck length and the adopted normalization is

$2 κ_{11}^{2} = (2 π)^{8} ℓ_{p}^{9} .$

Thus $[ℓ_{p}] = l e n g t h$ and $[κ_{11}^{2}] = {l e n g t h}^{9}$ . In coordinates of length dimension with dimensionless metric components, the action density is dimensionless after integration when $[C_{M N P}] = 1$ , $[F_{M N P Q}] = {l e n g t h}^{- 1}$ , and $[R] = {l e n g t h}^{- 2}$ . The form norm is signed Lorentzian norm $∣ F_{4} ∣_{g}^{2} = (1 / 4!) F_{M N P Q} F^{M N P Q}$ .

The CJS multiplet includes a 32-real-component Majorana gravitino and one 32-real-component local supersymmetry parameter; the displayed action omits all fermion terms. This adopted action is the source-free, two-derivative low-energy truncation. It omits brane currents and boundary terms, the higher-derivative $C_{3} \land X_{8}$ correction, and quantum corrections. The displayed Chern–Simons sign is tied to the stated potential convention: replacing $C_{3}$ by $- C_{3}$ reverses that sign along with $F_{4} = d C_{3}$ . The item asserts a model convention, not a universal law or a claim that every eleven-manifold admits these fields.

Postulate references

Hamilton, §5.1, Definition 5.1, and §8.2, Remark 8.26; Taylor, §1.4, equation (9); Gauntlett–Pakis, §2, equation (2.1).

Facts & Assumptions

Given: The action is adopted in the source-free, classical, two-derivative regime stated above.

[A1]

Supergravity Conventions And Normalizations: its current carrier declares natural units $ℏ = c = 1$ , Lorentzian signature $(- + \dots +)$ , the signed form norm $∣ F_{p} ∣^{2} = (1 / p!) F_{M_{1} \dots M_{p}} F^{M_{1} \dots M_{p}}$ , and one 32-real-component Majorana parameter for 11D CJS. Its current Step 1 record is ready at hash $41 d 207 e 61 d b e f 5573 f 48 b 5 c f 2 e 70 c 2 c 19 a 13696 e c 755 a 3 c 8047 c 832905 c 0 b 550$ . The current carrier does not state the $ℓ_{p}$ -to- $κ_{11}$ relation; that separate normalization is adopted here directly from Hamilton §8.2, Remark 8.26.

[F1]

Eleven-Dimensional Oriented Lorentzian Geometry defines the oriented metric volume form, signed Lorentzian form pairing, and Hodge star; Levi-Civita Connection on an Eleven-Dimensional Lorentzian Manifold supplies the unique torsion-free metric connection used for $R$ .

[F2]

Supergravity Conventions And Normalizations fixes the curvature convention $R^{P}_{Q M N} = \partial_{M} Γ^{P}_{N Q} - \partial_{N} Γ^{P}_{M Q} + Γ^{P}_{M R} Γ^{R}_{N Q} - Γ^{P}_{N R} Γ^{R}_{M Q}, R_{Q N} = R^{P}_{Q P N}, R = g^{Q N} R_{Q N} .$ The scalar curvature $R$ in the Postulate uses this convention and the Levi-Civita connection supplied in [F1].

Remarks

Variational equations are addressed in Source-Free Eleven-Dimensional CJS Bosonic Field Equations.

Taylor’s equation (9) is used as an independent low-energy field-content and action-form comparison. Its locator is not used to assert the exact $(2 π)^{8} ℓ_{p}^{9}$ normalization. The action is interpreted through differences under compactly supported variations when $M$ is noncompact; no finite total action is assumed.

Eleven-Dimensional CJS Supergravity Bosonic Action

Postulate

Postulate references

Facts & Assumptions

Remarks

Depends on

Dependency tree · 2 levels

Sources