Supersymmetric Background in Eleven-Dimensional CJS Supergravity

Use the following source-reported CJS convention. Let $U\subseteq M$ be a nonempty open set, with $M$ an eleven-manifold carrying the spin structure and 32-real-dimensional Majorana bundle $S$ defined in Eleven-Dimensional Spin Structure and Majorana Bundle. Let $(g,F_4)$ be a smooth source-free bosonic CJS solution on $U$, meaning the equations in Source-Free Eleven-Dimensional CJS Bosonic Field Equations hold there. Put $\eta =\mathrm{diag}(-1,1,\dots ,1)$. Let $(E_0,\dots ,E_{10})$ be the standard basis of $V={\mathbb{R}}^{11}$ and $i_{\eta }:V\to \mathrm{C}\mathrm{l}(V,\eta )$ its canonical Clifford generator map. In a local spin frame above an oriented, time-oriented $\eta$-orthonormal frame $e:V\to TM$, write $e_A=e(E_A)$ and set ${\mathrm{\Gamma }}_A:=\rho (i_{\eta }(E_A))$ and ${\mathrm{\Gamma }}^A:={\eta }^{AB}{\mathrm{\Gamma }}_B$, acting on the local spinor components as in the spin-bundle definition. The CJS representation choice is

${\mathrm{\Gamma }}_0{\mathrm{\Gamma }}_1\cdots {\mathrm{\Gamma }}_{10}=+I_{\mathrm{\Delta }}\mathrm{.}$

For each $p$-form $\omega$ and spinor section $s$, define ${\mathrm{\Gamma }}^{A_1\cdots A_p}:=\frac{1}{p!}{\sum }_{\sigma \in S_p}\mathrm{sgn}(\sigma ){\mathrm{\Gamma }}^{A_{\sigma (1)}}\cdots {\mathrm{\Gamma }}^{A_{\sigma (p)}}$ and $\omega \cdot s:=\frac{1}{p!}{\omega }_{A_1\cdots A_p}{\mathrm{\Gamma }}^{A_1\cdots A_p}s$, where $S_p$ is the permutation group of $\{1,\dots ,p\}$. These products are normalized antisymmetrizations, matching Hamilton's §3.1 convention; they obey $\{{\mathrm{\Gamma }}_A,{\mathrm{\Gamma }}_B\}=2{\eta }_{AB}I$. Set $X^{\mathrm{♭}}=g(X,\cdot )$ and let ${\iota }_X{F}_4$ denote contraction in the first argument. Call $(g,F_4)$ supersymmetric on $U$ exactly when there exists a smooth Majorana spinor $ϵ\in \mathrm{\Gamma }(S{\mathrm{\mid }}_U)$, not identically zero, whose spin covariant derivative ${\mathrm{\nabla }}^S$ induced from the Levi-Civita connection satisfies, for every smooth vector field $X$ on $U$,

${\mathrm{\nabla }}_X^Sϵ+\frac{1}{12}(X^{\mathrm{♭}}\wedge F_4-2{\iota }_X{F}_4)\cdot ϵ=0.$

The anticommutator alone does not fix the volume-element sign; because the flux terms use odd-degree $3$- and $5$-form products, the representation choice is part of this displayed Killing-spinor convention. This defines the classical Killing-spinor condition only. It claims no existence, uniqueness, global extension, or classification of solutions and no quantum or brane-source statement.

Hamilton, §§3.1, 3.6, and §5.2, Theorem 5.7; Gauntlett–Pakis, §2, equation (2.3), and Appendix A.

Hamilton's form equation and Gauntlett–Pakis's component equation agree in this convention: the first term has coefficient $1\mathrm{/}(12\cdot 4!)=1\mathrm{/}288$; the contraction term has coefficient $-2\mathrm{/}(12\cdot 3!)=-1\mathrm{/}36=-8\mathrm{/}288$, matching $[{\mathrm{\Gamma }}_{\mu }{}^{{\nu }_1\cdots {\nu }_4}-8{\delta }_{\mu }^{{\nu }_1}{\mathrm{\Gamma }}^{{\nu }_2{\nu }_3{\nu }_4}]\mathrm{/}288$. This is a convention cross-check, not a proof that a nonzero solution exists.