The warp factor of supersymmetric $D=11$ near-horizon geometries: single-point rigidity, the $Spin(7)$ perfect square, and global constraints

Abstract

On a compact connected section $\mathcal{S}$ of a supersymmetric $M$-horizon the Killing-spinor bilinears give a pointwise identity relating the warp factor $\Delta$, the rotation one-form $h$ and the two spinor norms, with nothing assumed about the norm of the Killing spinor. We derive three consequences of that unreduced identity. First, a single-point rigidity theorem: if $\Delta$ and $h$ vanish at one point of the section, then the flux vanishes identically, $\mathcal{S}$ is Ricci-flat and the horizon is ${\mathbb{R}}^{1,1}\times \mathcal{S}$. A single point replaces the usual global spinorial hypothesis. Second, the $\mathrm{Spin}(7)$ decomposition of the flux fixes $h$ algebraically and exhibits the invariant flux content of the scalar square $\Delta =4{\Phi }^2$ known in an adapted gauge, $\Delta =\frac{1}{108}\Vert w_7\pm \sigma ({\mathcal{Y}}_7){\Vert }^2$. Only the two $7$-summands $w_7$, ${\mathcal{Y}}_7$ reach the warp factor, and the square is degenerate rather than definite: it vanishes on the linear subspace $w_7=\mp \sigma ({\mathcal{Y}}_7)$ rather than at the origin, so positivity of $\Delta$ yields no case list. Third, the two combine into a pointwise budget: the single constant of supersymmetry is shared between the deviation from staticity and the $\mathrm{Spin}(7)$ mismatch of the flux, each bounded by that constant. The same identities characterise the constancy hypothesis, equivalent to $V=-fh$ for the bilinear one-form $V$, which fails on the static branch of known warped $Ad{S}_2$ solutions; they fix a weighted integral of the warp factor; and they reduce the Komar angular momentum of the horizon to a positive bilinear integral. Two no-go statements, for hidden symmetries built from the Killing spinor and for a second isometry from its bilinears, are stated with their hypotheses.

68 pages. Verification scripts and logs available from the author on request