超对称D=11近视界几何的翘曲因子：单点刚性定理、Spin(7)完全平方及整体约束
The warp factor of supersymmetric $D=11$ near-horizon geometries: single-point rigidity, the $Spin(7)$ perfect square, and global constraints
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中文总结 AI 辅助
该研究推导了超对称D=11近视界几何中翘曲因子的三个核心结论，含单点刚性定理、Spin(7)相关代数关系及逐点预算，还给出两个禁区定理，为相关几何研究提供关键约束。
中文摘要 AI 辅助
在超对称M视界的紧致连通截面S上，克利金旋量的双线态给出一个逐点恒等式，关联翘曲因子Δ、旋转1-形式h与两个旋量范数，且未对克利金旋量的范数做任何假设。我们从该未约化恒等式推导出三个结论：第一，单点刚性定理：若截面中某点的Δ和h均为零，则通量恒为零，S是里奇平坦的，且视界为ℝ^{1,1}×S，这里用单点取代了通常的整体旋量假设；第二，通量的Spin(7)分解代数地确定了h，并揭示了翘曲因子标量平方Δ=4Φ²的不变通量内容，该内容在适配规范下为Δ=(1/108)∥w₇±σ(𝒴₇)∥²，仅两个7-分量w₇、𝒴₇会影响翘曲因子，且该平方是退化的而非正定的：它在w₇=∓σ(𝒴₇)的线性子空间上为零，而非在原点处，因此Δ的正性不会产生情形列表；第三，前两个结论结合为逐点预算：超对称的单一常数在非静态偏差与通量的Spin(7)不匹配之间分配，两者均受该常数约束。这些恒等式还刻画了恒定性假设，其等价于双线态1-形式V满足V=-fh，该假设在已知的翘曲AdS₂解的静态分支上不成立；它们确定了翘曲因子的加权积分；并将视界的Komar角动量简化为正的双线态积分。我们还陈述了两个禁区定理，分别针对由克利金旋量构建的隐藏对称性及其双线态产生的第二个等距变换，并给出了它们的假设条件。
英文摘要
On a compact connected section ${\cal S}$ of a supersymmetric $M$-horizon the Killing-spinor bilinears give a pointwise identity relating the warp factor $Δ$, 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 $Δ$ and $h$ vanish at one point of the section, then the flux vanishes identically, ${\cal S}$ is Ricci-flat and the horizon is $\mathbb{R}^{1,1}\times{\cal S}$. A single point replaces the usual global spinorial hypothesis. Second, the ${\rm Spin}(7)$ decomposition of the flux fixes $h$ algebraically and exhibits the invariant flux content of the scalar square $Δ=4Φ^2$ known in an adapted gauge, $Δ=\frac{1}{108}\|w_{\bf 7}\pmσ({\cal Y}_{\bf 7})\|^2$. Only the two ${\bf 7}$-summands $w_{\bf 7}$, ${\cal Y}_{\bf 7}$ reach the warp factor, and the square is degenerate rather than definite: it vanishes on the linear subspace $w_{\bf 7}=\mpσ({\cal Y}_{\bf 7})$ rather than at the origin, so positivity of $Δ$ 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 ${\rm Spin}(7)$ 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 $AdS_2$ 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.
