度规-仿射Gauss--Bonnet引力中的一个主符号障碍
A principal-symbol obstruction in metric-affine Gauss--Bonnet gravity
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中文总结 AI 辅助
该研究比较了度规-仿射Gauss--Bonnet引力中两种延拓的联络主符号，发现度规缩并延拓存在非零主符号障碍，而双epsilon延拓则无此障碍，表明领头微分结构依赖于仿射延拓的选择。
中文摘要 AI 辅助
四维黎曼Gauss--Bonnet组合允许不止一种延拓到具有独立仿射联络的几何。两个自然的候选者，即度规缩并代表和双epsilon代表，在Levi-Civita极限下一致，我们询问它们是否也赋予独立联络相同的领头微分结构。在一个具有标量耦合$f(\phi)$到Gauss--Bonnet项的度规-仿射模型中，我们比较了它们在Levi-Civita轨迹附近的二次二导数联络主形式。对于度规缩并延拓，主形式在时间型主余向量处的二十四维无钩无挠见证子空间上求值，其秩为$18$，零度为$6$，且是不定的；因此，在同一余向量处的完整无挠形式是不定的，且秩至少为$18$。在$f(\phi)\neq0$的局部区域，线性化联络方程的二导数主符号非零，这阻碍了在激活方向上联络的纯代数局部消去。对于双epsilon延拓，相应的二次二导数形式在关于该轨迹的联络的任意扰动下恒为零，因此其主符号也随之消失，且该特定障碍不存在，尽管其不存在本身并不建立代数可消去性。因此，独立联络的领头微分结构取决于黎曼Gauss--Bonnet组合的仿射延拓。
英文摘要
The four-dimensional Riemannian Gauss--Bonnet combination admits more than one continuation to a geometry with an independent affine connection. Two natural candidates, the metric-contracted representative and the double-epsilon representative, agree in their Levi-Civita limit, and we ask whether they also give the independent connection the same leading differential structure. In a metric-affine model with a scalar coupling $f(ϕ)$ to the Gauss--Bonnet term, we compare their quadratic two-derivative connection principal forms about the Levi-Civita locus. For the metric-contracted continuation, the principal form, evaluated on a twenty-four-dimensional hook-free torsion-free witness subspace at a timelike principal covector, has rank $18$ and nullity $6$ and is indefinite; the full torsion-free form at the same covector is therefore indefinite with rank at least $18$. Locally where $f(ϕ)\neq0$, the two-derivative principal symbol of the linearized connection equations is nonzero, which obstructs purely algebraic local elimination of the connection in the activated directions. For the double-epsilon continuation, the corresponding quadratic two-derivative form vanishes identically for an arbitrary perturbation of the connection about that locus, so its principal symbol vanishes with it, and that particular obstruction is absent, though its absence does not by itself establish algebraic eliminability. Thus the leading differential structure of the independent connection depends on the affine continuation of the Riemannian Gauss--Bonnet combination.
