标准嵌入下物理heterotic $G_2$形变复形的链级解耦
Chain-level decoupling of the physical heterotic $G_2$ deformation complex at the standard embedding
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中文总结 AI 辅助
本文在标准嵌入下对heterotic $G_2$形变复形给出显式链级解耦，通过两个同伦消除瞬子与几何耦合，实现所有次数上同调分解，仅需和乐群包含于$G_2$。
中文摘要 AI 辅助
heterotic $G_2$紧化的物理形变复形将瞬子形变与几何场及其一阶$\alpha'$修正耦合在一起。我们在最小标准嵌入下（其中规范丛为切丛，规范连接为Levi-Civita连接，背景$G_2$结构无挠）给出了其一阶系数复形的显式链级解耦。第一个Atiyah耦合通过一个扩展诱导连接变化的斜协变导数是零同伦的。第二个同伦由端值上链的斜部分的协变余微分给出，它消除了从丛扇区到几何扇区的耦合。剩余的几何曲率项与第一个变换的贡献相抵消。这些恒等式在任意上链的每个典范次数上成立，并产生互逆的微分算子链映射。因此，上同调在所有次数上分解为瞬子系数上同调和几何系数上同调。该论证仅需$\mathrm{Hol}(g)\subseteq G_2$。据我们所知，此前没有显式的全次数分裂能消除这个物理约化的标准嵌入系数复形的内部规范耦合和几何耦合。
英文摘要
The physical deformation complex of heterotic $G_2$ compactifications couples instanton deformations to geometric fields and their first-order $α'$ corrections. We give an explicit chain-level decoupling of its first-order coefficient complex at the minimal standard embedding, where the gauge bundle is the tangent bundle, the gauge connection is the Levi--Civita connection, and the background $G_2$ structure is torsion-free. The first Atiyah coupling is null-homotopic through a skew covariant derivative extending the induced connection variation. A second homotopy, given by the covariant codifferential of the skew part of an endomorphism-valued cochain, removes the coupling from the bundle sector to the geometric sector. The residual geometric curvature term cancels against the contribution of the first transformation. These identities hold on arbitrary cochains in every canonical degree and yield mutually inverse differential-operator chain maps. Consequently, the cohomology decomposes into instanton and geometric coefficient cohomologies in all degrees. The argument requires only $\mathrm{Hol}(g)\subseteq G_2$. To our knowledge, no previous explicit all-degree splitting removes the internal gauge and geometric couplings of this physically reduced standard-embedding coefficient complex.
