标准嵌入附近的异质 $G_2$ 系统的导出形变理论
Derived deformation theory of heterotic $G_2$ systems near the standard embedding
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中文总结 AI 辅助
研究异质 $G_2$ 系统在标准嵌入附近的导出形变理论，通过相容性扩展和循环转移构造有效势，证明相对纤维同伦交换。
中文摘要 AI 辅助
我们研究在固定的自旋七维流形 $Y$ 上的弦 Courant 代数胚上的异质 $G_2$ 结构，在无挠标准嵌入附近，对 $\alpha'$ 一阶以及对形变的所有形式阶。可容许度量、旋量线和密度确定一个典范的广义 Dirac 泛函，其在物理分裂中的表达式是异质超势。我们在小通量扇区 $H^{[0]}=0$ 中工作。我们附加该条件及其相容性恒等式以获得一个可解物理理论，并构造相应的变分临界理论的相容性扩展。这两个解作为实局部形式 $Q$-理论是局部形式同构的。当 $Y$ 是闭的且 gerbe 和反常扇区固定时，超势的临界场与集中在零度的每个局部实 Artin 代数上的 BPS 场一致。平移余切哈密顿构造使变分临界理论成为循环的。混合阶椭圆性和同伦转移给出有限维最小模型，循环转移在这些零度 Artin 代数上对普通形式物理解轨迹产生一个有效势 $W_{\mathrm{eff}}$，在形式坐标变换之后。我们计算该相容性比较的相对上同调，它在 dg-Artin 代数上被探测到并被变分临界理论遗忘，并证明标准嵌入处的尖点相对纤维是同伦交换的。
英文摘要
We study heterotic $G_2$ structures on a fixed string Courant algebroid on a spin seven-manifold $Y$, near a torsion-free standard embedding, to first order in $α'$ and to all formal orders in the deformations. The admissible metric, spinor line and density determine a canonical generalized Dirac functional whose expression in a physical splitting is the heterotic superpotential. We work in the small-flux sector $H^{[0]}=0$. We adjoin this condition and its compatibility identities to obtain a resolved physical theory, and construct a corresponding compatibility extension of the variational critical theory. The two resolutions are locally formally isomorphic as real local formal $Q$-theories. When $Y$ is closed and the gerbe and anomaly sector is fixed, the critical fields of the superpotential coincide with the BPS fields over every local real Artin algebra concentrated in degree zero. The shifted-cotangent Hamiltonian construction makes the variational critical theory cyclic. Mixed-order ellipticity and homotopy transfer give finite-dimensional minimal models, and cyclic transfer yields an effective potential $W_{\mathrm{eff}}$ for the ordinary formal physical solution locus over these degree-zero Artin algebras, after a formal coordinate change. We compute the relative cohomology of this compatibility comparison, which is detected on dg-Artin algebras and forgotten by the variational critical theory, and prove that the pointed relative fibre at the standard embedding is homotopy abelian.
