异质 $G_2$ 形变理论中的循环 $L_3$ 模型与全纯约化
Cyclic $L_3$ Models and Holomorphic Reduction in Heterotic $G_2$ Deformation Theory
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中文总结 AI 辅助
本文构造异质G2形变理论的循环L3代数表示，通过辅助场约化恢复物理泛函，并在乘积轨迹上约化为全纯形式，证明其可由特征哈密顿约化得到，与已知全纯异质代数一致。
中文摘要 AI 辅助
我们构造了由异质 $G_2$ 超势在无扭转标准嵌入附近、至 $\alpha'$ 一阶所定义的变分形变理论的有限循环 $L_3$ 表示（即括号在三阶以上为零的循环 $L_\infty$ 代数）。逆余标架、旋量导数、通量和诱导切触联络的辅助场给出了次数至多为四的平移余切哈密顿量。消去辅助场后恢复了物理泛函。在平坦七环面上，该泛函在局部物理旋量坐标中具有非零的五次泰勒系数。对于 $Y=S^1\times X$（其中 $X$ 为卡拉比-丘三维流形），我们限制在具有水平规范和杰尔贝数据的乘积轨迹上。该泛函约化为 $ \frac L4\operatorname{Im}\int_X(H+i\\,d\omega)\wedge\Omega, $ 其中 $L$ 为圆长度，$\Omega=e^{-2\Phi}\Psi$ 为加权复体积。诱导的 $SU(3)$ 结构选取了一个 Hermitian 切触 Courant 子代数胚。其复化中的横向各向同性提升编码了全纯联络和弦方程，连同诱导切触联络、加权典范线和杰尔贝层级。它们的泰勒括号构成一个 $L_3$ 代数，其弦部分与已知的全纯异质代数一致。然后我们证明，该全纯理论是通过在每一个上链次数上对乘积变分理论的有限可缩稳定化进行特征哈密顿约化而得到的。虽然商空间捕获了全纯方程，但共形平衡、规范本原性和前导通量可容许性是实代表上的附加条件。
英文摘要
We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $α'$. Auxiliary fields for the inverse coframe, spinor derivative, flux and induced tangent connection give a shifted-cotangent Hamiltonian of degree at most four. Eliminating the auxiliary fields recovers the physical functional. On the flat seven-torus, this functional has a nonzero quintic Taylor coefficient in local physical spinor coordinates. For $Y=S^1\times X$, with $X$ a Calabi--Yau threefold, we restrict to the product locus with horizontal gauge and gerbe data. The functional reduces to $ \frac L4\operatorname{Im}\int_X(H+i\,dω)\wedgeΩ, $ where $L$ is the circle length and $Ω=e^{-2Φ}Ψ$ is the weighted complex volume. The induced $SU(3)$ structure selects a Hermitian tangent Courant subalgebroid. Transverse isotropic lifts in its complexification encode the holomorphic connection and string equations, together with the induced tangent connection, weighted canonical line and gerbe hierarchy. Their Taylor brackets form an $L_3$ algebra whose string part agrees with the known holomorphic heterotic algebra. We then show that this holomorphic theory is obtained by characteristic Hamiltonian reduction of a finite contractible stabilization of the product variational theory, in every cochain degree. While the quotient captures the holomorphic equations, conformal balance, gauge primitivity and leading-flux admissibility are additional conditions on a real representative.
