AI 中文总结
该研究针对共形耦合标量的宇宙学入入关联子，引入热带积分与弦论积分，推导了有限α'相关性质，还给出图关联多面体的几何实现及相关方程。
AI 中文摘要
我们引入了用于共形耦合标量宇宙学入入关联子的固定图贡献的热带积分与弦论积分。全时表示可分解为依赖于图的顶点空间拉普拉斯积分（每个图顶点对应一个实变量）与基本边空间拉普拉斯积分（每个内部边对应一个实变量）。顶点空间指数是与位点和相对边时间相关的绝对值之和；作为分段线性函数，它是入入 zonotope 的支撑函数。每个绝对值也是正 Laurent 二项式的热带极限。在热带化前保留这些二项式定义了有限 α' 的顶点空间弦论积分，因此多面体及其弦论积分均可直接从物理时间积分中读出。在 α'→0 极限下，该积分变为入入 zonotope 的归一化对偶体积，而边空间因子独立变形为 β 积分的乘积并恢复基本传播子归一化。我们从增广图腔中导出场论有理形式，以及精确的有限 α' 平行边约化、边能和部分能极点的因式分解公式，甚至后代塔。作为固定图关联子的替代几何实现，我们发现树图的图关联多面体的环境 Minkowski 和与弦论积分实现为其线图的所谓图立方体多面体。为完整起见，我们还记录了相关仿射除子排列的对数临界方程和通用参考次数。
英文摘要
We introduce tropical and stringy integrals for fixed-graph contributions to cosmological in-in correlators of conformally coupled scalars. The full-time representation factorizes into a graph-dependent vertex-space Laplace integral, with one real variable for each graph vertex, and an elementary edge-space Laplace integral, with one real variable for each internal edge. The vertex-space exponent is a sum of absolute values associated with sites and relative edge times; as a piecewise-linear function, it is the support function of the in-in zonotope. Each absolute value is also the tropical limit of a positive Laurent binomial. Retaining these binomials before tropicalization defines a finite-$α'$ vertex-space stringy integral, so both the polytope and its stringy integral are read directly from the physical time integral. In the $α' \to 0$ limit this integral becomes the normalized dual volume of the in-in zonotope, while the edge-space factor deforms independently into a product of beta integrals and restores the elementary propagator normalization. We derive the field-theory rational form from augmented-graph chambers, as well as exact finite-$α'$ parallel-edge reduction, factorization formulas for edge-energy and partial-energy poles, and even descendant towers. As an alternative geometric realization of fixed-graph correlators, we find an ambient Minkowski-sum and stringy-integral realization of the graph correlahedron for a tree graph as the so-called graph cubeahedron of its line graph. For completeness, we also record the logarithmic critical equations and generic reference degrees of the associated affine divisor arrangement.
Comments27+11 pages, 14 figures
