We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.

中文翻译：

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扩展稳定图边界中的橡胶环

我们研究了枚举几何背景下对数展开的环面作用。我们的主要结果是对出现在扩展稳定图空间边界中的更高阶橡胶环的内在且无坐标的描述。橡胶环是根据热带模量空间规范构建的，其对扩展目标的每个层的作用被编码在线性热带位置图中。通用目标力中 2-态射的存在扩展了稳定图，其与待识别的橡胶作用不同。这为扩展模空间边界的递归描述迈出了第一步，未来的应用包括定位和橡胶微积分。