使用同伦延拓求解离散引力中的面积-长度系统
Solving the area-length systems in discrete gravity using homotopy continuation
AI总结：
该研究利用数值代数几何工具求解离散引力中的面积-长度系统，证明单个4-单纯形最多有64组相容边长解，且通常无面积闭式表达，引入对称约束可得到边长表达式，为其在有效自旋泡沫模型中的应用迈出第一步。
摘要：
与度规表述中普遍存在的基本长度变量不同，面积变量是广义相对论联络表述的内禀变量。在四维离散引力中，尤其是基于三角剖分的离散引力框架里，面积-长度系统建立了与三角形关联的面积变量和边长变量之间的关系。该系统由从海伦公式推导而来的多项式方程构成，海伦公式将三角形的面积与其边长关联起来。\n 我们利用数值代数几何的工具研究面积-长度系统。具体而言，我们证明给定单个4-单纯形的十个三角形面积，最多可存在64组相容的边长集合。此外，通过分析该问题的伽罗瓦群（即单值群），我们证明这64组解通常不存在用面积表示的闭式表达式。我们还证明，通过引入额外的对称性约束，可以得到边长的此类表达式。我们迈出了将研究结果应用于离散量子引力的第一步，具体是在有效自旋泡沫模型（effective spin foam models）中的应用。
英文摘要：
Area variables are intrinsic to connection formulations of general relativity, in contrast to the fundamental length variables prevalent in metric formulations. Within 4D discrete gravity, particularly based on triangulations, the area-length system establishes a relationship between area variables associated with triangles and the edge length variables. This system is comprised of polynomial equations derived from Heron's formula, which relates the area of a triangle to its edge lengths. Using tools from numerical algebraic geometry, we study the area-length systems. In particular, we show that given the ten triangular areas of a single 4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover, we show that these 64 solutions do not, in general, admit formulae in terms of the areas by analyzing the Galois group, or monodromy group, of the problem. We show that by introducing additional symmetry constraints, it is possible to obtain such formulae for the edge lengths. We take the first steps toward applying our results within discrete quantum gravity, specifically for effective spin foam models.
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