AI 中文总结
本文在下降理论框架下，提出非几何R-空间的严格数学表述，将其视为双范畴值层，为兼具非交换与非结合结构的R-空间提供了全局几何定义。
AI 中文摘要
非交换与非结合几何在数学和物理学界均受到广泛关注，一个典型例子是基于底流形的弦理论R-空间，它兼具非交换与非结合结构，且被认为是局域非几何的。然而，以往研究主要通过局域弦位置坐标的代数关系描述R-空间，而非精确的全局几何定义。本文在下降理论框架下，提出了非几何R-空间的严格且系统的数学表述，核心思想是将非几何R-空间视为仅在底流形的每个局域坐标域上携带拟三角拟Hopf代数的局域数据，这些数据经连贯粘合后形成全局几何对象。我们证明，合适的全局结构并非普通空间或全局定义的代数，而是双范畴值层，这为R-空间提供了全局且几何的定义，同时保留了其局域非几何的固有特性。
英文摘要
Noncommutative and nonassociative geometry has attracted much attention in both mathematics and physics communities. A prominent example is a stringy $R$-space over an underlying manifold, which exhibits both noncommutative and nonassociative structures and is known to be locally nongeometric. However, previous studies have mainly described $R$-space through algebraic relations of local string position coordinates, instead of a precise global geometric definition. In this paper, we propose a rigorous and systematic mathematical formulation of nongeometric $R$-space within the framework of descent theory. The central idea is to regard nongeometric $R$-space as carrying only local data of a quasitriangular quasi-Hopf algebra on each local coordinate domain of the underlying manifold, which are then glued coherently together to produce a global geometric object. We show that the appropriate global structure is not an ordinary space or a globally defined algebra, but a bicategory-valued stack. This provides a global and geometric definition of $R$-space while still preserving its locally nongeometric intrinsic character.
Comments25 pages
