AI 中文总结
该研究为三维流形量子不变量开发新框架，构建(2 + 1)维TQFT，为曲面分配无序配置空间的扭曲同调，通过选择局部系统及额外数据满足TQFT公理，给出两个有用示例，方法纯拓扑且依赖配边多轨迹空间。
AI 中文摘要
我们通过将映射类群表示的同调构造扩展到配边，为三维流形的量子不变量开发了一个新框架。具体而言，我们构建了一个(2 + 1)维拓扑量子场论（TQFT），它为每个曲面分配其无序配置空间的扭曲同调。该构造需要在配置空间上选择局部系统以及额外数据。我们为这些数据制定了保证TQFT公理的充分条件，并表明至少有两个满足条件的有用示例。其中一个产生了射影Kerler - Lyubashenko TQFT的同调构造，另一个恢复了Frohman - Nicas - Donaldson TQFT。与量子不变量的经典代数构造不同，我们的方法是纯拓扑的，依赖于配边的多轨迹空间。
英文摘要
We develop a new framework for quantum invariants of $3$-manifolds by extending to cobordisms a homological construction of mapping class group representations. More specifically, we construct a $(2+1)$-dimensional topological quantum field theory (TQFT) that assigns to each surface the twisted homology of its unordered configuration space. The construction requires a choice of local systems on configuration spaces together with additional data. We formulate sufficient conditions on these data that guarantee the TQFT axioms, and we show that there are at least two useful examples satisfying them. One of them yields a homological construction of the projective Kerler--Lyubashenko TQFT, while the other recovers the Frohman--Nicas--Donaldson TQFT. In contrast to the classical algebraic constructions of quantum invariants, our approach is purely topological and relies on multi-trajectory spaces of cobordisms.
Comments53 pages
