论文标题
半圣经理论检测稳定的差异性
Semisimple Field Theories Detect Stable Diffeomorphism
论文作者
论文摘要
扩展了第一作者的作品，我们在任意维度中引入了半圣拓扑领域理论的概念，并表明这种领域理论必然导致稳定的差异不变。本文的主要结果是证明了这种“上限”是最佳的：为此，我们介绍并研究了一类半神经拓扑领域理论，从而推广了由Dijkgraaf-Witten构建的众所周知的有限规定理论，释放和Quinn。我们表明，满足某种有限条件的多种多样（包括具有有限基本组的4个模型）与这些现场理论且仅当它们具有稳定的差异性时，是无法区分的。因此，这种广义的Dijkgraaf-witten理论提供了最强大的半神经TFT不变性。这些结果适用于大量的环境切向结构。 我们讨论了许多应用程序，包括无定向的4维半密布场理论的构造，这些构建可以区分无定向的平滑结构和方向的高维半密布场理论，从而可以区分某些外来的球体。 在此过程中，我们表明，普遍的Dijkgraaf-witten理论的维度降低再次是普遍化的Dijkgraaf-Witten理论，我们在理性环境中利用了Ambidexterity，我们开发了与Galvez-Carrillo的Moebius Moebius逆向原则相关的技术。
Extending the work of the first author, we introduce a notion of semisimple topological field theory in arbitrary even dimension and show that such field theories necessarily lead to stable diffeomorphism invariants. The main result of this paper is a proof that this 'upper bound' is optimal: To this end, we introduce and study a class of semisimple topological field theories, generalizing the well known finite gauge theories constructed by Dijkgraaf-Witten, Freed and Quinn. We show that manifolds satisfying a certain finiteness condition -- including 4-manifolds with finite fundamental group -- are indistinguishable to these field theories if and only if they are stably diffeomorphic. Hence, such generalized Dijkgraaf-Witten theories provide the strongest semisimple TFT invariants possible. These results hold for a large class of ambient tangential structures. We discuss a number of applications, including the constructions of unoriented 4-dimensional semisimple field theories which can distinguish unoriented smooth structure and oriented higher-dimensional semisimple field theories which can distinguish certain exotic spheres. Along the way, we show that dimensional reductions of generalized Dijkgraaf-Witten theories are again generalized Dijkgraaf-Witten theories, we utilize ambidexterity in the rational setting, and we develop techniques related to the $\infty$-categorical Moebius inversion principle of Galvez-Carrillo--Kock--Tonks.
