非对易拓扑量子场论，非对易Floer同调，非对易Hodge理论
Noncommutative Topological Quantum Field Theory, Noncommutative Floer Homology, Noncommutative Hodge Theory
摘要：
本文提出了一种可能的非对易拓扑量子场论（NCTQFT）和非对易Floer同调（NCFH）的一些设想。我们的动机是双重的，它来自物理和数学：一方面，我们认为NCTQFT是正确的数学框架的量子场论的所有已知的相互作用在自然界（包括重力）。另一方面，我们希望一个可能的NCFH将适用于几乎所有的3-流形（而不仅仅是像普通的Floer同调一样适用于同调3-球面）。这两个动机是密切相关的，因为至少在交换的情况下，弗洛尔同调群构成了（3+1）维拓扑量子场论的量子观测空间。为了实现这一目标，我们提出了一些“非交换”版本的霍奇理论的非交换差分形式和切上同调叶理。
英文摘要：
We present some ideas for a possible Noncommutative Topological Quantum Field Theory (NCTQFT) and Noncommutative Floer Homology (NCFH). Our motivation is two-fold and it comes both from physics and mathematics: On the one hand we argue that NCTQFT is the correct mathematical framework for a quantum field theory of all known interactions in nature (including gravity). On the other hand we hope that a possible NCFH will apply to practically every 3-manifold (and not only to homology 3-spheres as ordinary Floer Homology currently does). The two motivations are closely related since, at least in the commutative case, Floer Homology Groups constitute the space of quantum observables of (3+1)-dim Topological Quantum Field Theory. Towards this goal we present some "Noncommutative" Versions of Hodge Theory for noncommutative differentail forms and tangential cohomology for foliations.
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