The Jones-Witten invariants can be generalized for non-singular smooth vector fields with invariant probability measure on 3-manifolds, giving rise to new invariants of dynamical systems [22]. After a short survey of cohomological field theory for Yang-Mills fields, Donaldson-Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. These invariants have the information of the flows and they are interpreted as the intersection number of these flow orbits and constitute invariants of smooth four-manifolds admitting global flows. We study the case of Kahler manifolds by using the Witten's consideration of the strong coupling dynamics of N=1 supersymmetric Yang-Mills theories. The whole construction is performed by implementing the notion of higher dimensional asymptotic cycles a la Schwartzman [18]. In the process Seiberg-Witten invariants are also described within this context. Finally, we give an interpretation of our asymptotic observables of 4-manifolds in the context of string theory with flows.
Noncommutative Cohomological Field Theories and Topological Aspects of Matrix models
我们研究矩阵模型和非交换上同调场论（NC CohFT）的拓扑方面。C. CohFT在任意无穷小非对易参数$θ$变形下具有对称性。这一事实意味着N. C. CohFT具有比K-理论更不敏感的拓扑性质，但N. C. CohFT对流形的分类有可能给出非对易流形整体特征的新观点。为了研究N. C. CohFT的性质，我们构造了一些模型，其不动点轨迹由投影算子集给出。特别地，通过使用矩阵模型计算Moyal平面上的配分函数。矩阵模型的模空间是格拉斯曼流形的并。使用格拉斯曼流形的欧拉数计算矩阵模型的配分函数。用矩阵模型对N. C. CohFT进行了辨识，得到了N. C. CohFT的配分函数。为了检验非交换参数的独立性，我们还研究了Moyal平面情形下的大$θ$极限和有限$θ$中的模空间。如果N. C. CohFT的配分函数在非对易几何意义下是拓扑的，那么它应该与K-理论有某种联系。因此，我们从K理论的角度研究了CohFT和N.C.CohFT的某些模型。这些观察给了我们CohFT和N.C.CohFT与K理论之间的类比。此外，我们还证明了在Moyal平面和非对易环面的情况下，我们的配分函数在不改变K-理论的变形下是不变的。最后，我们讨论了非对易上同调Yang-Mills理论。
Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for $2$-dimensional Calabi-Yau categories (with an appendix by Ben Davison)
