TQFTs in Spectra

这个提议的中心思想是对物理学家的拓扑引力概念（类似于西格尔对共形场论的数学定义）的数学定义，将其定义为以流形为对象的一元范畴的表示，使用这些对象之间的协射范畴的几何实现作为其态射空间；这些空间是协群的微分同态群的分类空间的并集。在二维中，得到的分类与西格尔考虑的非常相似，但它可以很自然地推广到四维，它与经典广义相对论有密切的联系；但也有一个版本的拓扑（即，非光滑）四流形，缺乏任何明确的经典类比。Donaldson和Seiberg-Witten理论很自然地符合这个框架，它预测这些不变量应该有高阶的“引力后代”，例如，越墙障碍物的高维版本。在二维中，这种形式主义一方面与Madsen和Tillmann对芒福德猜想的研究相吻合，另一方面与Kontsevich、Manin、Witten等人研究的“大”量子上同调理论相吻合。项目的一个具体目标是构建与三流形的Atiyah-Patodi-Singer不变量相关的上同调理论，因为Casson不变量与flower同调有关。经典力学研究的是光滑几何背景下点粒子的轨迹，弦理论中最近的许多工作也可以用类似的术语来表述，只不过背景被一些环境流形中光滑循环的（无限维）空间所取代。然而，这些自由环空间的数学是相当具有挑战性的，它们的拓扑（更不用说它们的几何）还没有被很好地理解。更复杂的是，量子场论中研究的模型以概念上固有的方式涉及拓扑变化，因此似乎经常不需要自由环空间本身，而是需要具有良好性质的适当补全——其性质仍在研究中。这一建议表明，对于自由环空间，期望的补全是有限维光滑流形对偶的模拟（如20世纪60年代Whitehead， Spanier， Atiyah等人所研究的）。这个观点似乎与Chas和Sullivan关于弦拓扑的工作，Cohen、Jones和Segal关于Floer同伦类型的工作，以及Ando和我关于Witten属的工作是一致的。

DMS-0116288Jack MoravaThe central idea of this proposal is a mathematical definition for the physicists' notion of topological gravity (analogous to Segal's mathematical definition of conformal field theory) as a representation of a monoidal category with manifolds as objects, using the geometric realization of a category of cobordisms between those objects as its morphism spaces; these spaces are unions of classifying spaces for the diffeomorphism groups of the cobordisms. In two dimensions, the resulting category is quite similar to that considered by Segal, but it generalizes very naturally, e.g. to four dimensions, where it has close connections with classical general relativity; but there is also a version for topological (i.e, non-smooth) four-manifolds, lacking any clear classical analog. Donaldson and Seiberg-Witten theory fit naturally into this framework, which predicts that such invariants should have higer-order `gravitational descendants', e.g. higher-codimension versions of the wall-crossing obstructions. In dimension two, this formalism fits in well on the one hand with work of Madsen and Tillmann on Mumford's conjecture, and on the other with the theory of a `large' quantum cohomology studied by Kontsevich, Manin, Witten, and others. One concrete goal of the projectis to construct a cohomological theory related to the Atiyah-Patodi-Singer eta-invariant of a three-manifold, as Casson's invariant is related to Floer homology. Classical mechanics studies the trajectories of point particles in a smooth geometric background, and much recent work in string theory can be formulated in similar terms, with the background replaced by the (infinite-dimensional) space of smooth loops in some ambient manifold. However, the mathematics of these free loopspaces is quite challenging, and their topology (not to mention their geometry) is not yet well-understood.An added complication is that the models studied in quantum field theory involve topology change in a conceptually intrinsic way, and thus seem often to call, not for the free loopspace itself, but for a suitablecompletion with nice properties -- whose nature is still being worked out. This proposal suggests that the desired completion is an analog, for free loopspaces, of the dual of a finite-dimensional smooth manifold (as studied in the 1960's by Whitehead, Spanier, Atiyah, and others). This point of view seems compatible with work of Chas and Sullivan on string topology, with work of Cohen, Jones, and Segal on Floer homotopy type, and with work of Ando and myself on the Witten genus.