TQFTs in Spectra

Spectra 中的 TQFT

• 批准号: 0116288
• 负责人: Jack Morava
• 金额: $ 17.02万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0116288&HistoricalAwards=false
• 关键词: TQFTs; Spectra

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.

这个提议的中心思想是物理学家对拓扑引力概念的数学定义（类似于西格尔的数学定义的共形场论）作为一个表示的monoidal范畴与流形作为对象，使用几何实现的范畴的协边之间的这些对象作为其态射空间;这些空间是配边的类同态群的分类空间的并。在二维中，所得到的范畴与西格尔所考虑的范畴非常相似，但它非常自然地推广到例如四维，在那里它与经典广义相对论有密切的联系;但也有一个拓扑（即非光滑）四维流形的版本，缺乏任何明确的经典类比。唐纳森和塞伯格-威滕理论很自然地符合这个框架，它预言这样的不变量应该有更高阶的“引力后裔”，例如越壁障碍物的更高阶余维版本。在二维中，这种形式主义一方面与马德森和蒂尔曼关于芒福德猜想的工作相吻合，另一方面与孔采维奇、马宁、维滕等人研究的“大”量子上同调理论相吻合。该项目的一个具体目标是构建一个与三元流形的Atiyah-Patodi-Singer η-不变量相关的上同调理论，因为Casson不变量与Floer同调相关。 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，只不过把背景换成某些环境流形中的光滑环（无限维）空间。 然而，这些自由回路空间的数学是相当具有挑战性的，它们的拓扑结构（更不用说它们的几何结构了）还没有得到很好的理解。一个额外的复杂性是，量子场论中研究的模型以一种概念上内在的方式涉及拓扑结构的变化，因此似乎经常要求的不是自由回路空间本身，而是一种具有良好性质的合适的完备性--其性质仍在研究之中。 这个建议表明，对于自由环空间，理想的完备化是有限维光滑流形的对偶的一个模拟（如Whitehead，Spanier，Atiyah等人在20世纪60年代所研究的）。这种观点似乎与查斯和沙利文关于弦拓扑的工作，科恩、琼斯和西格尔关于弗洛尔同伦类型的工作，安藤和我关于维滕属的工作是一致的。