Field Theories and Elliptic Cohomology

在之前与Peter Teichner的合作研究中，通过证明2维超对称场论的配分函数是一个积分模形式，成功地收集了椭圆上同调与超对称场论密切相关的证据；1维超对称场论的空间与由realk -理论谱决定的无限环空间是同伦等价的，流形上的0维超对称场论表示一个普通上同调类（具有实系数）。这些进展支持了首席研究员的信念，即他和Peter Teichner已经找到了将“超对称”和“局部性”结合到graeme Segal定义的量子场论中的正确方法。首席研究员乐观地认为，这可能是解决如何将场理论与椭圆上同调联系起来的二十年难题的关键一步。一个特别有趣的方面是，在这些生态同调理论中的“向前推进”映射似乎对应于物理学家通过泛函积分的量化过程，这在数学上还没有得到很好的理解。在过去的二十年里，理论物理学和拓扑学之间的相互作用卓有成效。在哲学层面上，这并不奇怪，因为拓扑学可以被广泛地理解为数学中“定性”方面的研究，并且可以预期，复杂量子系统的定性方面（即，当改变理论参数时不会改变的方面）应该是最容易分析的。这种相互作用深刻地影响了物理学家使用拓扑学的复杂工具和拓扑学家结合量子场论的思想这两个领域。首席研究员希望将0、1和2维的超对称量子场论与拓扑中已经研究得很好的对象联系起来，即“广义上同调理论”，将通过将代数拓扑的计算能力引入物理学和这些上同调类拓扑的物理/几何意义来显著加强这种相互作用。

AbstractAward: DMS-0707068Principal Investigator: Stephan StolzIn previous joint work with Peter Teichner, the principalinvestigator was successful in collecting evidence that ellipticcohomology is closely related to super symmetric field theoriesby showing that the partition function of a super symmetric fieldtheory of dimension 2 is an integral modular form; the space ofsuper symmetric field theories of dimension 1 is homotopyequivalent to the infinite loop space determined by the realK-theory spectrum and that a 0-dimensional super symmetric fieldtheory over a manifold represents an ordinary cohomology class(with real coefficients). These advances support the belief ofthe principal investigator that he and Peter Teichner have foundthe right way to incorporate `super symmetry' and `locality' inGraeme Segal's definition of quantum field theory. The principalinvestigator is optimistic that this might be the crucial stepnecessary to solve the two decade old problem of how to relatefield theories and elliptic cohomology. A particularlyinteresting aspect is that the `push forward' maps in thesecohomology theories seem to correspond to physicists'quantization procedures via functional integrals, which aremathematically not well understood.The last two decades have seen very fruitful interactions betweentheoretical physics and topology. On a philosophical level thisis not surprising since topology can be understood broadly as thestudy of `qualitative' aspects in mathematics, and it is to beexpected that it is the QUALITATIVE aspects of a complicatedquantum system (i.e., the aspects that don't change when varyingthe parameters of the theory) that should be easiest toanalyze. This interaction has deeply influenced both areas withphysicists using sophisticated tools from topology, andtopologists incorporating ideas from quantum field theory. Theprincipal investigator hopes that relating super symmetricquantum field theories of dimension 0,1 and 2 to well- studiedobjects in topology, namely `generalized cohomology theories'will significantly strengthen that interaction by bringing thecalculational power of algebraic topology to physics and thephysical/ geometric meaning of these cohomology classes totopology.