Field Theories and Elliptic Cohomology

AbstractAward：DMS-0707068首席研究员：Stephan Stolz在以前与Peter Teichner的合作工作中，首席研究员成功地收集了椭圆上同调与超对称场论密切相关的证据，证明了2维超对称场论的配分函数是一个积分模形式; 1维超对称场论空间同伦等价于由实K-理论谱确定的无限圈空间，流形上的维超对称场论表示一个普通的上同调类（具有真实的系数）。 这些进展支持了首席研究员的信念，即他和彼得·泰希纳已经找到了将“超对称性”和“定域性”纳入格雷姆·西格尔量子场论定义的正确方法。这位主要研究者乐观地认为，这可能是解决如何将场理论与椭圆上同调联系起来这个二十年前的问题所必需的关键一步。一个特别有趣的方面是，上同调理论中的“推进”映射似乎与物理学家通过泛函积分的量子化过程相对应，而泛函积分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取得了丰硕的成果。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，它是复杂量子系统的定性方面（即，当改变理论的参数时不变的方面），这应该是最容易分析的。这种相互作用深刻地影响了这两个领域，物理学家使用复杂的拓扑工具，拓扑学家结合量子场论的思想。主要研究者希望，将0维、1维和2维的超代数量子场论与拓扑学中研究得很好的上同调理论联系起来，即“广义上同调理论”，将通过把代数拓扑学的计算能力带到物理学中，并把这些上同调类的物理/几何意义带到拓扑学中，来显著地加强这种相互作用。

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.