Topological Field Theory and Integrable Systems

摘要奖：DMS-0505669首席研究员：Ezra Getzler在这个提议中，我们研究了弦理论提出的一个重要的数学问题：二级Gromov-Witten不变量理论的构造。换言之，Floer同调的有理同伦理论的一个类比。虽然有理同伦理论的思想在代数拓扑学中已经得到了广泛的应用，但在Gromov-Witten不变量的研究中，或者更广泛地说，拓扑场论的研究中，它们并没有受到太多的关注，本文试图纠正这一点。我们将研究类似的形式是否适用于紧凑的凯勒目标。我们的结果将根据我们的同伦Gerstenhaber代数理论和Tamarkin和Tsygan的同伦Batalin-Vilkovisky代数理论来表示。在某种意义上，这个建议解决了理解弦理论的数学表示的困难问题，而弦理论本身就是对自然界基本规律的一种非常有希望的描述。在其四十年的历史中，弦理论借鉴了现代数学的见解。但研究人员中有一种感觉，该理论的最终表述尚未达成：我们相信，这一提议将使我们更接近于找到这样一种描述所需的数学语言。

AbstractAward: DMS-0505669Principal Investigator: Ezra GetzlerIn this proposal, we study an important mathematical problemsuggested by string theory: the construction of a theory ofsecondary Gromov-Witten invariants. In other words, an analogueof rational homotopy theory for Floer homology. While the ideasof rational homotopy theory have been extensively applied inalgebraic topology, they have not received much attention in thestudy of Gromov-Witten invariants, or more generally, oftopological field theory; this proposal attempts to rectifythis. We will examine whether an analogue of formality holds forcompact Kaehler targets. Our results will be formulated in termsof our theory of homotopy Gerstenhaber algebras, and Tamarkin andTsygan's theory of homotopy Batalin-Vilkovisky algebras.In a sense, this proposal addresses the difficult problem ofunderstanding the mathematical formulation of string theory,which is itself a very promising descriptions of the fundamentallaws of nature. Throughout its forty year history, string theoryhas drawn on the insights of modern mathematics. But there is asense among researchers that the final formulation of the theoryhas not yet been attained: we believe that this proposal willbring us closer to finding the mathematical language which isneeded for such a description.