Topological Field Theory and Integrable Systems

AbstractAward：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数学问题：二级Gromov-Witten不变量理论的构建。换句话说，这是弗洛尔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已被广泛应用于代数拓扑，但它们在Gromov-Witten不变量的研究中，或更一般的拓扑场论中，并没有得到太多的关注;这个建议试图纠正这一点。我们将研究是否一个类似的形式持有紧凑的凯勒目标。我们的结果将根据我们的同伦Gerstenhaber代数理论以及Tamarkin和Tsygan的同伦Batalin-Vilkovisky代数理论来表述。从某种意义上说，这个提议解决了理解弦理论数学表述的难题，而弦理论本身就是对自然基本定律非常有希望的描述。在其40年的历史中，弦理论吸收了现代数学的深刻见解。但是，研究人员中有一种感觉，即理论的最终表述尚未实现：我们相信，这一提议将使我们更接近于找到这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.