Topological Field Theory and Integrable Systems

摘要/ abstract摘要：作者：Ezra getzler在本文中，我们研究了弦理论提出的一个重要数学问题：二阶Gromov-Witten不变量理论的构造。换句话说，这是花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中得到了广泛的应用，但在Gromov-Witten不变量的研究中，或者更一般地说，在拓扑场论中，它们并没有受到太多的关注；这项建议试图纠正这一点。我们将研究是否有类似的形式适用于紧凑的卡勒目标。我们的结果将用我们的同伦Gerstenhaber代数理论和Tamarkin和tsygan的同伦Batalin-Vilkovisky代数理论来表述。从某种意义上说，这个提议解决了理解弦理论的数学公式的难题，弦理论本身是对自然基本规律的一种非常有希望的描述。在其40年的历史中，弦理论吸取了现代数学的深刻见解。但是研究者们都意识到这个理论的最终形式还没有得到：我们相信这个提议将使我们更接近于找到这样一个描述所需要的数学语言。

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.