Mathematical Sciences: Properties of Quantum Invariants in 3-Dimensional Topology

9704893 Rozansky过去十年3D拓扑学的一个重要发展是发现了“量子不变量”，最著名的是链的琼斯多项式和三维流形的Reshetikhin-Turaev(RT)不变量。虽然这些不变量在分类纽结和3-流形方面很有效，但它们的拓扑性质仍然很模糊。这项研究的目的是通过将量子不变量分解成更简单的部分，称为有限类型不变量，并试图分别解释这些部分的性质，来研究量子不变量的拓扑起源。这可以通过使用量子场论的工具来实现，特别是通过费曼图解的路径积分的渐近展开。E.Witten将Jones多项式和RT不变量确定为3-流形上的SU(2)连通类上的某些路径积分。这种方法已经导致了亚历山大多项式和琼斯多项式中的米尔纳链接数的发现。RT不变量中最简单的“部分”(至少是一个猜想)被确定为平面连接的Chern-Simons不变量、Reidemister挠量和SU(2)Casson不变量。人们希望在RT不变量的其他部分中能找到更复杂的拓扑不变量，例如其他李群的Casson不变量。纽结的分类是一个开放的拓扑问题。拓扑学中的纽结是指打结的闭合绳索(即圆)。通过不断地使绳索变形而不切断它，可以解开一个特定的结吗？仅仅通过看一张结的照片就能确定这一点吗？如果找到足够多的纽结不变量，这个问题就可以解决。纽结不变量是一个可以通过检查纽结的图片来计算的数字。当一个结连续变形时，该数字不应改变。因此，如果一个结不变量在两个结上采用不同的值，则这些结是不同的，因为它们不能相互变形。在很长一段时间里，唯一有效的纽结不变量是亚历山大多项式。在过去的十年中，发现了许多新的不变量，如琼斯多项式。这些不变量在区分纽结方面相当有效，但它们的起源仍然是一个谜，因为它们的拓扑解释大多缺失。这个项目的目的是试图利用量子场论的工具，如路径积分和费曼图来填补这一空白。E.Witten发现了量子场论与低维拓扑的关联。他证明了琼斯多项式来自粒子理论，在粒子理论中，结看起来像是假想基本粒子的时空轨迹。这些粒子相互作用的方式类似于“真实生活”的粒子。这种方法已经产生了许多数学猜想，这些猜想可能会导致对琼斯多项式的拓扑性质的更好的理解，并最终导致纽结的分类。***

9704893 Rozansky An important development in 3d topology of the last decade was a discovery of "quantum invariants," most notably, the Jones polynomial of links and the Reshetikhin-Turaev (RT) invariant of 3-manifolds. Although these invariants are quite effective in classifying knots and 3-manifolds, their topological nature remains mostly obscure. The purpose of this research is to study the topological origin of quantum invariants by decomposing them into simpler pieces, called "finite type invariants," and trying to explain the nature of these pieces individually. This might be accomplished by using the tools of quantum field theory, in particular, the asymptotic expansion of path integrals through Feynman diagrams. E. Witten has identified the Jones polynomial and RT invariant as certain path integrals taken over the classes of SU(2) connections on a 3-manifold. This approach has already led to a discovery of the Alexander polynomial and Milnor linking numbers inside the Jones polynomial. The simplest "pieces" of the RT invariant were identified (at least, as a conjecture) with the Chern-Simons invariant of flat connections, Reidemeister torsion and SU(2) Casson invariant. One hopes that more complicated topological invariants, such as the Casson invariant of other Lie groups, will be found among other pieces of the RT invariant. A classification of knots is an open topological problem. A knot in topology is a closed rope (i.e., a circle) which is knotted. Can a particular knot be untangled by continuously deforming the rope without cutting it? How can one determine this just by looking at a picture of the knot? This problem may be solved if one finds enough knot invariants. A knot invariant is a number that can be calculated by examining a picture of a knot. The number should not change when a knot is continuously deformed. Thus if a knot invariant takes different values on two knots, then these knots are different, because they cannot be deformed into one another. For a long time the only effective knot invariant was the Alexander polynomial. A lot of new invariants, such as the Jones polynomial, were discovered during the last decade. These invariants are quite effective in distinguishing knots, but their origin is still a mystery, because their topological interpretation is mostly missing. The purpose of this project is to try to fill this gap by using the tools of quantum field theory, such as path integral and Feynman diagrams. The relevance of quantum field theory to low dimensional topology was discovered by E. Witten. He demonstrated that the Jones polynomial comes from a particle theory in which knots appear as space-time trajectories of hypothetical elementary particles. The particles interact with each other in ways that resemble the "real life" particles. This approach has yielded numerous mathematical conjectures that may lead to a better understanding of the topological nature of the Jones polynomial and ultimately to the classification of knots. ***