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

近十年来三维拓扑学的一个重要发展是“量子不变量”的发现，最值得注意的是链路的Jones多项式和3流形的Reshetikhin-Turaev （RT）不变量。虽然这些不变量在分类结和3流形方面非常有效，但它们的拓扑性质仍然很模糊。本研究的目的是通过将量子不变量分解为更简单的部分（称为“有限型不变量”）来研究量子不变量的拓扑起源，并试图单独解释这些部分的性质。这可以通过使用量子场论的工具来完成，特别是通过费曼图的路径积分的渐近展开。E. Witten将Jones多项式和RT不变量定义为3流形上SU(2)连接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米尔诺在琼斯多项式内连接数的发现。RT不变量的最简单“块”被识别（至少，作为一个猜想）与平连接的chen - simons不变量，Reidemeister扭转和SU(2) Casson不变量。人们希望在RT不变量的其他部分中发现更复杂的拓扑不变量，例如其他李群的Casson不变量。结点的分类是一个开放拓扑问题。拓扑学中的结是一根闭合的绳子（即一个圆），它被打结。一个特定的结可以通过不断地使绳子变形而不切断它来解开吗？一个人怎么能仅仅通过看结的图片就确定这一点呢？如果找到足够多的结不变量，这个问题就可以解决。结不变量是一个可以通过查看结的图片来计算的数字。当一个结连续变形时，这个数字不应该改变。因此，如果一个结不变量在两个结上取不同的值，那么这些结是不同的，因为它们不能变形成另一个。在很长一段时间里，唯一有效的结不变量是亚历山大多项式。许多新的不变量，如琼斯多项式，在过去十年中被发现。这些不变量在区分结方面非常有效，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。本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。量子场论与低维拓扑的关联是由E. Witten发现的。他证明了琼斯多项式来自于一个粒子理论，在这个理论中，结点表现为假设的基本粒子的时空轨迹。这些粒子以类似于“现实生活”粒子的方式相互作用。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理解，并最终导致结的分类。＊＊＊

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. ***