Field Theory, Link Invariants, and Higher Moduli

PI使用从粒子物理引入的方法来研究几何和拓扑问题。该项目分为两大部分。第一部分是关于“结不变量”的：给出两张弦的打结环的图片，这些方法可以确定是否有可能在不切割的情况下把一个变成另一个。PI和他的合作者正在开发一种新的方法，使用过去十年开发的粒子物理学的新工具来计算纽结不变量。第二部分是关于变形量子理论的新方法，PI希望这种方法在数学上有很多应用，包括微分方程式理论。这项工作的结果将在数学和高能物理界广泛传播，有助于将这两个领域更紧密地联系在一起。该项目还将有助于对这两个领域的研究生进行培训。该项目的第一部分涉及“Q-unabelian化”。这是定义R^3或更一般的3-流形中环的不变量的一种新方案。它是PI和合作者为研究平坦GL(N)-连通模空间而提出的利用谱网络的“非交换化”方法的Q-形变。PI和合作者的目的是在一般的3-流形和N的一般值上构造q-非可化化，从GL(2)的情况开始；在这种情况下，q-非可化化与Bonahon-Wong早些时候引入的构造密切相关。该项目的第二部分涉及到高等泰希穆勒理论的一种新方法。第一个关键思想是用S类超对称量子场论的边缘和无关形变的空间来识别“高阶TeichMuller空间”(也称为“希钦分量”)；第二个关键思想是发展S理论中表面缺陷的边缘和无关形变的模空间的相应图像，并将其应用于黎曼曲面上的希格斯丛理论。这个奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The PI studies problems of geometry and topology using methods imported from particle physics. This project is divided into two major parts. The first part concerns "knot invariants": these are ways of determining, given two pictures of knotted loops of string, whether it is possible to turn one into the other without cutting. The PI and his collaborators are developing a new method for calculating knot invariants using new tools from particle physics developed over the last decade. The second part concerns new ways of deforming quantum theories, which the PI expects to have many applications in mathematics, including to the theory of differential equations. The results of this work will be disseminated broadly both in the mathematics and high-energy physics communities, helping to bring these two areas closer together. The project will also contribute to the training of graduate students in both fields.The first part of the project concerns "q-nonabelianization". This is a new scheme for defining invariants of links in R^3 or more generally in 3-manifolds. It is a q-deformation of the method of "nonabelianization" using spectral networks, introduced earlier by the PI and collaborators for studying moduli spaces of flat GL(N)-connections. The PI and collaborators aim to construct q-nonabelianization on general 3-manifolds and for general values of N, beginning with the GL(2) case; in that case q-nonabelianization is closely related to constructions which have been introduced earlier by Bonahon-Wong. The second part of the project concerns a new approach to higher Teichmuller theory. The first key idea is to identify the "higher Teichmuller space" (also known as "Hitchin component") with a space of marginal and irrelevant deformations of a supersymmetric quantum field theory of class S; the second key idea is to develop the corresponding picture for the moduli spaces of marginal and irrelevant deformations of surface defects in the class S theory, with applications to the theory of Higgs bundles over Riemann surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.