Spectral Data and the Moduli Space of Higgs Bundles

本课题主要研究数学物理中出现的自对偶方程、微分方程的解空间。希格斯束，即这些方程在二维空间上的解，出现在几何、物理和表征理论的界面上。通过光谱数据来理解希格斯束，可以让人们看到物理物体和不同希格斯束性质之间美丽的相互作用。这种方法之所以吸引人，不仅是因为其内在的兴趣，还因为它非常适合数学和物理的几个分支之间的合作。该项目的主要目标是了解希格斯束及其光谱数据如何为研究各种几何和物理问题提供新的工具。值得注意的是，表面群表示（以及A-或b模型中的膜）的模空间的拓扑不变量在与相应希格斯束的光谱数据相关的组合和几何值方面通常有优雅和简单的表征。光谱数据方法可以扩展到具有奇点的希格斯束，PI目前正在执行一个这样做的计划。通过考虑希格斯束模空间的超kahler结构，作者和合作者构造了一个自然反全纯对合的三重，其不动点集是a模型和b模型中膜的新例子，其中一些与3流形的表示变化密切相关。更好地理解3流形拓扑与希格斯束子空间之间的关系是PI所追求的，特别是通过研究希格斯束与结和图互补的关系。最后，PI还考虑了Langlands对偶性，通过对膜族光谱数据进行傅里叶- mukai变换，从曲线和膜量化的角度为希格斯束的特征变化和模空间量化提供了一些必要的对象。

This research project is focused on the study of the space of solutions to the self-duality equations, differential equations arising in mathematical physics. Higgs bundles, the solutions to these equations over a two-dimensional space, appear at the interface of geometry, physics, and representation theory. Understanding Higgs bundles through spectral data allows one to see a beautiful interplay between physical objects and properties of different Higgs bundles. This approach is appealing not only because of its intrinsic interest, but also because it is wonderfully suited for collaborations among several branches of mathematics and physics. The main goal of the project is to understand how Higgs bundles and their spectral data can provide new tools to study various geometric and physical questions. Remarkably, there are often elegant and simple characterizations of topological invariants of the moduli space of surface group representations (and of branes in the A- or B-model) in terms of combinatorial and geometric values associated to the spectral data of the corresponding Higgs bundles. The spectral data approach may be extended to Higgs bundles with singularities, and the PI is currently carrying out a program to do so. By considering the hyperkahler structure of the moduli spaces of Higgs bundles, the PI and a collaborator constructed a triple of natural anti-holomorphic involutions whose fixed point sets are new examples of branes in the A-model and B-model, some of them closely related to the representation variety of 3-manifolds. Obtaining a better understanding of the relation between the topology of 3-manifolds and subspaces of Higgs bundles is something the PI is pursuing, in particular by studying the relation of Higgs bundles with knot and graph compliments. Finally, the PI is also considering Langlands duality by looking at the Fourier-Mukai transform on the spectral data for the families of branes, which should provide some of the necessary objects needed for quantization of character varieties and moduli spaces of Higgs bundles from the perspective of curve and brane quantization.