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解的空间。希格斯丛是这些方程在二维空间上的解，它出现在几何、物理和表象理论的界面上。通过光谱数据理解希格斯束可以让人们看到物理物体和不同希格斯束的性质之间的美丽相互作用。这种方法之所以吸引人，不仅是因为它本身的趣味性，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，根据与相应Higgs丛的光谱数据相关联的组合和几何值，经常有关于表面群表示(以及A或B模型中的膜)的模空间的拓扑不变量的优雅而简单的刻画。光谱数据方法可以扩展到具有奇点的希格斯束，PI目前正在执行一项这样做的计划。通过考虑Higgs丛的模空间的Hyperkahler结构，PI和一个合作者构造了一个自然的反全纯对合三元组，其不动点集是A-模型和B-模型中膜的新例子，其中一些与三维流形的表示种类密切相关。更好地理解3-流形的拓扑和Higgs丛的子空间之间的关系是PI所追求的，特别是通过研究Higgs丛与纽结和图补的关系。最后，PI还通过观察膜族的光谱数据的傅立叶-穆凯变换来考虑朗兰兹对偶，从曲线和膜量化的角度来看，这应该提供了从曲线和膜量化的角度量化Higgs丛的特征簇和模空间所需的一些必要对象。

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.