Spectral Data and the Moduli Space of Higgs Bundles

本研究项目主要研究数学物理中的自对偶方程、微分方程的解空间。希格斯束是这些方程在二维空间中的解，它出现在几何学、物理学和表象论的交界处。通过光谱数据理解希格斯束，可以看到物理对象和不同希格斯束属性之间的美丽相互作用。这种方法之所以吸引人，不仅是因为它的内在利益，而且还因为它非常适合数学和物理学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模型或B模型中的膜）的模空间的拓扑不变量，往往有优雅而简单的表征，它们是与相应希格斯束的谱数据相关联的组合和几何值。光谱数据的方法可能会扩展到希格斯束与奇点，和PI目前正在执行一个计划，这样做。通过考虑Higgs丛模空间的超kahler结构，PI和一个合作者构造了一个自然反全纯对合的三元组，其不动点集是A-模型和B-模型中膜的新例子，其中一些与3-流形的表示簇密切相关。获得更好的理解之间的关系的3流形和子空间的希格斯丛的拓扑是PI正在追求的东西，特别是通过研究的关系，希格斯丛与结和图形恭维。最后，PI也在考虑朗兰兹对偶性，通过观察膜族的谱数据的傅里叶-向井变换，这应该从曲线和膜量子化的角度为希格斯包的特征变量和模空间的量子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.