Spectral Data and the Moduli Space of Higgs Bundles

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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.