Fellows Seminar - Classifying flat and Higgs bundles, a tale of two spaces

By Florent Schaffhauser, 2018 Fellow

Equivalences and correspondences of varied kinds are everywhere in mathematics: we appreciate knowing how and when we can identify two objects, and this forms a subject of research known as moduli theory. A topic of particular interest in the field is the study of the shape of the parameter spaces that come up in classification problems and it is sometimes helpful to replace a given parameter space by another one, known to have the same shape but whose geometry is somehow richer and more amenable to computations. In fact, I will focus on one particular example of a correspondence between two different spaces, called the nonabelian Hodge correspondence. It was discovered in the 1980s, and has ramifications in many areas of mathematics. 

At its core, the nonabelian Hodge correspondence is differential-geometric in nature. It is about showing the existence of special metrics on two different types of objects; flat bundles on the one hand, and Higgs bundles on the other. As the name suggests, the latter objects originate in mathematical physics. They are reminiscent of the work of Peter Higgs, father of the famous Higgs boson in the standard model of particle physics. The gauge-theoretic equations that control the theory have a physical origin too: they were obtained by Nigel Hitchin from the self-duality equations in four dimensions. The flat bundles are more classical objects, in the sense that they are used to geometrically formalise the significance of the space of solutions of a linear differential equation.

After telling you about the origins of the nonabelian Hodge correspondence, I will lay out the main results of my USIAS project - run at IRMA in collaboration with Olivier Guichard - that consists of an extension of the nonabelian Hodge correspondence to a certain type of singular spaces, which are allowed to have hidden symmetry. Depending on the remaining time and the specific interests of the audience, I will mention a few applications or focus on the implications of this research for the cohomology of discrete groups.

 The talk is meant to be accessible to a large audience with a background in scientific research: I will therefore focus on the sequence of ideas that led the community to relate flat and Higgs bundles, and leave out the technical details.

• More information on Florent Schaffhauser and his USIAS project: The non-Abelian Hodge theory of an orbifold Klein surface.