GREAT · Deformations of fundamental Groups of REpresentATions

Deformations of fundamental Groups of REpresentATions

The interaction between Physics and Mathematics has always been a source of new and groundbreaking ideas for both sciences. In particular, the discovering of the Higgs bundles provided us with an extremely powerful tool in the mathematical understanding of Gauge theories. The Higgs bundles are particular solutions of the Yang-Mills equations which surprisingly make deep connections between Physics and the main branches of mathematics, such as Topology, Algebra and Analysis. From an algebraic point of view, a Higgs bundle consists of a bundle on a smooth complex projective algebraic curve (or a Riemann surface from an analytic point of view) together with a field, namely Higgs field. One of the wonders of these objects is that from a topological point of view a Higgs bundle is essentially the same as a fundamental topological object known as the representation of the fundamental group. The theory coming out of this relation is called non-abelian Hodge theory, and it presents a wonderful set of relations between essential objects in algebra, mathematical physics, and topology with the tools of geometry. This project aimed to consider the base space of the Higgs bundle X and a representation of its fundamental group. We studied the deformations of a representation when X degenerates into a singular curve with nodal singularities - notice that here we are understanding a Higgs bundle from its algebraic point of view as well as its topological point of view. This deformation theory question opens a brand new direction in the theory of representations of fundamental groups and Higgs bundles. The main tool to approach the problem was the non-abelian Hodge theory to deal with the topological ideas in geometrical terms. New algebraic objects, the so called generalised parabolic Higgs bundles, were the proposed tools to approach the problem. The main goal was to provide a deformation theory for Higgs bundles and for their associated objects from the algebraic to the topological side of the theory, namely harmonic bundles over X. The corresponding deformation theory should allow degenerations of the curve X to nodal like singularities. (See the figure provided: torus-to-nodal-higgs.png, where the degeneration of the base space is drawn together with the equations for a Higgs bundle. This figure was produced with Mathematica 11.1: Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL (2017).)

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