## Periodic Reporting for period 1 - GREAT (Deformations of fundamental Groups of REpresentATions)

Reporting period: 2016-01-01 to 2016-12-31

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).)

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).)

During the 12 months grant period I worked on the non-Abelian Hodge theory from the point of view of parabolic Higgs bundles and generalised parabolic Higgs bundles. I discovered that the non-abelian Hodge theory is not extendable in its actual terms to nodal like singularities. Therefore, a complete refurbishing of the theory is needed, and it is the work I started during the project. At the time of this report, I already made the first steps towards this further goal.

Moreover, I developed a full understanding of the newest tools in deformation theory and nodal curves. This knowledge also introduced me to related problems and helped me to establish new research networks.

I disseminated my work in many conferences and seminars in UK (where the grant was hosted) and abroad, so that the results coming out from the research done have been successfully broadcasted. Furthermore, I also gave a course on the theory of Higgs bundles on nodal curves to graduated students and postdocs. The lecture notes of this course will be published soon.

There is also a forthcoming publication directly related with the project, in which I develop the first step towards a non-abelian Hodge theory for nodal curves. Two more publications are in preparation, collateral to the project.

Moreover, I developed a full understanding of the newest tools in deformation theory and nodal curves. This knowledge also introduced me to related problems and helped me to establish new research networks.

I disseminated my work in many conferences and seminars in UK (where the grant was hosted) and abroad, so that the results coming out from the research done have been successfully broadcasted. Furthermore, I also gave a course on the theory of Higgs bundles on nodal curves to graduated students and postdocs. The lecture notes of this course will be published soon.

There is also a forthcoming publication directly related with the project, in which I develop the first step towards a non-abelian Hodge theory for nodal curves. Two more publications are in preparation, collateral to the project.

The main issue addressed in the project was that non-abelian Hodge theory, the theory connecting Higgs bundles with several branches of Mathematics and Physics, does not work for Higgs bundles when the base space has nodal singularities. There are partial results which use wild non-abelian Hodge theory (a non-abelian Hodge theory for logarithmic connections with higher order poles) to deform a solution to the Hitchin’s equations over a smooth compact Riemann surface to a solution (known as “fiducial solutions”) over a Riemann surface with a finite set of nodal singularities from an analytical point of view. Nevertheless those fiducial solutions are not yet connected with the algebraic notion of Higgs bundle on a nodal curve. Therefore there is still no link between both viewpoints in the nodal situation.

The research done for this project exposed the aforementioned issues and shed light on the depth of the problem. I realised, that even though those fiducial solutions are supposed to be the counterpart of a Higgs bundle on a nodal curve, the connection between the algebraic and the topological sides is not fully stated. This conclusion was achieved by studying generalised parabolic Higgs bundles as opposite to wild parabolic Higgs bundles.

Nevertheless, generalised parabolic Higgs bundles can still be used as a bridge between the algebraic and the analytical point of view in the case of nodal curves, but to unravel their role will require further study, which will be undertaken in the future. These objects are indeed opening new frontiers in the theory.

At present, the interest in this specific question has increased and more researchers are willing to apply generalised parabolic bundles Higgs bundles.

The research done for this project exposed the aforementioned issues and shed light on the depth of the problem. I realised, that even though those fiducial solutions are supposed to be the counterpart of a Higgs bundle on a nodal curve, the connection between the algebraic and the topological sides is not fully stated. This conclusion was achieved by studying generalised parabolic Higgs bundles as opposite to wild parabolic Higgs bundles.

Nevertheless, generalised parabolic Higgs bundles can still be used as a bridge between the algebraic and the analytical point of view in the case of nodal curves, but to unravel their role will require further study, which will be undertaken in the future. These objects are indeed opening new frontiers in the theory.

At present, the interest in this specific question has increased and more researchers are willing to apply generalised parabolic bundles Higgs bundles.