Hyperkähler Metrics on the Moduli Space of Weakly Parabolic Higgs Bundles

This thesis examines a construction of hyperkähler metrics on the moduli space of weakly parabolic SL(2,C)-Higgs bundles. Given a Hermitian vector bundle E of rank 2 over a compact Riemann surface C and some R>0, this space can be identified as the moduli space M of solutions (A,Φ) to the R-rescaled Hitchin equations.

In the weakly parabolic setting the Higgs field Φ has a finite set of simple poles and defines a meromorphic quadratic differential with double poles on C. On the subset of generic solutions of Hitchin’s equations this differential determines a family of triangulations of C with vertices at the poles of q.

Additionally a solution (A,Φ) defines a flat connection on E. Near each pole of q it defines a Volterra integral equation of the second kind on a non-compact interval. A corresponding theory is established and it is shown that for large R there exists a one dimensional space of solutions of the flatness equation for each pole, which exponentially decay along an edge of the triangulation.

The triangulation together with the sections is used to construct a set of coordinates on a subset M' of M. It is then proven that these coordinates are dominated by a leading term which is given by the coordinates for a corresponding simpler space of limiting configurations. Furthermore it is shown that the deviation from the leading term is given by a remainder that is exponentially suppressed in R.

A variation of the triangulation leads to the formulation of a Riemann-Hilbert problem that is shown to be solved by the coordinates. From this precise estimates of the exponential decay rate for large R are derived.

Finally it is explained how one can use twistor theory to construct a hyperkähler metric g on M' out of the coordinates. A comparison of this metric with the simpler semiflat metric then shows that their difference is exponentially decaying in R.


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