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Session Geometry and Topology of Manifolds
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The asymptotic geometry of the moduli space of Higgs bundles over a Riemann surface |
Jan Swoboda, Ludwig-Maximilians-Universität München and Max-Planck-Institut für Mathematik Bonn, Germany
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Joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt
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In this talk, I aim to give an overview of some known results and
several open questions concerning geometric and topological
properties of the moduli space $\mathcal M_{k,d}$ of stable Higgs
bundles of rank $k$ and degree $d$ on a compact Riemannian surface
$\Sigma$. I shall in particular discuss the construction of
$\mathcal M_{k,d}$ as the space of gauge equivalence classes of
solutions of the PDE
\[
\begin{cases}
0=&\bar\partial_A\Phi\\
0=&F_A+t^2[\Phi\wedge\Phi^{\ast}]
\end{cases}
\]
for some parameter $t>0$. Here $A$ denotes a unitary connection and
$\Phi$ a Higgs field on $\Sigma$. Some new analytical results
concerning the degeneration behaviour of $\mathcal M_{2,d}$ in the
limit $t\to\infty$ will be presented.
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