Members & Guests

Dr. Jan Swoboda

Project leader


Ludwig-Maximilians-Universität München

E-mail: swoboda(at)math.lmu.de
Telephone: +49 89 2180-4680
Homepage: http://www.mathematik.uni-muenchen.de/~s...

Publications within SPP2026

We study the asymptotics of the natural \(L^2\) metric on the Hitchin moduli space with group \(G=SU(2)\). Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

 

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We prove a gluing theorem for solutions \((A_0, \Phi_0)\)  of Hitchin's self-duality equations with logarithmic singularities on a rank-\(2\) vector bundle over a noded Riemann surface \( \Sigma\) representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface \( \Sigma_1\) carries a smooth solution \((A_1, \Phi_1)\) of the self-duality equations, which may be viewed as a desingularization of \((A_0, \Phi_0)\).

JournalAdv. Math.
PublisherElsevier
Volume322
Pages637-681
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \( \Sigma_R\) converging for \(R\searrow 0\) to a surface \( \Sigma_0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d \to \Sigma_R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

 

JournalAbh. Math. Semin. Univ. Hambg.
PublisherSpringer Berlin Heidelberg
Volume86
Pages189--201
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We associate to each stable Higgs pair \((A_0,\Phi_0)\) on a compact Riemann surface X a singular limiting configuration \((A_\infty,\Phi_\infty)\), assuming that \(\det\Phi\) has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions \((A_t,\Phi_t) \) to Hitchin's equations which converge to this limiting configuration as \(t\to\infty\). This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.  

 

JournalDuke Math. J.
PublisherDuke University Press
Volume165
Pages2227-2271
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

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