The large scale geometry of the Higgs bundle moduli space | Vidéo | Carmin.tv

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The large scale geometry of the Higgs bundle moduli space

Appears in collections : Gauge theory and complex geometry / Théorie de jauge et géométrie complexe, Exposés de recherche

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal{M}$ of rank-2 Higgs bundles over a Riemann surface $\Sigma$ as given by the set of solutions to the so-called self-duality equations $\begin{cases} &0 = \bar{\partial}_A \Phi \ & 0 = F_A + [ \Phi \wedge \Phi^²] \end{cases}$ for a unitary connection $A$ and a Higgs field $\Phi$ on $\Sigma$. I will show that on the regular part of the Hitchin fibration ($A$, $\Phi$) $\rightarrow$ det $\Phi$ this metric is well-approximated by the semiflat metric $G_{sf}$ coming from the completely integrable system on $\mathcal{M}$. This also reveals the asymptotically conic structure of $G_{L^2}$, with (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of $\mathcal{M}$. The analytic methods used there in addition yield a complete asymptotic expansion of the difference $G_{L^2} − G_{sf}$ between the two metrics.

Information about the video

Date of recording 20/06/2018
Date of publication 21/06/2018
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19417403
Cite this video Swoboda, Jan (20/06/2018). The large scale geometry of the Higgs bundle moduli space. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19417403
URL https://dx.doi.org/10.24350/CIRM.V.19417403

Domain(s)

Differential Geometry

Bibliography

Mazzeo, R., Swoboda, J., Weiss, H., & Witt, F. (2017). Asymptotic Geometry of the Hitchin Metric. <arXiv:1709.03433> - https://arxiv.org/abs/1709.03433
Mazzeo, R., Swoboda, J., Weiss, H., & Witt, F. (2016). Ends of the moduli space of Higgs bundles. Duke Mathematical Journal, 165(12), 2227-2271 - https://doi.org/10.1215/00127094-3476914
Gaiotto, D., Moore, G., & Neitzke, A. (2013). Wall-crossing, Hitchin systems, and the WKB approximation. Advances in Mathematics, 234, 239-403 - https://doi.org/10.1016/j.aim.2012.09.027

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