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H 1 The large scale geometry of the Higgs bundle moduli space

Auteurs : Swoboda, Jan (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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
$\begin{cases} &0 = \bar{\partial}_A \Phi \\ & 0 = F_A + [ \Phi \wedge \Phi^*] \end{cases}$
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.

Codes MSC :
14D20 - Algebraic moduli problems, moduli of vector bundles
14H60 - Vector bundles on curves and their moduli
53C07 - Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
53C26 - Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53D18 - Generalized geometries (à la Hitchin)

 Informations sur la Vidéo Langue : Anglais Date de publication : 21/06/2018 Date de captation : 20/06/2018 Collection : Research talks ; Algebraic and Complex Geometry ; Mathematical Physics Format : MP4 Durée : 01:01:24 Domaine : Algebraic & Complex Geometry ; Mathematical Physics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-06-20_Swoboda.mp4 Informations sur la rencontre Nom de la rencontre : Gauge theory and complex geometry / ​Théorie de jauge et géométrie complexeOrganisateurs de la rencontre : Bradlow, Steven B. ; Schmidt, Alexander ; Teleman, AndreiDates : 18/06/2018 - 22/06/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1747.htmlCitation Data DOI : 10.24350/CIRM.V.19417403 Cite this video as: Swoboda, Jan (2018). The large scale geometry of the Higgs bundle moduli space. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19417403 URI : http://dx.doi.org/10.24350/CIRM.V.19417403

### Voir aussi

Bibliographie

1. Mazzeo, R., Swoboda, J., Weiss, H., & Witt, F. (2017). Asymptotic Geometry of the Hitchin Metric. - https://arxiv.org/abs/1709.03433

2. 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

3. 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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