Exotic components of SO(<i>p</i>,<i>q</i>) surface group representations, and their Higgs bundle avatars

Marta Aparicio-Arroyo; Steven Bradlow; Brian Collier; Oscar García-Prada; Peter B. Gothen; André Oliveira

doi:10.1016/j.crma.2018.04.024

Algebraic geometry/Differential geometry

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

Presented by: Claire Voisin

Marta Aparicio-Arroyo ¹; Steven Bradlow ²; Brian Collier ³; Oscar García-Prada ⁴; Peter B. Gothen ⁵; André Oliveira ⁵

¹ Raet — HR software and services, Avenida de Bruselas 7, 28108 Alcobendas, Madrid, Spain
² Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
³ Department of Mathematics, University of Maryland, College Park, MD 20742, USA
⁴ Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera, 13–15, 28049 Madrid, Spain
⁵ Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 666-673.

Abstract
Résumé

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $SO (p, q)$ with $2 < p < q$ . These groups lie outside formerly know classes of groups associated with exotic components.

Pour les groupes de Lie semisimples, les espaces de modules de fibrés de Higgs sur une surface de Riemann sont en correspondance avec les variétés de représentations du groupe fondamental de la surface. Pour de nombreux groupes, les invariants topologiques naturels distinguent les composantes connexes de l'espace des modules. Les représentations de Hitchin dans un groupe réel déployé et des représentations maximales dans un groupe hermitien fournissaient les seuls exemples connus jusqu'ici dans lesquels les invariants primitifs étaient insuffisants. Cette note a pour objet d'annoncer l'existence de nouvelles composantes exotiques pour les espaces de modules pour les groupes $SO (p, q)$ , pour $2 < p < q$ .

Received: 2017-10-24
Accepted: 2018-04-26
Published online: 2018-05-10

DOI: 10.1016/j.crma.2018.04.024

Author's affiliations:

Marta Aparicio-Arroyo ¹; Steven Bradlow ²; Brian Collier ³; Oscar García-Prada ⁴; Peter B. Gothen ⁵; André Oliveira ⁵

¹ Raet — HR software and services, Avenida de Bruselas 7, 28108 Alcobendas, Madrid, Spain
² Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
³ Department of Mathematics, University of Maryland, College Park, MD 20742, USA
⁴ Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera, 13–15, 28049 Madrid, Spain
⁵ Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

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     author = {Marta Aparicio-Arroyo and Steven Bradlow and Brian Collier and Oscar Garc{\'\i}a-Prada and Peter B. Gothen and Andr\'e Oliveira},
     title = {Exotic components of {SO(\protect\emph{p},\protect\emph{q})} surface group representations, and their {Higgs} bundle avatars},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {666--673},
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     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.024},
     language = {en},
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Marta Aparicio-Arroyo; Steven Bradlow; Brian Collier; Oscar García-Prada; Peter B. Gothen; André Oliveira. Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 666-673. doi : 10.1016/j.crma.2018.04.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.024/

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Cited by

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