Asian Journal of Mathematics

Volume 21 (2017)

Number 5

Connectedness of Higgs bundle moduli for complex reductive Lie groups

Pages: 791 – 810



Oscar García-Prada (Instituto de Ciencias Matem´aticas, Madrid, Spain)

André Oliveira (Centro de Matemática da Universidade do Porto, Portugal; and Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal)


We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li in “The Space of Surface Group Representations” [Manuscripta Math., 78 (1993), pp. 223–243] of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.


semistable Higgs bundles, connected components of moduli spaces

2010 Mathematics Subject Classification

14D20, 14F45, 14H60

Received 20 August 2014

Accepted 17 March 2016

Published 9 February 2018