  We study polystable Higgs bundles twisted by a line bundle over a compact
K\"ahler manifold. These form a Tannakian category when the first and second
Chern classes of the bundle are zero. In this paper we identify the
corresponding Tannaka group in the case in which the line bundle is of finite
order. This group is described in terms of the pro-reductive completion of the
fundamental group of the manifold, and the character associated to the line
bundle.
