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We investigate parabolic Higgs bundles and Γ-Higgs bundles on a smooth complex projective variety.

We investigate principal bundles over a root stack. In the case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.We will recall that the classifying stack BG is the fibred category whose objects over a C-scheme U are principal G-bundles over U and whose morphisms are pullback diagrams of G-bundles. The following is not hard to verify.Lemma 1.1. The datum of a principal G-bundle over X in the above sense is equivalent to the datum of a morphism X → BG. Two G-bundles over X are isomorphic if and only if the corresponding morphisms X → BG are 2-isomorphic.Given X, G as above, we may now consider the fibred category whose objects over a C-scheme U are G-bundles E → X × U and whose morphisms are pullback diagrams of G-bundles. We will denote this category by Bun G X.

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