Character varieties and their relation with principal Higgs bundles

We begin by reviewing the notions of Lie groups and algebraic groups, both real and complex, as well as finitely generated groups, which are essential components in the definition of a character variety. We then analyze the space of representations of a finitely generated group into a Lie or an algebraic group, focusing on its topology, variety structure, tangent space, and the conjugation action by the target group. Next, we construct the quotient with respect to this action, namely the character variety, and present different approaches to this construction. We also examine the existence of deformation retractions between character varieties when considering maximal compact subgroups of the Lie group. Finally, we discuss the correspondence between representations of surface groups and principal bundles, with particular emphasis on Schottky representations, and relate these constructions to the geometric setting of principal Higgs bundles and branes. This perspective suggests further directions connecting representation theory, non-Abelian Hodge correspondence, and the geometry of moduli spaces.