Title: Higher Complex Structures and Flat Connections

Abstract: In 2018, Vladimir Fock and the author introduced a geometric structure on surfaces, called higher complex structure, whose moduli space shares several properties with Hitchin's component. Conjecturally, both spaces are canonically diffeomorphic, giving a new geometric approach to higher Teichm\"uller theory. In this paper, we prove several steps towards this conjecture and give a precise picture what has to be done.
We show that higher complex structures can be deformed to flat connections. More precisely we show that the cotangent bundle of the moduli space of higher complex structures can be included into a 1-parameter family of spaces of flat connections. Assuming some conjecture, we prove the equivalence with Hitchin's component.