Let X be a connected manifold. It is well known that there is an equivalence of categories between the representations of the fundamental group of X and the flat vector bundles over X.
The aim of the talk is to explain a higher version of this statement, where the fundamental group is replaced by the singular chains on the based loop space of X. Concatenation of loops equips the chains with a multiplication, called the Pontryagin product. The representations of the resulting algebraic structure turn out to be essentially equivalent to something comparably simple, namely the dg category of flat superconnections over X.
The talk is partly based on joint work with Camilo Arias Abad (Universidad Nacional de Colombia, Medellín).