Abstract: I will discuss a certain generalization of the notion of a flat connection on a vector bundle, known as a higher local system (or a flat superconnection). My main goal is to explain that if ordinary flat connections are representations of the fundamental group of X, higher local systems should be thought of as representations of the whole homotopy type of X. I will also discuss how higher local systems are related to loop spaces and briefly mention some (finished and unfinished) projects. This talk is based on joint work with F. Schaetz (Luxembourg).
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