I will begin by describing the Cheeger-Müller theorem. This is the fact that the Reidemeister torsion — a combinatorial invariant of odd dimensional manifolds — is equal to the analytic torsion defined by Ray and Singer. I will then explain how higher dimensional parallel transport, constructed via the $A_\infty$ version of de Rham\'s theorem, can be used to define Reidemeister torsion for flat superconnections.

This talk is based on joint work in progress with F. Schätz.