Reidemeister torsion for flat superconnections

Abstract

We use higher parallel transport—more precisely, the integration \(\mathsf{A }_\infty \)-functor constructed in Arias Abad and Schatz (The \(\mathsf{A}_\infty \) de Rham theorem and the integration of representations up to homotopy. Int Math Res Not, 2013) and Block and Smith (A Riemann–Hilbert correspondence for infinity local systems. arXiv:0908.2843, 2012)—to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger–Muller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu (Contemp Math 546, 199–212, 2011).