  We introduce volume forms on mapping stacks in derived algebraic geometry
using a parametrized version of the Reidemeister-Turaev torsion. In the case of
derived loop stacks we describe this volume form in terms of the Todd class. In
the case of mapping stacks from surfaces, we compare it to the symplectic
volume form. As an application of these ideas, we construct canonical
orientation data for cohomological DT invariants of closed oriented
3-manifolds.
