Legendrian and Lagrangian higher torsion

Let $M$ be a closed manifold. We introduce a family of Legendrian isotopy invariants for Legendrians in $J^1M$, which we collectively call Legendrian higher torsion. Given a choice of a class $\mathcal{F}$ of fibre bundles over $M$, equipped with suitable unitary local systems, the Legendrian higher torsion of a Legendrian $\Lambda \subset J^1M$ is the subset of $H^*(M;\mathbf{R})$ consisting of higher Reidemeister torsion cohomology classes of fibre bundles $W$ over $M$ in the class $\mathcal{F}$ such that $\Lambda$ admits a generating function on a stabilization of $W$. For the class of tube bundles in the sense of Waldhausen we call the invariant tube torsion. In particular, we show that the tube torsion of a nearby Lagrangian $L \subset T^*M$ is well-defined when the stable Gauss map $L \to U/O$ is trivial and consists of a union of cosets of a normalized version of the Pontryagin character. We also identify a distinguished coset, invariant under Hamiltonian isotopy of $L$, which we call nearby Lagrangian torsion. We do not know whether nearby Lagrangians must have trivial tube torsion, as would follow from the nearby Lagrangian conjecture. However, we show that there exist Legendrians $\Lambda \subset J^1M$ with nontrivial tube torsion whose projection $\Lambda \to M$ is homotopic to a diffeomorphism.