On the local constancy of regularized superdeterminants along special families of differential operators

We consider the flat-regularized determinant of families of operators of the form $D_{\tau }=[{\delta }_{\tau },d_{\mathrm{\nabla }}]$, where $\tau \to {\delta }_{\tau }$ are families of degree $-1$ maps in the twisted de Rham complex $({\mathrm{\Omega }}^{\bullet }(M,E),d_{\mathrm{\nabla }})$ generalizing the (twisted) Hodge codifferential. We show that under suitable assumptions, both geometrical and analytical in nature, the flat-regularized determinant of $D_{\tau }$, restricted to the subspace $\mathrm{i}\mathrm{m}({\delta }_{\tau })$, is constant in $\tau$. The general result we present implies both local constancy of the Ray–Singer torsion and of the value at zero of the Ruelle zeta function for a contact Anosov flow, upon choosing ${\delta }_{\tau }={\delta }_{g_{\tau }}$, the Hodge codifferential for a family of metrics, and ${\delta }_{\tau }={\iota }_{X_{\tau }}$, the contraction along a family of (regular, contact) Anosov vector fields, respectively.