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

Abstract

We consider the flat-regularized determinant of families of operators of the form $D_τ=[δ_τ,d_\nabla]$, where $τ\toδ_τ$ are families of degree $-1$ maps in the twisted de Rham complex $\left(Ω^\bullet(M,E),d_\nabla\right)$ generalizing the (twisted) Hodge codifferential. We show that under suitable assumptions, both geometrical and analytical in nature, the flat-regularized determinant of $D_τ$, restricted to the subspace $\mathrm{im}(δ_τ)$, is constant in $τ$. 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 $δ_τ= δ_{g_τ}$, the Hodge codifferential for a family of metrics, and $δ_τ=ι_{X_τ}$, the contraction along a family of (regular, contact) Anosov vector fields, respectively.