On the local constancy of regularized superdeterminants along special families of differential operators
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| Format: | Preprint |
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2025
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| author | Schiavina, Michele Stucker, Thomas |
| author_facet | Schiavina, Michele Stucker, Thomas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_03404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the local constancy of regularized superdeterminants along special families of differential operators Schiavina, Michele Stucker, Thomas Differential Geometry Mathematical Physics Algebraic Topology Dynamical Systems Spectral Theory 37C30, 58J52, 37D20, 58J10 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. |
| title | On the local constancy of regularized superdeterminants along special families of differential operators |
| topic | Differential Geometry Mathematical Physics Algebraic Topology Dynamical Systems Spectral Theory 37C30, 58J52, 37D20, 58J10 |
| url | https://arxiv.org/abs/2505.03404 |