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

Michele Schiavina, Thomas Stucker

TL;DR

The paper develops a unified framework for local constancy of regularized superdeterminants along smooth families of general codifferentials on a twisted de Rham complex. By introducing inner variations of codifferentials $\delta_\tau$ and the associated characteristic operators $D_\tau=[\delta_\tau,d_\nabla]$, the authors derive a general invariance result for the flat determinant $\mathrm{sdet}^\flat(D_\tau|_{L_\tau})$ under suitable analytic and wavefront assumptions. Special cases recover the classical invariance of analytic torsion under metric variation and, for regular contact Anosov flows, the local constancy of the value at zero of the Ruelle zeta function, linking topological and dynamical invariants through a BV-inspired gauge-fixing interpretation. The results provide a coherent method to understand Fried-type conjectures within a broader algebraic-analytic setting, and they offer a pathway to reformulate Fried's conjecture via families of codifferentials and their gauge-fixing data.

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.

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

TL;DR

The paper develops a unified framework for local constancy of regularized superdeterminants along smooth families of general codifferentials on a twisted de Rham complex. By introducing inner variations of codifferentials and the associated characteristic operators , the authors derive a general invariance result for the flat determinant under suitable analytic and wavefront assumptions. Special cases recover the classical invariance of analytic torsion under metric variation and, for regular contact Anosov flows, the local constancy of the value at zero of the Ruelle zeta function, linking topological and dynamical invariants through a BV-inspired gauge-fixing interpretation. The results provide a coherent method to understand Fried-type conjectures within a broader algebraic-analytic setting, and they offer a pathway to reformulate Fried's conjecture via families of codifferentials and their gauge-fixing data.

Abstract

We consider the flat-regularized determinant of families of operators of the form , where are families of degree maps in the twisted de Rham complex generalizing the (twisted) Hodge codifferential. We show that under suitable assumptions, both geometrical and analytical in nature, the flat-regularized determinant of , restricted to the subspace , is constant in . 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 , the Hodge codifferential for a family of metrics, and , the contraction along a family of (regular, contact) Anosov vector fields, respectively.
Paper Structure (17 sections, 27 theorems, 270 equations)

This paper contains 17 sections, 27 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. A remark concerning Fried's conjecture
  3. Geometric setup
  4. General codifferentials
  5. The field-theoretic perspective
  6. Flat superdeterminants of characteristic operators
  7. Structure of the general codifferential variation
  8. Local constancy of the flat superdeterminant
  9. Proof of Theorem \ref{['thm_invariance']}
  10. Local constancy of the analytic torsion
  11. The Hodge general codifferential
  12. Local constancy in the elliptic case
  13. Local constancy of the value at zero of the Ruelle zeta function
  14. The contact general codifferential
  15. Anosov flows and the Ruelle zeta function
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(-1,1)\mapsto \delta_\tau$ be a smooth family of degree $-1$ maps in $\Omega^\bullet(M,E)$ such that $\delta_\tau^2=0$ and $\delta_\tau$ is a regular general codifferential for all $\tau\in(-1,1)$ (Definitions def_general_codifferential and def_codifferential_types). Denote the associated chara

Theorems & Definitions (74)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Definition 2.1: Twisted topological data
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5: Codifferential and "gauge fixing"
  • Remark 2.6
  • ...and 64 more