Zeta invariants of Morse forms

TL;DR

This paper studies zeta functions associated with Morse forms on Riemannian manifolds, establishing their properties, limits, and connections to geometric structures and trace formulas, with implications for dynamical systems and topology.

Contribution

It introduces a new zeta function for Morse forms, analyzes its behavior at s=1, and links it to heat semigroups, instantons, and geometric currents, extending previous trace formula results.

Findings

 zeta function  smooth at s=1 and has a formula in terms of heat semigroup.

  as   converges to a real number  as    for Morse forms.

 for even n, any real value of  can be prescribed by perturbing geometric data.

Abstract

Let $\eta$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $\delta_z$ and $\Delta_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu+i\nu\in\mathbb C$ ($\mu,\nu\in\mathbb{R}$, $i=\sqrt{-1}$). Let $\zeta(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $\zeta(s,z)=\operatorname{Str}({\eta\wedge}\,\delta_z\Delta_z^{-s})$ for $\Re s\gg0$. We prove that $\zeta(s,z)$ is smooth at $s=1$ and establish a formula for $\zeta(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $\mu\to+\infty$, uniformly on $\nu$. We describe $\mathbf{z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class has a representative $\eta$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $\eta$ and $X$, and achieve the same limit as $\mu\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $\eta$. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem of C.~Deninger.