English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 typeperturbations 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 $\Res\gg0$. We prove that $\zeta(s,z)$ is smooth at $s=1$ and establish a formulafor $\zeta(1,z)$ in terms of the associated heat semigroup. For a class ofMorse 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 theinstantons of an auxiliary Smale gradient-like vector field $X$ and theMathai-Quillen current on $TM$ defined by $g$. Any real 1-cohomology class hasa representative $\eta$ satisfying the hypothesis. If $n$ is even, we canprescribe 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 anddescribe certain tempered distributions induced by $g$ and $\eta$. Thesedistributions appear in another publication as contributions from the preservedleaves in a trace formula for simple foliated flows, giving a solution to aproblem of C.~Deninger.

Zeta invariants of Morse forms