Zeta invariants of Morse forms
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| Format: | Preprint |
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2021
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| author | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| author_facet | López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric |
| contents | Let $η$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $δ_z$ and $Δ_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=μ+iν\in\mathbb C$ ($μ,ν\in\mathbb{R}$, $i=\sqrt{-1}$). Let $ζ(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $ζ(s,z)=\operatorname{Str}({η\wedge}\,δ_zΔ_z^{-s})$ for $\Re s\gg0$. We prove that $ζ(s,z)$ is smooth at $s=1$ and establish a formula for $ζ(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $ζ(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $μ\to+\infty$, uniformly on $ν$. 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 $η$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $η$ and $X$, and achieve the same limit as $μ\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $η$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_03191 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zeta invariants of Morse forms López, Jesús A. Álvarez Kordyukov, Yuri A. Leichtnam, Eric Differential Geometry Geometric Topology 58A12, 58A14, 58J20, 57R58 Let $η$ be a closed real 1-form on a closed Riemannian $n$-manifold $(M,g)$. Let $d_z$, $δ_z$ and $Δ_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian, parametrized by $z=μ+iν\in\mathbb C$ ($μ,ν\in\mathbb{R}$, $i=\sqrt{-1}$). Let $ζ(s,z)$ be the zeta function of $s\in\mathbb{C}$, defined as the meromorphic extension of the function $ζ(s,z)=\operatorname{Str}({η\wedge}\,δ_zΔ_z^{-s})$ for $\Re s\gg0$. We prove that $ζ(s,z)$ is smooth at $s=1$ and establish a formula for $ζ(1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $ζ(1,z)$ converges to some $\mathbf{z}\in\mathbb{R}$ as $μ\to+\infty$, uniformly on $ν$. 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 $η$ satisfying the hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf{z}$ by perturbing $g$, $η$ and $X$, and achieve the same limit as $μ\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $η$. 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. |
| title | Zeta invariants of Morse forms |
| topic | Differential Geometry Geometric Topology 58A12, 58A14, 58J20, 57R58 |
| url | https://arxiv.org/abs/2112.03191 |