ZETA INVARIANTS OF MORSE FORMS
Let $\eta $ be [-11pc] [-7pc]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 \...
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| Vydáno v: | Journal of the Institute of Mathematics of Jussieu Ročník 24; číslo 2; s. 411 - 480 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cambridge, UK
Cambridge University Press
01.03.2025
|
| Témata: | |
| ISSN: | 1474-7480, 1475-3030 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
$\eta $
be [-11pc] [-7pc]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
$\Re s\gg 0$
. 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 stated by C. Deninger. |
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| ISSN: | 1474-7480 1475-3030 |
| DOI: | 10.1017/S1474748024000343 |