ZETA INVARIANTS OF MORSE FORMS - Álvarez López, Jesús A.; Kordyukov, Yuri A.; Leichtnam, Eric (Journal of the Institute of Mathematics of Jussieu, 2025)
Let $η $ be [-11pc] [-7pc]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 $\Re s\gg 0$ . 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 $z\in \mathbb {R}$ as $μ → +∞ $ , uniformly on $ν $ . We describe $z$ 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 $z$ by perturbing g, $η $ and X and achieve the same limit as $μ → -∞ $ . 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 stated by C. Deninger.