ZETA INVARIANTS OF MORSE FORMS

Abstract 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.