Zeta invariants of Morse forms

Abstract

Let 𝜂 be a closed real 1-form on a closed Riemannian n-manifold (𝑀,𝑔). Let 𝑑𝑧, 𝛿𝑧 and Δ𝑧 be the induced Witten’s type perturbations of the de Rham derivative and coderivative and the Laplacian, parametrized by 𝑧=𝜇+𝑖𝜈∈ℂ ( 𝜇,𝜈∈ℝ, 𝑖=√−1). Let 𝜁(𝑠,𝑧) be the zeta function of 𝑠∈ℂ, defined as the meromorphic extension of the function 𝜁(𝑠,𝑧)=Str(𝜂∧𝛿𝑧Δ−𝑠𝑧) for ℜ𝑠≫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∈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.