RT Journal Article
T1 Zeta invariants of Morse forms
A1 Álvarez López, Jesús Antonio
A1 Kordyukov, Yuri A.
A1 Leichtnam, Eric
K1 Witten’s perturbation
K1 Morse form
K1 Morse complex
K1 zeta function of operators
K1 heat invariant
K1 Ray–Singer metric
AB 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.
PB Cambridge University Press
SN 1474-7480
YR 2025
FD 2025-03
LK https://hdl.handle.net/10347/39686
UL https://hdl.handle.net/10347/39686
LA eng
NO J.A. Alvarez López; Y.A. Kordyukov; E. Leichtnam. 2025. Zeta invariants of Morse forms. J. Inst. Math. Jussieu 24 (2), 411-480.
NO AEI/FEDER, UE (grants MTM2017-89686-P and PID2020-114474GB-I00).
NO Xunta de Galicia, FEDER (grant ED431C 2019/10)
DS Minerva
RD 4 may 2026