Witten deformation of the analytic torsion and the spectral sequence of a filtration

1994/11/27 by Maxim Braverman, Braverman, Maxim
Computer Science · Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #High Energy Physics - Theory (hep-th) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #dg-ga #hep-th #math.DG

paper · pdf · doi:10.48550/arxiv.dg-ga/9411013

23 pages; AMS-LaTeX; to appear in GAFA

openalex publication_date 1994/11/27 · arxiv created 1995/06/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a flat vector bundle over a compact Riemannian manifold M and let f be a Morse function. Let g be a smooth Euclidean metric on F, let g_t=e^-2tfg and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g_t. Assuming that the vector field grad(f) satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log(ρ(t)) for t→ +∞ of the form a₀+a₁t+blog(\frac tπ)+o(1), where the coefficient b is a half-integer depending only on the Betti numbers of F. In the case where all the critical values of f are rational, we calculate the coefficients a₀ and a₁ explicitly in terms of the spectral sequence of a filtration associated to the Morse function. These results are obtained as an applications of a theorem by Bismut and Zhang.

Related

• Witten deformation of Ray-Singer analytic torsion
• Witten deformation using Lie groupoids
• Witten deformation for non-Morse functions and gluing formula for analytic torsions
• Witten deformation and the spectral package of a Riemannian manifold
• Adiabatic limit, Witten deformation and analytic torsion forms