On the full asymptotics of analytic torsion

2017/05/31 by Siarhei Finski · 15 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Asymptotic expansion #Bundle #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Line bundle #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Orbifold #Pure mathematics #Torsion (gastropod) #math-ph #math.DG #math.MP

paper · pdf · doi:10.1016/j.jfa.2018.06.012

published in Journal of Functional Analysis 275(12), 3457-3503 (Elsevier BV) · Published in Journal of Functional Analysis

openalex created_date 2017/05/19 · openalex publication_date 2018/07/03 · arxiv created 2018/10/16 · arxiv updated 2018/10/18 · openalex updated_date 2026/08/05

Abstract

The purpose of this article is to study the asymptotic expansion of Ray-Singer analytic tosion associated with increasing powers p of a given positive line bundle. Here we prove that the asymptotic expansion associated to a manifold contains only the terms of the form p^n-i log p, p^n-i for i-natural. For the two leading terms it was proved by Bismut and Vasserot in 1989. We will calculate the coefficients of the terms p^n-1 log p, p^n-1 in the Kahler case and thus answer the question posed in the recent work of Klevtsov, Ma, Marinescu and Wiegmann about quantuum Hall effect. Our second result concerns the general asymptotic expansion of Ray-Singer analytic torsion for an orbifold.

Citations

• Lectures on Arakelov Geometry
• Analytic torsion and holomorphic determinant bundles
• Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion
• Heat Kernels and Dirac Operators
• An arithmetic Riemann-Roch theorem
• Arithmetic intersection theory
• The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds
• Riemann-Roch isometries in the non-compact orbifold setting
• Analytic torsion and holomorphic determinant bundles