Witten deformation of the analytic torsion and the Reidemeister torsion

1998 ◽  
pp. 23-39
Author(s):  
Dan Burghelea ◽  
Leonid Friedlander ◽  
Thomas Kappeler
Keyword(s):  
Analytic Torsion ◽  
Reidemeister Torsion ◽  
Witten Deformation

scholarly journals A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS

2015 ◽  
Vol 16 (4) ◽  
pp. 673-743 ◽  
Author(s):  
Jonathan Pfaff
Keyword(s):  
Asymptotic Behaviour ◽  
Eisenstein Series ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Analytic Torsion ◽  
Arithmetic Groups ◽  
Reidemeister Torsion ◽  
Cohomology Of Arithmetic Groups ◽  
Gluing Formula

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems, we define the Reidemeister torsion of the Borel–Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula, and here our approach is heavily inspired by a recent paper of Lesch.

scholarly journals Asymptotic Expansion of the Witten Deformation of the Analytic Torsion

1996 ◽  
Vol 137 (2) ◽  
pp. 320-363 ◽  
Author(s):  
D. Burghelea ◽  
L. Friedlander ◽  
T. Kappeler
Keyword(s):  
Asymptotic Expansion ◽  
Analytic Torsion ◽  
Witten Deformation

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

2021 ◽  
pp. 1-38
Author(s):  
Xianzhe Dai ◽  
Junrong Yan
Keyword(s):  
Essential Role ◽  
Morse Function ◽  
Noncompact Manifold ◽  
Morse Inequalities ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Witten Laplacian ◽  
Relative Cohomology ◽  
Witten Deformation ◽  
Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

Sign in / Sign up

Export Citation Format

Share Document