### Abstract

Given a compact Riemannian manifold (M^{d}, g), a finite dimensional representation ρ:π_{1}(M) → GL(V) of the fundamental group π_{1}(M) on a vector space V of dimension l and a Hermitian structure μ on the flat vector bundle ℰ → ^{p} M associated to ρ, Ray-Singer [RS] have introduced the analytic torsion T = T(M,ρ,g,μ) > 0. Witten's deformation d_{q}(t) of the exterior derivative d_{q}, d_{q}(t) = e^{-ht}d_{q}e^{ht}, with h: M → R a smooth Morse function, can be used to define a deformation T(h, t) > 0 of the analytic torsion T with T(h, 0) = T. The main results of this paper are to provide, assuming that grad _{g}h is Morse Smale, an asymptotic expansion for log T(h, t) for t → ∞ of the form Σ^{d+1}_{j=0} a_{j}t^{j} + b log t + O(1/√t) and to present two different formulae for a_{0}. As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation script capital T sign = (h, g) and the dual triangulation script capital T sign _{script D} = (d-h, g).

Original language | English (US) |
---|---|

Pages (from-to) | 320-363 |

Number of pages | 44 |

Journal | Journal of Functional Analysis |

Volume | 137 |

Issue number | 2 |

DOIs | |

State | Published - May 1 1996 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*137*(2), 320-363. https://doi.org/10.1006/jfan.1996.0049

**Asymptotic expansion of the Witten deformation of the analytic torsion.** / Burghelea, D.; Friedlander, Leonid; Kappeler, T.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 137, no. 2, pp. 320-363. https://doi.org/10.1006/jfan.1996.0049

}

TY - JOUR

T1 - Asymptotic expansion of the Witten deformation of the analytic torsion

AU - Burghelea, D.

AU - Friedlander, Leonid

AU - Kappeler, T.

PY - 1996/5/1

Y1 - 1996/5/1

N2 - Given a compact Riemannian manifold (Md, g), a finite dimensional representation ρ:π1(M) → GL(V) of the fundamental group π1(M) on a vector space V of dimension l and a Hermitian structure μ on the flat vector bundle ℰ → p M associated to ρ, Ray-Singer [RS] have introduced the analytic torsion T = T(M,ρ,g,μ) > 0. Witten's deformation dq(t) of the exterior derivative dq, dq(t) = e-htdqeht, with h: M → R a smooth Morse function, can be used to define a deformation T(h, t) > 0 of the analytic torsion T with T(h, 0) = T. The main results of this paper are to provide, assuming that grad gh is Morse Smale, an asymptotic expansion for log T(h, t) for t → ∞ of the form Σd+1j=0 ajtj + b log t + O(1/√t) and to present two different formulae for a0. As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation script capital T sign = (h, g) and the dual triangulation script capital T sign script D = (d-h, g).

AB - Given a compact Riemannian manifold (Md, g), a finite dimensional representation ρ:π1(M) → GL(V) of the fundamental group π1(M) on a vector space V of dimension l and a Hermitian structure μ on the flat vector bundle ℰ → p M associated to ρ, Ray-Singer [RS] have introduced the analytic torsion T = T(M,ρ,g,μ) > 0. Witten's deformation dq(t) of the exterior derivative dq, dq(t) = e-htdqeht, with h: M → R a smooth Morse function, can be used to define a deformation T(h, t) > 0 of the analytic torsion T with T(h, 0) = T. The main results of this paper are to provide, assuming that grad gh is Morse Smale, an asymptotic expansion for log T(h, t) for t → ∞ of the form Σd+1j=0 ajtj + b log t + O(1/√t) and to present two different formulae for a0. As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation script capital T sign = (h, g) and the dual triangulation script capital T sign script D = (d-h, g).

UR - http://www.scopus.com/inward/record.url?scp=0030140010&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030140010&partnerID=8YFLogxK

U2 - 10.1006/jfan.1996.0049

DO - 10.1006/jfan.1996.0049

M3 - Article

AN - SCOPUS:0030140010

VL - 137

SP - 320

EP - 363

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -