A geometric description of the heat kernel of the Witten Laplacian and the Cheeger-Müller theorem
Vasu, Karthik
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https://hdl.handle.net/2142/129410
Description
Title
A geometric description of the heat kernel of the Witten Laplacian and the Cheeger-Müller theorem
Author(s)
Vasu, Karthik
Issue Date
2025-04-18
Director of Research (if dissertation) or Advisor (if thesis)
Albin, Pierre
Doctoral Committee Chair(s)
Dunfield, Nathan M
Committee Member(s)
Laugesen, Richard
La Nave, Gabriele
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Cheeger Muller theorem
Blow up
heat kernels
Language
eng
Abstract
In 1935, Reidemeister introduced the Reidemeister torsion, an invariant on cochain complexes with fixed basis. These invariants were used to distinguish homotopy equivalent manifolds that are not homeomorphic. Ray and Singer in 1971, introduced an analytic analogue of the torsion on smooth manifolds, the Ray-Singer analytic torsion. They noticed these two torsions exhibit similar properties and conjectured that they must be equal. The conjecture was proved independently by Cheeger and Müller in 1979. Later in 1994, Bismut and Zhang introduced a new proof of the Cheeger-Müller theorem using the heat kernel of the Witten Laplacian. The proof was based on the study of the asymptotic expansion of the heat kernel. In this thesis, we provide a geometric description of the heat kernel of the Witten Laplacian by using Melrose’s blow up techniques. We construct a blown up heat space and show that the heat kernel is polyhomogeneous on it. We then use Getzler’s rescaling argument to get a explicit expression for the anomaly term when the representation is not unimodular.
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