A geometric description of the heat kernel of the Witten Laplacian and the Cheeger-Müller theorem

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Vasu, Karthik

Contributors dc:contributor

• Albin, Pierre
• Dunfield, Nathan M
• Laugesen, Richard
• La Nave, Gabriele

• Cheeger Muller theorem
• Blow up
• heat kernels

Statement dc:rights

• Copyright 2025 Karthik Vasu

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129410

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129410