Abstract
Starting from Scott's recent work on Logarithmic Topological Quantum Field Theories (LogTQFTs), we will show that the Euler characteristic of a manifold with boundary is another instance of a topological invariant arising as a character of a LogTQFT. Along the way, we will prove a classification theorem for 2-dimensional LogTQFTs and study the additivity (with respect to gluing) of the index of Dirac operators from the point of view of the boundary integrals.In Part II, we will generalize the ideas and concepts in Part I and introduce Higher LogTQFTs, which will be defined as log-functors on subcategories of Cobn, the category of n-dimensional cobordisms. Such log-functors take values in the cyclic homology of a representation of Cobn and will be, in most cases, obtained by composition with Chern characters. This generalization appears natural in the light of the functorial construction of a LogTQFT and provides a tool to capture finer additive invariants of manifolds which arise from the presence of additional data, such as a fibering of the manifold or a group action on a covering. The family and Novikov signatures will be shown to be two key examples of characters of higher logTQFTs and their additive nature will arise as a consequence of this.
Finally, in Part III, we will define a new log-structure called residue analytic torsion, in analogy with Ray-Singer analytic torsion, and introduced for the first time by Scott in his last work. It is defined via Wodzicki residue trace, hence the name. We will show a classification theorem for residue torsion on manifolds (with and without boundary) and relate this results to Index Theory and LogTQFTs. Moreover, it will also be possible to extend such torsion to fibre bundles and characterize it in terms of Higher LogTQFTs, in the spirit of Part II.
| Date of Award | 1 Jul 2018 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | Simon Scott (Supervisor) & Alexander Pushnitski (Supervisor) |