The Local Index Theorem and the Tangent Groupoid
Open Access
- Author:
- Haj Saeedi Sadegh, Ahmad Reza
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 16, 2021
- Committee Members:
- Murat Gunaydin, Outside Unit & Field Member
Ping Xu, Major Field Member
Nigel Higson, Chair & Dissertation Advisor
Nathanial Brown, Major Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- Index Theorem
Groupoid
Spinors
Dirac Operator
Cyclic Cohomology
Rescaled Bundle
Rescaling
Harmonic Oscillator
Heat Equation - Abstract:
- The primary goal of this thesis is to study the Getzler calculus of differential operators acting on Clifford modules using the tangent groupoid. In this view, the family of heat kernels associated with a Dirac operator extends to a smooth section of a smooth vector bundle over the tangent groupoid. Over the zero fiber of the tangent groupoid this section is the heat operator of a family of geometric harmonic oscillators on the fibers of the tangent bundle. This leads quickly to a geometric proof of the “local” Atiyah-Singer index theorem. First we shall study in detail the construction of the tangent groupoid and, more generally, the construction of the deformation to the normal cone. Then we shall show that the heat kernel of any Dirac operator gives a Schwartz-class section of the rescaled spinor bundle of Higson and Yi over the tangent groupoid. Finally we shall show that we may apply a generalized Getzler’s method to Dirac operators corresponding to connections that have non-zero torsion. Bismut’s Dirac operator is one such operator. It corresponds to any three-form over the manifold. When the three-form is closed there is a local formula for the index of Bismut Dirac operator. We will show, using residue cocycles, that we may obtain similar results in low dimensions even when the three-form is not closed.