Local Index Theory and Cyclic Cohomology
Open Access
- Author:
- Sanchez, Jesus
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 13, 2022
- Committee Members:
- Radu Roiban, Outside Unit & Field Member
Nigel Higson, Chair & Dissertation Advisor
Alberto Bressan, Major Field Member
Jeffrey Case, Major Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- Differential Geometry
Cyclic Cohomology
Index Theory - Abstract:
- In Alain Connes' version of noncommutative differential geometry, a spin Riemannian manifold is replaced by a spectral triple. An important achievement of Connes--Moscovici was the extension of local index theory to spectral triples. This includes the case of convolution algebras generated by discrete Lie groups, foliation groupoids, and manifolds with singularities. In this thesis we focus on both the foundational and computational aspects of the noncommutative local index theorem. We first construct out of a given spectral triple the complex powers of curvature and derive transgression relations. These are used to derive a homotopy between the Connes-Moscovici residue cocycle and the Connes-Chern character. Next, we calculate the Connes-Moscovici residue cocycle for a Riemannian manifold equipped with a 3-form. We calculate the cocycle in full generality when the 3-form is closed and in low dimensions when the 3-form is not necessarily closed. This work was done jointly with Ahmad Reza Haj Saeedi Sadegh, Northeastern University, and Yiannis Loizides, Cornell University. Next, we give a calculation of the Mehler kernel associated to the Getzler-symbol of the Dirac operator squared. We lift the Getzler calculus to the principal spin bundle and compute the rescaled heat kernel there. The geometry of the principal spin bundle is utilized to give a new proof of the local index theorem. The final piece of this thesis focuses on a departure from spectral geometry and begins to explore the realm of algebraic index theory, in the sense of Dennis Perrot's work on the index theorem in cyclic cohomology. We focus on constructing a natural signature (n,n) Dirac operator on the cotangent bundle of a manifold. We use this operator to build a periodic cyclic cocycle which calculates the Todd class of the manifold pulled back to the cosphere bundle. This work was done jointly with Jonathan Block, University of Pennsylvania, and Nigel Higson, Penn State University. The author was partially supported by NSF grant DMS-1952669.