Bundles of elementary Clifford modules

Goal: Understand the classification of bundles of Clifford modules on a pseudo-Riemannian manifold and their description as vector bundles associated to Lipschitz structures. Describe the classification of elementary Lipschitz structures and the topological obstructions to their existence in arbitrary dimension and signature.

Supervisor: Mirela Babalic

Scheme:

A. Basic theory of Clifford algebras over the real and complex
numbers. Real and complex Bott periodicity

B. Basic theory of modules over noncommutative rings.

C. Representation theory of real and complex Clifford
algebras. Invariant bilinear forms.

D. The Clifford bundle of a pseudo-riemannian manifold. Bundles of
Clifford modules.

E. The basic theory of Lipschitz groups. Application to Spin and Pin
groups.

F. The equivalence of categories between bundles of Clifford modules
and Lipschitz bundles.

G. Topological obstructions to elementary Lipschitz bundles in
arbitrary dimension and signature.

References:

Bourbaki, Elements de Mathematique

Lawson & Michelson: "Spin Geometry", Princeton U.P., 1990

Bourguignon et al, A spinorial approach to Riemannian and conformal geometry

Rosen, Geometric multivector analysis: from Grassmann to Dirac

Lazaroiu & Shahbazi: "Real pinor bundles and real Lipschitz
structures", Asian Journal of Mathematics, Vol. 23, No. 5 (2019),
pp. 749-836

Lazaroiu & Shahbazi: "Complex Lipschitz structures and bundles of
complex Clifford modules", Differential Geometry and its Applications,
Vol. 61, Dec. 2018, Pages 147-169

Lazaroiu, Babalic & Coman: "The geometric algebra of Fierz identities
in arbitrary dimensions and signatures", JHEP09(2013)156

Pre-requisites:

a. Characteristic classes (see Subject 3)

b. Basic category theory (see S. Mac Lane, "Categories for the working
mathematician").