From Clifford Bundles to Spinor Classification: Algebraic and Geometric Approaches, and New Spinor Fields in Flux Compactifications

By bridging geometric and algebraic concepts, this dissertation lays the groundwork for a comprehensive study of the Clifford structures on bundles and spinor fields. We delve into the K\"ahler-Atiyah bundle, which encapsulates the essence of Clifford algebras and provides profound insights into the algebraic structures underlying geometric frameworks. The algebraic and classical definitions of spinors within Clifford algebras are concerned, as well as their global realisation as sections of the bundle of spinors constructed within a spin structure on a manifold from the K\"ahler-Atiyah bundle background which serves as an effective framework for applications involving spinors such as their classification based on their bilinear covariants and Fierz identities. Homogeneous differential forms playing the role of bilinear covariants can vanish due to algebraic obstructions in a warped flux compactification AdS$_3 \times M_8$ yielding the identification of new spinor field classes.