Clifford Geometric Algebras, Spin Manifolds, and Group Actions in Mathematics and Physics

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The main purpose of this paper is to highlight some striking features of the relationship between Clifford geometric algebras, spin manifolds and group actions. In particular, we shall stress the fundamental importance of these concepts for a deep understanding of the connections between mathematics and physics. There is more and more evidence that spinors—and the structure underlying them, such as Clifford algebras, spin groups and their representations, and spin structures on manifolds—are profoundly involved in the structure and dynamics of most fundamental physical theories like supergravity, superstring theory and non-commutative geometry. Furthermore, Clifford algebras and bundles, spin groups and their representations seem to be deeply linked with the most significant topics in geometry and topology, as the Hopf homotopy groups, the Bott periodicity theorem and the Atiyah-Singer index theorem show it clearly. We shall address some relevant aspects of these subjects. That small portions of space [i.e., in our present language, quantum space] are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.W. K. Clifford. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon.H. Weyl As mathematician I would find it a pity if God has not found some use for all the beautiful ideas[e.g.: string theory and quantum field theory, twistor theory, and non-commutative geometry] that have been put forward. M. Atiyah