Group actions on spinors. Lecture Notes.

*(English)*Zbl 0703.53001
Monographs and Textbooks in Physical Science, Lecture Notes, 9. Napoli: Bibliopolis. 116 p. (1988).

This is a set of rather mathematically sophisticated lecture notes based on courses given for theoretical physicists in Trieste and Naples. It consists of three parts dedicated respectively to algebra, geometry, and group actions. Part I (written in collaboration with G. Landi) is essentially a prĂ©cis of the Clifford algebra approach to spinors - the author describes it as a “concise dictionary of facts on Clifford algebras” - and contains sections on Clifford algebras; spinor groups; spinors; spinors and spinor groups for \(p+q\leq 5\); and charged spinors. Part II is concerned with spin structures and especially, bundle prolongations on manifolds and conditions for their existence/description of inequivalent prolongations. It includes sections on spin structures; prolongation of a structure group to its central extension; inequivalent prolongations; and homogeneous spaces: bundle of frames and spin structures. Part III is based on the author’s Ph.D. thesis (1986) and is devoted to group actions and physical considerations. Contents include sections on gauge transformations, diffeomorphisms and Weyl rescalings; diffeomorphisms and spinors on a torus and a compactified Minkowskian space-time; spaces quotiented by a discrete group; orientable and non- orientable surfaces; connections and covariant derivatives; and the Dirac operator. The notes conclude with an extensive set of references. These notes, while probably not suitable for an introduction to the subject, are valuable and will be useful for people with some prior knowledge of Clifford algebras/spinors.

Reviewer: J.D.Zund

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C27 | Spin and Spin\({}^c\) geometry |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |