Spinors have played a crucial role in both the physics and the mathematics of the 20th century. Discovered in 1913 by Cartan in his investigations of the representation theory of the orthogonal groups, spinors first appeared in physics in the 1920's in the guise of Pauli's spin matrices and in Dirac's relativistic theory of electron spin. Since that time, spinors, spin structures and their attendant Dirac operators have remained of fundamental importance in quantum physics and in many areas of mathematics, especially those dealing with the relation between geometry, topology and analysis. In mathematics they provide key insights into many questions, including index theorems for elliptic operators, the integrality of characteristic numbers, existence of metrics of positive scalar curvature, twistor spaces, and (most recently) Seiberg-Witten theory.
The goal of this course is to survey the role of spinors in mathematics and physics with emphasis on topics of modern importance. We will emphasize the interaction between the physical and mathematical points of view. Topics to be covered include:
- Clifford algebras and representations of Spin groups
- spin structures on manifolds and Dirac operators
- the role of spin in physical applications
- index theory and anomalies in quantum field theory
- supersymmetry, Seiberg-Witten theory, invariants of four manifolds
- applications of spin geometry in superstring theory; dualities
- spinors and twistors in mathematics and physics
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