Spinors in Geometry and Physics

Math 425 and Physics 498C

S.B. Bradlow and R.G. Leigh

Table Of Contents:

Course Abstract

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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How to find us

Steve Bradlow

238 Illini Hall
Rob Leigh

431 Loomis Lab

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How to find the classroom

3:00-4:00 PM
144 Loomis Lab

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How to make the grade

  • come to class
  • participate in tutorials
  • do a term paper

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Web resources and learned societies

If you just can't get enough of spinors, Clifford algebras, etc., there are people just like you out there on the 'net. Check 'em out:

John Baez (UC-Riverside) produces "This Week's Finds in Mathematical Physics." You may enjoy reading these, and the latest issues are directly relevant to this course.

Simple reviews of some of the physics covered in the first few weeks can be found in many books, but also on the web

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Reading List

Please note that we do not REQUIRE you to purchase any of these books, although several of them may be found in the bookstore. However, you will find that Gilkey in particular is a good reference book on this and other topics. Many of them will also be found on reserve at the Math Library in Altgeld Hall.

Core Texts:

P.B. Gilkey
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
2nd edition, CRC Press, 1995.

J.W. Morgan
The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds
Princeton, 1996.

H.B. Lawson and M.-L. Michelson
Spin Geometry
Princeton, 1989.

Other Texts and Sources:

M.Atiyah, R.Bott and A.Shapiro
Clifford Modules
Topology, Vol 3 1964 pp 3-38

F. Reese Harvey
Spinors and Calibrations
Academic, 1990.

I.M. Benn and R.W. Tucker
An Introduction to Spinors and Geometry with Applications in Physics
Hilger, 1987

C. Chevalley
The Algebraic Theory of Spinors and Clifford Algebras
Springer, 1991.

P. Lounesto
Clifford Algebras and Spinors
Cambridge, 1997.

I.R. Porteous
Clifford Algebras and the Classical Groups
Cambridge, 1995.

E. Cartan
The Theory of Spinors
Dover, 1966.

Wm. E. Baylis, ed.
Clifford (Geometric) Algebras
Birkhauser, 1996.

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©1997 R.G. Leigh