Speaker: Casey Cartwright

Title: A SUSY proof of the Dirac index theorem. Part 1: A toy theorem, and the necessary geometry

Abstract: The solution to a set of differential equations is an essential part of the study of physics. It is often the case that these solutions are difficult to come by. Although it may be difficult to find solutions, the number of solutions to this system of equations can be found. Surprisingly this number does not require knowledge of the solutions to begin with. Rather, one needs only knowledge of the topology of the manifold. This is the content of the Atiyah-Singer index theorem which states that “the analytic index of a differential operator is equal to its topological index.” A differential equation of utmost importance to theoretical physics is the Dirac equation on a general manifold. In this set of seminars I will discuss a limiting case of Atiyah-Singer index theorem when applied to the Dirac operator. In this case the theorem can be proved by investigating the path integral expression for a supersymmetric system on a general manifold. Since the Atiyah-Singer index theorem and its related material could be the subject of a semester long course this set of seminars will not constitute an in-depth treatment of the theorem. Rather, I will develop only the essential preliminaries needed to gain a basic understanding of the setting and content of the index theorem as it applies to the Dirac operator.

References: Video of Atiyah discussing the path taken to the index theorem (history fun to watch)

https://www.webofstories.com/play/michael.atiyah/46;jsessionid=80C32FE66A406E380E1F46B3717CC8D3

I will follow M. Nakahara’s treatment of this topic closely (Geometry, Topology and Physics). Additional references for the geometry and topology of fiber bundles, as well as their characteristic classes and the related homology theory, include:

Kobayashi and Nomizu: Foundations of Differential Geometry Vol 1 (Older notation but still good)

Husemoller: Fiber Bundles (Good for K-theory and Characteristic Classes of Bundles)

Milnor and Stasheff: Characteristic Classes (Good obviously for characteristic classes and bundles)

Hatcher: Vector Bundles and K-theory (Bit heavy on the K-theory still very interesting. Begining has a good passage on bundles. Written very well.)

Frankel: The Geometry of Physics: An introduction (HUGE BOOK. Has a LOT of topics also has good discussion of bundles.)

If You Are Brave…..

Deligne, Kazhdan, Etingof, Morgan, Freed, Morrison, Jeffery, Witten: Quantum Fields and Strings: A Course for Mathematicians (pg 475 starts a discussion of the index theorem for the Dirac operator and some related theorems. This book IS INTENSE, very good if you are mathematically inclined. i.e. you’ve studied math at/beyond a/the graduate level)

Notes: HEPSeminar_F2017_Cartwright_NotesTalk1.pdf

Exercises: HEPSeminar_F2017_Cartwright_Homework1