M392C: Topics in Geometry and Quantum Physics


Announcements

I will hold office hours Monday, November 23, 3:30-5:00 and Tuesday, November 24, 3:30-4:30. No office hours the rest of the week.

Projects are due at the beginning of class December~1. When writing think about explaining the theorem(s) or other results you have been studying to a classmate. This may include background (definitions, previous work), the statement, examples, (aspects of) proofs, etc.

Basic Information

Professor: Dan Freed, RLM 9.162

Class Meetings: TTh 11:00-12:30, RLM 10.176

Office Hours: W 2:00-3:00, Th 10:00-11:00

First Day Handout

This is the first semester of a year-long topics course. I will treat topics from geometry and topology which are relevant for quantum field theory and string theory. The emphasis is on the mathematics, though I will also explain some basics about the physics as well. The course is designed to be of interest to all geometry students. Roughly, the first semester will cover geometry which enters into semi-classical descriptions of quantum systems. These include fiber bundles and connections, symplectic and Poisson geometry, Riemannian geometry, spin geometry, etc. Lectures will introduce the basic ideas and prove some theorems about them in geometry as well as discuss their use in the physics. Topics for the second semester will probably include the Heisenberg representation (free bosonic particles and fields) and the spin representation (free fermionic fields), both in finite and infinite dimensions; anomalies and the Atiyah-Singer index theorem; supersymmetry; and perhaps others. The formal work for the course is a term paper each semester.

Notes

The lecture numbers in the notes do not correspond to the lectures; there is a monotone map which relates them. Please mark corrections, comments, questions, etc. about the notes. Do NOT email them to me, however. Come and talk to me during office hours instead.

A password is necessary to access these notes.

Lecture #1: Affine spaces and spacetimes

Problem Sets

Problem Set #1

Problem Set #2

Problem Set #3

Problem Set #4

Problem Set #5

Problem Set #6

Problem Set #7

Problem Set #8

Problem Set #9

Problem Set #10

Projects

Project ideas and guidelines

Readings

A password is necessary to access these readings.

Notes on basic geometry of Minkowski spacetime by John Milnor: Part 1, Part 2

Gauge theory, a survey article for the "general public".

Old lecture notes on connections leading up to the classical Yang-Mills equations.

Notes by Pavel Etingof on mathematical ideas in quantum field theory.

Chapter about lagrangians and symmetries from "Five Lectures on Supersymmetry".

Classical Field Theory (joint with P. Deligne) from "Classical Fields and Strings: A Course for Mathematicians".

Notes on elliptic theory

Summer Reading (Prerequisites)

Some students have asked what they can do to prepare for the course. I strongly recommend strong grounding in basics smooth manifolds, particularly calculus on manifolds. This includes some topics beyond the prelim class, such as Lie derivatives (including forms), the Frobenius theorem, basics about Lie groups, etc. Here are some suggested texts:

Volume 1 of Michael Spivak's "Comprehensive Introduction to Differential Geometry"
Chapters 1-4 of Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups"
Jack Lee's book on smooth manifolds
Notes on smooth manifolds by Nigel Hitchin: 1 2 3 4 Appendix

If you'd like to go further, you can learn some algebraic topology. The classic "Differential Forms in Algebraic Topology" by Bott and Tu is highly recommended. You should also know some algebraic topology from a more traditional point of view, as in Hatcher's book.

Finally, if you'd like to read up on the physics background I recommend learning about special relativity and electromagnetism. The Feynman Lectures on Physics make great reading, including the third volume on quantum mechanics.

Please do not feel you need to read all of this to attend and follow the lectures! These are suggestions. Also, it will be more fun if you form study groups with other students; email me if you need help finding other students registered for the course.

Have a great summer!