M392C: Topics in Geometry and Quantum Physics


Announcements

PLEASE NOTE: The course is moving to RLM 9.166 effective immediately.

Basic Information

Professor: Dan Freed, RLM 9.162

Class Meetings: TTh 11:00-12:30, RLM 9.166

Office Hours: W 2:00-3:30

First Day Handout

This is the second 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. The first semester is not strictly necessary to follow the lectures in the second semester, but it is useful. 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.

Problem Sets

Problem Set #1

Problem Set #1

Readings

A password is necessary to access these readings.

Atiyah-Bott-Shapiro on Clifford algebras and Clifford modules.

Deligne on Clifford algebras and spinors.

Atiyah-Singer index theorem part 1.

Atiyah-Singer index theorem part 3.

Hatcher on vector bundles and K-theory.

Chern-Simons paper on primary and secondary characteristic forms.

Section 1 of my paper on classical Chern-Simons covers the theory of connections and a bit about characteristic forms.

My more recent Bulletin survey on Chern-Simons has a brief appendix on the Chern-Simons-Weil theory of connections and characteristic forms.

First Semester Class Projects

3-dimensional abelian gauge theory with a Chern-Simons term (M. Sohaib Alam)

Adjoint orbits as Hamiltonian systems (Mio Alter and Michael Williams)

The singularity theorems of general relativity (Braxton Collier)

Mirror symmetry (Orit Davidovich)

On the classification of hyperkahler ALE spaces (Anindya Dey)

Geometric quantization (Syed Asif Hassan and Thomas Mainiero)

Knot polynomials from Chern-Simons theory (Junting Huang and Pavel Safranov)

Spinors and the positive energy theorem (Jason Jo)

Berry's phase: how quantum adiabatic evolution leads to abelian gauge theory (Bill Kalahurka)

Non-self-dual solutions to the Yang Mills equations on R-4 (David Rosen)

2-dimensional TQFT with finite gauge group (Aaron Royer)

Volume forms and diffeomorphism (Sean Simmons)

The Yang-Mills equations studied by Atiyah and Bott (Yuecheng Zhu)

First 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

First Semester 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

Problem Set #11

First Semester Projects

Project ideas and guidelines

First Semester 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 (revised)

Note on the spectral theorem for positive compact operators.

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!