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!