Physics 229: Advanced Mathematical Methods in Physics

Course Information for Fall 2016 / Winter 2017

We will learn modern geometric methods in physics.

Notice:

• Class meets at 469 Lauritsen, 10:30 - 12:00 on Tuesdays and Thursdays.
• TA of this class is Petr Kravchuk. He can be reached at pkravchuk at caltech.edu.

Topics to be covered:

Fall 2016

• Differential Manifolds, Riemannian Manifolds – Lecture Note 1
• Differential Forms – Lecture Note 2
• Cohomologies – Lecture Note 3
• Complex Manifolds, Kaehler Manifolds – Lecture Note 4
• Vector Bundles, Gauge Theory – Lecture Note 5
• Homology – Lecture Note 6
• Characteristic Classes – Lecture Note 7
• Supersymmetry and Index Theorems
  Lecture Note 8
  Note on Witten Index
  Supersymmetry and Morse Theory
  Supersymmetric Sigma Model

Winter 2017

• Homotopy – Lecture Note 9
  Ref: Chapter 4 of Nakahara
• Geometry of Lie Groups – Lecture 10
  Ref: Sections 5.6 and 5.7 of Nakahara
• Geometry of Gauge Theory – Lecture 11
• Random Matrix Models – Lecture 12
  Ref: Lectures on Matrix Model by M. Marino, Chapter 2 Quantum Field Theory Techniques in Graphical Enumeration by D. Bessis, C. Itzykson, and J.-B. Zuber
• Chern-Simon Gauge Theory – Lecture 13
  Quantum Field Theory and the Jones Polynomials by E. Witten

1. The default grade is pass/fail since this is an advanced graduate course.
2. I can provide letter grades on request.
3. There will be a take home exam distributed at the last class of each quarter.
4. There will be no time limit, and you are allowed to use my lecture notes and your hand-written notes.
5. You can expect that exam problems to be at the level of questions in my lecture notes.

Text Book:
I will distribute my notes every week. The notes are accessible from the Caltech network.

Recommended books:

• "Geometry, Topology and Physics" 2nd edition by M. Nakahara.
• "Gravitation, Gauge Theories and Differential Geometry" by T. Eguchi, P. B. Gilkey and A. J. Hanson.
• "Geometry of Differential Forms" by S. Morita.
• "Geometrical Methods of Mathematical Physics" by B. Schutz.