Topology in Physics

Spring 2022: PHYS 601 Methods of Theoretical Physics II

Tue-Thu 12:00-1:15 pm; Room PHYS 331. The videos of the lectures are on posted YouYube here. If you have signed up for the course or planning to sign up please send me an email so that I can invite you to the SLACK channel.

Prerequisites: Familiarity with graduate-level quantum mechanics and statistical physics. The target audience is advanced graduate students in physics, with an emphasis on condensed matter and high energy. To register, you must be enrolled in a graduate program, however, advanced undergrads are welcome to audit the class.

Course description: In this course, we will introduce basic notions in topology such as homology, homotopy, and cohomology focusing on applications in topological phases of matter, supersymmetric quantum mechanics, and topological field theory.

References:

We will closely follow these lecture notes by John McGreevy, occasionally switching to Nakahara‘s book to explain concepts in topology.

Other useful references that include more advanced material are these notes by Greg Moore on topological field theories.

We will use the application Slack to post course announcements, reports, discuss ideas, and share interesting papers we have come across. If you are planning to attend the class please send me an email and I will add you to the course channel on Slack. All lectures are going to be recorded and posted on YouTube.

Homework:

The students should read the relevant section of the lecture notes before coming to class and submit two questions on slack. Every lecture will have homework problems that will be assigned in class and have to be submitted on Slack up to a week after the lecture.

Syllabus:

Click on the lecture to download the lecture note. The syllabus is subject to change as the course progresses

Part I: Homology and Physics

Lecture 0: Overview of the course
Lecture 1: Gapped phases of matter (reading: McGreevy’s notes “Introductory remarks”)
Lecture 2: Toric Code and Homology I (reading: McGreevy’s notes “Toric code and homology” up to the beginning of 1.1)
Lecture 3: Toric Code and Homology II (reading: McGreevy’s notes sec 1.1)
Lecture 4: Cell Complexes and Homology I
Lecture 5: Cell Complexes and Homology II (reading: McGreevy’s notes Sec 1.2 and 1.3)
Lecture 6: p-form Z_N Toric Code (reading: Nakahara sections 3.2 and 3.3)
Lecture 7: Some examples
Lecture 8: Higgsing (reading: McGreevy’s notes Sec 1.4)
Lecture 9: Gapped Boundaries and Relative Homology (reading: McGreevy’s notes Sec 1.5, 1.6)
Lecture 10: Duality (reading: McGreevy’s notes Sec 1.7)
Lecture 11: Summary and Questions

Part II: Cohomology and Physics

Lecture 12: Supersymmetric quantum mechanics I (reading: McGreevy’s notes Sec 2.1 up to pg55)
Lecture 13: Supersymmetric quantum mechanics II (reading: McGreevy’s notes remainder of Sec 2.1)
Lecture 14: Differential forms I (reading: McGreevy’s notes Sec 2.2)
Lecture 15: Differential forms II (reading: Nakahara Sec 5.4 and 5.5)
Lecture 16: Morse theory I (reading: McGreevy’s notes Sec 2.3)
Lecture 17: Morse theory II
Lecture 18: Global information from local information (reading: McGreevy’s notes Sec 2.4)
Lecture 19: Homology and Cohomology (reading: McGreevy’s notes Sec 2.5)
Lecture 20: Cech cohomology (reading: McGreevy’s notes Sec 2.6 and 2.7)
Lecture 21: Summary and Questions

Part III: Homotopy and Physics

Lecture 22: Homotopy groups I (reading: Nakahara 4.2 up to 4.3)
Lecture 23: Homotopy groups II (reading: Nakahara 4.3 to 4.5)
Lecture 24: Homotopy equivalence and Homology (reading: McGreevy’s notes Sec 3 to the beginning of 3.4)
Lecture 25: Homotopy groups III (reading: McGreevy’s notes Sec 3.4 and 3.5)
Lecture 26: Quantum double model (reading: McGreevy’s notes Sec 3.6)
Lecture 27: Fiber bundles and Covering maps (reading: McGreevy’s notes Sec 3.7)
Lecture 28: Vector bundles and Connections (reading: McGreevy’s notes Sec 3.8)
Lecture 29: Quantum double model and Fundamental group (reading: McGreevy’s notes Sec 3.9)
Lecture 30: Summary and Questions

The term project/take-home final exam Term Project take home exam.pdf
The deadline is May 8th.

If you have any questions/comments or suggestions please do not hesitate to send me a message.