PHYS 570: Fusion Categories for Physics

Course Description

An introduction to fusion categories and their applications to physics. Students will learn to work with fusion categories and their algebraic precursors with an emphasis on graphical and computational methods with the goal of preparing for research in theoretical/mathematical physics. Applications to be covered may include but are not limited to: anyon models and topological phases of matter in (2+1)D, topological quantum field theory, generalized symmetries in (1+1)D, quantum error correction, topological quantum computation, superselection theory, and (1+1)D conformal field theory.

Syllabus

Click here to download the public-facing syllabus. See Brightspace for the student-facing syllabus.

Course Schedule

The pace of the course will be based on the participants and the schedule will be adjusted accordingly. Suggestions for further reading to reinforce each week's lecture content will be posted as the course progresses.

Tuesday Thursday Further Reading
Week 1

8/26. Course overview

Lecture notes

8/28. Groups, Invertible Symmetries, and Categories (Part I)

Lecture notes

I.1-I.2 in Zee's Group Theory in a Nutshell for Physicists

Chapters 1-3 in Abstract Algebra, Dummit and Foote

Chapter 1 in Category Theory in Context, Riehl

Week 2

9/2. Groups, Invertible Symmetries, and Categories (Part II)

Lecture notes

9/4. The \( \mathbb{Z}_2 \) toric code (Part I)

Section 1 and 2 of Kitaev's Fault-tolerant quantum computation by anyons

Section 3.1 of Kong and Zhang's An invitation to topological orders and category theory

Section 2 of Purdue Professor Shawn Cui's lecture notes on Topological Quantum Computation

Chapter 27 of Simon's Topological Quantum (Chapter 24 in the Proto-Book)

Week 3

9/9. The \( \mathbb{Z}_2 \) toric code (Part II)

Lecture notes

9/11.Representation theory, Fusion Rings, and Noninvertible Symmetries (Part I)

Lecture notes

Unit 1 Exercises Due, TeX source

II.1-I.3 in Zee's Group Theory in a Nutshell for Physicists

Chapters 1-2 in Representation Theory; A First Course, Fulton and Harris

Chapters 18-19 in Abstract Algebra, Dummit and Foote

Week 4

9/16. Representation theory, Fusion Rings, and Noninvertible Symmetries (Part II)

Lecture notes

9/18. Representation theory, Fusion Rings, and Noninvertible Symmetries (Part III)

Lecture notes

Week 5

9/23. Kitaev's Quantum Double Model (Part I)

Lecture notes

9/25. Kitaev's Quantum Double Model (Part II)

Lecture notes

Unit 2 Exercises Due, TeX source

Section 5 and 6 of Kitaev's Fault-tolerant quantum computation by anyons

Sections 3 and 4 of Purdue Professor Shawn Cui's lecture notes on Topological Quantum Computation

Chapter 31.1-31.3 of Simon's Topological Quantum (Chapter 25.0 only in the Proto-Book)

Week 6

9/30. Kramers-Wannier Duality

10/2. Fusion Categories, TQFTs, and Anyons (Part I)

Section 3.3.1 in Shu-Heng Shao's TASI Lectures on Non-Invertible Symmetries

Seiberg, Seifnashri, and Shao (2024) Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

Week 7

10/7. Fusion Categories, TQFTs, and Anyons (Part II)

10/9. Fusion Categories, TQFTs, and Anyons (Part III)

Unit 3 Exercises Due, TeX source

Week 8

10/14. No class (Fall Break)

10/16. Levin-Wen string-net Models

Week 9

10/21. Turaev-Viro-Barrett-Westbury state-sum TQFT

10/23. Fusion Categories, TQFTs, and Anyons (Part IV)

Unit 4 Exercises Due

Week 10

10/28. Fusion Categories, TQFTs, and Anyons (Part V)

10/30. Topological Quantum Computation (Part I)

Week 11

11/4. Topological Quantum Computation (Part II)

11/6. Module categories, domain walls, boundaries, defects, and SymTFT (Part I)

Unit 5 Exercises Due

Week 12

11/11. Module categories, domain walls, boundaries, defects, and SymTFT (Part II)

11/13. Module categories, domain walls, boundaries, defects, and SymTFT (Part III)

Week 13

11/18. Module categories, domain walls, boundaries, defects, and SymTFT (Part IV)

11/20. Guest Lecture: Computational methods for fusion categories and anyon models

Unit 6 Exercises Due

Week 14

11/25. Module categories, domain walls, boundaries, defects, and SymTFT (Part V)

11/27. No class (Thanksgiving Break)

Week 15

12/2. Advanced Topics Presentations

12/4. Advanced Topics Presentations

Week 16

12/9. Advanced Topics Presentations

12/11. Advanced Topics Presentations