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.
Click here to download the public-facing syllabus. See Brightspace for the student-facing syllabus.
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 |
8/28. Groups, Invertible Symmetries, and Categories (Part I) |
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)
|
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)
|
9/11.Representation theory, Fusion Rings, and Noninvertible Symmetries (Part I)
|
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)
|
9/18. Representation theory, Fusion Rings, and Noninvertible Symmetries (Part III)
|
|
| Week 5 |
9/23. Kitaev's Quantum Double Model (Part I)
|
9/25. Kitaev's Quantum Double Model (Part II)
|
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 (Part I)
|
10/2. Fusion Categories, TQFTs, and Anyons (Part I) Lecture notes (still under construction)
|
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)
|
Sections 1.1 and 2.1 in Gert Vercleyen's PhD Thesis Section IIA in Barkeshli , Bonderson, Cheng, and Wang (2014) Symmetry Fractionalization, Defects, and Gauging of Topological Phases Appendix E.1 and E.2 in Kitaev's Anyons in an exactly solved model and beyond |
| Week 8 |
10/14. No class (Fall Break) |
10/16. Levin-Wen String-Net Model |
Hahn and Wolf's Generalized string-net model for unitary fusion categories without tetrahedral symmetry Levin and Wen's original paper String-net condensation:A physical mechanism for topological phases |
| Week 9 |
10/21. Atiyah-Segal axioms, braided monoidal categories, and TQFT |
10/23. Turaev-Viro-Barrett-Westbury state-sum TQFT |
Atiyah's 1988 paper Topological Quantum Field Theory Turaev and Viro's 1992 State sum invariants of 3-manifolds and quantum 6j-symbols |
| Week 10 |
10/28. Fusion Categories, TQFTs, and Anyons (Part IV) |
10/30. Anyon Models |
|
| Week 11 |
11/4. Topological Quantum Computation |
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 |