Fusion Categories and SymTFT
Graduate, State University of Londrina, Physics Department, 2026
This lecture series provides a minimal yet practical toolkit for understanding and working with generalized symmetries from a categorical perspective. We will cover, among other topics, topological phases of matter in (2+1)D, topological quantum field theories, generalized symmetries in (1+1)D, and their concrete realizations in lattice models.
Tentative Syllabus
- Groups, Representation Theory, and Fusion Rings;
- Unitary Fusion and Braided Categories;
- TQFTs, Turaev-Viro-Barrett-Westbury TQFTs, and Levin-Wen String Nets;
- Drinfeld Center, SymTFTs
Lecture #1: Group and Representation Theory
Lecture #2: Fusion Rings and Categories
Recording (in Portuguese) with passcode: 3KJ&6n3!
Lecture #3: Fusion Rings and Fusion Categories
Recording (in Portuguese) with passcode: JJF2B*gV
Lecture #4: Fusion Categories and F-symbols
Recording (in Portuguese) with passcode: %iP^fx01
Lecture #5: Fusion Categories and Gauge transformations
Recording (in Portuguese) with passcode: Ad4?JnYf
Lecture #6: Non-invertible symmetries in 1+1d Ising and Anyonic chains
Recording (in Portuguese) with passcode: *5&qn7rJ
Lecture #7: Anyonic chains: Hilbert Space and Hamiltonian
Homework: See Sec. 3.4 of Shu-Heng Shao’s TASI lecture notes and Sec. 1.2 of Inamura & Ohmori (2023)
Recording (in Portuguese) with passcode: 25VV^kE!
Lecture #8: Braided Fusion Categories
Recording (in Portuguese) with passcode: U%#3TTVA
Lecture #9: Modular S and T Matrices
Homework: See Sec. I- IV of Xiao-Gang Wen’s paper.
See Table I of the paper above. Can you identify any familiar phases listed there? For example, the \(\mathbb{Z}_2\) toric code, Fibonacci, Ising, or similar topological orders.
Recording (in Portuguese) with passcode: 3R?rN61?
Lecture #10: Levin-Wen String-net Models
Guest lecturer: Lucas Q. Silveira (Northeastern)
Lecture notes and homework are courtesy of our guest lecturer
Recording (in Portuguese) with passcode: Q6%%CJf%
Lecture #11: Quantum Double Models
Guest lecturer: Bruno O. Araújo (Universidade Estadual de Londrina)
Lecture notes and homework are courtesy of our guest lecturer
Recording (in Portuguese) with passcode: M!6U2X.8
Lecture #12: Quantum Double Models and (Intro to) TQFTs
Guest lecturer: Bruno O. Araújo (Universidade Estadual de Londrina)
Part 1: Lecture Notes on quantum doubles [courtesy of our guest lecturer]
Part 2: Lecture Notes on TQFTs
Recording (in Portuguese) with passcode: 7!#8TVD#
Lecture #13: Turaev-Viro-Barrett-Westbury TQFTs
Homework: Unit 5 Exercises of Colleen Delaney’s course at Purdue on Fusion Categories for Physics (Fall 2025)
Recording (in Portuguese) with passcode: ?3+Blghg
Lecture #14: K-matrix Chern Simons TQFTs
Guest lecturer: Rieli Tainá Gomes dos Santos (Universidade Estadual de Londrina)
Lecture notes and homework are courtesy of our guest lecturer
Recording (in Portuguese) with passcode: &.2#n+mn
Lecture #15: Sandwich Construction of (2+1)d Toric Code
Guest lecturer: Giovanna Pimenta (Universidade Estadual de Londrina / Southern Denmark University)
Lecture notes are courtesy of our guest lecturer
Recording 1 (in Portuguese) with passcode: W#XJR%4J
Recording 2 (in Portuguese) with passcode: Qcq@&m8F