Tensor categories and 2d topological orders
Student No.:50
Time:Tuesday 19:00-21:00, Thursday 15:20-16:55 & 19:00-21:00, 2017.9.19-2017.10.19, except Oct.2-6
Instructor:Kong Liang  [Tsinghua University]
Place:Lecture Hall, Floor 3, Jinchunyuan West Building
Starting Date:2017-9-19
Ending Date:2017-10-17

Category theory, a branch of mathematics, is notorious for its abstractness (and uselessness even for many mathematicians). It is surprising that the mathematical theory of tensor category has entered the field of topological phases of matter at its full strength. Almost all ingredients of the representation theory of unitary fusion categories have their physical meanings.
This short course is an introductory course of category theory and its application in the study of 2d topological orders. Among many important results in 2d topological orders, the theory of anyon condensation provides a unique way to demonstrate the power of category theory. More importantly, it serves as an indispensable base for many other important results and further developments. For example, it naturally leads us to a classification of all topological defects of all codimensions, including gapped edges and domain walls, and bulk-edge duality, etc. The main goal of this course is to explain this theory in details [9]. The course is mainly designed for students in condensed matter physics. Since we will also have a few audiences from string theory and mathematical physics community, I will try to connect it to other topics as well, such as topological field theories in mathematics.
A tentative but detailed syllabus is available at
If you have any questions or comment, please feel free to ask there.

Linear algebra and some basic knowledge of quantum mechanics are needed. A little bit representation theory of finite group will be helpful. A good background in condensed matter physics or quantum field theory is not necessary but certainly helpful.

1. Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik, The Witt group of non-degenerate braided fusion categories,  Journal für die reine und angewandte Mathematik (Crelles Journal), 2013 (677), pp. 135-177 [arXiv:1009.2117]
2. F.A. Bais, J.K. Slingerland, Condensate induced transitions between topologically ordered phases, Phys. Rev. B 79 (2009) 045316, [arXiv:0808.0627]
3. Maissam Barkeshli, Chao-Ming Jian, Xiao-Liang Qi, Theory of defects in Abelian topological states, Phys. Rev. B 88, 235103 (2013) [arXiv:1305.7203]
4. Pavel Etingof, Dmitri Nikshych, Shlomo Gelaki, Victor Ostrik, Tensor categories, a book, available at
5. Yidun Wan, Ling-Yan Hung, Generalized ADE Classification of Gapped Domain Walls, JHEP 1507 (2015) 120 [arXiv:1502.02026]
6. Michael Levin, Protected edge modes without symmetry, Physical Review X 3, 021009 (2013) [arXiv:1301.7355]
7. Michael Levin, Xiao-Gang Wen, String-net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005) 045110. [arXiv:cond-mat/0404617]
8. Alexei Kitaev, Liang Kong,  Models for gapped boundaries and domain walls, Commun. Math. Phys. 313 (2012) 351-373 [arXiv:1104.5047]
9. Liang Kong, Anyon condensation and tensor categories, Nuclear Physics B 886 (2014) 436-482, [arXiv:1307.8244]
10. Liang Kong, Hao Zheng, Gapless edges of 2d topological orders and enriched monoidal categories, [arXiv:1705.01087]