I will give a short course on “tensor categories and topological orders” starting roughly from the middle of September to the end of October at Yau’s center of mathematics at Tsinghua University. The precise date, time and location will be announced later.

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.

The following is a tentative syllabus of this course.

- Lecture 1: An overview of the field of 2d topological orders.
- I will give some historical remarks, and review some basic facts of 2d topological orders. Then I will use some physical intuition to motivate the notion of a unitary modular tensor categories (UMTC). I will state some important results of 2d topological orders in terms of categorical language. This sets the goal that we want to reach at the end of the course.

- Lecture 2: Toric code model
- Toric code model is the simplest 2d lattice model that realize a topological order. It is so simply. Yet, it contains almost all the non-trivial phenomena that could occur in much more general topological orders. In this lecture, I will show how to classify the topological defects of all codimensions

- Lecture 3: Basics of category theory
- introduce the notion of a category, a functor and a natural transformation;
- examples: Sets, Groups, Vecter spaces, Representations of groups, Algebras, Modules over an algebra, etc.
- finite semisimple abelian categories and examples.
- unitary categories:
- more non-trivial examples:

- Lecture 4: Monoidal categories
- the notion of a monoidal category, a monoidal functor,
- examples:
- the notion of unitary fusion category and examples
- expressing the data in basis

- Lecture 5: Unitary modular tensor categories
- the notion of a braided monoidal category, a braided monoidal functor,
- examples:
- the notion of a (unitary) modular tensor category, S-, T-matrices
- examples: toric code, Ising UMTC, …
- return to physics

- Lecture 6: the mathematical theory of anyon condensation, I
- the new vacuum and condensable algebras in a UMTC.
- examples

- Lecture 7: the mathematical theory of anyon condensation, II
- deconfined topological excitations
- confined topological excitations and gapped domain wall
- modules and local modules with examples
- the complete results

- Lecture 8: the mathematical theory of anyon condensation, III
- Example: revisit toric code
- open projects.

- Lecture 9: the mathematical theory of anyon condensation, IV
- gapped edges and Lagrangian algebras
- boundary-bulk duality
- gapped domain walls
- Witt equivalence and Witt group
- More condensation results and open questions

- Lecture 10: Modules over a monoidal category and Drinfeld center
- left/right/bi-module categories, module functors,
- Morita equivalence
- Drinfeld center
- Mathematical results of Mueger, Kitaev, Etingof-Nikshych-Ostrik

- Lecture 11: Levin-Wen models enriched by defects
- Briefly sketch the construction and results
- boundary-bulk duality

- Lecture 12: Conclusions and outlooks
- GSD and factorization homology
- topological order with symmetries
- rational conformal field theories
- gapless edges and enriched monoidal categories

Based on these materials, one can move on to more advanced topics: such as topological orders with symmetries, gapless edges and rational conformal field theories.

The contents are not completely fixed. If you have any questions, comments and suggestions, please feel free to post it here.

**Participants**:

- 清华大学：姚顺宇，陈一鸣，冒家豪，刘峥，宁上强，林海，周杉，黄施霖
- 北京师范大学：寇謖鹏，孔潇，郭翠仙， 杨梦蕾
- 理论物理所：苏宁，陈晋
- 人民大学：王久才，刘正鑫，
- 物理所：
- 北大：

**Prerequisite: **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. However, if you have too much knowledge in condensed matter physics, it might also prevent you from absorbing the new language easily.

**References (will be added)**

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 http://www-math.mit.edu/~etingof/egnobookfinal.pdf

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]

孔老师 这些课可以直播吗 如果可以的话能上午上午上吗 照顾 一下北美的同学

感谢留言，这个真不知道。感觉我讲的东西不值得大家看直播啊。呵呵。过去很多年从未有机会在任何凝聚态物理的会议上做过报告，除了一两次在中国。倒是有一些机会参加数学物理会议，对数学物理学家来说，这是近20年前的结果。所以这东西没地方去讲啊，可能结果是也没有地方去听。直播有些复杂，我也不清楚，需要问问。希望我们能整理出来笔记帮助大家学习掌握。