﻿ 范畴学与拓扑序系列报告-清华丘成桐数学科学中心  ## Activities

1. YMSC Courses Mini Courses Seminars Lectures Conferences

## Upcoming Talks

Isomorphic operator algebra and holographic equivalent symmetry

Symmetry is really determined by the algebra of local symmetric operators. Such an operator algebra is described by a braided fusion higher category. So a symmetry is really a topological order in one higher dimension. This point of view reveals that many seemingly different (anomalous) symmetries are actually equivalent.

Speaker: 文小刚
Affiliation: Massachusetts Institute of Technology

Higher Dimensional Topological Order, Higher Category and A Classification in 3+1D

Topological orders are gapped quantum liquid states without any symmetry. Most of their properties can be captured by investigating topological defects and excitations of various dimensions. Topological defects in n dimensions naturally form a (weak) n-category. In particular, anomalous topological order (boundary theory) is described by fusion n-category and anomaly-free topological order (bulk) is described by non-degenerate braided fusion n-category. Holographic principle works for topological orders: boundary always has a unique bulk. Another important property in 3+1D or higher is that point-like excitations must have trivial statistics; they must carry representations of a certain group. Such a "gauge group" is hidden in every higher dimensional topological order. In 3+1D, condensing point-like excitations leads to a canonical boundary which in turn determines the bulk topological order. By studying this boundary, a rather simple classification is obtained: 3+1D topological orders are classified by the above "gauge group" together with some cocycle twists. These ideas would also play an important role in dimensions higher than 3+1D and in the study of higher categories, topological quantum field theories and other related subjects.

Speaker: 兰天
Affiliation: 香港中文大学

Enforced symmetry breaking by invertible topological order

It is well known that two-dimensional Chern insulator must break the time reversal symmetry, as manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topological order and symmetry can occur more generally. In this talk, I will discuss enforced breaking of general discrete internal symmetries by a class of fermionic topological orders, namely invertible topological orders. General criteria and examples will be discussed for 0D, 1D and 2D systems. These results rely on a general exploration of symmetry-enriched invertible topological orders and quantum anomaly.

Speaker: 王晨杰
Affiliation: 香港大学

Topological phases in interacting fermionic systems

In this talk, I will introduce several exactly solvable lattice models for fermionic topological phases. These models include fermionic string-net models constructed from super-fusion categories, and fermionic symmetry-protected topological models from group super-cohomologies. If time allows, I will discuss some nontrivial properties of these models and list some unsolved related problems.

Speaker: 王晴睿
Affiliation: 清华大学丘成桐数学科学中心, 北京雁栖湖应用数学研究院

## Past Talks

Geometry and CFT’s from quantum Hall systems

In this talk we shall discuss matrix models arising from quantum Hall effects in 2+1 dimension and in 4+1 dimension. By taking large N limit we could reproduce background geometry and conformal field theories. In particular it gives a mathematical construction of non commutative Chern-Simons theory. Localization techniques could be applied to get data of quantum groups. In this setting gauge/gravity correspondence are naturally implemented as Koszul duality on which further investigations are underway.

Speaker: 胡森

Affiliation: 中国科技大学

Video kink：https://meeting.tencent.com/user-center/shared-record-info?id=12a64aba-1d45-45e3-9629-d84f3c3d73c7&from=3

One dimensional gapped quantum phases and enriched fusion categories

The notion of a quantum phase is a macroscopic one. In this talk, I will show that the macroscopic observables in two gapped quantum phases realized by the Ising chain form two enriched fusion categories. Generalizing to arbitrary finite onsite symmetries, we immediately recover the well-known classification of all 1-dimensional gapped quantum phases. This is a joint work with Xiao-Gang Wen and Hao Zheng.

Speaker: 孔良

Affiliation: 南方科技大学量子研究院

The Modular Extension Characterization of SPT/SET phases

I will discuss the structures of "symmetry" and "stacking" in quantum phases, and correspondingly, in braided fusion categories. We are particularly interested in the 2+1D symmetry protected topological phases, with bosonic finite unitary onsite symmetry G, which under stacking form the third cohomology group of G valued in U(1). This case is simple but nontrivial enough for us to pin down the correct categorical description of topological phases with symmetry, where the notion of minimal modular extension is an essential ingredient.

Speaker: 兰天

Affiliation: 香港中文大学

An introduction to 2d bosonic anyon condensation

Anyon condensation is an interesting phenomenon of topological phase transitions. In this talk, we will introduce the categorical description of 2d bosonic anyon condensation. We start from the bootstrap analysis of the condensed phase by natural physical requirements. We will show the condensation process is controlled by a condensable algebra in the original phase. After the introduction of the category of condensed phase and 1d gapped domain wall, we will give the boundary-to-wall map which is a functor. At the end of this talk, there will be some explicit examples of 2d bosonic anyon condensation.

Speaker: 杨圣宇

Affiliation: 南方科技大学量子研究院

Solvable Lattice Hamiltonians with Fractional Hall Conductivity

We construct a class of bosonic lattice Hamiltonians that exhibit fractional Hall conductivity. These Hamiltonians, while not being exactly solvable, can be controllably solved in their low energy sectors through a combination of perturbative and exact techniques. Our construction demonstrates a systematic way to circumvent the Kapustin-Fidkowski no-go theorem, and is generalizable to fermionic abelian topological orders coupled to a spin-c background. We hope this work can shed light on the study of more general symmetry enriched topological orders with symmetry protected gapless boundary.
References: Zhaoyu Han and Jing-Yuan Chen, [2107.02817]; Jing-Yuan Chen, [1902.06756]

Speaker: 陈静远
Affiliation: 清华大学高等研究院

What is the direct sum of two anyons?

This talk is focused on the descriptive power of the category theory. After reviewing the illuminating history of irrational numbers, we make sense of the direct sum of two anyons in terms of functors. At the end of this talk, we will announce a mathematical definition of topological orders and other quantum liquid phases.

Speaker: 郑浩
Affiliation: 北京雁栖湖应用数学研究院, 清华大学丘成桐数学科学中心

The centers of enriched categories

Boundary-bulk duality is one of the most important guiding principles in the study of topological orders. It states that the bulk phase is determined by an edge as its center. Recent progress in the study of topological orders revealed that the gapless boundaries of topological orders can be described by enriched categories. This observation and the boundary-bulk correspondence principle urge us to study the centers of enriched categories. In this talk, I will first review some basic facts of enriched categories, e.g., the definition and the canonical construction of enriched categories. Then I will introduce the notion of the E_0-center (E_1-center or E_2-center) of an enriched (monoidal or braided monoidal) category, and compute the centers for enriched categories obtained from the canonical constructions.

Speaker: 袁巍
Affiliation: 中科院数学所

Introduction to symmetry-protected topological phases through concrete examples

Symmetry-protected topological (SPT) phases are gapped invertible quantum states that cannot be deformed into trivial state while preserving some symmetries, even though there is no intrinsic topological order (anyons) in the system. In this talk, we will introduce the basic ideas and properties of SPT phases through some concrete examples such as 1D Haldane chain and 2D Levin-Gu model.

Speaker: 王晴睿
Affiliation: 清华大学丘成桐数学科学中心, 北京雁栖湖应用数学研究院

Boundary-bulk relation in topological orders

The purpose of this talk is to show that abstract nonsense arguments may lead to surprising results in physical problems. First, we formulate the universal property of center by reviewing the center of a group. Then we introduce a new kind of morphisms between topological orders and verify that the bulk of a boundary topological order satisfies the universal property of center hence draw the conclusion ”the bulk is the center of a boundary”.

Speaker: 郑浩

Affiliation: 北京雁栖湖应用数学研究院, 清华大学丘成桐数学科学中心

Categories in the toric code model (II)

Category theory plays an important role in the study of topological orders. In this talk, I will introduce two boundary theories of the toric code model as basic examples of the boundary-bulk relation. Mathematically, the boundary-bulk relation can be expressed by the notion of a Drinfeld center and a central functor, which will be explained by physical intuitions. The boundary-bulk relation also helps us to study the general domain walls between topological orders. I will show that all topological defects in the toric code model, including 1d domain walls and 0d defects, form a fusion 2-category.

Speaker: 张智浩
Affiliation: 中国科技大学, 南方科技大学量子研究院

Categories in the toric code model (I)

Category theory plays an important role in the study of topological orders. In this talk, I will biefly explain the notion of a topological order and why a topological order can be described by a (higher) category. The main idea will be illustrated by the toric code model, which is the simplest lattice model of a 2+1D topological order. I will explicitly calculate the point-like topological defects of the toric code model and show that they form a unitary modular tensor category. If time permits, I will also introduce the 3+1D toric code model and show that all topological defects of codimension-2 and higher form a braided fusion 2-category.

Speaker: 张智浩
Affiliation: 中国科技大学, 南方科技大学量子研究院