FRG Seminar

Organizers: Richard Ng, Eric Rowell, Zhenghan Wang, Xiao-Gang Wen

Zoom Link
Meeting ID: 915 9272 6993
Passcode: 481134

4/30/2021, 10am (EDT) Video of the talk
Speaker: Vincentas Mulevicius (Hamburg)
Title: Exploring modular categories via 3-dimensional defect TQFTs
Abstract: A modular fusion category can be seen as an input to the Reshetikhin-Turaev construction to obtain 3-dimensional TQFTs. An extension of this construction also allows one to consider line and surface defects. In this talk I will look into how some notions involving modular categories have a natural interpretation in terms of such defect TQFTs. In particular, the construction of generalised orbifolds on defect TQFTs provides a way to build new modular categories out of a given one.

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Past Talks

2021

1/15/2021, 10am (EDT) Video of the talk
Speaker: Samuel Wilson (Louisiana State University)
Title: Representations of SL(2, Z/nZ) and Applications to Modular Categories
Abstract: In this talk, we describe the irreducible representations of the groups SL(2, Z/nZ) and how they may be constructed as submodules of quadratic modules. We also discuss properties of such representations that are relevant to the study and classification of modular categories.

1/22/2021, 10am (EDT) Video of the talk
Speaker: Shawn Cui (Purdue University)
Title: From Three Manifolds to Modular Categories
Abstract: We outline a program to construct modular tensor categories from three dimensional manifolds, that was first proposed in (JHEP 2020, 115(2020) ) using M theory. The classical Chern-Simons invariant and the adjoint Reidemeister torsion provide the T-matrix and quantum dimensions of simple objects. The modular S-matrix is produced by local operators based on a guess-and-trial process. We made a number of improvements based on extensive computations of two infinite families of three manifolds, namely, the Seifert fibered spaces and the torus bundles over the circle. From the two families, we obtained premodular categories that are related to the Temperley-Lieb-Jones categories and metaplectic modular categories. The program reveals a deep connection between two parallel universes of 3-manifolds: the classical Thurston world of geometric topology and the quantum Jones world of topological quantum field theories.

2020

9/11/2020, 10am (EDT) Video of the talk
Speaker: Dongmin Gang (APCTP)
Title: M-theoretic genesis of topological phases
Abstract: I will talk about a novel way of constructing (2+1)d topological phases using M-theory. They emerge as macroscopic world-volume theories of M5-branes wrapped on non-hyperbolic 3-manifolds. After explaining the algorithm of extracting modular structures of the topological phase from topological data of the 3-manifold, I will discuss the possibility of full classification of topological orders via the geometrical construction.

9/18/2020, 10am (EDT) Video of the talk
Speaker: Yilong Wang (Louisiana State University)
Title: Modular categories with transitive Galois actions
Abstract: Modular categories enjoy interesting arithmetic properties. An important example is the action of the absolute Galois group of Q on the simple objects, which plays prominent roles in many classification programs of modular categories. In this talk, we will discuss the classification of modular categories with transitive Galois group actions. We will start by the prime decomposition theorem of transitive modular categories. Then we will use the representations of SL(2, Z/nZ) associated with transitive modular categories and the Verlinde formula to completely classify prime transitive modular categories. Finally, we will give the full classification of transitive modular categories by combining the aforementioned results. This talk is based on the joint work with Siu-Hung Ng and Qing Zhang.

9/25/2020, 10am (EDT) Video of the talk
Speaker: Yang Qiu (University of California--Santa Barbara)
Title: Representation of Motion Group from Djikgraaf-Witten TQFT and Related Calculation
Abstract: In this talk, I will talk about a kind of representation of motion group of link in S³. This representation can be derived from extended Dijkgraaf-Witten TQFT. I will show calculation of this representation for torus link by computing the fundamental group of complement of link in S³ and try to relate it with representation of surface braid group from extended Dijkgraaf-Witten. This talk is based on the work joint with my advisor Professor Zhenghan Wang.

10/2/2020, 10am (EDT) Video of the talk
Speaker: Tian Lan (Institute for Quantum Computing, University of Waterloo)
Title: Matrix Formulation for Non-Abelian Family of Topological Orders
Abstract: I will introduce the notion of non-Abelian family of topological orders and the matrix formulation for them. Abelian topological orders belong to the trivial family. Topological orders differing by Abelian ones belong to the same non-Abelian family and share similar non-Abelian properties. Here differing by Abelian ones means being connected by reversible generalized hierarchy construction, which can be viewed as adding or removing Abelian topological orders. The data of the construction can be efficiently described by a series of Abelian anyons together with a K matrix. Such matrix formulation, generalizing the previous one for only Abelian topological orders to any non-Abelian family, provides an efficient way to organize, present, and construct known and unknown topological orders.

10/9/2020, 10am (EDT) Video of the talk
Speaker: Cesar Galindo (Universidad de los Andes)
Title: Braided Zesting and its applications
Abstract: In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. I will present a complete obstruction theory and parameterization approach to the construction and illustrate its utility with some examples. The talk is based on the manuscript https://arxiv.org/abs/2005.05544 a joint work with Colleen Delaney, Julia Plavnik, Eric C. Rowell, and Qing Zhang.

10/16/2020, 10am (EDT) Video of the talk
Speaker: Colleen Delaney (Indiana University)
Title: Zesting link invariants goes beyond modular data
Abstract: Following the discovery of modular categories not determined by their modular data there have been several invariants proposed as alternatives, like the Whitehead matrix, Borromean tensor, traces of higher genus mapping class group representations, and many others. This proliferation of invariants which go beyond modular data suggests there may be nothing special about any particular invariant and begs the question of how knot theory can be best applied for classifying modular categories. By realizing Mignard and Schauenburg's "modular isotopes" as examples of the ribbon zesting construction, we are able to explain the phenomenology of link invariants that go beyond modular data and identify specific knots and links which are powerful for studying modular categories related by zesting.

10/23/2020 and 10/30/2020, 10am (EDT) Video of the first talk , Video of the second talk
Speaker: Feng Xu (University of California--Riverside)
Title: On operator algebraic approach to QFT
Abstract: This is an introduction to operator algebraic approach to QFT, with emphasis on chiral CFT and relations to MTC.