Quantum Algebra and Quantum Topology Seminar, OSU

The quantum algebra and quantum topology seminar meets Thursdays at 1:50-3:00 PM (usually) in CH 240.

Fall 2019

12/05- Mathew Harper, The Ohio State University.

Knot Invariants from Unrolled Quantum $\mathfrak{sl}_3$

In this talk, we discuss knot invariants from unrolled $\mathfrak{sl}_3$ and compare them with other polynomial invariants. We use unrolled $\mathfrak{sl}_2$ as a starting point to motivate the higher rank case. Unrolled quantum groups admit natural Borel induced representations, which are reducible at some degenerate points. Time permitting, we will sketch a proof that the two-variable $\mathfrak{sl}_3$ invariant coincides with the $\mathfrak{sl}_2$ invariant for knots precisely when evaluated at degenerate points.

11/07- Andrew Schopieray, MSRI.

In the context of conformal field theory, Moore and Seiberg claimed the study of modular tensor categories "should be viewed as a generalization of group theory". In this analogy the order of the group is the category's dimension, now taking values outside the natural numbers, which begs the question: what is the set of possible dimensions? This question is almost entirely open, but we provide a partial solution in the case the dimension lies in a quadratic extension of the rational numbers.  Despite being a major source of open research questions, this material will be approachable to anyone with modest knowledge of undergraduate algebra.

10/31- Hyeran Cho, The Ohio State University.

Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

A genus one 1-bridge knot (simply called a (1, 1)-knot) is a knot that can be decomposed into two trivial arcs embed in two solid tori in a genus one Heegaard splitting of a lens space. A (1,1)-knot can be described by a (1,1)-diagram D(a, b, c, r) determined by four integers a, b, c, and r. It is known that every 2-bride knot is a (1, 1)-knot and has a (1, 1)-diagram of the form D(a, 0, 1, r). In this talk, we give the dual diagram of D(a, 0, 1, r) explicitly and present how to derive a Schubert normal form of a 2-bridge knot from the dual diagram. This gives an alternative proof of the Grasselli and Mulazzani’s result asserting that D(a, 0, 1, r) is a (1, 1)-diagram of 2-bridge knot with a Schubert normal form b(2a+1, 2r).

10/24- Giovanni Ferrer, University of Puerto Rico Mayaguez.

Classifying Unitary Modules for Generalized TLJ

Generalized Temperley-Lieb-Jones (TLJ) *-2-categories associated to weighted graphs were introduced in unpublished work of Morrison and Walker. We will go over the construction of these graph-generated TLJ *-2-categories and introduce a corresponding notion of unitary modules. In joint work with Hern\'andez-Palomares, we provide a classification of these modules in terms of weighted graphs in the spirit of Yamagami's classification of fiber functors on TLJ categories and DeCommer and Yamashita's classification of unitary modules for Rep(SUq(2))Rep(SUq(2)). We will discuss these classification projects and the relations between them.

10/17- Peter Huston, The Ohio State University.

Nets of Categories

Various constructions in physics construct a tensor category from local algebraic data, such as the DHR category associated to a net of algebras in relativistic quantum field theory, or the modular tensor category associated to a topological phase of matter. Nets of categories are a general framework for understanding the construction of a tensor category from local information. We will motivate the notion of a net of categories and explore how the data of a tensor category can be extracted from a suitable net of categories. We will also draw connections between the coherence theory of nets of categories and configuration spaces.

10/03- Puttipong Pongtanapaisan, University of Iowa.

Differences of Jones polynomials for links caused by a local move

A local move on a link diagram is the substitution of a given subdiagram for another. Local moves generate many important equivalence relations in knot theory. For instance, Murakami and Nakanishi showed that two links are equivalent by a sequence of delta moves if and only if they have the same pairwise linking numbers. In this talk, I will discuss joint work with Paul Drube, where we analyze the effect of various local moves of rotation type by showing that the difference of the Jones polynomials of two links related by such moves satisfies a particular divisibility condition.

9/19- Micah Chrisman, The Ohio State University.

Concordances to prime hyperbolic virtual knots

n 1979, Kirby and Lickorish proved that every knot in S^3 is concordant to a prime knot. Later, in 1981, Livingston proved that every knot is concordant to a prime satellite knot. Myers proved (1983) that every knot is concordant to a hyperbolic knot. Furthermore, these concordances can be realized so that the Alexander polynomial is preserved (Bleiler 1982, Nakanishi 1983). There has been much recent work on hyperbolicity of knots in thickened surfaces (see e.g. Adams et al. 2018, Champanekar-Kofman-Purcell 2018). We will sketch a proof extending the above mentioned results to this context. It will be shown that every knot in a thickened surface is (virtually) concordant to a prime satellite virtual knot and a prime hyperbolic virtual knot. Moreover, if a knot is homologically trivial in some thickened surface, then it is concordant to a prime satellite virtual knot and a prime hyperbolic virtual knot having the same Alexander polynomial.  The main difficulty in each case lies in proving primeness. Indeed, there are hyperbolic knots in thickened surfaces that are not prime as virtual knots. This talk is based on the paper at https://arxiv.org/pdf/1904.05288.pdf

9/5- David Green, The Ohio State University.

Classification of rank 6 MTC’s

It’s of interest to classify modular tensor categories (MTC’s) by rank. I will explain some current classification techniques and illustrate their application to a specific subset of the rank 6 classification. Based on https://arxiv.org/abs/1908.07128, which was completed with help from Eric Rowell.

8/22- Marcel Bischoff, Ohio University.

Computing Fusion Rules for G-Extensions of Fusion Categories

A G-graded extension of a fusion category C yields a categorical action of G by braided autoequivalences on the Drinfel'd center Z(C) of C.
I will explain a method for recovering the fusion rules of the G-graded extension in terms of the above-mentioned categorical action. The goal is to discuss several examples which can be computed using this method.

Based on joint work in progress with Corey Jones.

Spring 2019

1/29- Dror Bar-Natan, University of Toronto.

Computation without Representation

A major part of "quantum topology" (you don't have to know what's that) is the definition and computation of various knot invariants by carrying out computations in quantum groups (you don't have to know what are these). Traditionally these computations are carried out "in a representation", but this is very slow: one has to use tensor powers of these representations, and the dimensions of powers grow exponentially fast.

In my talk I will describe a direct-participation method for carrying out these computations without having to choose a representation and explain why in many ways the results are better and faster. The two key points we use are a technique for composing infinite-order "perturbed Gaussian" differential operators, and the little-known fact that every semi-simple Lie algebra can be approximated by solvable Lie algebras, where computations are easier.

This is joint work with Roland van der Veen and continues work by Rozansky and Overbay

2/5- Yilong Wang, Louisiana State University.

Classification of spherical fusion categories of Frobenius-Schur exponent 2

In this talk, we will introduce the notion of the Frobenius-Schur exponent of a spherical fusion category. Then we will classify spherical fusion categories of Frobenius-Schur exponent 2. Finally, we will show that if the category is in addition modular, it is completely determined by the Arf invariant associated to the quadratic form that defines the categories.

This is a joint work with Zheyan Wan.

2/26- Jacob Bridgeman, Perimeter Institute.

Anomalies and entanglement renormalization

We study 't Hooft anomalies of discrete groups in the framework of  (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.

Joint work with Dominic Williamson (Yale), arXiv:1703.07782

3/19- Dominic Williamson, Yale University.

Symmetry-enriched topological tensor network states and graded fusion categories

I will describe how studying symmetries of 2D tensor network wavefunctions with topological order led us to the theory of graded fusion categories. I will then move on to explain how one can extract emergent physical properties of a tensor network wavefunction from its symmetries.

(Joint with NCGOA Seminar).

4/2- Alissa Crans, Loyola Marymount University.

Knots and Links with Finite n-Quandles

A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group and algebraically encode the three Reidemeis- ter moves from classical knot theory. Thus, quandles are a fruitful source of applications to knots and knotted surfaces; in particular they provide a complete invariant of knots. An n-quandle is a quandle that, roughly speaking, satisfies the additional axiom that applying the quandle operation n times with the same element is trivial. We will consider the collection of knots and links having finite n-quandles, describe many of these quandles, and identify their automorphism, inner auto- morphism, and transvection groups. This is joint work with Jim Hoste, Blake Mellor and Patrick Shanahan.

4/16- Micah Chrisman, The Ohio State University.

Virtually slice knots and the generalized Alexander polynomial

In 1994, Jaeger, Kauffman, and Saleur introduced a determinant formulation for the Alexander-Conway polynomial based on the the free fermion model in statistical mechanics. This can be used to define an invariant of knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented . The usual Alexander-Conway polynomial for knots in the $3$-sphere can be recovered from this construction. For knots in $\Sigma \times [0,1]$, with $\Sigma \ne S^2$, the JKSpolynomial gives something new. Sawollek further showed that the JKS polynomial is an invariant of virtual knots. This invariant has been studied from many different perspectives (e.g. using biquandles, the extended knot group, and the virtual knot group) and is now commonly known as the generalized Alexander polynomial.

A knot $K \subset \Sigma \times [0,1]$ is said to be virtually slice if there is a compact connected oriented $3$-manifold $W$ and a disc $D$ smoothly embedded in $W \times [0,1]$ such that $\partial W=\Sigma$ and $\partial D=K$. This definition is due to Turaev. Here we show that the generalized Alexander polynomial is vanishing on all virtually slice knots. To do this, we also prove that Bar-Natan's Zh'' correspondence and Satoh's Tube map are both functorial under concordance. The result is applied to determining the slice status of many low crossing number virtual knots. This project is joint work with H. U. Boden (https://arxiv.org/pdf/1903.08737.pdf).

# Fall 2018

8/28- Marcel Bischoff, Ohio University.

Automorphisms of Lagrangian algebras.

A commutative algebra in a non-degenerate braided category is called Lagrangian if the category of local modules is trivial. Lagrangian algebras arise e.g. in gapped boundaries of topological phases of matter or in conformal field theory.  Motivated by orbifolds of holomorphic conformal field theories andtheir global symmetries, I will discuss automorphism groups of Lagrangian algebras in Drinfel'd centers of finite groups and their associated 3-cocycles (anomalies). The talk is based on joint work in progress with A. Davydov and D.A. Simmons.

9/11- Josh Edge, Indiana University Bloomington.

Classification of spin models on Yang-Baxter planar algebras.

After the discovery of the Jones polynomial in the 1980s, many mathematicians were interested in finding sources for more invariants of knots and links. One promising method pursued by Kauffman, Jaeger, de la Harpe, and Kuperberg among others was via so-called spin models, whose original purpose was to explain magnetism in certain physical models. The classification of such models for the Jones polynomial was first noted by Kauffman in 1986, which Jaeger then generalized to the classification of spin models for the Kauffman polynomial (or BMW algebra) in 1995 by connecting the existence of such a model to the existence of graphs satisfying certain properties. In 2015, Liu finished the classification began by Bisch and Jones of so-called Yang-Baxter planar algebras (YBPAs), planar algebras that satisfy a generalization of the Reidermeister moves. In this talk, we will use the classification of YBPAs to generalize Jaeger's result about spin models of the Kauffman polynomial (which itself is a YBPA) to classify all spin models of Yang-Baxter planar algebras by making a connection to graphs similar to Jaeger. In particular, we will demonstrate that aside from the spin models arising from BMW classified by Jaeger, the only other YBPAs giving spin models are the Bisch-Jones algebra and the Jones polynomial at a discrete sets of values.

9/13- Zhenghan Wang, University of California Santa Barbara.

On Generalized Symmetries and Structures of Modular Categories

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases.  As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.

This is a joint work with Shawn Cui and Modjtaba Shokrian Zini.

10/2- Adu Vengal, The Ohio State University.

Biquandle Knot Invariants.

Biquandles give a convenient way to algebraically represent the Reidemeister moves in knot theory. We'll first define biquandles, and then introduce a few knot invariants relating to them. These include biquandle counting invariants, 2-cocycle state sums, and biquandle brackets. Time permitting, we'll further explore the connections between these invariants.

10/23- Colleen Delaney, University of California Santa Barbara.

When the spacetime trajectories of anyons in (2+1)D topological phases of matter trace out knots or links, the probability amplitudes of these physical processes is given by a knot or link invariant. These invariants can be computed from the algebraic theory of anyons, which is given by a unitary modular tensor category (UMTC). An interesting question is when a family of UMTCs can be distinguished by the invariants they produce for a finite set of knots and links. I will report on some recent progress in this direction based on joint work with Alan Tran, Parsa Bonderson, Cesar Galindo, Eric Rowell, and Zhenghan Wang.

I will introduce UMTCs, explain how they give rise to link invariants, and interpret our results in the context of topological phases of matter and their application to quantum computation

11/8-  Cain Edie-Michell, Vanderbilt University.

(Joint with Noncommutative Geometry and Operator Algebras, in Room MA 105)

Title: Classifying fusion categories generated by an object of small dimension.

Abstract: One of the main goals in the study of fusion categories is to provide a complete classification result. However at the current time not even a conjectural description of all fusion categories has been given. Because a complete classification is completely out of reach with current techniques, current research on fusion categories focuses on classifying small” fusion categories, where small can have a variety of different meanings. For this talk, small will mean generated by an object of small Frobenius-Perron dimension. I will discuss a recent partial classification result of mine along these lines. I assume the generating object has dimension less than 2, along with a mild assumption on the commutativity of the generating object. This classification contains some very surprising categories, including some exceptional quantum subgroups.

11/13- Micah Chrisman, The Ohio State University.

Estimating the virtual slice genus of classical knots in $S^3$.

For a compact connected oriented surface $\Sigma$, a knot $K$ in $\Sigma \times [0,1]$ is said to be virtually slice if there is a compact connected oriented $3$-manifold $W$ and a disc $D$ smoothly embeddded in $W \times [0,1]$ such that $\partial D=K$. The  virtual slice genus of $K$ is the smallest genus orientable surface that $K$ bounds in some $W \times [0,1]$, where the minimum is taken over all compact connected oriented $3$-manifolds $W$. A classical knot in $S^3$ can be considered as a knot in the thickened surface $S^2 \times [0,1]$. A conjecture of Kauffman states that the slice genus of a knot in $S^3$ is equal to its virtual slice genus. Boden and Nagel proved the conjecture to be true for slice knots in $S^3$. In this talk, we provide some positive evidence for Kauffman's conjecture in both the smooth and topological categories. In particular, we will prove Kauffman's conjecture is true for all classical knots up to crossing number 11 and for 2150 of the 2175 clasical knots with crossing number 12. The first portion of this talk will be an introduction to virtual knot concordance. This project is joint work with H. U. Boden and R. Gaudreau.

11/27- Theo Johnson-Freyd, Perimeter Institute.

Holomorphic SCFTs of small index

Stolz and Teichner have conjectured that the moduli space of D=1+1, N=(0,1) QFTs provides a geometric model for Topological Modular Forms. Some important building blocks in this moduli space are the holomorphic superconformal field theories, and the conjecture leads to predictions about the possible values the supersymmetric index of such SCFTs. Specifically, the conjecture leads one to predict the existence of SCFTs of small nonzero index, and where the minimal possible index depends in an interesting way on the central charge of the SCFT. I will explain a construction of some SCFTs with indexes equal to the predicted minimal values. The construction leads to a new divisibility result in the seemingly unrelated field of algebraic coding theory. Based on joint work with Davide Gaiotto.

12/4- Matthew Harper, The Ohio State University.

Introduction to the Involutory Kuperberg Invariant

Given a compact oriented 3-manifold M and a finite dimensional involutory Hopf algebra H (or Hopf super-algebra, or a Hopf object in a suitable monoidal category) we may define the Kuperberg invariant Ku(M,H).
For this talk we give an overview of Heegaard diagrams, including basic examples of presentations for 3-manifolds. We then cover diagram equivalences as is necessary to define Ku(M,H), and compute the invariant for different choices of H.
Time permitting, we remark on relations between the Kuperberg invariant and other quantum topological invariants.