# QT Seminar

Welcome to the Quantum Topology seminar at IU! This is organized by Colleen Delaney (IU) and me.

Topics: Tensor categories, topological quantum field theories (TQFTs), quantum invariants of knots and 3-manifolds.

When: Mondays 3-4 pm.
Where: Zoom.

Past talks

April 25. Helen Wong (Claremont McKenna College). Skein algebras and cluster algebras of punctured surfaces.

In joint work with Han-Bom Moon, we establish a connection between two different algebras based on the decorated Teichmuller space of a punctured surface. One, defined by Roger and Yang, is a generalization of the Kauffman bracket skein algebra that includes arcs that go from puncture to puncture. The other, due to Fomin, Shapiro, and Thurston, is a cluster algebra of tagged arcs on the punctured surface. The connection between them naturally provides a geometric interpretation of the tags on the tagged arcs. It will also allow us to resolve an earlier conjecture, that the skein algebra is a quantization of decorated Teichmuller space for punctured surfaces.

April 18. Jacob Bridgeman (Ghent University). Computing data for endomorphism fusion categories.

For many applications of fusion categories, particularly in physics, it is necessary to fix bases for all Hom spaces, and provide all morphisms explicitly with respect to these. Practically, finding this data generally involves the challenging task of solving highly over-complete sets of polynomial equations.

Once we have a solution, it’s desirable to maximise the payback. I will discuss an algorithm to make maximal use of data already obtained by computing that of the full Morita equivalence class.

Based on work 2110.03644 with Daniel Barter and Ramona Wolf

April 11. Andrew Schopieray (PIMS). Modular tensor categories of prime categorical dimension. (Slides)

Abstract: In the late 1980's Edward Witten coined the term "topological quantum field theory" and is oft-quoted as comparing the subject to the more classical group theory both in importance and structure. In a particularly small dimension, and with suitable definitions, we now know that a topological quantum field theory is equivalent to a "modular tensor category", a less topological and very algebraically-structured object. In many ways these are the finite groups of the future. To organize the futuristic finite groups, modular tensor categories possess a categorical notion of dimension akin to computing the dimension of a representation of a finite group by taking the trace of the identity matrix. The difference is that categorical dimensions are not necessarily (rational) integers. Trivial exercises from your undergraduate class on finite groups now become research papers. In this talk I'll discuss my pandemic-age classification of modular tensor categories of prime categorical dimension contained at the very end of this paper: https://doi.org/10.1016/j.jalgebra.2020.10.014

April 4. No talk.

Mar. 28. Jin-Cheng Guu (Stony Brook). TQFT and invariants of manifolds and algebras

We will provide a crash course of topological quantum field theory, explain how it produces interesting invariants for both spaces and algebras, and provide a few examples. If time permits, I will talk about my work on categorical invariants from the Crane-Yetter model.

Mar. 21. Colleen Delaney (IU). Zesting and Witten-Reshetikhin-Turaev invariants.

I’ll introduce the ribbon zesting construction on pre-modular categories from a diagrammatic point of view and show that Witten-Reshetekhin-Turaev invariants of framed knots and links decouple under zesting. Time permitting I will explain how the Mignard-Schauenburg modular isotopes” can be understood as an application of zesting.

This talk is based on joint work with Cesar Galindo, Julia Plavnik, Eric Rowell, and Qing Zhang as well as Sung Kim.

Mar. 14. Spring Break.

Mar. 7. Efstratia Kalfagianni (MSU). On the asymptotic behavior of quantum invariants and representations.

The generalization of the Jones polynomial for links and 3-manifolds, due to Witten-Reshetiking-Turaev in the late 90’s, led to constructions of Topological Quantum Field Theory in dimensions (2+1).

These theories also include representations of surface mapping class groups. The question of how much of the Thurston geometric picture of 3-manifolds is reflected in these theories is open.

I will talk on some work in this direction with emphasis on the corresponding mapping class group representations.

Feb. 28. Christine Ruey Shan Lee (South Alabama). Normal surfaces and colored Khovanov homology.

The colored Jones polynomial is a generalization of the Jones polynomial from the representation theory of Uq(sl2). One motivating question in quantum topology is to understand how the polynomial and its categorification, colored Khovanov homology, relate to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the boundary slopes of essential surfaces of a knot. Motivated by recent progress on the conjecture, we show a connection between the colored Khovanov homology of a knot with normal surface theory of its complement. We relate certain generators of colored Khovanov homology to normal surfaces of a triangulation of the knot complement, and we show that if such a normal surface is essential, then the corresponding homology class is nontrivial. We will discuss the implications of this result for the strong slope conjecture and possible applications to low-dimensional topology.

Jan. 31. Sunghyuk Park (Caltech). R-matrix and a q-series invariant of 3-manifolds. (Slides)

A couple of years ago, S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be resummed into a two-variable series F_K(x,q), as part of a bigger program to construct a 3d TQFT \hat{Z} that had been predicted from physics. In this talk, I will explain how to prove their conjecture for a big class of links by inverting" an R-matrix state sum.

Jan. 24. Eric Samperton (UIUC). Topological quantum computation is hyperbolic.

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams one compiles are hyperbolic. Furthermore, the diagrams can be arranged to have additional nice properties, such as being alternating with minimal crossing number. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.

Dec. 6. Qing Zhang (Purdue). From torus bundles to particle-hole equivariantization.

We continue the program of constructing (pre)modular categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\operatorname{SL}(2, \mathbb{C})$ characters on a given manifold which serves as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions play a key role in the construction, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular $S$-matrix is computed from local operators and follows a trial-and-error procedure. It is currently unknown how to produce data beyond the modular $S$- and $T$-matrices. There are also several subtleties in the construction which remain to be solved. In this paper, we consider an infinite family of 3-manifolds, torus bundles over the circle. We show that the modular data produced by such manifolds are realized by the $\mathbb{Z}_2$-equivariantization of certain pointed premodular categories. This is a joint work with Xingshan Cui, Paul Gustafson and Yang Qiu.

Nov. 29. Fiona Torzewska (U. Leeds). Motion groupoids & mapping class groupoids.

The braiding statistics of point particles in 2-dimensional

topological phases are given by representations of the braid groups.

One approach to the study of generalised particles in topological

phases, loop particles in 3-dimensions for example, is to generalise

(some of) the several different realisations of the braid group.

In this talk I will construct for each manifold $M$ its motion

groupoid $\Mot$, whose object class is the power set of $M$.

I will also give a construction of a mapping class groupoid $\mcg$

associated to a manifold $M$ with the same object class.

For each manifold $M$ I will construct a functor $F\colon \Mot \to \mcg$ and prove that this is an isomorphism if

$\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of

$M$ is trivial. In particular there is an isomorphism in

the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in \N$.

I will give some examples demonstrating the utility of this functor.

Nov. 22. No talk.

Nov. 15. Eric Samperton (UIUC). Rescheduled for 2nd semester.

Nov. 8. Wade Bloomquist (Georgia Tech). Generalizations and Properties of Stated Skein Algebras.

In this talk, we will start by giving an introduction to stated skein algebras of surfaces. Where ordinary skein algebras are defined on links in thickened surfaces, stated skein algebras allow for tangles that have a state (plus or minus) assigned to each endpoint. This extension, introduced by Thang Le in 2016, allows for a splitting homomorphism to be defined along ideal arcs in punctured bordered surfaces. We will review some of the algebraic properties of stated skein algebras with a focus on connections to the quantum group Oq(SL(2)) and quantum tori.

Following this introduction, we will discuss some generalizations of stated skein algebras of punctured bordered surfaces to other topologically motivated situations (for example allowing interior marked points). Our focus will be on which techniques and results carry over into these new contexts. We will also describe some conclusions that can be drawn from studying these generalizations that motivate their study both as a tool and in their own right. This is joint work with Hiroaki Karuo and Thang Le.

Nov. 1. Cris Negron (USC Dornsife). Sightings of the Springer resolution in tensor categories of quantum group representations.

I will discuss upcoming work with Julia Pevtsova, where we apply the geometry of the Springer resolution to study tensor categories of quantum group representations. As one points of interest, we classify thick tensor ideals in the derived category of quantum group representations via closed subsets in the Springer resolution. I will present this result, then discuss further (at present conjectural) applications to geometric representation theory and to logarithmic topological quantum field theory.

Oct. 25. Cristina Palmer-Anghel (U. Geneva). Uq(sl(2))−Quantum invariants for links and 3-manifolds from Lagrangian intersections in configuration spaces.

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of invariants. In this context, the quantum group Uq(sl(2)) gives the sequence of coloured Jones polynomials. The quantum group at roots of unity gives the sequence of coloured Alexander polynomials. We construct a unified topological model for these two sequences of quantum invariants for links. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum of Lagrangian intersections (defined over 3 variables) in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space. In the second part of the talk we discuss a topological model for the Witten-Reshetikhin-Turaev invariants for 3-manifolds, presenting the N-th WRT invariant as a state sum of Lagrangian intersections in a fixed configuration space in the punctured disk.

Oct. 18. Marco de Renzi (U. Zurich). Quantum Invariants of 4-Dimensional 2-Handlebodies.

A connected 4-dimensional 2-handlebody is a 4-manifold with boundary obtained from the 4-ball by attaching a finite number of handles of index 1 and 2. A 2-deformation is a diffeomorphism implemented by a finite sequence of handle moves that never introduce handles of index 3 and 4. Whether there exist diffeomorphisms which are not 2-deformations is still an open question, mainly due to the lack of invariants for detecting them. We will explain how to construct quantum invariants of 4-dimensional 2-handlebodies up to 2-deformations using unimodular ribbon categories, like the category of representations of a unimodular ribbon Hopf algebra. We will discuss the case of the small quantum group of 𝔰𝔩2, and how these invariants could potentially be used to investigate group theoretical questions like the (weak) Andrews-Curtis conjecture. This is a joint work with Anna Beliakova.

Oct. 11. Sean Sanford (IU). Fusion Categories over Non–Algebraically Closed Fields.

Much of the early work on Fusion Categories was inspired by physicists desire for rigorous foundations of topological quantum field theory. One effect of this was that base fields other than the complex numbers were rarely considered, if at all. The relevant features of $\mathbb C$ that make the theory work are the fact that it is characteristic zero, and algebraically closed.

This talk will focus on the interesting things that can be found when the algebraically closed requirement is removed. The content will start with lots of examples, and slowly accelerate into higher categorical implications.

Oct. 4. Renaud Detcherry (U. Bourgogne). A quantum obstruction to purely cosmetic surgeries.

The cosmetic surgery conjecture asks whether it is possible that two Dehn-surgeries on the same non-trivial knot in S³ give the same oriented 3-manifolds. We will present new obstructions for a knot to admit purely cosmetic surgeries, using Witten-Reshetikhin-Turaev invariants. In particular, we will show that if a knot admits purely cosmetic surgeries, then the slopes of the surgery are ±1/5k unless the Jones polynomial of K is 1 at the fifth root of unity.

Sept. 27. Calvin McPhail-Snyder (Duke/UNC). Quantum invariants from unrestricted quantum groups.

Quantum groups are a central part of the construction of quantum invariants of knots, links, and 3-manifolds. Existing work focuses mainly on the case where the quantization parameter q is generic, or on the semisimplified theory at q a root of unity. In this talk, I will discuss how to construct invariants using the non-semisimple part of unrestricted quantum sl_2 at a root of unity. These "holonomy invariants" turn out to be very geometric: they depend on the extra data of a map from π_1 of the knot complement to SL_2(C), which is essentially a choice of hyperbolic structure.

Sept. 20. Roland van der Veen (U. Groningen). An algebra quantization of the SL(2) character variety of knots.

The skein module of a knot complement is often said to be a quantization of

the SL(2) character ring of the knot group. By introducing a notion of base point in

skein theory we construct a different quantization of the same character ring that may

have better properties, in particular it is an algebra. We will give some concrete examples

and show how our algebras relate to the classical algebra of Brumfiel and Hilden.

This is joint work in progress with Jun Murakami and extends our previous work

https://arxiv.org/abs/1812.09539.

Sept. 13. Daniel López Neumann (IU). A Fox-calculus-twisted Kuperberg model for quantum invariants from twisted Drinfeld doubles.

In this talk we will show that universal quantum invariants of knots obtained from "twisted" Drinfeld doubles (an Aut(H)-graded extension of the usual Drinfeld double) can be computed via a Kuperberg-style tensor in which the tensors are twisted by doing Fox calculus on a certain presentation of the fundamental group of the complement. As a corollary, the twisted Reidemeister torsion of a knot complement is realized as a Reshetikhin-Turaev invariant. This suggests that G-crossed ribbon categories might be relevant to geometric topology. This is work in progress.