University Quantum Symmetries Lectures

( UQSL )

This is an online seminar organized by Corey Jones, David Penneys, and Julia Plavnik. Topics of interest range widely over "quantum" mathematics, and include (but are certainly not limited to):

  • Tensor categories

  • Subfactors and operator algebras

  • Hopf algebras and quantum groups

  • Representation theory

  • Higher Categories

  • TQFT and low dimensional topology

  • Categorification

  • Topological phases of matter

  • Conformal field theory

  • Algebraic quantum field theory

 

We typically meet Thursdays from 2:00 pm - 3:00 pm, US Eastern Time Zone. If you would like to attend, please email Corey Jones at cormjones88@gmail.com to be added to the Google Group for weekly announcements of abstracts and Zoom links.

Here's a link to our schedule from previous semesters.

Below is a list of currently scheduled talks for this semester (which is updated frequently!):

Spring 2022:

1/13/2022- Ying-Hsuan Lin, Harvard: Numerical Evidence for a Haagerup conformal field theory.

We numerically study an anyon chain based on the Haagerup fusion category, and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ~2.

1/20/2022- Emily Peters, Loyola University Chicago: Conway's Rational Tangles and the Thompson group.

In the process of studying Thompson's group F (of piecewise linear homeomorphisms from the closed unit interval [0,1] to itself, which are differentiable except at finitely many dyadic rational numbers), Vaughan Jones observed a map from F to knots.  He proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element.  Jones' algorithm to achieve this is rather piecemeal, and he asked if there was a better one.

In a project with undergraduate student Ariana Grymski, we approach this question through the lens of Conway's rational tangles.  We are able to give methods to constructs any product or concatenation of simple tangles, and we hope these are seeds for a more skein-theoretic approach to the construction question.

1/27/2022- Ana Ros Camacho, Cardiff University: On generalizing a tensor equivalence within the Landau-Ginzburg/conformal field theory correspondence

 

The Landau-Ginzburg/conformal field theory correspondence is a physics result from the late 80s and early 90s predicting some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. At present we lack an explicit mathematical statement for this result, yet we have examples available. The only example of a tensor equivalence in this context was proven back in 2014 by Davydov-Runkel-RC, for representations of the N=2 unitary minimal model with central charge 3(1-2/d) (where d integer bigger than 2) and matrix factorizations of the potential x^d-y^d. This equivalence was proven back in the day only for d odd, and in this talk we explain how to generalize this result for any d. Joint work with T. Wasserman (University of Oxford).

2/3/2022- Kevin Walker, Microsoft Research: Going from n+ε to n+1 in non-semisimple oriented TQFTs.

In the 1990's, Lyubashenko, Kuperberg, and several others gave constructions of Reshetikhin-Turaev-like and Turaev-Viro-like TQFTs starting from non-semisimple input categories.  An outstanding problem has been to understand these examples as fully extended TQFTs that can be constructed in the same way as more familiar semisimple examples such as Dijkgaaf-Witten, Turaev-Viro, and Crane-Yetter TQFTs.  In this talk I'll present significant progress toward that goal.  This is joint work with David Reutter.

2/10/2022- Ramona Wolf, ETH Zurich: Computing F-symbols of endomorphism fusion categories.

Applications of fusion categories often require the F-symbols to be known explicitly, for example, for constructing lattice models in physics. Although, in principle, these matrices can always be determined by solving the pentagon equation, this task is often difficult in practice since it corresponds to solving a vast system of coupled polynomial equations in a large number of variables. This is especially true for categories with multiplicities, and there are only a handful of such categories whose F-symbols are known. In this talk, I will present an algorithm that allows the F-symbols for some category to be computed from a Morita equivalent category with known data. This algorithm utilizes the representation theory of the tube algebra constructed from a module over the known category to compute the unknown associator data. This is joint work with Daniel Barter and Jacob Bridgeman.

2/17/2022- Kristin Courtney, University of Münster: Nuclearity and generalized inductive limits.


One of Alain Connes' seminal results establishes that any von Neumann algebra which can be well-approximated by finite dimensional von Neumann algebras (i.e., is semi-discrete) can actually be built from them via a direct limit construction (i.e., is hyperfinite). A direct C*-analogue to this theorem is impossible: most nuclear C*-algebras are not AF. Nonetheless, as is usually the case in C*-theory, there is great incentive in developing an appropriate C*-analogue to this powerful result in von Neumann algebras, i.e., a characterization of nuclear C*-algebras as those arising from finite dimensional C*-algebras via an inductive limit construction. Great strides in this direction were taken by Blackadar and Kirchberg, who were able to characterize quasidiagonal nuclear C* algebras as those arising as so-called generalized inductive limits of finite dimensional C* algebras. However, though many interesting classes of nuclear C*-algebras are covered by this result, many others are not. Building on structural results of orthogonality preserving (order zero) maps, Wilhelm Winter and I are able to give an inductive limit description of all separable nuclear C*-algebras.
 

2/24/2022- Bruno Nachtergaele, University of California, Davis: The stability of gapped phases and automorphic equivalence

Gapped ground state phases of infinite quantum many-body systems can be defined as classes of ground states equivalent under automorphisms of the algebra of local observables with good quasi-locality properties. We will review recent progress on the stability of such phases, in particular those that exhibit topological order.

3/3/2022- Luca Giorgetti, University of Rome, Tor Vergata: A planar algebraic description of conditional expectations

 

Jones’ notion of index was introduced for II_1 subfactors and soon after generalized to unital inclusions of arbitrary von Neumann algebras in several ways. Such an inclusion N < M is said to have finite Jones index if it admits at least one normal faithful conditional expectation E of M onto N with finite Kosaki index. In the talk, I will report on a representation formula for such finite index expectations and their dual expectations by means of the solutions of the conjugate equations for the inclusion morphism from N to M and its conjugate morphism from M to N. This provides a 2-C*-categorical (or C* 2-categorical) formulation of the theory of index in this general setting. As a consequence, we have a double loop diagram picture for Ind(E). Another consequence is that an arbitrary pair N < M, E as before can be described by a Q-system. Both these results are originally due to Longo in the subfactor case. Based on https://arxiv.org/abs/2111.04488 and supported by EU MSCA-IF beyondRCFT grant n. 795151

3/10/2022- Roberto Conti, Sapienza Università di Roma: Symmetries of noncommutative spaces: a guided tour through the Cuntz algebra case

Multiplets of isometries with orthogonal ranges summing up to 1 were systematically used by Doplicher and Roberts in the early 70's  in their study of the superselction structure of Quantum Field Theory. Nowadays, the C*-algebra generated by any such multiplet is known as Cuntz algebra O_n, where n is the cardinality of the given multiplet. Since their introduction, Cuntz algebras have been the subject of endless investigations, from many different points of view. They are perhaps the most studied/used class of C*-algebras ever. Notably, the study of their automorphisms presents many challenging facets, where operator algebras meet Lie theory, dynamical systems and combinatorics.. We will present an overview of recent results, along with some open questions.

4/7/2022- Bojko Bakalov, North Carolina State University: An operadic approach to vertex algebras and Poisson vertex algebras

I will start by reviewing the notions of vertex algebra, Poisson vertex algebra, and Lie conformal algebra, and their relations to each other. Then I will present a unified approach to all these algebras as Lie algebras in certain pseudo-tensor categories, or equivalently, as morphisms from the Lie operad to certain operads. As an application, I will introduce a cohomology theory of vertex algebras similarly to Lie algebra cohomology, and will show how it relates to the cohomology of Poisson vertex algebras and of Lie conformal algebras. The talk is based on joint work with Alberto De Sole, Reimundo Heluani, Victor Kac, and Veronica Vignoli.

4/14/2022- Richard Ng, Louisiana State University: Reconstruction of modular data from representations of SL(2,Z)

Modular data is the most important invariant of a modular tensor category. Associated to a modular data is a family of projectively equivalent linear representations of SL(2,Z), which are symmetric and congruence. One would naturally ask whether the representation type of these representations of SL(2,Z) could determine the underlying modular data. Since every congruence representation is symmetrizable. One would like to understand which congruence representation type of SL(2,Z) could be realized by modular tensor categories. We have shown that for any congruence SL(2,Z) representation of dimension 6, it is either not realizable, or realized by a Galois conjugate of the modular data of a Deligne product of some quantum group modular tensor categories. This reconstruction process can be implemented for computer automation for higher dimensional congruence representations. The talk is based on some joint work with Eric Rowell, Zhenghan Wang, Xiao-Gang Wen.

 

4/21/2022- Laura Colmenarejo, North Carolina State University: An insertion algorithm on multiset partitions with applications to diagram algebras

In algebraic combinatorics, the Robinson-Schensted-Knuth algorithm is a fundamental correspondence between words and pairs of semistandard tableaux illustrating identities of dimensions of irreducible representations of several groups. 

In this talk, I will present a generalization of the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to new enumerative results that have representation-theoretic interpretation as decomposition of centralizer algebras and the spaces they act on. I will also present a variant of this algorithm for diagram algebras that has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. 

4/28/2022- Shawn Cui, Purdue University: From Three Manifolds to Modular Categories 
 

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 somewhat mysterious 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. This is joint work with P. Gustafson, Y. Qiu,  Z. Wang, and Q. Zhang.

5/5/2022- Jamie Vicary, University of Cambridge: Introducing homotopy.io: A proof assistant for geometrical higher
category theory


Abstract: Weak higher categories can be difficult to work with algebraically, with the weak structure potentially leading to considerable bureaucracy. Conjecturally, every weak infty-category is equivalent to a "semistrict" one, in which unitors and associators are trivial; such a setting might reduce the burden of constructing large proofs. In this talk, I will present the proof assistant homotopy.io, which allows direct construction of composites in a finitely-generated semistrict (infty,infty)-category. The terms of the proof assistant have an interpretation as string diagrams, and interaction with the proof assistant is entirely geometrical, by clicking and dragging with the mouse, completely unlike traditional computer algebra systems. I will give an outline of the underlying theoretical foundations, and demonstrate use of the proof assistant to construct some nontrivial homotopies, rendered in 2d, 3d, and in 4d as movies. I will close with some speculations about the possible interaction of such a system with more traditional type-theoretical approaches. (Joint work with Nathan Corbyn, Calin Tataru, Lukas Heidemann, Nick Hu and David Reutter.)