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!):

Fall 2021:

8/26/2021- Xiao-Gang Wen, MIT:

Generalized symmetry, local fusion higher category, and their holographic point of view.

 

Symmetry (including higher symmetry) is usually defined via the (higher) group formed by the symmetry transformations. Due to Tannaka duality, a symmetry can also be defined via the symmetric fusion category formed by the representations of the symmetry group. The fusion category point of view is more general. In particular local fusion n-category can be used to describe generalized symmetry in n-dimensional space, that can go beyond the higher group description. I will also describe a holographic point of view of symmetry, from which we can see the equivalence between symmetries described by different  local fusion n-categories.

9/2/2021- Ingo Runkel, Universität Hamburg:

Defects and orbifolds in Reshetikhin-Turaev TQFTs

In their original form, RT-TQFTs are defined on bordisms with embedded Wilson lines, or, more generally, ribbon graphs, coloured by objects and morphisms of the underlying modular fusion category. In this talk I will discuss an enhancement of these TQFTs to more general stratified bordisms which include surface defects, and which allow different three-dimensional regions to be labelled by different modular categories form a given Witt class. An application of these enhanced TQFTs is a generalisation of the orbifold construction which produces new TQFTs out of a given one. It turns out that RT-TQFTs close under this operation, that is, generalised orbifolds of RT-TQFTs are again RT-TQFTs. All these structures and operations have explicit algebraic counterparts in modular fusion categories.

9/9/2021- Amanda Young, TU Munich:

A bulk gap in the presence of edge states for a truncated Haldane psueodpotential

One of the fundamental quantities for classifying quantum phases of matter is the existence or non-existence of a spectral gap above the ground state energy that is uniform in the system size. While the importance of a spectral gap is very well known, it is notoriously difficult to prove. This task becomes even more arduous for models with edge states, where the finite volume Hamiltonians have low-lying excitations that do not appear in the thermodynamic limit. The Haldane pseudopotentials, which are expected to faithfully describe all important features of fractional quantum Hall systems or rotating Bose gases, is one class of models long conjectured to have a nonvanishing gap. In this talk, we discuss a recent result establishing a nonvanishing gap for a truncated version of the 1/2-filled bosonic Haldane pseudopotential. Our proof relies on decomposing the Hilbert space into invariant subspaces to which we apply spectral gap methods previously only developed for quantum spin Hamiltonians. By customizing the gap method to the invariant subspace, we are able to circumvent edge states and prove a more accurate estimate on the bulk gap.

9/16/2021- Henry Tucker, UC Riverside:

Frobenius-Schur indicators for quadratic fusion categories and their Drinfel’d centers

Quadratic categories are fusion categories with a unique non-trivial orbit from the tensor product action of the group of invertible objects. Familiar examples are the near-groups (with one non-invertible object) and the Haagerup-Izumi cate- gories (with one non-invertible object for each invertible object). Frobenius-Schur indicators are an important invariant of fusion categories generalized from the theory of finite group representations. These indicators may be computed for objects in a fusion category C using the modular data of the Drinfel’d center Z(C) of the fusion category, which is itself a modular tensor category. Recently, Izumi and Grossman provided new (conjectured infinite) families of modular data that include the modular data of Drinfel’d centers for the known quadratic fusion categories. We use this information to compute the FS indicators; moreover, we consider the relationship between the FS indicators of objects in a fusion category C and FS indicators of objects in that category’s Drinfel’d center Z(C).

9/23/2021- Ben Hayes, University of Virginia:

Property (T) and strong 1-boundedness for von Neumann algebras

 

The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung in 2007. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. I will discuss recent work, joint with Jekel and Kunnawalkam Elayavalli, proving that if M is either a Property (T) von Neumann algebra (in the sense of Connes-Jones, Popa) with finite dimensional center, or a group von Neumann algebras of a Property (T) group, then M is strongly 1-bounded. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. Prior knowledge of Property (T) will not be assumed.

9/30/2021- David Jordan, University of Edinburgh:

Higher symmetries in skein theory

Skein modules are quantizations of character varieties defined using the representation theory of quantum groups: specifically they depend on a ribbon braided tensor category, such as the category Rep_q(G) consisting of finite dimensional representations of U_q(g) whose weights lie in the weight lattice of G.  Given two algebraic groups with the same Lie algebra, we can ask, how are their skein modules related?  In the classical setting of character varieties, this relationship is elementary:  for example if G_{ad} and G_{sc} denote the adjoint and simply connected form of the group, then the G_{ad}-character variety of M is the disjoint union of G_{sc}-twisted character varieties of M, by an action of H_1(M,Z(G_{sc})).

In fact this paradigm quantizes to skein modules as an instance of higher form symmetry, and allows us to compute skein modules for G=PGL_2 in terms of skein modules for G=SL_2, together with some classical geometry of character varieties due to Hitchin.  In the talk I'll outline this computation (joint with Gunningham and Safronov), and I'll explain how it confirms a special case of the Langlands duality conjecture for skein modules, which we have recently formulated with Ben-Zvi, Gunningham, and Safronov.

10/7/2021- Alexei Davydov, Ohio University:

 

Moduli spaces of tensor categories

As mathematical structures effectively defined by systems of polynomial equations tensor categories and tensor functors naturally form algebro-geometric objects, their moduli spaces. Locally these geometric objects are controlled by the deformation cohomology of tensor categories and tensor functors. Free symmetric tensor categories and symmetric tensor functors out of them will be used as examples.

10/14/2021- Pieter Naaijkens, Cardiff University:

 

Long-range entanglement and the split property

In this talk I consider a Doplicher-Haag-Roberts approach to the superselection sectors of gapped 2D quantum spin systems. I will first briefly outline how this allows us to recover the full tensor category describing the anyonic excitations of a topologically ordered ground state. It is well-known in the topological phases of matter community that the existence of non-trivial anyonic excitations is due to the ground state having long-range entanglement. I will outline how this can be proven rigorously in this operator-algebraic "DHR" setting. In particular, we prove that if the ground state is not long-range entangled, then the category of superselection sectors is isomorphic to Vect. Based on joint work with Yoshiko Ogata.

10/21/2021- Dan Freed, University of Texas at Austin:

 

Boundaries and 3-dimensional topological field theories

Just as differential equations often boundary conditions of various types, so too do quantum field theories often admit boundary theories.  I will explain these notions and then discuss a theorem proved with Constantin Teleman which characterizes certain 3-dimensional topological field theories which admit nonzero boundary theories.  One application is to gapped systems in condensed matter physics.  Another characterizes fusion categories among tensor categories.

10/28/2021- Ellen Kirkman, Wake Forest University:

 

Reflection Hopf algebras

Let k be an algebraically closed field of characteristic zero. When H is a semisimple Hopf algebra that acts inner faithfully and homogenously on an Artin-Schelter algebra A so that the subalgebra of invariants A^H is also Artin-Schelter regular, we

call H a reflection Hopf algebra for A; when H=k[G] and A =k[x_1, ... ,x_n], then H is a reflection Hopf algebra for A if and only if G is a reflection group. We provide examples of reflection groups and reflection Hopf algebras for noncommutative Artin-Schelter algebras. We show that in this noncommutative context there exist notions of the Jacobian, reflection arrangement, and discriminant that extend the definitions used for reflection groups actions on polynomial algebras.

11/4/2021- Chris Heunen, University of Edinburgh:

Axioms for the category of Hilbert spaces

We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure such as probabilities, convexity, complex numbers, continuity, or dimension. We'll discuss the axioms, sketch the proof of the theorem, and survey open questions, further directions, and context.  (Based on joint work with Andre Kornell arxiv:2109.07418.)

11/11/2021- No Talk: Veteran's Day (Nicolle González  talk rescheduled for next semester).

11/18/2021- Peter Schauenburg, Université de Bourgogne:

Prime factorization of modular categories

A modular category is prime if it admits no proper modular subcategory. By a result of Müger, a modular category can always be written as a Deligne product of prime modular categories. The factorization is unique if there are no invertible objects. If the category is pointed on the other hand, the factorization is not unique and this was known long before modular categories were first defined. We discuss what happens in between, that is, to what extent the factorization may be unique in the general case of a non-pointed category with a nontrivial pointed part.