Titles, Abstracts and Slides

Titles and abstracts

Daniel Barter (Australian National University, Canberra, Australia)

Title: Computing bimodule associators in the Brauer-Picard 3-category

Abstract: The Brauer-Picard 3-category has objects fusion categories, 1-morphisms bimodules, 2-morphisms bimodule funtors and 3-morphisms natural transformations. The bimodule associator is a bimodule functor and we shall demonstrate how to compute it explicitly.

Dietmar Bisch (Vanderbilt University, USA)

Title: Degrees of Noncommutativity for Subfactors

Abstract. The standard representation of a subfactor generates a nice

unitary rigid C*-tensor category, or planar algebra, that is a complete

invariant for amenable, hyperfinite subfactors. However, generic

subfactors are not amenable, and it is open how to distinguish them.

I will explain a notion of ``noncommutativity'' for a subfactor that is

not captured by the planar algebra and explain a theorem that gives

the first examples of ``very noncommutative'' subfactors. Perhaps this

property should be viewed as a quantum symmetry.

Zsuzsanna Dancso (University of Sydney, Australia)

Title: Loops on a punctured disk and knotted tubes in R^4

Abstract: Homotopy classes of loops on a twice-punctured disk (more generally a surface with boundary) admit a Lie bi-algebra structure called the Goldman-Turaev Lie bi-algebra. The Lie bracket and co-bracket are defined in terms of intersections and self-intersections of curves. Combining results of Alekseev - Kawazumi - Kuno - Naef with results of the speaker and Bar-Natan, one obtains a surprising statement: "well-behaved universal finite type invariants" of the (enhanced) Goldman-Turaev Lie bi-algebra are in bijection with the same invariants of a very different topological structure: a class of knotted tubes in R^4. However, the bijection goes through representing both classes of invariants as solutions to certain equations in Lie theory (the Kashiwara-Vergne equations). This is clearly the wrong proof of a worthwhile theorem. But what is the right proof?

Colleen Delaney (UCSB, USA)

Title: Quantum computing with permutation extensions of modular categories

Abstract: In this talk I'll motivate the application of symmetry defects to topological quantum computing and discuss some recent progress in constructing their explicit algebraic models as certain module categories over modular categories.

Michael Freedman (Microsoft Quantum - Santa Barbara, USA)

Title: Linearity of the GNVW index in High Dimensions

Abstract: Topogical tricks are employed to show that if P and Q are codimension one cycles, the index satisfies I(P+Q) = I(P) + I(Q).

Terry Gannon (University of Alberta, Canada)

Title: Fantastic beasts and where to find them

Abstract: In recent years, subfactor methods have constructed hundreds of new fusion categories, with strong evidence these live in infinite families. Taking their doubles, we obtain hundreds (and probably infinite families) of new modular tensor categories. These modular tensor categories have a distinctive appearance. Abstracting this appearance, it is natural to guess that there is a new construction, which we call the smashed-sum, combining old modular tensor categories into new ones. It is tempting to guess the smashed-sum construction also lives in the VOA world, in fact this may be its natural home. An example of such a VOA would be the mythical Haagerup VOA.

Iva Halacheva (University of Melbourne, Australia)

Title: Diagrammatic categories and the periplectic Lie superalgebra

Abstract: We relate the representation theory of the periplectic Lie superalgebra p(n) to the diagrammatic affine VW supercategory sVW, with a view towards higher Schur-Weyl duality. Namely, we establish commuting actions of p(n) and the affine VW superalgebra sVW(a) on the tensor product of a p(n)-module M with a copies of the standard representation. More generally, we define a superfunctor from sVW to p(n)-mod which allows us to transfer information between the two supercategories. In particular, we establish bases for the morphism spaces of sVW and determine the center Z(sVW(a)).

Vaughan Jones (Vanderbilt University, USA)

Title: A couple of things that didn’t work-so far

Yasuyuki Kawahigashi (University of Tokyo, Japan)

Title: Topological phases of matter, subfactors and the relative Verlinde formula

Abstract: We present an operator algebraic approach to modular tensor categories appearing in studies of topological phases of matter. We give the relative Verlinde formula in the context of a relative version of the boundary-bulk duality and the relative Drinfeld commutants of a fusion subcategory.

Gus Lehrer (University of Sydney, Australia)

Title: A category equivalence between Temperley-Lieb of type B and a category of infinite

dimensional representations of quantum $sl_2$

Abstract: I shall present an equivalence of categories between the type B Temperley-Lieb category, realised as a subquotient of the classical category of unoriented tangles, and a category of projective infinite dimensional representations of quantum $sl_2$. This is joint work with Ruibin Zhang and Kenji Iohara.

Galina Levitina (UNSW, Sydney, Australia)

Title: Derivations with values in ideals of von Neumann algebras

Abstract: A derivation on an algebra is a linear mapping which satisfies the Leibniz rule. A very special type of derivations are so-called inner derivations which are of the form [ . , a] for some fixed element a of the algebra. In this talk we study derivation defined of a von Neumann subalgebra A of a von Neumann algebra M with values in ideals of M. We show that for specific ideals of M derivations from A with values in these ideals are necessarily inner.

Roberto Longo (University of Rome-Tor Vergata, Italy)

Title: Modular time and entropy/energy bounds

Magdalena Musat (University of Copenhagen, Denmark)

Title: Von Neumann Algebras meet Quantum Information Theory

Abstract: The study of quantum correlations arising under two different assumptions of commutativity of observables, initiated by Tsirelson in the 80’s, has proven over the last decade to have deep interconnections with important problems in operator algebras theory, including various reformulations of the Connes Embedding Problem. In very recent work with M. Rørdam, we show that in every dimension n ≥ 11, the set of n×n matrices of correlations arising from unitaries in finite dimensional von Neumann algebras is not closed. As a consequence, in each such dimension there are quantum channels that admit type II1-von Neumann algebras as ancillas, but not finite dimensional ones, thus witnessing new infinite dimensional phenomena in quantum information theory.

Arun Ram (University of Melbourne, Australia)

Title: A point of view on Conformal Field Theory

Abstract: I will give a proposal for studying conformal field theory by combinatorics of crystals (in the sense of Kashiwara and Lusztig), more precisely, via crystals of level 0 integrable representations of affine Lie algebras.

David Ridout (University of Melbourne, Australia)

Title: Modularity beyond rationality

Abstract: It is well known that rational conformal field theories are excellent sources of modular tensor categories. However, rationality is extremely special and many physical applications require greater generality. It is natural then to ask how modularity generalises beyond rationality. I shall review some of what is known in this direction and why people only interested in vanilla modular tensor categories should care.

Andrew Schopieray (UNSW Sydney, Australia)

Title: Numerical invariants of modular tensor categories

Abstract: The research literature of fusion categories (and their spiritual ancestor, finite groups) is full of elementary questions which are disappointingly difficult to answer. Are these categories equivalent in any way? Can I construct this category from another? Does this category even exist? The dream would be to study these questions by associating to categories sets of complex numbers which are easier to compare. These numerical invariants must be robust in the sense that they are immune, or at least predictably change under standard constructions. In this talk I will describe a family of numerical invariants (higher Gauss sums and central charges) associated to modular tensor categories which are robust in this sense, and allow meaningful conclusions to be drawn about categories using basic arithmetic.

Alexander Stottmeister (University of Münster, Germany)

Title: Gauge theory and Jones’ actions of Thompson’s groups

We present some recent work combining Jones’ construction of actions of Thompson’s groups and ideas from lattice gauge theory and loop quantum gravity. Starting from a compact group G, we construct models of 1+1-dimensional gauge theories in terms of a spatially local net of C*-algebras, the time-zero fields, together with a state determined by a family of probability measures on the unitary dual of G. A particularly interesting family of measures is directly related to the Kogut-Susskind Hamiltonian.

We discuss the problem of extending Jones’ actions to the von Neumann closure of the net, and the possibility to implement time-evolution via Tomita-Takesaki theory. For abelian G, we state precise results on both issues.

This is joint work with Arnaud Brothier.

James Tener (Australian National University, Canberra, Australia)

Title: Subfactors via vertex operator algebras

Abstract:Vertex operator algebras and conformal nets are mathematical formalisations of the notion of 2d chiral conformal field theory. They are expected to be essentially equivalent, which provides the opportunity to link different looking areas of mathematics. In this talk I will describe how subfactors arise from conformal nets, and how to build them out of the geometric data of vertex operator algebras.

Makoto Yamashita (University of Oslo, Norway)

Title: Structure of tube algebra from categorical viewpoint

Abstract:

In this talk I will present a survey of our recent works on tube algebras. The tube algebra was introduced by Ocneanu to describe the structure of quantum double in the framework of finite depth subfactors, and it provides the most combinatorial model to capture the concept of quantum double of tensor categories. In the recent years it reemerged in the analytic study of rigid C*-tensor categories, in which one regards the quantum double of tensor category C as the category of "representations of C".

Expanding on works of Izumi, Muger, and Bruguieres-Virelizier, we describe how the tube algebra naturally appears as (dense nonunital subring of) End(X) for a suitable object X in the Drinfeld center of the ind-completion. This has implication to understanding of Morita equivalence of tensor categories and also to Hopf algebraic model of quantum double, the Drinfeld double construction.

In another direction, when C is graded over a discrete group G, the tube algebra becomes a Fell bundle over the action groupoid of G with respect to the adjoint action on itself. When one considers a twisting of C by a 3-cocycle on G, the tube algebra is twisted by an induced groupoid 2-cocycle. This generalizes the quasi-Hopf algebra for quantum double of pointed categories by Bantay. One notable feature is that this twisting is cohomologically trivial when G is a cyclic group.

Combining the above, we obtain one explanation of why we see the same parametrization between Pusz's classification of irreducible unitary representations of quantum Lorentz group on the one hand, and the Ghosh-Jones classification of irreducible annular representations of Temperley-Lieb-Jones planar algebra on the other.

Based on joint works with S. Neshveyev, J. Bhowmick, S. Ghosh, N. Rakshit.

Andrzej Zuk (University of Paris Diderot, France)

Title：From PDEs to groups

Abstract : We present a construction which associates to a KdV equation the lamplighter group. In order to establish this relation we use automata and random walks on ultra discrete limits. It is also related to the von Neumann dimension and L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.