Workshop on Operator Algebras: Subfactors, K-Theory and Conformal Field Theory
Lead Research Organisation:
Cardiff University
Department Name: Sch of Mathematics
Abstract
Research in von Neumann algebras, originally studied by Murray and von Neumann in order to set up a mathematically rigorous formulation of quantum mechanics, received a major boost with the study of subfactors initiated by Vaughan Jones in the early 1980's. Subfactor theory rapidly led to connections with link and 3-manifold invariants, quantum groups and exactly solvable models in statistical mechanics, reinforcing the connections with physics. Subsequently deep applications and connections have been uncovered with algebraic, topological and conformal quantum field theory (CFT), with impressive progress in recent years in these applications. Free probability and planar algebra techniques have been combined to not only construct subfactors but derive matrix model computations in loop models of statistical mechanics.
The focus of this workshop is on shaping the future research directions in operator algebras, bringing together experts from such diverse disciplines as subfactor theory, K-theory and CFT, who will consider solutions to identified problems through a combination of lectures and participative sessions of breakout groups. The topics of the workshop include:
- The connections between the different formulations of chiral CFTS (local conformal nets, vertex operator algebras, Segal framework).
- The reconstruction programme - realisation of a chiral CFT which reproduces a given modular tensor category.
- Quantum symmetries on C*-algebras.
- Higher equivariant twists as equivariant Fell bundles.
- Connections between von Neumann relative entropy in CFT, subfactors, and mathematical physics.
The workshop will bring together researchers at the beginning of their careers with experts who were instrumental in finding applications of von Neumann algebras, including Jones and Voiculescu who have already agreed to speak. Von Neumann algebras, subfactors, K-theory and CFT have had a huge influence on the study of group representations, algebraic topology and across theoretical physics in the past. In a future in which quantum theory is rapidly becoming a fundamental part of everyday technology, the applications of operator algebras is expanding and can be expected to develop new applications for a new generation.
The focus of this workshop is on shaping the future research directions in operator algebras, bringing together experts from such diverse disciplines as subfactor theory, K-theory and CFT, who will consider solutions to identified problems through a combination of lectures and participative sessions of breakout groups. The topics of the workshop include:
- The connections between the different formulations of chiral CFTS (local conformal nets, vertex operator algebras, Segal framework).
- The reconstruction programme - realisation of a chiral CFT which reproduces a given modular tensor category.
- Quantum symmetries on C*-algebras.
- Higher equivariant twists as equivariant Fell bundles.
- Connections between von Neumann relative entropy in CFT, subfactors, and mathematical physics.
The workshop will bring together researchers at the beginning of their careers with experts who were instrumental in finding applications of von Neumann algebras, including Jones and Voiculescu who have already agreed to speak. Von Neumann algebras, subfactors, K-theory and CFT have had a huge influence on the study of group representations, algebraic topology and across theoretical physics in the past. In a future in which quantum theory is rapidly becoming a fundamental part of everyday technology, the applications of operator algebras is expanding and can be expected to develop new applications for a new generation.
Organisations
| Description | Operator Algebras: Subfactors, K-theory, Conformal Field Theory The conference "Operator Algebras: Subfactors, K-theory, Conformal Field Theory" was held in honour of the 70th birthday of David Evans from Cardiff University. It was organised as an in-person event from 27th July to 2nd August 2022 at Gregynog Hall in Wales after it had to be postponed in 2021 due to the pandemic. The organisers gratefully acknowledge the financial support received from the LMS, EPSRC, the Learned Society of Wales, and the Clay Mathematics Institute. The conference was focused on a wide range of interconnected themes, among them operator algebras and subfactor theory, fusion categories, topological and operator-algebraic K-theory and applications of all these areas in conformal field theory. This list of topics reflects the rich body of research David Evans has contributed to and cultivated over the years. The occasion brought together high-profile established speakers as well as early career researchers and featured talks by two Clay lecturers: Dan-Virgil Voiculescu (University of California, Berkeley) and Sorin Popa (UCLA). Voiculescu, who is an elected member of the National Academy of Sciences, spoke about the quasicentral modulus in functional analysis. This measure of how well n-tuples of operators commute with finite-rank approximate units leads to a non-commutative generalisation of the condenser capacity. Popa, who currently holds the Takesaki Endowed Chair in Operator Algebras, gave an account of W*-representation theory for subfactors. Stuart White (Oxford) reviewed the recent breakthroughs in the Elliot classification programme for simple separable nuclear Z-absorbing C*-algebras, a topic that he also discussed as an invited speaker at the recent virtual ICM 2022. Another former ICM speaker (of which there were eight present in total) was Constantin Teleman (University of California, Berkeley) who presented his work on equivariant higher twists of K-theory and how it connects with Coulomb branches in gauge theory. Roberto Longo (Tor Vergata, Rome) spoke about his research programme on the role of Tomita-Takesaki theory in quantum field theory (QFT). The algebraic framework of QFT was also the topic of the talk by Kasia Rejzner (York), who outlined a C*-algebraic approach to QFT with interactions. Yasuyuki Kawahigashi (University of Tokyo) reported on his work on producing subfactors of finite depth. Subfactor theory was developed by Vaughan Jones and its main object of study are inclusions of "building block" von Neumann algebras, called factors. Surprisingly, finite-dimensional invariants that are called commuting squares already capture a lot of information about such inclusions and have recently also appeared in condensed matter physics in connection to topological orders. The Thompson group, whose amenability has been a long-standing open problem, was the subject of two talks: Emily Peters (Loyola University Chicago) discussed Conway's rational tangles. Her talk was based on work of Vaughan Jones that has built a bridge between the theory of knots and links and that of the Thompson group. Alina Vdovina (Newcastle University) outlined its higher-dimensional analogues that can be described using buildings, i.e. geometric structures with a strong combinatorial flavour. The intricate connections between stable homotopy theory and topological quantum field theories were also a recurring theme: Apart from Constantin Teleman's presentation it featured in a talk by Mayuko Yamashita (RIMS, Kyoto) about invertible QFTs and differential cohomology. The topological view on quantum field theory was shared by Katrin Wendland (Trinity College Dublin) in her talk about elliptic genera for orbifolds with simple singularities. Terry Gannon (University of Alberta) spoke about modular invariants for rational conformal field theories. In two dimensions the restriction of a CFT to its chiral halves has enough symmetries to help classify them. In particular, those theories have a modular invariant partition function. This condition is quite restrictive and allowed Terry Gannon to find a solution to the problem up to rank 7. This diversity of topics and speakers reflects David Evans' widespread scientific interests and relations to colleagues around the world - including several former students and PostDocs - and the central role played by operator algebras in connecting these different branches of mathematics. Providing an opportunity to discuss and connect with other researchers after the long "communication drought" and "Zoom fatigue" caused by the pandemic was one of the main aims of the conference. The venue, Gregynog Hall in Wales, was chosen with this goal in mind. Acquired by the Davies sisters in 1920 it first became a hub to bring art, music and creative skills to the people of Wales in the aftermath of the First World War. After its wartime use as a Red Cross convalescent home, it was bequeathed by the sisters to the University of Wales as a conference centre and in 2019 transferred to the Gregynog Trust. Their substantial collection of French impressionist and post-impressionist art is now in the National Museum in Cardiff. Gregynog has a history of serving as a location for mathematics colloquia, such as the annual meeting of mathematicians in Wales, and workshops including satellite meetings for the Isaac Newton Institute programmes, EPSRC networks and an LMS regional meeting and workshop. Greygnog has a comfortable atmosphere and all participants had accommodation on site during the event. We were happy to witness that this sparked many mathematical discussions among participants and hope that a lot of interesting future discoveries in the diverse research areas covered by the speakers will be traced back to a conversation that began over tea in the Blayney room or in the evening after the talks under the stars in the courtyard. Masaki Izumi (Kyoto) Gandalf Lechner (Erlangen) Ulrich Pennig (Cardiff) Mathew Pugh (Cardiff) |
| Exploitation Route | There are many possible ways how this workshop will affect new research and collaborations. Please see the above article for more detail. |
| Sectors | Other |
| URL | https://oa-gregynog-2021.weebly.com/ |