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Timetable (OASW03)

Subfactors, K-theory and conformal field theory

Monday 12th June 2017 to Friday 16th June 2017

Monday 12th June 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Vaughan Jones
Phase transitions in the semicontinuous limit of a quantum spin chain
A quest for the construction of a conformal field theory directly from a subfactor has taken an unexpected turn involving a "semicontinuous limit” Hilbert space, with Thompson group symmetry, that might be as relevant to critical quantum spin chains as CFT itself. Models give spin chains with various phases governed by the value of a spectral parameter, and a holomorphic dynamical system. The phases are the Fatou connected components of the dynamical system and the phase transistions occur when the spectral parameter crosses the Julia set from one Fatou component to another. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Arthur Jaffe
On Picture Language
We introduce some recent work on pictures and the development of the quon language (joint work with Zhengwei Liu and Alex Wozniakowski). We describe a pictorial journey from planar algebras and parafermions, through a problem in quantum information.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Yasu Kawahigashi
The relative Drinfeld commutant and alpha-induction
We study relative Drinfeld commutants of a fusion category in another fusion category in terms of half-braidings. We identify half-braidings with minimal central projections of the relative tube algebra and certain sectors related to the Longo-Rehren subfactors. We apply this general machinery to various fusion categories arising from alpha-induction applied to a modular tensor category and compute the relative Drinfeld commutants explicitly.
14:30 to 15:30 Roberto Longo
Discussion about the Landauer principle (and bound)
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Stefaan Vaes
Rothschild Lecture: Classification of von Neumann algebras
The theme of this talk is the dichotomy between amenability and non-amenability. Because the group of motions of the three-dimensional Euclidean space is non-amenable (as a group with the discrete topology), we have the Banach-Tarski paradox. In dimension two, the group of motions is amenable and there is therefore no paradoxical decomposition of the disk. This dichotomy is most apparent in the theory of von Neumann algebras: the amenable ones are completely classified by the work of Connes and Haagerup, while the non-amenable ones give rise to amazing rigidity theorems, especially within Sorin Popa's deformation/rigidity theory. I will illustrate the gap between amenability and non-amenability for von Neumann algebras associated with countable groups, with locally compact groups, and with group actions on probability spaces.

17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 13th June 2017
09:00 to 10:00 Dietmar Bisch
Subfactors with infinite representation theory
10:00 to 11:00 Zhengwei Liu
Synergy on quon language
We will talk about the discovery of the quon language, which was inspired by ideas in different areas: subfactors, quantum information, CFT, TQFT. New proofs and results were carried out by synergy on the quon language. The Fourier analysis on quons turns out to be powerful to study modular tensor categories. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Hubert Saleur
Associative algebras and conformal field theories
I will review in this talk the relationships physicists observe/conjecture  between the associative algebras
(such as Temperley-Lieb) that appear in lattice models, and the  conformal field theories (CFT)  that  describe
their  continuum limits. I will then discuss in more detail a possible way to perform "fusion" for affine Temperley-Lieb
modules, and what this may have to do with Operator Product Expansion (OPE) in CFT. 
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Paul Fendley
Baxterising using conserved currents
Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess an unusual sort of conserved current. Such currents have been constructed by utilising quantum-group algebras, fermionic and parafermionic operators, and ideas from ``discrete holomorphicity''. I define them generally and naturally using a braided tensor category, a structure familiar from the study of knot invariants and from conformal field theory.  Requiring the existence of the currents provides a simple way of ``Baxterising'', i.e. building a solution of the Yang-Baxter equation out of topological data.  This approach allows many new examples of conserved currents to be found, for example in height models.  Although integrable models found by this construction are critical, I find one non-critical generalisation: requiring a ``shift'' operator in the chiral clock chain yields precisely the Hamiltonian of the integrable chiral Potts chain.

14:30 to 15:30 Gandalf Lechner
Yang-Baxter representations of the infinite symmetric group
The Yang-Baxter equation (YBE) lies at the heart of many subjects, including quantum statistical mechanics, QFT, knot theory, braid groups, and subfactors. In this talk, I will consider involutive solutions of the YBE ("R-matrices"). Any such R-matrix defines a representation and an extremal character of the infinite symmetric group as well as a corresponding tower of subfactors. 

Using these structures, I will describe how to find all R-matrices up to a natural notion of equivalence (given by the character and the dimension), how to completely parameterize the set of solutions, and how to decide efficiently whether two given R-matrices are equivalent. Joint work with U. Pennig and S. Wood.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Kasia Rejner
The Quantum Sine-Gordon model in perturbative AQFT
Co-author: Dorothea Bahns (University of Goettingen)

In this talk I will present recent results on the convergence of the formal S-matrix and interacting currents in the Sine-Gordon model in 2 dimensions, obtained directly in Minkowski signature, using a class of Hadamard states. In our approach one starts with a perturbation series obtained from the formalism of perturbative AQFT and then one can prove the convergence of the series by some simple estimates. Our result opens the posibility to use pAQFT methods to study integrable models in 2 dimensions and to construct local observables in such models.

Related Links
Wednesday 14th June 2017
09:00 to 10:00 Constantin Teleman
Kramer-Wannier and electro-magnetic duality in field theory
A classical duality (Kramer-Wannier) relates the low and high temperature of the 2-dimensional Ising model. It has been generalized to other dimensions and groups other than Z/2 and distilled into Poincare duality combined with the Abelian Fourier transform. In this talk, I describe a vast generalization in the language of topological field theories, which includes non-Abelian examples. Via the notion of boundary field theory, thus is related to a duality of TQFTs, specifically electro-magnetic duality in 3 dimensions. There arises a natural speculation about invertibility of gapped phases in a large class of lattice models. This is joint work (in progress) with Dan Freed. 
10:00 to 11:00 Pedram Hekmati
An application of T-duality to K-theory
Co-author: David Baraglia (The University of Adelaide)

T-duality is a discrete symmetry that was discovered by physicists in the context of string theory, but has now matured into a precise mathematical statement. In this talk I will give a brief overview of topological T-duality and explain how it can be used to give a new, surprisingly simple proof of Hodgkin’s famous theorem on the K-theory of compact simply connected Lie groups.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Giovanni Landi
Line bundles over noncommutative spaces
We give a Pimsner algebra construction of noncommutative lens spaces as `direct sums of line bundles' and exhibit them as `total spaces' of certain principal bundles over noncommutative weighted projective spaces. For each quantum lens space one gets an analogue of the classical Gysin sequence relating the KK theory of the total space algebra to that of the base space one. This can be used to give explicit geometric representatives of the K-theory classes of the lens spaces. 
12:30 to 13:00 Sandwich Lunch at INI
14:00 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Trinity College
Thursday 15th June 2017
09:00 to 10:00 Antony Wassermann
Conformal Field Theory, Operator algebras and symmetric Fuchsian equations
10:00 to 11:00 Robin Hillier
Loop groups and noncommutative geometry
I describe a way of expressing loop groups and their representation theories (the Verlinde fusion ring) in terms of spectral triples and KK-theory. The ideas come from operator algebraic conformal field theory and extend to many other conformal field theoretical models.

Co-author: Sebastiano Carpi.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Ralf Meyer
Induced C*-hulls for *-algebras
Let A be a *-algebra that is graded by a group G with fibres A_g for g in G.  Assume that we have found a C*-algebra B_e whose “representations” are “equivalent” to the “integrable” “representations” of the unit fibre A_e.  Call a “representation” of A “integrable”, if its restriction to A_e is “integrable”.  Under some assumptions, the “integrable” “representations” of A are “equivalent ” to the “representations” of a certain C*-algebra B constructed from B_e and the graded *-algebra A.  The C*-algebra B is the section C*-algebra of a Fell bundle over G.  The words in quotation marks have to be interpreted carefully to make this true.  In particular, representations must be understood to take place on Hilbert modules, not just Hilbert spaces, and the equivalence is required natural with respect to induction of representations and isometric intertwiners.  Under some commutativity assumptions, the main result of my lecture has been proved by Savchuk and Schmüdgen, who also give several examples.  A sample of the result concerns Weyl algebras and twisted Weyl algebras in countably many generators.  These come with a canonical grading by the free Abelian group on countably many generators.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Christian Voigt
The string group and vertex algebras
I will describe a categorification of complex Clifford algebras arising from certain categories of twisted modules over fermionic vertex superalgebras. Along the way I'll discuss some background from the theory of unitary vertex algebras, and how the String 2-group fits into the picture.
14:30 to 15:30 Andre Henriques
Bicommutant categories
Bicommutant categories are higher categorical analogs of von Neumann algebras. There exist currently two sources of examples of bicommutant categories: unitary fusion categories, and completely rational unitary conformal field theories.

I will give a general introduction on what and why bicommutant categories are. Then I will talk about a recent result with Dave Penneys: Morita equivalent unitary fusion categories have isomorphic associated bicommutant categories.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Lilit Martirosyan
Affine centralizer algebras
I will describe the representations of (affine) centralizer algebras for quantum groups in terms of paths. I will talk about generators for these algebras. As an example, we will consider the Lie algebra $G_2$ and its centralizer algebras. 
Friday 16th June 2017
09:00 to 10:00 Feng Xu
On questions around reconstruction program
10:00 to 11:00 Ching Hung Lam
On the Classification of holomorphic vertex operator algebras of central charge 24
In 1993, Schellekens obtained a list of possible Lie algebra structures for the weight one subspaces of holomorphic vertex operator algebras (VOA) of central charge 24. It was also conjectured that the VOA structure of a holomorphic VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one space. Recently, all 71 cases in Schellekens' list have been constructed.  

In this talk, we will discuss the recent progress on the classification of holomorphic vertex operator algebras of central charge 24. In particular, we will discuss the construction of holomorphic VOAs using various types of orbifold constructions. A technique, which we call ``Reverse orbifold construction", will also be discussed. This technique may be used to prove the uniqueness of holomorphic VOAs of central charge 24 if the weight one subspace is not zero. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Simon Wood
What to expect from logarithmic conformal field theory
Logarithmic conformal field theory is a generalisation of ordinary conformal field theory that allows for logarithmic singularities in correlation functions. This implies the existence of reducible yet indecomposable modules on which the action of the Virasoro L_0 operator is not diagonalisable. In this talk I will recall some of what is known about rational conformal field theory and contrast it with what has been achieved so far in logarithmic conformal field theory.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Sebastiano Carpi
Conformal nets, VOAs and their representations
We discuss some recent results on the connection between conformal nets, VOAs and their representation theories. 
14:30 to 15:30 Terry Gannon
The truth about finite group orbifolds
Chiral CFTs (VOAs or conformal nets) are interesting for their representation theory. Orbifolds are a standard method for constructing new chiral CFTs from old ones. Start with a chiral theory with trivial representation theory, and orbifold it by a finite group; the result (called a holomorphic orbifold) has the representation theory given by the twisted Drinfeld double of that finite group, where the twist is a 3-cocycle. In practise it is hard to identify that twist. 

I'll begin my talk by giving some examples of orbifolds. I'll identify a well-known class of holomorphic orbifolds where we now know the twist. I'll relate holomorphic orbifolds to KK-theory as well as the PhD thesis of a certain Vaughan Jones. Then I'll explain how any choice of finite group and 3-cocycle is realized by a chiral CFT. This is joint work with David Evans.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons