Program

Schedule

Lecture Series

Christoph Schweigert

Lecture series on state-sum construction and applications

Invited Talks

Kenny De Commer

Title: Unitary Doi-Koppinen representations

Abstract: We will give an overview of the theory of Doi-Koppinen modules in the setting of operator algebras, and will illustrate the theory with the example of quantum SL(2,R).

David Evans

Title: Quantum Symmetries

Abstract: This talk is part of a programme to understand  quantum symmetries through subfactors and twisted equivariant K-theory and their applications in conformal field theory. Here I discuss the question of constructing actions of these quantum symmetries on the irrational rotation algebras and more generally noncommutative tori. This is based on joint work with Corey Jones.

Jürgen Fuchs

Title: Grothendieck-Verdier module categories and Frobenius algebras

Abstract: A Grothendieck-Verdier category is a monoidal category having a duality structure more general than rigidity. It comes with a right exact and with a left exact monoidal structure, which are related by non-invertible morphisms. I will describe crucial aspects of a natural class of module categories over Grothendieck-Verdier categories. Such a Grothendieck-Verdier module category has two distinguished subcategories on which certain structure morphisms are invertible. The module category can be realized as the category of modules over the internal Hom of suitable objects in those subcategories. Moreover, it admits a partial relative Serre functor which furnishes an equivalence between the two subcategories. The internal Hom of a fixed point of the relative Serre functor carries a natural structure of a Grothendieck-Verdier Frobenius algebra.

Lucas Hataishi

Title: Integrating Drinfeld doubles of compact quantum groups over surfaces

Abstract: In this talk I will discuss an approach (work in progress) to analyze a class of framed 2-dimensional TQFTs where the symmetries are encoded by Drinfeld doubles of  compact quantum groups. The TQFTs are constructed via factorization homology with values in C*-categories. As an intermediate step, we obtain Yetter-Drinfeld C*-algebras carrying actions of modular groups of surfaces.

Robin Hillier

Title: Vertex operator algebras, conformal nets, their representations and some classifications.

Abstract: For chiral conformal quantum field theory, I provide an overview to the algebraic framework of vertex operator algebras and the functional analytic framework of conformal nets. I explain what is known and conjectured about the relation between these two approaches. I will then move on to their representations and extensions using methods from the framework of C*-tensor categories and explain some results and conjectures in this case. As a corollary, we obtain classifications of some particular families of conformal field theories in the two frameworks.

Corey Jones

Title: Dualities and DHR bimodules of quantum spin chains

Abstract: Dualities, in the spirit of Kramers-Wannier, are defined for quantum spin chains as bounded spread isomorphisms between subalgebras of the quasi-local algebra that exchange local Hamiltonians. Dualities lead to insight into the symmetries of phase diagrams and phase transitions. This motivates the mathematical problem of classifying bounded spread isomorphisms between abstract spin chains, generalizing the GNVW classification of 1D quantum cellular automata. We will discuss how to use the braided tensor category of DHR bimodules, in conjunction with the Jones index for subfactors, to shed light on this problem. We then use these tools to give strong classification results for G-dualities, where G is a finite group acting on-site.

Gandalf Lechner

Title: Quantum Field Theory and Subfactors on a Nichols Algebra

Abstract: Any braided vector space V defines a Nichols algebra as the direct sum of the quotients of the tensor powers of V by the kernels of the quantum symmetrizers. In this talk we will start from a Hilbert space version of this construction and then use standard subspaces H of V to generate von Neumann algebras L(H) on the Nichols algebra Hilbert space. It is shown that the vacuum of the Nichols algebra is cyclic and separating for L(H) under natural assumptions related to an operator-algebraic version of the crossing symmetry of elementary particle physics, generalizing known results about second quantization factors and free group factors. This construction gives rise to a large family of inclusions of von Neumann algebras (often subfactors of type II or III) and can be used to construct quantum field theories. Joint work with Ricardo Correa da Silva.

Pieter Naaijkens

Title: Local topological order and boundary algebras

Abstract: The study of topologically ordered quantum phases has has led to interesting connections with, for example, the study of tensor categories or subfactor theory. In this talk I will introduce a new axiomatisation of such quantum models defined on d-dimensional square lattices in terms of nets of projections. These local topological order axioms are satisfied by known 2D models such as the toric code and Levin-Wen models built on a unitary fusion category. We show that these axioms lead to a definition of boundary algebras naturally living on a hyperplane. This boundary algebra encodes information about the excitations in the bulk theory, leading to a bulk-boundary correspondence. In this talk I will outline the main points. Based on joint work with Corey Jones, David Penneys and Daniel Wallick (arXiv:2307.12552).

Iordanis Romaidis

Title: On the holonomicity of skein modules

Abstract: Skein theory takes a ribbon tensor category C and forms a once-categorified 3d TQFT, assigning skein algebras to surfaces and skein modules to 3-manifolds. Motivated by physics, for quantum group representations, these modules are expected to satisfy a certain holonomicity property, generalizing Witten's finiteness conjecture of skein modules. In this talk, we will revisit skein theory as a deformation quantization theory and proceed to state and present the generalized Witten's finiteness conjecture.

Qing Zhang

Title: Modular Data of Non-Semisimple Modular Categories  
 
A braided finite tensor category is called modular if it is nondegenerate and ribbon. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case using representations of factorizable ribbon Hopf algebras as a case study. In this talk, we will discuss the Cohen-Westreich modular data, which is obtained from the Lyubashenko-Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen-Westreich S-matrix diagonalizes the mixed fusion rules and reduces to the usual S-matrix in the semisimple case. We discuss small quantum groups u_q(sl₂) and the Drinfeld doubles of Nichols Hopf algebras, especially the SL(2,Z)-representation on their centers, Cohen-Westreich modular data, and the validity of the congruence kernel theorem.

Contributed Talks

Maria Stella Adamo

Title: On the construction of full 2D CFTs with pointed representation category

Abstract: In this talk, we present explicit constructions for full two-dimensional conformal quantum field theories (2D CFTs) with pointed representation categories, i.e., those with chiral components whose representation categories possess enough automorphisms. First, for such theories we construct local nets of operator algebras by introducing charged primary fields acting as shifts between different charged sectors. On the other hand, we produce Wightman fields by gluing together charged primary fields on each chiral component. In the case of a rational 2D CFTs, we show that under reasonable regularity assumptions on the Wightman fields such constructions turn out to be equivalent.
This talk is mainly based on joint work with L. Giorgetti, Y. Tanimoto, CMP 2023.

Ricardo Correa da Silva

Title: Crossing symmetry and Q-systems

Abstract: This talk aims to introduce and discuss some properties of the "crossing map", a transformation of operators in Hilbert spaces defined in terms of modular theory and inspired by "crossing symmetry" from elementary particle physics, and discuss the connection between crossing-symmetric and Q-systems. We will also mention the connection of the crossing symmetry with T-twisted Araki-Woods algebras, endomorphisms of standard subspaces, and Fourier transforms.

Anupam Datta

Title: Equivariant KK-theory and model categories

Abstract: KK-theory can be seen as a homotopy theory for C*-algebras. It is
plausible to ask whether it can be cast in the framework of abstract
homotopy theory. In the non-equivariant case, Joachim-Johnson showed that
KK-groups are in fact the homotopy groups of a stable model
structure on pro-C*-algebras. I will discuss similar results in the
equivariant case, with a locally compact group action on the algebras
involved. Some new techniques involving universal equivariant algebras
were needed in the construction which would also be discussed. Lastly, a
comparison could be made to the more recent infinity categorical
equivariant KK-theory à la Bunke.

This is joint work with Michael Joachim.

Giovanni Ferrer

Title: Steps towards higher functional analysis

Abstract: In this talk, we describe a research program centered around higher operator algebras and higher Hilbert spaces based on recent work in arXiv:2403.01651, arXiv:2404.05193, arXiv:2411.01678 and more specifically arXiv:2410.05120.

David Green

Title: Levin-Wen is a Gauge Theory

Abstract: We will examine a class of SPT's protected by the tube algebra of a unitary fusion category C and show that gauging this symmetry produces the standard Levin-Wen model associated to C. Mathematically, the main tool we introduce are the spine coral skein modules which correspond graphically to the Day convolution products of tube algebra representations. This talk is based off of arXiv:2401.13838, joint with Kyle Kawagoe, Corey Jones, Sean Sanford, and David Penneys.

Roberto Hernández Palomares

Title: Quantum graphs, subfactors and tensor categories

Abstract: Graphs and their noncommutative analogues are interesting objects of study from the perspectives of operator algebras, quantum information and category theory. In this talk we will introduce equivariant graphs with respect to a quantum symmetry along with examples such as classical graphs, Cayley graphs of finite groupoids, and their quantum analogues. We will also see these graphs can be constructed concretely by modeling a quantum vertex set by an inclusion of operator algebras and the quantum edge set by an equivariant endomorphism that is an idempotent with respect to convolution/Schur product. Equipped with this viewpoint and tools from subfactor theory, we will see how to obtain all these idempotents using higher relative commutants and the quantum Fourier transform. Finally, we will state a quantum version of Frucht's Theorem, showing that every quasitriangular finite quantum groupoid arises as certain automorphisms of some categorified graph.

Cristian Ivanescu

Title: The Cuntz Semigroup and its Applications

Kan Kitamura

Title: Actions of tensor categories on Kirchberg algebras

Abstract: We give a systematic strategy for constructing outer actions of tensor categories on nuclear, simple, purely infinite C*-algebras. In particular, every countable unitary tensor category admits an outer action on the Cuntz algebra generated by two isometries. If time permits, we will discuss when module structures over a fusion ring on K-groups can lift to actions of a tensor category.

Ting Lu

Titlle: Computing the norm of the matrix coefficients of the Peter-Weyl decomposition on O(SL_q(3))

Abstract: The quantized algebra of functions O(SL_q(3)) is a Hopf algebra that forms a pairing of Hopf algebras with the Drinfeld-Jimbo quantim group U_q(sl(3)). O(SL_q(3)) is a pre-Hilbert space with its Haar state as the inner product. Hence, it is possible to study the quantum group O(SL_q(3)) from an analytic perspective. However, there is no orthonormal basis known on this group yet. The work of Noumi, Yamada, and Mimachi shows that a good candidate for an orthonormal basis on O(SL_q(3)) is the matrix coefficients of the Peter-Weyl decomposition. In this talk, I will introduce my on-going research on computing the norm of the matrix coefficients of the Peter-Weyl decomposition on O(SL_q(3)).

Melody Molander

Title: Index 4 Subfactor Planar Algebras and Their Corresponding Categories

Abstract: Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have invariants called an index and a principal graph. In this talk, I will discuss subfactor planar algebras and find presentations for subfactor planar algebras of index 4 associated with the affine A Dynkin diagram. Then I will show that these planar algebras have corresponding pointed fusion categories and representation categories.

Mizuki Oikawa

Title: Center construction for group-crossed tensor categories

Abstract: In this talk, I will talk about my recent generalization of the Drinfeld center construction for group-crossed tensor categories. A group-crossed tensor category is a tensor category with compatible group action and grading, which naturally appear in two-dimensional conformal theory as categories of twisted modules. Indeed, my construction for such categories yields categories "braided for a matched pair of groups", which is a notion introduced recently by Natale.

Sergio Giron Pacheco

Title: G-kernels and topological crossed modules

Abstract: For a group G and a C*-algebra A, a G-kernel on A is a homomorphism from $G$ to outer automorphism group of A. To any G-kernel, one can associate a cohomological obstruction to lifting it to a group action on A. In a recent work with Evington and subsequent work of Izumi, new cohomological invariants of G-kernels were introduced. These invariants place restrictions to the possible values of lifting obstructions on C*-algebras. In this talk, I will discuss how these obstructions can be interpreted as cohomology with coefficients in crossed modules. This is joint work with Masaki Izumi and Ulrich Pennig.

Sachin Valera

Title: Momentum-Space Braiding and Topological Order

Abstract: Nodal points are momentum-space defects in the Brillouin zone of 2d topological semimetals. Recent years have seen an interest in the possibility to braid such points. We aim here to take some steps towards extending this programme to make contact with TQFTs. This runs parallel to the work of Sati and Schreiber in arXiv 2206.13563.

Hao Zhang

Title: Conformal blocks and coends

Abstract: Conformal blocks are central objects in conformal field theory. In this talk, I will present the Sewing-Factorization (SF) theorem, a powerful result about the conformal blocks of C2-cofinite vertex operator algebras. Additionally, I will explore the connection between various coends and the SF theorem. This is based on an ongoing project with Bin Gui, initiated in arXiv:2305.10180, arXiv:2411.07707.