Yasuyuki Kawahigashi

The theory of subfactors plays an important role in the discovery and analysis of quantum symmetries that seem to be ubiquitous in mathematics and physics. Subfactors profoundly interact with a wide range of different areas such as quantum topology, vertex operator algebras, quantum groups, free probability theory, quantum computing and quantum information theory, quantum field theory, conformal field theory, tensor categories, condensed matter physics, and, of course, operator algebras.The aim of this workshop was to bring together an international group of researchers from these fields to disseminate recent results and to stimulate new collaborations. The focus was on operator algebraic, vertex operator algebraic and categorical aspects of quantum symmetries and their applications to open questions in mathematical physics. A substantial group of young mathematicians attended the workshop and were given the opportunity to present their work.

We review the physical meaning of modular invariance for unitary conformal quantum field theories in d = 2 . For quantum field theory models, while T invariance is necessary for locality, S invariance is not mandatory. S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational conformal field theories (CFTs), the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual information. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index. Published by the American Physical Society 2024

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.

This paper surveys the long-standing connections and impact between Vaughan Jones’s theory of subfactors and various topics in mathematical physics, namely statistical mechanics, quantum field theory, quantum information, and two-dimensional conformal field theory.

This paper surveys the long-standing connections and impact between Vaughan Jones's theory of subfactors and various topics in mathematical physics, namely statistical mechanics,quantum field theory,quantum information and two-dimensional conformal field theory.

A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck–Mariën–Williamson–Şahinoğlu-Haegeman–Verstraete on two-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of lengthkis naturally identified with thekth higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between two-dimensional topological order and subfactor theory.

We give a characterization of a finite-dimensional commuting square of C*-algebras with a normalized trace that produces a hyperfinite type II_1 subfactor of finite index and finite depth in terms of Morita equivalent unitary fusion categories. This type of commuting squares were studied by N. Sato, and we show that a slight generalization of his construction covers the fully general case of such commuting squares. We also give a characterization of such a commuting square that produces a given hyperfinite type II_1 subfactor of finite index and finite depth. These results also give a characterization of certain 4-tensors that appear in recent studies of matrix product operators in 2-dimensional topological order.