# Overview

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#### Quantum symmetries: Tensor categories, Topological quantum field theories, Vertex algebras

October 10 - November 4, 2022

This concentration month is dedicated to applications of category theory to various flavors of quantum field theory.

**For more information, including schedules, titles and abstracts, please click here**

The mathematics of conformal field theory

The Lie algebra of conformal transformations in two dimensions is infinite-dimensional, which has remarkable consequences for 2D conformal field theories (Belavin, Polyakov, Zamolodchikov). Such theories are, in many cases, almost entirely determined by how the quantum fields transform under conformal transformations, that is with respect to the action of Virasoro algebra. More generally, the algebraic structure underlying conformal field theory is known as a vertex operator algebra. Well known examples include those corresponding to the Virasoro algebra, affine Kac–Moody algebras and W-algebras.

A significant part of conformal field theory is then the representation theory of vertex operator algebras and how their structures interplay with physical applications. Initially, researchers working in this area mostly focused on the so-called rational conformal field theories which are built from a finite semisimple category of modules over the vertex operator algebra. However, there are many physical applications that do not satisfy the requirements of rationality. Since the mid-90s, classes of non-rational theories have been explored, including the logarithmic conformal field theories on which the hamiltonian is not diagonalizable. This research is still very active in the mathematics and mathematical physics communities and will feature prominently in this concentration period.

Quantum symmetries and tensor categories

The significance of tensor categories in mathematical physics emerged in the late 80s, with Moore and Seiberg’s work on conformal field theory and the Verlinde formula, and the early 90s with Reshetikhin and Turaev’s combinatorial constructions of invariants of links and 3-manifolds. In this sense, tensor categories encode the symmetries of both conformal field theories and topological field theories. So, as groups encode classical symmetries, tensor categories provide a unifying language to encode the symmetries that arise in many quantum contexts, including those mentioned above.

The study of tensor categories, especially those relating to the above applications (fusion categories and modular tensor categories), has since become a hugely important field in its own right with connections to the Hopf algebra (and quantum groups) and subfactor communities. The relation of these categories to link and knot invariants, Temperley-Lieb-Jones theories and braid group representations makes them the essential mathematical tool to properly describe such physical applications as topological quantum computation and topological phases of matter.

These classes of tensor categories are finite and semisimple, hence they are relevant in the study of rational conformal field theories and semisimple topological field theories. An important question that is currently being actively pursued is to identify more general classes of categories whose structures will advance our understanding of the logarithmic and non-semisimple versions of these physical theories. By bringing together international experts from both communities, this concentration period will catalyze collaborations that aim to solve this important question. This progress is also expected to lead to new results for the original applications, for example non-semisimple categories also yield new and interesting invariants of links and 3-manifolds.