Continuous Tensor Categories and Direct Integrals

Abstract: Finitely semisimple tensor categories are ubiquitous in quantum algebra: they appear in the representation theory of Hopf algebras, quantum groups, TQFTs, CFTs, and others. However, one usually needs extra adjectives to ensure that the categories one comes across satisfy the necessary finite properties of finitely semisimple tensor categories. Without enough adjectives, one sometimes encounters tensor categories which are still “semisimple”, but have continuously many simple objects, and a generic object is a direct integral of such simple objects, as opposed to a direct sum. In this talk, we will introduce a new model to treat these type of categories: continuous tensor categories; and provide some basic examples. Time permitting, we will explore how continuous tensor categories allow us to compute certain categories appearing in the study of non-rational 2d conformal field theories such as the non-compact boson and related theories.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home