Hopf algebras and finite tensor categories in conformal field theory

Abstract

In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about representation categories of chiral symmetry algebras that are not semisimple. We focus on generalizations of the Verlinde formula, for which certain finite-dimensional complex Hopf algebras are used as a tool, and on the structural importance of the presence of a Hopf algebra internal to finite tensor categories.

arXiv comment46 pages, several figures. v2: missing text added after (4.5), references added, and a few minor changes. v3: typos corrected, bibliography updated

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Cited by in corpus (5)

• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• The Nichols algebra of screenings
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• Module categories over finite pointed tensor categories