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24 January 2021
 
  » arxiv » math.CT/0411507

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Picard groups in rational conformal field theory
Jürg Fröhlich ; Jürgen Fuchs ; Ingo Runkel ; Christoph Schweigert ;
Date 23 Nov 2004
Subject Category Theory; Mathematical Physics; Quantum Algebra MSC-class: 81T40,18D10,18D35,81T45 | math.CT hep-th math-ph math.MP math.QA
AbstractAlgebra and representation theory in modular tensor categories can be combined with tools from topological field theory to obtain a deeper understanding of rational conformal field theories in two dimensions: It allows us to establish the existence of sets of consistent correlation functions, to demonstrate some of their properties in a model-independent manner, and to derive explicit expressions for OPE coefficients and coefficients of partition functions in terms of invariants of links in three-manifolds. We show that a Morita class of (symmetric special) Frobenius algebras $A$ in a modular tensor category $calc$ encodes all data needed to describe the correlators. A Morita-invariant formulation is provided by module categories over $calc$. Together with a bimodule-valued fiber functor, the system (tensor category + module category) can be described by a weak Hopf algebra. The Picard group of the category $calc$ can be used to construct examples of symmetric special Frobenius algebras. The Picard group of the category of $A$-bimodules describes the internal symmetries of the theory and allows one to identify generalized Kramers-Wannier dualities.
Source arXiv, math.CT/0411507
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