Volume 18, issue 2 (2018)

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Conformal nets IV: The $3$-category

Arthur Bartels, Christopher L Douglas and André Henriques

Algebraic & Geometric Topology 18 (2018) 897–956

Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. We previously introduced a notion of composition, called fusion, between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal 3-category. This 3-category encodes the algebraic structure of the possible interactions among conformal field theories.

conformal net, defect, sector, fusion, conformal field theory, topological field theory, von Neumann algebra, Connes fusion, $3$-category
Mathematical Subject Classification 2010
Primary: 18D05, 81T05
Received: 4 January 2017
Revised: 28 June 2017
Accepted: 10 July 2017
Published: 12 March 2018
Arthur Bartels
Mathematisches Institut
Westfälische Wilhelms-Universität Münster
Christopher L Douglas
Mathematical Institute
University of Oxford
Oxford, United Kingdom
André Henriques
Mathematisch Instituut
Universiteit Utrecht
Utrecht, Netherlands