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Fusion of defects
About this Title
Arthur Bartels, Westfälische Wilhelms-Universität Münster, Christopher L. Douglas, University of Oxford and André Henriques, Universiteit Utrecht
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 258, Number 1237
ISBNs: 978-1-4704-3523-3 (print); 978-1-4704-5065-6 (online)
DOI: https://doi.org/10.1090/memo/1237
Published electronically: February 20, 2019
Keywords: Conformal net,
defect,
sector,
fusion,
conformal field theory,
soliton,
vacuum,
conformal embedding,
Q-system,
von Neumann algebra,
Connes fusion
MSC: Primary 81T05, 46L37, 46M05, (Primary), 81T40, 46L60, 81R10, (; Secondary )
Table of Contents
Chapters
- Acknowledgments
- Introduction
- 1. Defects
- 2. Sectors
- 3. Properties of the composition of defects
- 4. A variant of horizontal fusion
- 5. Haag duality for composition of defects
- 6. The $1 \boxtimes 1$-isomorphism
- A. Components for the 3-category of conformal nets
- B. Von Neumann algebras
- C. Conformal nets
- D. Diagram of dependencies
Abstract
Conformal nets provide a mathematical model for conformal field theory. We define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. We introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. Our most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, we construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.- Arthur Bartels, Christopher L. Douglas, and André Henriques, Dualizability and index of subfactors, Quantum Topol. 5 (2014), no. 3, 289–345. MR 3342166, DOI 10.4171/QT/53
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