Fusion of Defects.
Bartels, Arthur.
Fusion of Defects. - 1st ed. - 1 online resource (114 pages) - Memoirs of the American Mathematical Society Series ; v.258 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Acknowledgments -- Introduction -- OT1OT1cmrcmrmmnnnsca. Conformal nets -- OT1OT1cmrcmrmmnnnscb. Defects -- OT1OT1cmrcmrmmnnnscc. Sectors -- OT1OT1cmrcmrmmnnnscd. The vacuum sector of a defect -- OT1OT1cmrcmrmmnnnsce. Composition of defects -- OT1OT1cmrcmrmmnnnscf. Fusion of sectors and the interchange isomorphism -- OT1OT1cmrcmrmmnnnscg. The 1⊠1-isomorphism -- OT1OT1cmrcmrmmnnnsch. Construction of the 1⊠1-isomorphism -- Chapter 1. Defects -- 1.OT1OT1cmrcmrmmnnnsca. Bicolored intervals and circles -- 1.OT1OT1cmrcmrmmnnnscb. Definition of defects -- 1.OT1OT1cmrcmrmmnnnscc. Examples of defects -- 1.OT1OT1cmrcmrmmnnnscd. The category \CN₁ of defects -- 1.OT1OT1cmrcmrmmnnnsce. Composition of defects -- 1.OT1OT1cmrcmrmmnnnscf. Associativity of composition -- Chapter 2. Sectors -- 2.OT1OT1cmrcmrmmnnnsca. The category \CN₂ of sectors -- 2.OT1OT1cmrcmrmmnnnscb. Horizontal fusion -- 2.OT1OT1cmrcmrmmnnnscc. Vertical fusion -- Chapter 3. Properties of the composition of defects -- 3.OT1OT1cmrcmrmmnnnsca. Left and right units -- 3.OT1OT1cmrcmrmmnnnscb. Semisimplicity of the composite defect -- Chapter 4. A variant of horizontal fusion -- 4.OT1OT1cmrcmrmmnnnsca. The keyhole and keystone fusion -- 4.OT1OT1cmrcmrmmnnnscb. The keyhole fusion of vacuum sectors of defects -- 4.OT1OT1cmrcmrmmnnnscc. The keystone fusion of vacuum sectors of defects -- 4.OT1OT1cmrcmrmmnnnscd. Comparison between fusion and keystone fusion -- Chapter 5. Haag duality for composition of defects -- 5.OT1OT1cmrcmrmmnnnsca. The dimension of the Haag inclusion -- 5.OT1OT1cmrcmrmmnnnscb. The double bridge algebra is a factor -- 5.OT1OT1cmrcmrmmnnnscc. The dimension of the bridge inclusions -- Chapter 6. The 1⊠1-isomorphism -- 6.OT1OT1cmrcmrmmnnnsca. The 1⊠1-map is an isomorphism -- 6.OT1OT1cmrcmrmmnnnscb. The 1⊠1-isomorphism for an identity defect. 6.OT1OT1cmrcmrmmnnnscc. Unitors for horizontal fusion of sectors -- 6.OT1OT1cmrcmrmmnnnscd. The interchange isomorphism -- Appendix A. Components for the 3-category of conformal nets -- Appendix B. Von Neumann algebras -- B.I. The Haagerup ²-space -- B.II. Connes fusion -- B.III. Cyclic fusion -- B.IV. Fusion and fiber product of von Neumann algebras -- B.V. Compatibility with tensor products -- B.VI. Dualizability -- B.VII. Statistical dimension and minimal index -- B.VIII. Functors between module categories -- B.IX. The split property -- B.X. Two-sided fusion on ²-spaces -- Appendix C. Conformal nets -- C.I. Axioms for conformal nets -- C.II. The vacuum sector -- C.III. Gluing vacuum sectors -- C.IV. Finite-index conformal nets -- C.V. Sectors and the Hilbert space of the annulus -- C.VI. Extension of conformal nets to all 1-manifolds -- Appendix D. Diagram of dependencies -- Bibliography -- Back Cover.
Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. They 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. The authors' 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, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
9781470450656
Topological fields.
Generalized spaces.
Topology.
Electronic books.
QA611 .B378 2019
530.14/3
Fusion of Defects. - 1st ed. - 1 online resource (114 pages) - Memoirs of the American Mathematical Society Series ; v.258 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Acknowledgments -- Introduction -- OT1OT1cmrcmrmmnnnsca. Conformal nets -- OT1OT1cmrcmrmmnnnscb. Defects -- OT1OT1cmrcmrmmnnnscc. Sectors -- OT1OT1cmrcmrmmnnnscd. The vacuum sector of a defect -- OT1OT1cmrcmrmmnnnsce. Composition of defects -- OT1OT1cmrcmrmmnnnscf. Fusion of sectors and the interchange isomorphism -- OT1OT1cmrcmrmmnnnscg. The 1⊠1-isomorphism -- OT1OT1cmrcmrmmnnnsch. Construction of the 1⊠1-isomorphism -- Chapter 1. Defects -- 1.OT1OT1cmrcmrmmnnnsca. Bicolored intervals and circles -- 1.OT1OT1cmrcmrmmnnnscb. Definition of defects -- 1.OT1OT1cmrcmrmmnnnscc. Examples of defects -- 1.OT1OT1cmrcmrmmnnnscd. The category \CN₁ of defects -- 1.OT1OT1cmrcmrmmnnnsce. Composition of defects -- 1.OT1OT1cmrcmrmmnnnscf. Associativity of composition -- Chapter 2. Sectors -- 2.OT1OT1cmrcmrmmnnnsca. The category \CN₂ of sectors -- 2.OT1OT1cmrcmrmmnnnscb. Horizontal fusion -- 2.OT1OT1cmrcmrmmnnnscc. Vertical fusion -- Chapter 3. Properties of the composition of defects -- 3.OT1OT1cmrcmrmmnnnsca. Left and right units -- 3.OT1OT1cmrcmrmmnnnscb. Semisimplicity of the composite defect -- Chapter 4. A variant of horizontal fusion -- 4.OT1OT1cmrcmrmmnnnsca. The keyhole and keystone fusion -- 4.OT1OT1cmrcmrmmnnnscb. The keyhole fusion of vacuum sectors of defects -- 4.OT1OT1cmrcmrmmnnnscc. The keystone fusion of vacuum sectors of defects -- 4.OT1OT1cmrcmrmmnnnscd. Comparison between fusion and keystone fusion -- Chapter 5. Haag duality for composition of defects -- 5.OT1OT1cmrcmrmmnnnsca. The dimension of the Haag inclusion -- 5.OT1OT1cmrcmrmmnnnscb. The double bridge algebra is a factor -- 5.OT1OT1cmrcmrmmnnnscc. The dimension of the bridge inclusions -- Chapter 6. The 1⊠1-isomorphism -- 6.OT1OT1cmrcmrmmnnnsca. The 1⊠1-map is an isomorphism -- 6.OT1OT1cmrcmrmmnnnscb. The 1⊠1-isomorphism for an identity defect. 6.OT1OT1cmrcmrmmnnnscc. Unitors for horizontal fusion of sectors -- 6.OT1OT1cmrcmrmmnnnscd. The interchange isomorphism -- Appendix A. Components for the 3-category of conformal nets -- Appendix B. Von Neumann algebras -- B.I. The Haagerup ²-space -- B.II. Connes fusion -- B.III. Cyclic fusion -- B.IV. Fusion and fiber product of von Neumann algebras -- B.V. Compatibility with tensor products -- B.VI. Dualizability -- B.VII. Statistical dimension and minimal index -- B.VIII. Functors between module categories -- B.IX. The split property -- B.X. Two-sided fusion on ²-spaces -- Appendix C. Conformal nets -- C.I. Axioms for conformal nets -- C.II. The vacuum sector -- C.III. Gluing vacuum sectors -- C.IV. Finite-index conformal nets -- C.V. Sectors and the Hilbert space of the annulus -- C.VI. Extension of conformal nets to all 1-manifolds -- Appendix D. Diagram of dependencies -- Bibliography -- Back Cover.
Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. They 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. The authors' 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, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
9781470450656
Topological fields.
Generalized spaces.
Topology.
Electronic books.
QA611 .B378 2019
530.14/3