Fusion of defects / Arthur Bartels, Christopher L. Douglas, Andre Henriques.

Format:

Book

Author/Creator:

Bartels, Arthur, 1971- author.

Douglas, Christopher L., author.

Henriques, André G. (André Gil), 1977- author.

Series:

Memoirs of the American Mathematical Society ; Number 1237.

Memoirs of the American Mathematical Society ; Number 1237

Language:

English

Subjects (All):

Topological fields.

Generalized spaces.

Topology.

Physical Description:

1 online resource (114 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2019]

Summary:

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.

Contents:

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.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-5065-8