2024 Fall, Topics in Operator Algebras: Algebraic Conformal Field Theory

• Course language: Chinese

• Time: Friday 9:50-12:15

• Location: 六教6B312

• Course Videos

Course description

In this course, we study 2d conformal field theory (CFT) in the framework of algebraic quantum field theory (AQFT). In AQFT, a quantum field theory is formulated as a family of operator algebras acting on a fixed Hilbert space H satisfying certain axioms. In this setting, we will construct rigorous models of 2d chiral CFT. Our main goal is to give rigorous proofs for the PCT symmetry of these models, the Bisognano-Wichmann theorem (which relates the dilation symmetry and the PCT symmetry to the Tomita-Takesaki theory of von Neumann algebras), and the Haag duality. Although I initially intended to cover the construction of braided tensor categories from these models and to explain their connection to Jones’ subfactor theory—which fundamentally relies on Haag duality—I was unable to include these topics due to time constraints.

We assume that the readers are familiar with the basic Hilbert space techniques such as the spectral theorem of bounded self-adjoint operators. Some familiarty with the basic notion of unbounded closed operators is also helpful. Important theorems about unbounded operators (spectral theorem, polar decomposition, Borel functional calculus, Stone’s theorem) and probably their proofs will be reviewed in class. The reference for this part is the following notes, which do not require any prior knowledge of operator algebras.

Notes ST: Spectral Theory for Strongly Commuting Normal Closed Operators

Lecture notes

Notes ACFT: Topics in Operator Algebras: Algebraic Conformal Field Theory

Schedule

• 9/13 Introduction to the CPT symmetry in QFT, and its relationship to the Tomita-Takesaki theory in operator algebras. (Notes ACFT, Sec. 1)

• 9/20 Spectral theorem and Borel functional calculus for adjointly commuting bounded normal operators. Basics of unbounded operators. Spectral theorem for unbounded positive operators. (Notes ST, Sec. 2-4)

• 9/27 Uniqueness of positive square roots. Polar decomposition. Strong commutativity. (Notes ST, Sec. 4 (last lemma)-Sec. 6)

• 10/11 Strong commutativity, spectral theorem for strongly commuting normal closed operators. (Notes ST, Sec. 6, 7)

• 10/18 Bounding projections, unbounded Borel functional calculus. (Notes ST, Sec. 8, 9)

• 10/25 2d Conformal field theory. Introduction to the mathematical theory of chiral fields. (Notes ACFT, Sec. 2, Subsec. 3.1-3.4)

• 11/1 Local fields. (Notes ACFT, Subsec. 3.5-3.13)

• 11/8 Chiral algebras and examples, energy-bounded fields, smeared fields. (Notes ACFT, Subsec. 3.14-3.15, 4.1-4.5)

• 11/15 Criteria for energy bounds, product of smeared fields. (Notes ACFT, Subsec. 4.6-4.12) Stone’s theorem. (Notes ST, Sec. 10)

• 11/22 Stone’s theorem. (Notes ST, Sec. 10) Domains of positive operators and analytic continuation, criteria for cores and self-adjointness. (Notes ACFT, Subsec. 5.1-5.7)

• 11/29 Criteria for strong commutativity, flows of Möbius transforms. (Notes ACFT, Subsec. 5.8-5.11, 6.1-6.2)

• 12/6 Möbius covariance of smeared fields and pointed fields, the Reeh-Schlieder theorem for smeared fields and pointed fields. (Notes ACFT, Subsec. 6.3-6.8)

• 12/13 The PT and PCT theorems. (Notes ACFT, Subsec. 7.1-7.9)

• 12/20 The PCT theorem, Tomita-Takesaki theory for unbounded operators. (Notes ACFT, Subsec. 7.10-7.11, 8.1-8.9)

• 12/27 Uniqueness of the modular operators, Bisognano-Wichmann theorem, Haag duality, and their applications. (Notes ACFT, Subsec. 8.10-8.11, Sec. 9)

References

The most relevant references are:

• Buchholz, Schulz-Mirbach (1990). Haag duality in conformal quantum field theory

• Carpi-Kawahigashi-Longo-Weiner (2018). From vertex operator algebras to conformal nets and back

Other references that might be helpful:

Books and monographs

• Haag. Local Quantum Physics

• Halvorson, Müger. Algebraic Quantum Field Theory

• Gui. Lectures on Vertex Operator Algebras and Conformal Blocks

Articles

• Kawahigashi (2015). Conformal field theory, tensor categories and operator algebras

• Rehren (2015). Algebraic conformal quantum field theory in perspective