# Algebraic and Topological Quantum Field Theory

## Math 9600-02 - Fall 2015 - Vanderbilt University

Lecture: TR 2:35pm-3:50pm

Course Description

### Contact Information

Instructor: Marcel Bischoff

Office: SC 1418

Email: marcel.bischoff @ ...

### Files

### Lectures

Lecture | Day | Topic | Notes |
---|---|---|---|

1 | Thu 08/27 | Quantum theory | |

2 | Tue 09/01 | Weyl algebra and representations | |

3 | Tue 09/03 | Relativity, Minkowski space, Poincaré group | |

4 | Tue 09/08 | Minkowski space, Poincaré group, Wigner's theorem, One-particle space | |

5 | Thu 09/10 | Positive enegery representations of the Poincaré group, Wigner's classification | |

6 | Tue 09/15 | Massive scalar free field, Standard subspaces | The material on standard subspaces were mostly taken from Longo's lecture notes Section 2.1 |

7 | Thu 09/17 | KMS (Kubo-Martin-Schwinger) condition I, Bisognano-Wichman property (exercise), Reeh-Schlieder property | The proof of the KMS condition in the first formulation is Prop 2.1.7 in [Lo1]. The alternative formulation is given and proven in 11. Prop 2 in [Wa]. |

8 | Tue 09/22 | Reeh-Schlieder property, Modular Localization, Borchers' theorem | The Reeh-Schlieder property proof is similar to the one for nets here |

9 | Thu 09/24 | Standard pairs, Longo--Witten unitaries, Second Quantization (CCR) I | |

10 | Tue 09/29 | Second quantization (CCR) II, type III_{1} property, CAR I
| The type III_{1} property is Thm. 3 in [Lo79] |

11 | Thu 10/01 | CAR II, PER of Möbius group, free (complex) Fermion net on the circle | The proof of the modular properties for CAR can be found in [Fo83], [Wa], the proofs are similar to CCR. |

12 | Tue 10/06 | Handout: Fermi Nets. The Boson--Fermion correspondence. | The Fermion-Bose correspondence is well-known, but the statement for nets was not really contained in the literature. So we had to give a proof in Section 3 [BiTa2013] |

13 | Thu 10/08 | Dirac Sea model. The Boson--Fermion correspondence II. | |

14 | Tue 10/13 | The Boson--Fermion correspondence, Exponentiating fields. | |

15 | Tue 10/20 | Localized Endomorphisms I. | |

16 | Thu 10/22 | Localized Endomorphisms II. | |

17 | Tue 10/27 | Braiding | |

18 | Thu 10/29 | Braiding 2, Dimensions | |

19 | Tue 11/03 | Dimensions 2 | |

20 | Thu 11/05 | Statistics, Permuation Symmetry | |

21 | Tue 11/10 | Reconstruction of the field net | |

22 | Thu 11/12 | Representation of free boson | |

23 | Tue 11/17 | Extension of free bosons by even lattices | |

24 | Thu 11/19 | Non-degenerate Sectors, Modular Tensor Categories | |

25 | Tue 12/01 | Extension of nets by Q-systems | The standard reference is Longo, Rehren: Nets of Subfactors. |

### Exercises

### References

Other Lecture Notes:- H. Halvorson, M. Müger:
*Algebraic quantum field theory*, In J. Butterfield & J. Earman (eds.), Handbook of the philosophy of physics. Kluwer (2006). Available at: arXiv, Princeton. - W. Dybalski:
*Algebraic Quantum Field Theory*, Unpublished Lecture Notes 2012. Available at: TUM - [Lo1,2] R. Longo:
*Lecture Notes on Conformal Field Theory. Part 1 & 2*, Part 1 published as Real Hilbert subspaces, modular theory, SL(2,R) and CFT, Von Neumann algebras in Sibiu, 2008, pp. 33–91. Available at: Uniroma2 - Y. Kawahigashi:
*Conformal field theory, tensor categories and operator algebras*, J. Phys. A 48 (2015), 303001. Available at: arXiv, IOP (published version) - [Wa] A. Wassermann:
*Operator algebras and conformal field theory III. Fusion of positive energy representations of LSU(N) using bounded operators*. Available here. - [Lo79] R. Longo,
*Notes on Algebraic Invariants for Non-commutative Dynamical Systems*, Comm. Math. Phys. Volume 69, Number 3 (1979), 195-207. Available here. - [Fo83] J. Foit,
*Abstract Twisted Duality for Quantum Free Fermi Fields*, Publ. RIMS, Kyoto Univ. 19 (1983), 729-741. Available here

### Course Description

The goal of this course is to give an introduction to Algebraic Quantum Field Theory and Topological Quantum Field Theory.

Quantum field theory (QFT) was invented to describe particle physics at high energies, but also gives rise to interesting mathematics. The plan of the lecture is to start with a mild introduction to quantum theory and symmetries. I will discuss the axiomatization of quantum field theory due to Haag--Kastler. In this approach we will study free field examples and relation to the usual definition of quantum fields as operator valued distributions and discuss the theory of superselection sectors, which gives rise to tensor categories.

In the second part, I will give an introduction to topological quantum field theory (TQFT), which is of a bit different flavor. TQFTs are used in mathematics to calculate topological invariants of spaces, while in physics it describes low energy effective theories in condensed matter physics. We will focus on how TQFTs are obtained from tensor categories.

Finally, we will be studying conformal field theory (CFT) in low dimensions from an algebraic point of view. Here, in contrast to higher dimensions, many interesting and non-trivial models can be rigorously constructed. We will encounter subfactors and see that so-called rational CFTs give rise to TQFTs and topological invariants, like the Jones polynomial. If time permits we will talk about classification results in rational CFT.

Last update: 2016-10-04 13:57:06.