Algebraic and Topological Quantum Field Theory

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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 @ ...

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Lectures

Lecture Day Topic Notes
1Thu 08/27Quantum theory
2Tue 09/01Weyl algebra and representations
3Tue 09/03Relativity, Minkowski space, Poincaré group
4Tue 09/08Minkowski space, Poincaré group, Wigner's theorem, One-particle space
5Thu 09/10Positive enegery representations of the Poincaré group, Wigner's classification
6Tue 09/15Massive scalar free field, Standard subspacesThe material on standard subspaces were mostly taken from Longo's lecture notes Section 2.1
7Thu 09/17KMS (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].
8Tue 09/22Reeh-Schlieder property, Modular Localization, Borchers' theoremThe Reeh-Schlieder property proof is similar to the one for nets here
9Thu 09/24Standard pairs, Longo--Witten unitaries, Second Quantization (CCR) I
10Tue 09/29Second quantization (CCR) II, type III1 property, CAR I The type III1 property is Thm. 3 in [Lo79]
11Thu 10/01CAR II, PER of Möbius group, free (complex) Fermion net on the circleThe proof of the modular properties for CAR can be found in [Fo83], [Wa], the proofs are similar to CCR.
12Tue 10/06Handout: 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]
13Thu 10/08Dirac Sea model. The Boson--Fermion correspondence II.
14Tue 10/13The Boson--Fermion correspondence, Exponentiating fields.
15Tue 10/20Localized Endomorphisms I.
16Thu 10/22Localized Endomorphisms II.
17Tue 10/27Braiding
18Thu 10/29Braiding 2, Dimensions
19Tue 11/03Dimensions 2
20Thu 11/05Statistics, Permuation Symmetry
21Tue 11/10Reconstruction of the field net
22Thu 11/12Representation of free boson
23Tue 11/17Extension of free bosons by even lattices
24Thu 11/19Non-degenerate Sectors, Modular Tensor Categories
25Tue 12/01Extension of nets by Q-systemsThe standard reference is Longo, Rehren: Nets of Subfactors.

Exercises

References

Other Lecture Notes:

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.

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