Categorical tools in low-dimensional quantum field theory
This course will present various categorical tools that are, or may become, relevant for the study of low-dimensional quantum field theories, in particular for two-dimensional conformal field theories and for topological field theories, and thereby also for the understanding of exotic phases of matter. Among the applications are the description of non-invertible symmetries with the help of topological domain walls, variants of string-net models, and constructions of correlators in two-dimensional conformal field theories.
Lecturer
Date
20th August ~ 26th September, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 14:20 - 16:55 | A3-3-301 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
The course addresses audience from different backgrounds in mathematics and theoretical physics. Only basic familiarity with algebra and representation theory as well as some elementary topology is assumed. No background in physics is formally required, but for appreciating the applications, a previous exposure to concepts of quantum mechanics and quantum field thepry will be of avail.
Syllabus
1. Basics of categories and of bi- and double categories
2. Monoidal categories, rigidity and braiding
3. The graphical string calculus for monoidal categories and for bicategories
4. Grothendieck-Verdier duality
5. Frobenius algebra objects
6. Module categories and their realization as categories of modules over algebra objects
7. Modular functors and three-dimensional topological field theories
8. Pivotal and spherical structures for monoidal and module categories
9. Finite tensor and ribbon categories, modular categories
10. Nakayama functors, relative Serre functors and Eilenberg-Watts calculus
11. Domain walls and non-invertible symmetries
12. String-net models and their connection with state-sum modular functors
13. Constructions of correlators in two-dimensional conformal field theories
2. Monoidal categories, rigidity and braiding
3. The graphical string calculus for monoidal categories and for bicategories
4. Grothendieck-Verdier duality
5. Frobenius algebra objects
6. Module categories and their realization as categories of modules over algebra objects
7. Modular functors and three-dimensional topological field theories
8. Pivotal and spherical structures for monoidal and module categories
9. Finite tensor and ribbon categories, modular categories
10. Nakayama functors, relative Serre functors and Eilenberg-Watts calculus
11. Domain walls and non-invertible symmetries
12. String-net models and their connection with state-sum modular functors
13. Constructions of correlators in two-dimensional conformal field theories
Reference
Categories:
[1] Adamek, Herrlich, and Strecker: "Abstract and Concrete Categories"
[2] Riehl: "Category Theory in Context"
[3] Etingof, Gelaki, Nikshych, Ostrik: "Tensor Categories"
Modular functors and TFT:
[4] Bakalov, Kirillov: "Lectures on Tensor Categories and Modular Functors"
[5] Kock: "Frobenius Algebras and 2D Topological Quantum Field Theories"
[6] Turaev, Virelizier: "Monoidal Categories and Topological Field Theory"
Graphical string calculus:
[7] Selinger: arXiv:0908.3347
Grothendieck-Verdier duality:
[8] Boyarchenko, Drinfeld: arXiv:1108.6020
[9] F, Schaumann, Schweigert, Wood: arXiv:2405.20811
Eilenberg-Watts calculus:
[10] F, Schaumann, Schweigert: arXiv:1612.04561
String-net models and conformal field theory:
[11] Kirillov: arXiv:1106.6033
[12] F, Schweigert, Yang: arXiv:2112.12708
[13] F, Schweigert, Wood, Yang: arXiv:2305.02773
[1] Adamek, Herrlich, and Strecker: "Abstract and Concrete Categories"
[2] Riehl: "Category Theory in Context"
[3] Etingof, Gelaki, Nikshych, Ostrik: "Tensor Categories"
Modular functors and TFT:
[4] Bakalov, Kirillov: "Lectures on Tensor Categories and Modular Functors"
[5] Kock: "Frobenius Algebras and 2D Topological Quantum Field Theories"
[6] Turaev, Virelizier: "Monoidal Categories and Topological Field Theory"
Graphical string calculus:
[7] Selinger: arXiv:0908.3347
Grothendieck-Verdier duality:
[8] Boyarchenko, Drinfeld: arXiv:1108.6020
[9] F, Schaumann, Schweigert, Wood: arXiv:2405.20811
Eilenberg-Watts calculus:
[10] F, Schaumann, Schweigert: arXiv:1612.04561
String-net models and conformal field theory:
[11] Kirillov: arXiv:1106.6033
[12] F, Schweigert, Yang: arXiv:2112.12708
[13] F, Schweigert, Wood, Yang: arXiv:2305.02773
Audience
Graduate
, Postdoc
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
Jürgen Fuchs is a professor of theoretical physics at Karlstad University, Sweden. He has obtained his PhD in 1985 at Heidelberg University, Germany. Jürgen's research interests are low-dimensional quantum field theories and the mathematical structures needed for their investigation. For a CV see https://jfuchs.hotell.kau.se/gen/cv_5.html.