Categorical tools for topological phases
These lectures continue and extend those given in 2024 (Categorical tools in low-dimensional quantum field theory). They present further categorical tools that are, or may become, relevant for low-dimensional conformal and topological field theories and thereby for the study of exotic phases of matter.
Lecturer
Date
4th September ~ 30th October, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 13:30 - 16:05 | Shuangqing | ZOOM 09 | 230 432 7880 | BIMSA |
Syllabus
1 Categories
Reminder about categories, monoidal categories, rigid and braided categories, functors, adjointness, module categories etc, with emphasis on the graphical string calculus.
A few additional categorical notions.
2 Conformal field theory
Reminder about basics of chiral and full conformal field theory (CFT).
More about vertex operator algebras and their representation categories.
More about conformal blocks, modular functors and topological field theory (TFT).
3 The TFT construction of RCFT correlators
Detailed description of the construction of consistent systems of correlators of rational CFTs with the help of RT-type TFT, including explicit formulas for specific situations.
Treatment of unoriented world sheets and of topological defects.
Pertinent aspects of Frobenius algebras in monoidal categories.
4 Domain walls and symmetries
Domain walls and symmetries in rational CFT.
Domain walls in three-dimensional TFT.
5 Module categories versus categories of modules
Reminder about internal Hom functors.
The case of module categories over semisimple rigid monoidal categories.
Generalizations beyond semisimplicity.
The case of module categories over Grothendieck-Verdier categories.
6 Bicategories and double categories
Reminder and more details about bicategories.
Double categories.
Pivotal module categories and spherical Morita contexts.
7 String nets
Reminder and more details about string nets and their connection with state-sum modular functors.
Graphical string calculus for pivotal categories and bicategories.
String-net models based on pivotal bicategories.
8 The string-net construction of RCFT correlators
Reminder and more details about the construction of RCFT correlators with the help of categorical and bicategorical string nets.
A double categorical picture of the construction.
A relation with monoidal vertical transformations.
9 Outlook
Miscellaneous topics to be determined during the course
Reminder about categories, monoidal categories, rigid and braided categories, functors, adjointness, module categories etc, with emphasis on the graphical string calculus.
A few additional categorical notions.
2 Conformal field theory
Reminder about basics of chiral and full conformal field theory (CFT).
More about vertex operator algebras and their representation categories.
More about conformal blocks, modular functors and topological field theory (TFT).
3 The TFT construction of RCFT correlators
Detailed description of the construction of consistent systems of correlators of rational CFTs with the help of RT-type TFT, including explicit formulas for specific situations.
Treatment of unoriented world sheets and of topological defects.
Pertinent aspects of Frobenius algebras in monoidal categories.
4 Domain walls and symmetries
Domain walls and symmetries in rational CFT.
Domain walls in three-dimensional TFT.
5 Module categories versus categories of modules
Reminder about internal Hom functors.
The case of module categories over semisimple rigid monoidal categories.
Generalizations beyond semisimplicity.
The case of module categories over Grothendieck-Verdier categories.
6 Bicategories and double categories
Reminder and more details about bicategories.
Double categories.
Pivotal module categories and spherical Morita contexts.
7 String nets
Reminder and more details about string nets and their connection with state-sum modular functors.
Graphical string calculus for pivotal categories and bicategories.
String-net models based on pivotal bicategories.
8 The string-net construction of RCFT correlators
Reminder and more details about the construction of RCFT correlators with the help of categorical and bicategorical string nets.
A double categorical picture of the construction.
A relation with monoidal vertical transformations.
9 Outlook
Miscellaneous topics to be determined during the course
Video Public
Yes
Notes Public
Yes
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.