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.

Introduction

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.

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

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