This is a 20-lecture course, with each lecture being about 45 minutes or so, given by Thomas Creutzig. It gives an introduction to vertex operator algebras from the point of view of quantum mechanics.
Vertex operator algebras (VOAs) first appeared in the 1980s as the rigorous notion of chiral algebras (the symmetry algebras) of two-dimensional conformal quantum field theories. Since then, they have been employed as key ingredients in many modern problems of mathematical physics and pure mathematics, ranging from monstrous moonshine to knot theory and geometry. The older problems have been mostly concerned with the simplest type of VOAs, so‐called rational theories.
In the last few years, it has been realized that VOAs and their representation theories yield rich invariants of three and four‐dimensional supersymmetric quantum field theories. This provides new insights into low‐dimensional topology and the quantum geometric Langlands programme. Involved VOAs are however not rational (often called logarithmic) and so their representation theory is rich and exciting.
These lectures will be a very modern introduction to the theory of VOAs. We will use techniques from representation theory (especially Lie theory), geometry and topology; no knowledge of VOAs is needed. The lectures will be a mix of general theory and illustrating it with the most important examples, that is free field theories, affine and W‐algebras; and the school will end with an exposition of the very recent use and appearance of VOAs in physics, geometry, and low‐dimensional topology.
- Formal calculus and definition of VOA, modules, fields and intertwiners 1
- Formal calculus and definition of VOA, modules, fields and intertwiners 2
- Formal calculus and definition of VOA, modules, fields and intertwiners 3
- Illustrating these in the simplest examples of free field theories ‐ free boson, free fermion, symplectic boson and symplectic fermion 1
- Illustrating these in the simplest examples of free field theories ‐ free boson, free fermion, symplectic boson and symplectic fermion 2
- Finiteness conditions (Zhu’s algebra and associated varieties) and their implications to representation theory 1
- Finiteness conditions (Zhu’s algebra and associated varieties) and their implications to representation theory 2
- Affine Lie algebras and WZW theories with a focus on admissible levels 1
- Affine Lie algebras and WZW theories with a focus on admissible levels 2
- Affine Lie algebras and WZW theories with a focus on admissible levels 3
- Affine Lie algebras and WZW theories with a focus on admissible levels 4
- Affine Lie algebras and WZW theories with a focus on admissible levels 5
- W‐algebras 1
- W‐algebras 2
- Cosets and Orbifolds 1
- Cosets and Orbifolds 2
- VOAs in physics, geometry and topology 1
- VOAs in physics, geometry and topology 2
- VOAs in physics, geometry and topology 3
- VOAs in physics, geometry and topology 4
These videos are of a lecture course by Thomas Creutzig at the Centre for Quantum Mechanics at the University of Southern Denmark in 2022.
