Lecture #1: Vertex Operator Algebras (1/20); Formal calculus and definition of VOA...

Описание: QM Masterclass by Thomas Creutzig (University of Alberta) held at University of Southern Denmark. Organized and funded by Centre for Quantum Mathematics.

Vertex operator algebras (VOAs) first appeared in the 1980's 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 program. 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.

Lecture content:
Lecture 1-3: Formal calculus and definition of VOA, modules, fields and intertwiners
Lecture 4-5: Illustrating these in the simplest examples of free field theories ‐ free boson, free fermion, symplectic boson and symplectic fermion
Lecture 6-7: Finiteness conditions (Zhu's algebra and associated varieties) and their implications to representation theory
Lecture 8-12: Affine Lie algebras and WZW theories with a focus on admissible levels
Lecture 13-14: W‐algebras
Lecture 15-16: Cosets and Orbifolds
Lecture 17-20: VOAs in physics, geometry and topology

Read more: https://www.sdu.dk/en/qm-masterclass-voa

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