From vertex operator algebras to modular tensor categories

Around 2d CFTs

Conformal field theories and topological quantum field theories in physics led in mathematics to various axiomatizations, which have many interrelations. In this talk (October 6) and the next one (October 27), we aim at explaining some of them, from the algebraic (neither physical, nor field-theoretic) perspective.

October 6 - From vertex operator algebras to modular tensor categories

I am a complete beginner in this topic, but will try to describe:
- what is a vertex operator algebra (VOA) ;
- what is a modular tensor category (MTC) ;
- and how does the category of representations of a VOA gives rise to a MTC
We shall put in logical place in this picture the notion of characters, of S-matrix, of fusion rule, and meet the Verlinde formula in its simplest form. "Modular" here means that we will meet representations of SL(2,Z), and if time allows, of mapping class groups of surfaces of any genus.