nLab
vertex operator algebra

Idea

The usual definition of vertex operator algebra is long and unenlightning. But due to work by Huang and Kong it is known now that vertex operator algebras are nothing but certain FQFTs:

There is a monoidal category or operad whose morphisms are conformal spheres with $n$-punctures marked as incoming and one puncture marked as outgoing (each puncture equipped with a conformally parametrized annular neighborhood). Composition of such spheres is by gluing along puctures. This can be regarded as a category $2{\mathrm{Cob}}_{\mathrm{conf}}^0$ of 2-dimensional genus-0 conformal cobordisms.

As shown by theorems by Huang and Kong, a vertex operator algebra is precisely a holomorphic representation? of this category, or algebra over an operad for this operad i.e. an genus-0 conformal FQFT, hence a monoidal functor

$$V:2{\mathrm{Cob}}_{\mathrm{conf}}^0\to \mathrm{Vect}$$V : 2Cob_{conf}^0 \to Vect

such that its component $V_1$ is a holomorphic function on the moduli space of conformal punctured spheres.

References

The original article with the interpretation of vertex operator algebras as holomorphic algebras over the holomorphic punctured sphere operad is

• Yi-Zhi Huan, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA, Vol 88. (1991) pp. 9964-9968

A standard textbook summarizing these results is

• Huang, Two-dimensional conformal geometry and vertex operator algebras

As mentioned in the acknowledgements there, Todd Trimble and Jim Stasheff had a hand in making the operadic picture manifest itself here.

More recently Huang’s student Liang Kong has been developing these results further, generalizing them to open-closed CFTs and to non-chiral, i.e. completely general CFTs. See

• Liang Kong, Open-closed field algebras Commun. Math. Physics. 280, 207-261 (2008) (arXiv).