Contents

Idea

The usual definition of vertex operator algebra (VOA) is long and unenlightning. But due to work by Huang and Kong it is now known that vertex operator algebras are equivalent to certain FQFTs (see also CFT):

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 Yi-Zhi Huang and Lian 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

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

Examples

• Moonshine

References

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

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

A standard textbook summarizing these results is

• Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras, Progr. in Math. Birkhauser 1997, gbooks

As mentioned in the acknowledgements there, Todd Trimble and Jim Stasheff had a hand in making the operadic picture manifest itself here. Other operadic approaches are known, e.g. the earlier one in

• Bojko Bakalov, Alessandro D’Andrea, Victor G. Kac, Theory of finite pseudoalgebras, Adv. Math. 162 (2001), no. 1, 1–140, MR2003c:17020

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), math.QA/0610293.

See also:

• Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics 134, Academic Press, New York 1998.

• Edward Frenkel, David Ben-Zvi: Vertex algebras and algebraic curves, Math. Surveys and Monographs 88, AMS 2001, xii+348 pp. (Bull. AMS. review, ZMATH entry)

• Victor Kac: Vertex algebras for beginners, Amer. Math. Soc. (ZMATH entry)

An algebrogeometric version is due Beilinson and Drinfel’d and called the chiral algebra.

There is an interesting theory of deformation quantization of VOAs from

• Pavel Etingof, David Kazhdan, Quantization of Lie bialgebras. V. Quantum vertex operator algebras, Selecta Math. (N.S.) 6 (2000), no. 1, 105–130, MR2002i:17022

• E. Frenkel, N. Reshetikhin, Towards deformed chiral algebras, q-alg/9706023

and of chiral algebras due Dmitry Tamarkin.

• Ruthi Hortsch, Igor Kriz, Ales Pultr, A universal approach to vertex algebras, arxiv/1006.0027

Much algebraic insight to algebaric structures in CFT is in unfinished notes

• A. Beilinson?, B. Feigin, B. Mazur, Notes on conformal field theory, http://www.math.sunysb.edu/~kirillov/manuscripts.html