nLab
vertex operator algebra

Idea

The usual definition of vertex operator algebra (VOA) is long and unenlightening. 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 punctures. 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 Liang 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.

Standard definition

(under construction) A vertex algebra consists of the following data:

• nonnegatively graded complex vector space $V={\oplus }_{n=0}^{\mathrm{\infty }}{V}_n$
• vacuum vector $\mid 0⟩\in V_0$
• a shift operator $T:V\to V$
• operation $Y=Y(-,z):V\to \mathrm{End}V[[z,z^{-1}]]$

This data satisfy three axioms:

(vacuum)

(translation axiom)

(locality)

In practice, the most important case is when the shift is $T$ is related to a conformal vector “the energy momentum tensor” $\omega$ whose Fourier components are related to the generators $L_i$ of the Virasoro Lie algebra $\mathrm{Vir}$. In that case we talk about conformal vertex algebra.

There is a notion of a module over a vertex algebra. A conformal vertex algebra $A$ is said to be rational if every $A$-module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple $A$-modules.

Properties

Category of vertex operator algebras

Vertex operator algebras naturally form a category (see section 2.4 of (FrenkenHuangLepowsky). This is naturally a monoidal category with respect to tensor product of VOAs (section 2.5).

This is equivalent to

• the category of algebras over the holomorphic punctured sphere operad (Huang);

• the category of vertex operator coalgebras (Hubbard).

Modular category of modules over a VOA

The category of modules/representations over a given vertex operator algebra is a modular tensor category, (Huang)

Relation to $A_{\mathrm{\infty }}$-algebras and RG fixed points

A functor from the category of BRST-VOAs to that of A-infinity algebras is described in

• Anton Zeitlin, Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory (arXiv)

and argued to algebraically encode the effective string theory background encoded by the CFT given by the VOA.

Examples

A class of examples are current algebras .

A database of examples is given by (GannonHöhn).

The Moonshine module over the Griess algebra? admits the structure of a vertex operator algebra, which has

• rank 24;

• is a self-dual object in the category of VOAs;

• has trivial degree-1 subspaces.

A conjecture by Frenkel, Lepowsky and Meurman says that the Moonshine VOA is, upt to isomoprhism the unique VOA with these properties.

References

General

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

• 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)

• Igor Frenkel, Yi-Zhi Huang, James Lepowsky, On Axiomatic approaches to Vertex Operator Algebras and Modules , Memoirs of the AMS Vol 104, No 494 (1993)

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

As algebras over the holomorphic sphere operad

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. an earlier one in

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

and even earlier treatments via coloured operads and generalized multicategories, for references (Snydal, Soibelman, Beilinson-Drinfeld) see relaxed multicategory.

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.

Relation to modular tensor categories

• Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories (arXiv:math/0412261)

• Terry Gannon, Gerald Höhn, Hiroshi Yamauchi, The online database of Vertex Operator Algebras and Modular Categories

As chiral algebras

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

• 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

Relation to 2d conformal field theory

• Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras Birkhäuser (1997)

Category of vertex operator algebras

• Keith Hubbard, The duality between vertex operator algebras and coalgebras, modules and comodules (pdfuality between vertex operator algebras and coalgebras, modules and comodules))

Relation to sporadic groups

• John Duncan, Vertex operator algebras and sporadic groups (pdf)

• John Duncan, Moonshine for Rudvalis’s sporadic group I (pdf)

Deformations

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

Further references

• what-is-the-motivation-for-a-vertex-algebra - a MathOverflow discussion on the motivation
• The online database of Vertex Operator Algebras and Modular Categories