# nLab vertex operator algebra

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Vertex operator algebras (or “vertex algebras”, for short) are algebras with a product-operation parameterized by points in the complex plane.

Vertex operator algebras equipped with an action of the Virasoro algebra encode the local (genus-0 behaviour) of 2-dimensional conformal field theories. Here one may think of the complex plane as the Riemann sphere and of the $z$-parameterized product operation in the vertex algebras as being the 3-point function of the 2d CFT with field insertions at the points 0, $z$ and $\infty$. In fact one vertex operator algebra enodes (only) one chiral/holomorphic half of such a genus-0 2d CFT; a full genus-0 2d CFT is given by the combination of two vertex operator algebras called a full field algebra.

The traditional definition of vertex operator algebra (VOA) is long and tends to be somewhat unenlightening. But due to (Huang 91) it is now known that vertex operator algebras have equivalenlty an FQFT-type characterization which manifestly captures this relation to n-point functions in the 2d 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 $2Cob_{conf}^0$ 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

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

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 $V = \oplus_{n =0}^\infty V_n$
• vacuum vector $|0\rangle\in V_0$
• a shift operator $T: V\to V$
• operation $Y = Y(-,z) : V\to End V [ [z,z^{-1}] ]$

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 $Vir$. 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 $A_\infty$-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.

### Relation to conformal nets

Subject to some conditions, from a vertex operator algebra one may induce a conformal net and conversely (Capri-Kawahigahshi-Longo-Weiner 15).

## Examples

A class of examples are current algebras .

A database of examples is given by (Gannon-Höhn).

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, up to isomorphism the unique VOA with these properties.

See at monster vertex algebra.

## References

### General

• 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

### As factorization algebras

Discussion of vertex operator algebras as factorization algebras of observables is in section 6.3 and section 6.5 of

• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

### Relation to modular tensor categories

The representation categories of (rational) vertex operator algebras (modular tensor categories) are discussed in

### 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.

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

### Relation to 2d conformal field theory

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

Relation specifically to conformal nets is discussed in

### Category of vertex operator algebras

• Keith Hubbard, The duality between vertex operator algebras and coalgebras, modules and comodules, pdf