nLab vertex operator algebra

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 encodes (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 equivalently 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 $T$ 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 a 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)

Goddard-Thorn theorem

• Goddard-Thorn theorem

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 (Carpi-Kawahigahshi-Longo-Weiner 15, Carpi 16) also (Gui 18).

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 due to

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

says that the Moonshine VOA is, up to isomorphism the unique VOA with these properties. See at monster vertex algebra.

Variants

• Poisson vertex algebra

• rational vertex operator algebra

• super vertex operator algebra

• sheaf of vertex operator algebras, vertex operator algebroid

Related concepts

• operator product expansion, conformal bootstrap

• affine Lie algebra

• conformal field theory,

• factorization algebra,

• automorphism of a vertex operator algebra

References

General

• Victor Kac, Vertex algebras for beginners, Amer. Math. Soc., Univ. Lecture Series 10 (1998) [ams:ulect-10-r, ZMATH entry]

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

Review:

• Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, pp. 4 of: Algebraic structures in two-dimensional conformal field theory, Encyclopedia of Mathematical Physics [arXiv:2305.02773]

• Chenjing Bu: Vertex algebras done right, blog post (2024)

See also:

• what-is-the-motivation-for-a-vertex-algebra - a MathOverflow discussion on the motivation

• The online database of Vertex Operator Algebras and Modular Categories

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

• Bong H. Lian, Andrew R. Linshaw: Vertex algebras and commutative algebras [arXiv:2107.03243]

Relation to sporadic groups:

• John Duncan, Vertex operator algebras and sporadic groups, pdf; Moonshine for Rudvalis’s sporadic group I, pdf

Vertex operator algebras over non-archimedean fields:

• Victor Kac, Non-Archimedean vertex algebras [arXiv:2304.09651]

Classification of strongly rational holomorphic vertex operator algebras of central charge $\leq 24$ (of relevance in heterotic string theory and monstrous moonshine):

• Adrian Norbert Schellekens, Meromorphic $c = 24$ Conformal Field Theories, Commun. Math. Phys. 153 (1993) 159-186 [doi:10.1007/BF02099044]

• Chongying Dong, Geoffrey Mason, Holomorphic vertex operator algebras of small central charges, Pacific Journal of Mathematics 213 2 (2004) 253–266 [arXiv:math/0203005, doi:10.2140/pjm.2004.213.253]

• Philip Boyle Smith, Ying-Hsuan Lin, Yuji Tachikawa, Yunqin Zheng, Classification of chiral fermionic CFTs of central charge $\leq 16$ [arXiv:2303.16917]

• Brandon C. Rayhaun, Bosonic rational conformal field theories in small genera, chiral fermionization, and symmetry/subalgebra duality [arXiv:2303.16921]

• Gerald Höhn, Sven Möller, Classification of self-dual vertex operator superalgebras of central charge at most 24 [arXiv:2303.17190]

On equivalence of VOAs:

• Sven Möller, Brandon C. Rayhaun: Equivalence Relations on Vertex Operator Algebras, I: Genus [arXiv:2408.07117]

On orbifolds of VOAs

• Terry Gannon, Andrew Riesen: Orbifolds of Pointed Vertex Operator Algebras I [arXiv:2410.00809]

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 (doi:10.1073/pnas.88.22.9964)

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.

See also:

• Yi-Zhi Huang, Meromorphic open-string vertex algebras, J. Math. Phys. 54, 051702 (2013) [doi:10.1063/1.4806686]

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.

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)

• Emily Cliff, Universal factorization spaces and algebras, arxiv:1608.08122

We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an 'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.

As coalgebras over a comonad

On vertex operator algebras as coalgebras over a comonad:

• Jethro van Ekeren, A quantum field comonad [arXiv:2305.18223]

Relation to modular tensor categories

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

• Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Nat. Acad. Sci. 102 (2005) 5352-5356 $[$arXiv:math/0412261, doi:10.1073/pnas.0409901102$]$

• 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

• 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

Relation to 2d CFT:

• Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras Birkhäuser (1997) (doi:10.1007/978-1-4612-4276-5)

For orbifolds:

• Yi-Zhi Huang, Representation theory of vertex operator algebras and orbifold conformal field theory (arXiv:2004.01172)

Relation specifically to conformal nets:

• Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, Mihály Weiner, From vertex operator algebras to conformal nets and back, Memoirs of the American Mathematical Society 254 1213 (2018) [doi:10.1090/memo/1213, arXiv:1503.01260]

• Sebastiano Carpi, Operator algebras and vertex operator algebras, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) [arXiv:1603.06742, doi:10.1142/9789813226609_0508]

and for super vertex operator algebras:

• Sebastiano Carpi, Tiziano Gaudio, Robin Hillier, From vertex operator superalgebras to graded-local conformal nets and back [arXiv:2304.14263]

See also

• James E. Tener, Representation theory in chiral conformal field theory: from fields to observables (arXiv:1810.08168)

Relation of the corresponding ribbon categories:

• Bin Gui, Categorical extensions of conformal nets (arXiv:1812.04470)

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

Deformation quantization of chiral algebras are studied by

• Dmitry Tamarkin, Deformations of chiral algebras, Proceedings of the ICM, Beijing 2002, vol. 2, 105–118

A class of “free” vertex algebras are also quantized using Batalin-Vilkovisky formalism, with results important for understanding mirror symmetry, in

• Si Li, Vertex algebras and quantum master equation (arxiv/1612.01292)