Algebras and modules
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States and observables
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 -parameterized product operation in the vertex algebras as being the 3-point function of the 2d CFT with field insertions at the points 0, and . 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 -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 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
such that its component is a holomorphic function on the moduli space of conformal punctured spheres.
(under construction) A vertex algebra consists of the following data:
- nonnegatively graded complex vector space
- vacuum vector
- a shift operator
This data satisfy three axioms:
In practice, the most important case is when the shift is related to a conformal vector “the energy momentum tensor” whose Fourier components are related to the generators of the Virasoro Lie algebra . 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 is said to be rational if every -module is completely reducible. Then it follows that there are only finitely many nonisomorphic simple -modules.
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
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 -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).
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
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.
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
As factorization algebras
Discussion of vertex operator algebras as factorization algebras of observables is in section 6.3 and section 6.5 of
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.
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
Relation to sporadic groups
- John Duncan, Vertex operator algebras and sporadic groups, pdf; Moonshine for Rudvalis’s sporadic group I, pdf
There is an interesting theory of deformation quantization of VOAs from
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