nLab automorphism of a vertex operator algebra

Idea

Given a vertex operator algebra $\mathcal{V}$ (VOA) or super vertex operator algebra (sVOA), or more generally a full 2d CFT (of which the VOA is the local and chiral data) or 2d SCFT (hence a “2-spectral triple”) one may ask (as for any object in any category) for its automorphisms, hence the homomorphisms

$$\mathcal{V} \overset{\simeq}{\longrightarrow} \mathcal{V}$$

in the corresponding category of sVOA-s/2d SCFTs, from $\mathcal{V}$ to itself, which are invertible and hence constitute a symmetry of $\mathcal{V}$.

In perturbative string vacua and Connes-Lott models

If $\mathcal{V}$ is a direct summand of a 2d CFT/2-spectral triple encoding a perturbative string theory vacuum, such as, typically, a rational 2d CFT such as a Gepner model encoding a “non-geometric” KK-compactification, then the automorphisms of $\mathcal{V}$ are the formal duals to symmetries of that KK-compactification-fiber space (see the references below)

In the point-particle limit where the 2d SCFT/2-spectral triple becomes an ordinary spectral triple (see there) this hence reduces to the automorphisms of internal algebras as discussed in Connes-Lott-Chamseddine-Barrett models.

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Examples

Moonshine

moonshine-examples:

• The Conway group $CO_{0}$ is the group of automorphisms of a super VOA of the unique chiral N=1 super vertex operator algebra of central charge $c = 12$ without fields of conformal weight $1/2$

  (Duncan 05, see also Paquette-Persson-Volpato 17, p. 9)

• similarly, there is a super VOA, the Monster vertex operator algebra, whose group of of automorphisms of a VOA is the monster group

  (Frenkel-Lepowski-Meurman 89, Griess-Lam 11)

References

General

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As symmetries of non-geometric string compactifications

Automorphisms of vertex operator algebras regarded as symmetries of non-geometric perturbative string theory vacua (e.g. Gepner models):

• Chris Hull, Dan Israel, Alessandra Sarti, Non-geometric Calabi-Yau Backgrounds and K3 automorphisms, JHEP11(2017)084 (arXiv:1710.00853)

Moonshine automorphism groups

• Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics 134, Academic Press, New York 1998. liv+508 pp. MR0996026

• John F. Duncan, Super-moonshine for Conway’s largest sporadic group (arXiv:math/0502267)

• Robert Griess Jr., Ching Hung Lam, A new existence proof of the Monster by VOA theory (arXiv:1103.1414)

• Shamit Kachru, Natalie Paquette, Roberto Volpato, 3D String Theory and Umbral Moonshine (arXiv:1603.07330)

• Natalie Paquette, Daniel Persson, Roberto Volpato, Monstrous BPS-Algebras and the Superstring Origin of Moonshine (arXiv:1601.05412)

• Miranda Cheng, Sarah M. Harrison, Roberto Volpato, Max Zimet, K3 String Theory, Lattices and Moonshine (arXiv:1612.04404)

• Natalie Paquette, Daniel Persson, Roberto Volpato, BPS Algebras, Genus Zero, and the Heterotic Monster (arXiv:1701.05169)

• Shamit Kachru, Arnav Tripathy, The hidden symmetry of the heterotic string (arXiv:1702.02572)