Vertex Operator Algebras with central charge 8 and 16

TL;DR

This paper classifies certain vertex operator algebras with central charges 8 and 16 by analyzing their character spaces, linking them to specific lattice and affine vertex operator algebras, and discusses related differential equations and classifications.

Contribution

It provides a partial classification of vertex operator algebras with central charges 8 and 16 based on their character spaces and associated modular differential equations.

Findings

Character spaces match those of specific lattice VOAs and affine VOAs.

Unique determination of character spaces for certain lattices under mild conditions.

Classification extends to c=4 vertex operator algebras in the appendix.

Abstract

We will partially classify spaces of characters of vertex operator algebras $V$ with central charges 8 and 16, such that the spaces of characters is 3-dimensional and the characters forms a basis of the solution space of a third order monic modular linear differential equation with rational indicial roots. Assuming a mild arithmetic condition, we show that the space of characters of $V$ coincides with the space of characters of lattice vertex operators associated with integral lattices $\sqrt{2}E_8$ or the affine vertex operator algebra of type $D_{20}^{(1)}$ for $c=8$, and the Barnes--Wall lattice $\Lambda_{16}$, the affine vertex operator algebras of type $D_{16}^{(1)}$ with level 1 and type $D_{28}^{(1)}$ with level 1 for $c=16$. (The central charge of the affine vertex operator algebra of type $D_{28}^{(1)}$ with level 1 is 28, but the space of characters satisfies the differential equations for $c=16$.) Supposing a mild condition on characters of $V$, then it uniquely determines (up to isomorphism) the spaces of characters of the lattice $\sqrt{2}E_8$ and the Barnes--Wall lattice $\Lambda_{16}$, respectively. The reason why vertex operator algebras with central charges 8 and 16 are intensively studied is that there are solutions which do not depend on extra parameters (which represent conformal weights). This fact is well understood using the hypergeometric function $3F2$. Hence we cannot apply our standard method to classify vertex operator algebras in which we are interested. In appendix we classify $c=4$ vertex operator algebras with the same conditions mentioned above.