## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Vertex operator algebras with central charge 1/2 and $-68/7$

#### Abstract

In this article we study \textit{simple vertex operator algebras} whose spaces of characters are 3-dimensional, and satisfy \textit{3rd order (modular) linear differential equations}. We classify such vertex operator algebras with central charge 1/2 or $-68/7$. One of the main results is that these vertex operator algebras have \textit{conformal weights} $\{0,1/2,1/16\}$ or $\{0,-2/7,-3/7\}$, respectively, and are isomorphic to the minimal models of central charge $c=c_{3,4}=1/2$ and $c_{2,7}=-68/7$. Moreover, we give the explicit formulas representing characters of the minimal models with $c=1/2$, $-68/7$ by using the classical Weber functions.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 2 (2016), 33-37.

Dates
First available in Project Euclid: 28 January 2016

https://projecteuclid.org/euclid.pja/1453993552

Digital Object Identifier
doi:10.3792/pjaa.92.33

Mathematical Reviews number (MathSciNet)
MR3455133

Zentralblatt MATH identifier
1370.17027

#### Citation

Nagatomo, Kiyokazu; Sakai, Yuichi. Vertex operator algebras with central charge 1/2 and $-68/7$. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 2, 33--37. doi:10.3792/pjaa.92.33. https://projecteuclid.org/euclid.pja/1453993552

#### References

• G. Anderson and G. Moore, Rationality in conformal field theory, Comm. Math. Phys. 117 (1988), no. 3, 441–450.
• N. D. Elkies, The Klein quartic in number theory, in The eightfold way, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999, pp. 51–101.
• T. Ibukiyama, Modular forms of rational weights and modular varieties, Abh. Math. Sem. Univ. Hamburg 70 (2000), 315–339.
• K. Iohara and Y. Koga, Representation theory of the Virasoro algebra, Springer Monographs in Mathematics, Springer, London, 2011.
• M. Kaneko, K. Nagatomo and Y. Sakai, Modular forms and second order ordinary differential equations: applications to vertex operator algebras, Lett. Math. Phys. 103 (2013), no. 4, 439–453.
• M. Kaneko, K. Nagatomo and Y. Sakai, The 3rd order modular linear differential equations. (Preprint).
• G. Köhler, Eta products and theta series identities, Springer Monographs in Mathematics, Springer, Heidelberg, 2011.
• G. Mason, Vector-valued modular forms and linear differential operators, Int. J. Number Theory 3 (2007), no. 3, 377–390.
• S. D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988), no. 3, 303–308.
• M. Miyamoto, Modular invariance of vertex operator algebras satisfying $C_{2}$-cofiniteness, Duke Math. J. 122 (2004), no. 1, 51–91.
• N. Yui and D. Zagier, On the singular values of Weber modular functions, Math. Comp. 66 (1997), no. 220, 1645–1662.
• Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302.