##
**Vertex operator algebras with central charge \(1/2\) and \(-68/7\).**
*(English)*
Zbl 1370.17027

This paper classifies certain order three differential equations whose space of solutions is invariant under the standard modular slash action of \(\mathrm{SL}_2 (\mathbb{Z})\). In particular, the authors consider order three ‘modular linear differential equations (MLDEs)’ whose solutions are characters of irreducible modules of vertex operator algebras (VOAs) with specific central charges. In the case that a VOA has central charge \(1/2\) or \(-68/7\), has precisely three inequivalent irreducible modules (including itself), and the leading exponents of the module characters satisfy an equation known as the non-zero Wronskian condition, the authors show that the VOA must be isomorphic to the minimal model of central charge \(1/2\) or \(-68/7\), respectively. Some additional expected assumptions are also made. Namely, the VOA is assumed to have no nonnegative weight spaces, the weight zero space is one-dimensional, and the VOA is \(C_2\)-cofinite (to ensure the conformal weights are rational and that the space of characters is modular-invariant).

To obtain the results mentioned above, the authors begin with a generic order three MLDE whose solutions would be of the form given by a formal character of a VOA module (which is a \(q\)-series). An important restriction of such solutions is that the coefficients of the \(q\)-series must be nonnegative integers, since they represent the dimensions of the graded spaces of the module. Additionally, such characters contain the parameters of the central charge of the VOA, as well as the conformal weight of the module. Indeed, these two parameters are then expressed in the coefficients of the terms in the MLDE. Restricting to the cases that the central charge must be \(1/2\) or \(-68/7\), the authors explain that the possible conformal weights required to satisfy the MLDE can be controlled by one parameter. They then develop a list of possible values for this parameter.

In the case that the central charge is \(1/2\), this list first consists of eight possible values which could satisfy the MLDE. However, from these eight, four are redundant as pairs of these values produce the same conformal weights. From the four remaining possible values, the authors consider the \(q\)-series (that is, the form of the character) that would result from these values as solutions to the MLDE. Here, they are able to exclude all but one value as three of the four values would give rise to series that have coefficients that are not nonnegative integers, thus contradicting the required form of the character. After showing that the only functions which satisfy these restrictions are the same functions as the characters of the minimal model \(L(1/2,0)\), the authors are able to conclude that a VOA whose characters satisfy these conditions must be isomorphic to \(L(1/2,0)\). The proof of the central charge \(-68/7\) case is similar.

To obtain the results mentioned above, the authors begin with a generic order three MLDE whose solutions would be of the form given by a formal character of a VOA module (which is a \(q\)-series). An important restriction of such solutions is that the coefficients of the \(q\)-series must be nonnegative integers, since they represent the dimensions of the graded spaces of the module. Additionally, such characters contain the parameters of the central charge of the VOA, as well as the conformal weight of the module. Indeed, these two parameters are then expressed in the coefficients of the terms in the MLDE. Restricting to the cases that the central charge must be \(1/2\) or \(-68/7\), the authors explain that the possible conformal weights required to satisfy the MLDE can be controlled by one parameter. They then develop a list of possible values for this parameter.

In the case that the central charge is \(1/2\), this list first consists of eight possible values which could satisfy the MLDE. However, from these eight, four are redundant as pairs of these values produce the same conformal weights. From the four remaining possible values, the authors consider the \(q\)-series (that is, the form of the character) that would result from these values as solutions to the MLDE. Here, they are able to exclude all but one value as three of the four values would give rise to series that have coefficients that are not nonnegative integers, thus contradicting the required form of the character. After showing that the only functions which satisfy these restrictions are the same functions as the characters of the minimal model \(L(1/2,0)\), the authors are able to conclude that a VOA whose characters satisfy these conditions must be isomorphic to \(L(1/2,0)\). The proof of the central charge \(-68/7\) case is similar.

Reviewer: Matthew Krauel (Sacramento)

### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

11F11 | Holomorphic modular forms of integral weight |

11F22 | Relationship to Lie algebras and finite simple groups |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

PDFBibTeX
XMLCite

\textit{K. Nagatomo} and \textit{Y. Sakai}, Proc. Japan Acad., Ser. A 92, No. 2, 33--37 (2016; Zbl 1370.17027)

### References:

[1] | G. Anderson and G. Moore, Rationality in conformal field theory, Comm. Math. Phys. 117 (1988), no. 3, 441-450. · Zbl 0647.17012 · doi:10.1007/BF01223375 |

[2] | 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. · Zbl 0991.11032 |

[3] | T. Ibukiyama, Modular forms of rational weights and modular varieties, Abh. Math. Sem. Univ. Hamburg 70 (2000), 315-339. · Zbl 1009.11028 · doi:10.1007/BF02940923 |

[4] | K. Iohara and Y. Koga, Representation theory of the Virasoro algebra , Springer Monographs in Mathematics, Springer, London, 2011. · Zbl 1222.17001 · doi:10.1007/978-0-85729-160-8 |

[5] | 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. · Zbl 1283.11070 · doi:10.1007/s11005-012-0602-5 |

[6] | M. Kaneko, K. Nagatomo and Y. Sakai, The 3rd order modular linear differential equations. (Preprint). · Zbl 1283.11070 · doi:10.1007/s11005-012-0602-5 |

[7] | G. Köhler, Eta products and theta series identities , Springer Monographs in Mathematics, Springer, Heidelberg, 2011. |

[8] | G. Mason, Vector-valued modular forms and linear differential operators, Int. J. Number Theory 3 (2007), no. 3, 377-390. · Zbl 1197.11054 · doi:10.1142/S1793042107000973 |

[9] | 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. · doi:10.1016/0370-2693(88)91765-0 |

[10] | M. Miyamoto, Modular invariance of vertex operator algebras satisfying \(C_{2}\)-cofiniteness, Duke Math. J. 122 (2004), no. 1, 51-91. · Zbl 1165.17311 · doi:10.1215/S0012-7094-04-12212-2 |

[11] | N. Yui and D. Zagier, On the singular values of Weber modular functions, Math. Comp. 66 (1997), no. 220, 1645-1662. · Zbl 0892.11022 · doi:10.1090/S0025-5718-97-00854-5 |

[12] | Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237-302. · Zbl 0854.17034 · doi:10.1090/S0894-0347-96-00182-8 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.