Vertex operator algebras with central charge 8 and 16. (English) Zbl 1479.17056
Krauel, Matthew (ed.) et al., Vertex operator algebras, number theory and related topics. International conference, California State University, Sacramento, CA, USA, June 11–15, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 753, 157-186 (2020).
Summary: In this paper we will partly classify spaces of characters of vertex operator algebras with central charges 8 and 16, whose spaces of characters are 3-dimensional and each space of characters forms a basis of the space of solutions of a third order monic modular linear differential equation with rational indicial roots.
Under the condition that the dimension of the lowest weight space, say \(d\), divides the dimension of the space of weight \(d+1\), we clarify that the spaces of characters of those vertex operator algebras coincide with the spaces of characters of lattice vertex operators associated with integral lattices \(\sqrt{2}E_8\) or the affine vertex operator algebra of type \(D_{20}^{(1)} \) (denoted by \(L_{D_{16}^{(1)},1}\)) with level 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 \(c=16\).)
Moreover, if we suppose a mild condition on characters of vertex operator algebras, then it uniquely determines (up to some isomorphisms) the spaces of characters of the lattice \(\sqrt{2}E_8\) and the Barnes-Wall lattice \(\Lambda_{16}\), respectively.
Finally, the reason why vertex operator algebras with central charges 8 and 16 are intensively studied is that there are solutions such that they does not depends on extra parameters (which represent conformal weights). More precisely, \(E_4/\eta^8\) and \(E_4^2/\eta^{16}\) are solutions of MLDEs for central charges 8 and 16, respectively. This fact is well understood by using hypergeometric function \({}_3F_2\). 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.
For the entire collection see [Zbl 1452.17002].
Under the condition that the dimension of the lowest weight space, say \(d\), divides the dimension of the space of weight \(d+1\), we clarify that the spaces of characters of those vertex operator algebras coincide with the spaces of characters of lattice vertex operators associated with integral lattices \(\sqrt{2}E_8\) or the affine vertex operator algebra of type \(D_{20}^{(1)} \) (denoted by \(L_{D_{16}^{(1)},1}\)) with level 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 \(c=16\).)
Moreover, if we suppose a mild condition on characters of vertex operator algebras, then it uniquely determines (up to some isomorphisms) the spaces of characters of the lattice \(\sqrt{2}E_8\) and the Barnes-Wall lattice \(\Lambda_{16}\), respectively.
Finally, the reason why vertex operator algebras with central charges 8 and 16 are intensively studied is that there are solutions such that they does not depends on extra parameters (which represent conformal weights). More precisely, \(E_4/\eta^8\) and \(E_4^2/\eta^{16}\) are solutions of MLDEs for central charges 8 and 16, respectively. This fact is well understood by using hypergeometric function \({}_3F_2\). 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.
For the entire collection see [Zbl 1452.17002].
MSC:
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 11F22 | Relationship to Lie algebras and finite simple groups |
| 11F11 | Holomorphic modular forms of integral weight |
References:
| [1] | Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi, Affine vertex operator algebras and modular linear differential equations, Lett. Math. Phys., 106, 5, 693-718 (2016) · Zbl 1338.81344 · doi:10.1007/s11005-016-0837-7 |
| [2] | Arike, Yusuke; Nagatomo, Kiyokazu; Sakai, Yuichi, Vertex operator algebras, minimal models, and modular linear differential equations of order 4, J. Math. Soc. Japan, 70, 4, 1347-1373 (2018) · Zbl 1406.81081 · doi:10.2969/jmsj/74957495 |
| [3] | Arike, Yusuke; Nagatomo, Kiyokazu; Sakai, Yuichi, Characterization of the simple Virasoro vertex operator algebras with 2 and 3-dimensional space of characters. Lie algebras, vertex operator algebras, and related topics, Contemp. Math. 695, 175-204 (2017), Amer. Math. Soc., Providence, RI · Zbl 1414.17017 · doi:10.1090/conm/695/14002 |
| [4] | Conway, J. H.; Sloane, N. J. A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 290, lxxiv+703 pp. (1999), Springer-Verlag, New York · Zbl 0915.52003 · doi:10.1007/978-1-4757-6568-7 |
| [5] | Dong, Chongying; Lin, Xingjun; Ng, Siu-Hung, Congruence property in conformal field theory, Algebra Number Theory, 9, 9, 2121-2166 (2015) · Zbl 1377.17025 · doi:10.2140/ant.2015.9.2121 |
| [6] | Adamovi\'{c}, Dra\v{z}en; Milas, Antun, An analogue of modular BPZ-equation in logarithmic (super)conformal field theory. Vertex operator algebras and related areas, Contemp. Math. 497, 1-17 (2009), Amer. Math. Soc., Providence, RI · Zbl 1225.17029 · doi:10.1090/conm/497/09765 |
| [7] | Frenkel, Igor; Lepowsky, James; Meurman, Arne, Vertex operator algebras and the Monster, Pure and Applied Mathematics 134, liv+508 pp. (1988), Academic Press, Inc., Boston, MA · Zbl 0674.17001 |
| [8] | Hiramatsu, Toyokazu; K\"{o}hler, G\"{u}nter, Coding theory and number theory, Mathematics and its Applications 554, xii+148 pp. (2003), Kluwer Academic Publishers Group, Dordrecht · Zbl 1143.94017 · doi:10.1007/978-94-017-0305-5 |
| [9] | Griess, Robert L., Jr., Pieces of \(2^d\): existence and uniqueness for Barnes-Wall and Ypsilanti lattices, Adv. Math., 196, 1, 147-192 (2005) · Zbl 1076.11043 · doi:10.1016/j.aim.2004.08.014 |
| [10] | Kaneko, Masanobu; Koike, Masao, On modular forms arising from a differential equation of hypergeometric type, Ramanujan J., 7, 1-3, 145-164 (2003) · Zbl 1050.11047 · doi:10.1023/A:1026291027692 |
| [11] | Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi, The third order modular linear differential equations, J. Algebra, 485, 332-352 (2017) · Zbl 1416.11058 · doi:10.1016/j.jalgebra.2017.05.007 |
| [12] | Marks, Christopher, Irreducible vector-valued modular forms of dimension less than six, Illinois J. Math., 55, 4, 1267-1297 (2013) (2011) · Zbl 1343.11052 |
| [13] | Mason, Geoffrey, Vector-valued modular forms and linear differential operators, Int. J. Number Theory, 3, 3, 377-390 (2007) · Zbl 1197.11054 · doi:10.1142/S1793042107000973 |
| [14] | Milas, Antun, On certain automorphic forms associated to rational vertex operator algebras. Moonshine: the first quarter century and beyond, London Math. Soc. Lecture Note Ser. 372, 330-357 (2010), Cambridge Univ. Press, Cambridge · Zbl 1227.11065 |
| [15] | Milas, Antun, Modular invariance, modular identities and supersingular \(j\)-invariants, Math. Res. Lett., 13, 5-6, 729-746 (2006) · Zbl 1165.11043 · doi:10.4310/MRL.2006.v13.n5.a5 |
| [16] | Milas, Antun, Characters, supercharacters and Weber modular functions, J. Reine Angew. Math., 608, 35-64 (2007) · Zbl 1132.17015 · doi:10.1515/CRELLE.2007.052 |
| [17] | Milas, Antun, Virasoro algebra, Dedekind \(\eta \)-function, and specialized Macdonald identities, Transform. Groups, 9, 3, 273-288 (2004) · Zbl 1112.17028 · doi:10.1007/s00031-004-7014-2 |
| [18] | Nagatomo, Kiyokazu; Sakai, Yuichi, Vertex operator algebras with central charge \(1/2\) and \(-68/7\), Proc. Japan Acad. Ser. A Math. Sci., 92, 2, 33-37 (2016) · Zbl 1370.17027 · doi:10.3792/pjaa.92.33 |
| [19] | Zhu, Yongchang, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc., 9, 1, 237-302 (1996) · Zbl 0854.17034 · doi:10.1090/S0894-0347-96-00182-8 |
| [20] | Franc, Cameron; Mason, Geoffrey, Hypergeometric series, modular linear differential equations and vector-valued modular forms, Ramanujan J., 41, 1-3, 233-267 (2016) · Zbl 1418.11064 · doi:10.1007/s11139-014-9644-x |
| [21] | Franc, Cameron; Mason, Geoffrey, Three-dimensional imprimitive representations of the modular group and their associated modular forms, J. Number Theory, 160, 186-214 (2016) · Zbl 1335.11034 · doi:10.1016/j.jnt.2015.08.013 |
| [22] | Marks, Christopher, Irreducible vector-valued modular forms of dimension less than six, Illinois J. Math., 55, 4, 1267-1297 (2013) (2011) · Zbl 1343.11052 |
| [23] | Shimakura, Hiroki, On isomorphism problems for vertex operator algebras associated with even lattices, Proc. Amer. Math. Soc., 140, 10, 3333-3348 (2012) · Zbl 1307.17030 · doi:10.1090/S0002-9939-2011-11167-5 |
| [24] | Stein, William, Modular forms, a computational approach, Graduate Studies in Mathematics 79, xvi+268 pp. (2007), American Mathematical Society, Providence, RI · Zbl 1110.11015 · doi:10.1090/gsm/079 |
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.
