Abstract
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which play an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.
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Borcherds, R.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Nat. Acad. Sci. USA 83(10), 3068–3071 (1986)
Borcherds, R.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992)
Carnahan, S.: Building vertex algebras from parts. Commun. Math. Phys. 373(1), 1–43 (2020)
Creutzig, T., Kanade, S., Linshaw, A.: Simple current extensions beyond semi-simplicity. Commun. Contemp. Math. 22(1), 1950001, 49 (2020)
Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, 2nd edn. Springer, New York (1993)
Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators, Progress in Mathematics, vol. 112. Birkhäuser Boston Inc, Boston (1993)
Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math. 132(1), 148–166 (1997)
Dong, C., Mason, G.: Holomorphic vertex operator algebras of small central charge. Pac. J. Math. 213(2), 253–266 (2004)
Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. 56, 2989–3008 (2004)
Dong, C., Ren, L.: Representations of the parafermion vertex operator algebras. Adv. Math. 315, 88–101 (2017)
van Ekeren, J., Möller, S., Scheithauer, N.: Construction and classification of holomorphic vertex operator algebras. J. Reine Angew. Math. 759, 61–99 (2020)
Frenkel, I., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, 494 (1993)
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press Inc, Boston (1988)
Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)
Huang, Y.-Z., Lepowsky, J., Zhang, L.: Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and Their Generalized Modules, Conformal Field Theories and Tensor Categories, Mathematical Lectures, pp. 169–248. Peking University, Beijing (2014)
Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9. Springer, New York (1978). (Second printing, revised)
Höhn, G., Scheithauer, N.: A natural construction of Borcherds’ fake Baby Monster Lie algebra. Am. J. Math. 125(3), 655–667 (2003)
Höhn, G., Scheithauer, N.: A generalized Kac–Moody algebra of rank 14. J. Algebra 404, 222–239 (2014)
Höhn, G.: Genera of vertex operator algebras and three-dimensional topological quantum field theories, Vertex operator algebras in mathematics and physics (Toronto, ON, 2000). Fields Inst. Commun. 39, 89–107 (2000)
Höhn, G.: On the Genus of the Moonshine Module, q-alg:1708.05990
Kitaoka, Y.: Arithmetic of Quadratic Forms, Cambridge Tracts in Mathematics, vol. 106. Cambridge University Press, Cambridge (1993)
Kneser, M., Puppe, D.: Quadratische Formen und Verschlingungsinvarianten von Knoten. Math. Z. 58, 376–384 (1953)
Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96(3), 279–297 (1994)
Li, H.: The physics superselection principle in vertex operator algebra theory. J. Algebra 196(2), 436–457 (1997)
Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, vol. 227. Birkhäuser Boston Inc, Boston (2004)
Lam, C., Shimakura, H.: Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24. Proc. Lond. Math. Soc. (3) 104(3), 540–576 (2012)
Lam, C., Shimakura, H.: Classification of holomorphic framed vertex operator algebras of central charge 24. Am. J. Math. 137(1), 111–137 (2015)
Mason, G.: Vertex rings and their Pierce bundles, Vertex algebras and geometry. Contemp. Math. 711, 45–104 (2018)
Roitman, M.: Combinatorics of free vertex algebras. J. Algebra 255(2), 297–323 (2002)
Schellekens, A.: Meromorphic \(c=24\) conformal field theories. Commun. Math. Phys. 153(1), 159–185 (1993)
Acknowledgements
The author would like to express his gratitude to Professor Atsushi Matsuo, for his encouragement throughout this work and numerous advices to improve this paper. He is also grateful to Hiroki Shimakura and Shigenori Nakatsuka for careful reading of this manuscript and their valuable comments. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.
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Moriwaki, Y. Genus of vertex algebras and mass formula. Math. Z. 299, 1473–1505 (2021). https://doi.org/10.1007/s00209-021-02702-0
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DOI: https://doi.org/10.1007/s00209-021-02702-0