Michigan Mathematical Journal

A new existence proof of the Monster by VOA theory

Robert L. Griess Jr. and Ching Hung Lam

Full-text: Open access

Article information

Source
Michigan Math. J., Volume 61, Issue 3 (2012), 555-573.

Dates
First available in Project Euclid: 7 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1347040259

Digital Object Identifier
doi:10.1307/mmj/1347040259

Mathematical Reviews number (MathSciNet)
MR2975262

Zentralblatt MATH identifier
1279.17005

Subjects
Primary: 17B69: Vertex operators; vertex operator algebras and related structures 20D08: Simple groups: sporadic groups 20C10: Integral representations of finite groups
Secondary: 20D05: Finite simple groups and their classification

Citation

Griess Jr., Robert L.; Lam, Ching Hung. A new existence proof of the Monster by VOA theory. Michigan Math. J. 61 (2012), no. 3, 555--573. doi:10.1307/mmj/1347040259. https://projecteuclid.org/euclid.mmj/1347040259


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References

  • J. H. Conway, A simple construction for the Fischer–Griess monster group, Invent. Math. 79 (1985), 513–540.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford Univ. Press, Oxford, 1985.
  • J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1999.
  • L. Dolan, P. Goddard, and P. Montague, Conformal field theory, triality and the Monster group, Phys. Lett. B 236 (1990), 165–172.
  • –––, Conformal field theory of twisted vertex operators, Nuclear Phys. B 338 (1990), 529–601.
  • C. Y. Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), 245–265.
  • –––, Representations of the moonshine module vertex operator algebra, Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, 1992), Contemp. Math., 175, pp. 27–36, Amer. Math. Soc., Providence, RI, 1994.
  • C. Dong and R. L. Griess Jr., Automorphism groups and derivation algebras of finitely generated vertex operator algebras, Michigan Math. J. 50 (2002), 227–239.
  • C. Dong, R. L. Griess Jr., and G. Höhn, Framed vertex operator algebras, codes and the moonshine module, Comm. Math. Phys. 193 (1998), 407–448.
  • C. Dong, R. L. Griess Jr., and C. H. Lam, Uniqueness results for the moonshine vertex operator algebra, Amer. J. Math. 129 (2007), 583–609.
  • C. Dong, H. Li, and G. Mason, Simple currents and extensions of vertex operator algebras, Comm. Math. Phys. 180 (1996), 671–707.
  • –––, Modular-invariance of trace functions in orbifold theory and generalized moonshine, Comm. Math. Phys. 214 (2000), 1–56.
  • C. Dong and G. Mason, Nonabelian orbifolds and the boson–fermion correspondence, Comm. Math. Phys. 163 (1994), 523–559.
  • –––, Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 56 (2004), 2989–3008.
  • C. Dong, G. Mason, and Y. Zhu, Discrete series of the Virasoro algebra and the moonshine module, Algebraic groups and their generalizations: Quantum and infinite-dimensional methods (University Park, 1991), Proc. Sympos. Pure Math., 56, part 2, pp. 295–316, Amer. Math. Soc., Providence, RI, 1994.
  • I. Frenkel, Y. Z. Huang, and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, Amer. Math. Soc., Providence, RI, 1993.
  • I. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, Pure Appl. Math., 134, Academic Press, Boston, 1988.
  • J. Fuchs and D. Gepner, On the connection between WZW and free field theories, Nuclear Phys. B 294 (1998), 30–42.
  • R. L. Griess Jr., The structure of the “Monster” simple group, Proceedings of the conference on finite groups (Park City, 1975), pp. 113–118, Academic Press, New York, 1976.
  • –––, A construction of $F_1$ as automorphisms of a 196,883-dimensional algebra, Proc. Natl. Acad. Sci. USA 78 (1981), 689–691.
  • –––, The friendly giant, Invent. Math. 69 (1982), 1–102.
  • –––, A vertex operator algebra related to $E_{8}$ with automorphism group $O^+(10,2),$ The Monster and Lie algebras (Columbus, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7, pp. 43–58, de Gruyter, Berlin, 1998.
  • R. L. Griess Jr., U. Meierfrankenfeld, and Y. Segev, A uniqueness proof for the Monster, Ann. of Math. (2) 130 (1989), 567–602.
  • Y. Z. Huang, A nonmeromorphic extension of the moonshine module vertex operator algebra, Moonshine, the Monster and related topics (South Hadley, 1994), Contemp. Math., 193, pp. 123–148, Amer. Math. Soc., Providence, RI, 1996.
  • A. A. Ivanov, The Monster group and Majorana involutions, Cambridge Tracts in Math., 176, Cambridge Univ. Press, Cambridge, 2009.
  • C. H. Lam and H. Shimakura, Ising vectors in the vertex operator algebra $V^{+}_{\Lambda}$ associated with the Leech lattice $\Lambda,$ Int. Math. Res. Not. 2007 (2007), no. 24.
  • C. H. Lam and H. Yamauchi, A characterization of the moonshine vertex operator algebra by means of Virasoro frames, Int. Math. Res. Not. 2007 (2007), no. 1.
  • –––, On the structure of framed vertex operator algebras and their pointwise frame stabilizers, Comm. Math. Phys. 277 (2008), 237–285.
  • H. S. Li, Extension of vertex operator algebras by a self-dual simple module, J. Algebra 187 (1997), 236–267.
  • M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, J. Algebra 179 (1996), 523–548.
  • –––, Binary codes and vertex operator (super)algebras, J. Algebra 181 (1996), 207–222.
  • –––, Representation theory of code vertex operator algebra, J. Algebra 201 (1998), 115–150.
  • –––, The moonshine VOA and a tensor product of Ising models, The Monster and Lie algebras (Columbus, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7, pp. 99–110, de Gruyter, Berlin, 1998.
  • –––, 3-State Potts model and automorphisms of vertex operator algebras of order 3, J. Algebra 239 (2001), 56–76.
  • –––, A new construction of the moonshine vertex operator algebra over the real field, Ann. of Math. (2) 159 (2004), 535–596.
  • S. Sakuma, 6-Transposition property of $\tau$-involutions of vertex operator algebras, Int. Math. Res. Not. 2007 (2007), no. 9.
  • S. Sakuma and H. Yamauchi, Vertex operator algebra with two Miyamoto involutions generating $S_{3},$ J. Algebra 267 (2003), 272–297.
  • A. N. Schellekens and S. Yankielowicz, Extended chiral algebras and modular invariant partition functions, Nuclear Phys. B 327 (1989), 673–703.
  • H. Shimakura, The automorphism group of the vertex operator algebra $V_{L}^{+}$ for an even lattice $L$ without roots, J. Algebra 280 (2004), 29–57.
  • –––, Lifts of automorphisms of vertex operator algebras in simple current extensions, Math. Z. 256 (2007), 491–508.
  • –––, An $E_{8}$ approach to the moonshine vertex operator algebra, J. London Math. Soc. (2) 83 (2011), 493–516.
  • S. Smith, Large extraspecial subgroups of widths 4 and 6, J. Algebra 58 (1979), 251–281.
  • J. Tits, Résumé de cours, Annuaire du Collège de France, 1982/83, pp. 89–102.
  • –––, Remarks on Griess' construction of the Griess–Fischer sporadic group, I, II, III, IV, mimeographed letters, December 1982 – July 1983.
  • –––, Le Monstre (d'après R. Griess, B. Fischer et al.), Séminaire Bourbaki, 26, exp. no. 620, pp. 105–122.
  • –––, On R. Griess' “friendly giant”, Invent. Math. 78 (1984), 491–499.