Duke Mathematical Journal

Super-moonshine for Conway's largest sporadic group

John F. Duncan

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Abstract

We study a self-dual N=1 super vertex operator algebra and prove that the full symmetry group is Conway's largest sporadic simple group. We verify a uniqueness result that is analogous to that conjectured to characterize the Moonshine vertex operator algebra (VOA). The action of the automorphism group is sufficiently transparent that one can derive explicit expressions for all the McKay-Thompson series. A corollary of the construction is that the perfect double cover of the Conway group may be characterized as a point-stabilizer in a spin module for the Spin group associated to a 24-dimensional Euclidean space

Article information

Source
Duke Math. J., Volume 139, Number 2 (2007), 255-315.

Dates
First available in Project Euclid: 31 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1185891824

Digital Object Identifier
doi:10.1215/S0012-7094-07-13922-X

Mathematical Reviews number (MathSciNet)
MR2352133

Zentralblatt MATH identifier
1171.17011

Subjects
Primary: 17B69: Vertex operators; vertex operator algebras and related structures
Secondary: 20D08: Simple groups: sporadic groups

Citation

Duncan, John F. Super-moonshine for Conway's largest sporadic group. Duke Math. J. 139 (2007), no. 2, 255--315. doi:10.1215/S0012-7094-07-13922-X. https://projecteuclid.org/euclid.dmj/1185891824


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References

  • M. Aschbacher, Finite Group Theory, 2nd ed., Cambridge Stud. Adv. Math. 10, Cambridge Univ. Press, Cambridge, 2000.
  • R. E. Borcherds and A. J. E. Ryba, Modular moonshine, II, Duke Math. J. 83 (1996), 435--459.
  • J. H. Conway, A group of order $8,315,553,613,086,720,000$, Bull. London Math. Soc. 1 (1969), 79--88.
  • —, ``Three lectures on exceptional groups'' in Finite Simple Groups (Oxford, 1969), Academic Press, London, 1971, 215--247.
  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Eynsham, England, 1985.
  • J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308--339.
  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2nd ed., Grundlehren Math. Wiss. 290, Springer, New York, 1993.
  • L. Dixon, P. Ginsparg, and J. Harvey, Beauty and the beast: Superconformal symmetry in a Monster module, Comm. Math. Phys. 119 (1988), 221--241.
  • C. Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), 245--265.
  • C. Dong, R. L. Griesso Jr., and G. HöHn, Framed vertex operator algebras, codes and the moonshine module, Comm. Math. Phys. 193 (1998), 407--448.
  • C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progr. Math. 112, Birkhäuser, Boston, 1993.
  • C. Dong, H. Li, and G. Mason, Simple currents and extensions of vertex operator algebras, Comm. Math. Phys. 180 (1996), 671--707.
  • —, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571--600.
  • C. Dong and G. Mason, Nonabelian orbifolds and the boson-fermion correspondence, Comm. Math. Phys. 163 (1994), 523--559.
  • —, Holomorphic vertex operator algebras of small central charge, Pacific J. Math. 213 (2004), 253--266.
  • —, Rational vertex operator algebras and the effective central charge, Int. Math. Res. Not. 2004, no. 56, 2989--3008.
  • —, Integrability of $C\sb 2$-cofinite vertex operator algebras, Int. Math. Res. Not. 2006, no. 80468.
  • C. Dong and K. Nagatomo, ``Automorphism groups and twisted modules for lattice vertex operator algebras'' in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, N.C., 1998), Contemp. Math. 248, Amer. Math. Soc., Providence, 1999, 117--133.
  • C. Dong and Z. Zhao, Modularity in orbifold theory for vertex operator superalgebras, Comm. Math. Phys. 260 (2005), 227--256.
  • A. W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. (2) 102 (1975), 291--325.
  • A. J. Feingold, I. B. Frenkel, and J. F. X. Ries, Spinor Construction of Vertex Operator Algebras, Triality, and $E\sp (1)\sb 8$, Contemp. Math. 121, Amer. Math. Soc., Providence, 1991.
  • I. B. Frenkel, Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981), 259--327.
  • I. B. Frenkel, Y.-Z. Huang, and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. Amer. Math. Soc. 104 (1993), no. 494.
  • I. B. Frenkel, J. Lepowsky, and A. Meurman, ``A moonshine module for the Monster'' in Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ. 3, Springer, New York, 1985, 231--273.
  • —, Vertex Operator Algebras and the Monster, Pure Appl. Math. 134, Academic Press, Boston, 1988.
  • A. FröHlich and M. J. Taylor, Algebraic Number Theory, Cambridge Stud. Adv. Math. 27, Cambridge Univ. Press, Cambridge, 1993.
  • G. HöHn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Math. Schriften 286, Mathematisches Institut, Univ. Bonn, Bonn, 1996.
  • —, personal communication, May 2005.
  • Y.-Z. Huang, ``A nonmeromorphic extension of the moonshine module vertex operator algebra'' in Moonshine, the Monster, and Related Topics (South Hadley, Mass., 1994), Contemp. Math. 193, Amer. Math. Soc., Providence, 1996, 123--148.
  • H.-K. Hwang, Limit theorems for the number of summands in integer partitions, J. Combin. Theory Ser. A 96 (2001), 89--126.
  • V. Kac and W. Wang, ``Vertex operator superalgebras and their representations'' in Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, Mass., 1992), Contemp. Math. 175, Amer. Math. Soc., Providence, 1994, 161--191.
  • A. Kapustin and D. Orlov, Vertex algebras, mirror symmetry, and D-branes: The case of complex tori, Comm. Math. Phys. 233 (2003), 79--136.
  • G. Meinardus, Asymptotische Aussagen über Partitionen, Math. Z. 59 (1954), 388--398.
  • G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, Des. Codes Cryptogr. 24 (2001), 99--121.
  • R. A. Rankin, Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977.
  • N. R. Scheithauer, Vertex algebras, Lie algebras, and superstrings, J. Algebra 200 (1998), 363--403.
  • J.-P. Serre, Linear Representations of Finite Groups, Grad. Texts in Math. 42, Springer, New York, 1977.
  • P. H. Tiep, Globally irreducible representations of finite groups and integral lattices, Geom. Dedicata 64 (1997), 85--123.
  • Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237--302.
  • —, Vertex operator algebras, elliptic functions, and modular forms, Ph.D. dissertation, Yale University, New Haven, 1990.