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THE MOONSHINE MODULE FOR CONWAY’S GROUP

Part of: Lie algebras and Lie superalgebras Representation theory of groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 June 2015

JOHN F. R. DUNCAN  and
SANDER MACK-CRANE
JOHN F. R. DUNCAN
Affiliation:
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; john.duncan@case.edu, mack-crane@case.edu
SANDER MACK-CRANE
Affiliation:
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; john.duncan@case.edu, mack-crane@case.edu

Abstract

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We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11F22: Relationship to Lie algebras and finite simple groups 17B69: Vertex operators; vertex operator algebras and related structures 20C34: Representations of sporadic groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e10
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Aspinwall, P. S., ‘K3 surfaces and string duality’, inFields, Strings and Duality (Boulder, CO, 1996) (World Scientific, River Edge, NJ, 1997), 421–540.Google Scholar
Borcherds, R., ‘Vertex algebras, Kac–Moody algebras, and the monster’, Proc. Natl Acad. Sci. USA 83(10) (1986), 3068–3071.Google Scholar
Borcherds, R. E., ‘Monstrous moonshine and monstrous Lie superalgebras’, Invent. Math. 109(2) (1992), 405–444.Google Scholar
Borcherds, R. E. and Ryba, A. J. E., ‘Modular moonshine II’, Duke Math. J. 83(2) (1996), 435–459.Google Scholar
Bridgeland, T., ‘Stability conditions on K3 surfaces’, Duke Math. J. 141(2) (2008), 241–291.Google Scholar
Carnahan, S., ‘Generalized moonshine I: genus-zero functions’, Algebra Number Theory 4(6) (2010), 649–679.Google Scholar
Carnahan, S., ‘Generalized moonshine, II: Borcherds products’, Duke Math. J. 161(5) (2012), 893–950.Google Scholar
Carnahan, S., ‘Generalized moonshine IV: monstrous lie algebras’, Preprint, 2012, arXiv:1208.6254.Google Scholar
Carnahan, S., ‘Monstrous lie algebras’, RIMS Kôkyûroku 1872 (2014).Google Scholar
Cheng, M. C. N., ‘K3 surfaces, N = 4 dyons and the Mathieu group M 24’, Commun. Number Theory Phys. 4(4) (2010), 623–657.Google Scholar
Cheng, M. C. N., Dong, X., Duncan, J. F. R., Harrison, S., Kachru, S. and Wrase, T., ‘Mock modular Mathieu moonshine modules’, Preprint, 2014, arXiv:1406.5502 [hep-th].CrossRefGoogle Scholar
Cheng, M. C. N. and Duncan, J. F. R., ‘The largest Mathieu group and (mock) automorphic forms’, inString-Math 2011, Proceedings of Symposia in Pure Mathematics, 85 (American Mathematical Society, Providence, RI, 2012), 53–82.CrossRefGoogle ScholarPubMed
Cheng, M. C. N. and Duncan, J. F. R., ‘On Rademacher sums, the largest Mathieu group, and the holographic modularity of moonshine’, Commun. Number Theory Phys. 6(3) (2012), 697–758.Google Scholar
Cheng, M. C. N., Duncan, J. F. R. and Harvey, J. A., ‘Umbral moonshine’, Commun. Number Theory Phys. 8(2) (2014), 101–242.Google Scholar
Cheng, M. C. N., Duncan, J. F. R. and Harvey, J. A., ‘Umbral moonshine and the Niemeier lattices’, Res. Math. Sci. 1(3) (2014), 1–81.CrossRefGoogle Scholar
Conway, J. H., ‘A perfect group of order 8, 315, 553, 613, 086, 720, 000 and the sporadic simple groups’, Proc. Natl Acad. Sci. USA 61 (1968), 398–400.Google Scholar
Conway, J. H., ‘A characterisation of Leech’s lattice’, Invent. Math. 7 (1969), 137–142.Google Scholar
Conway, J. H., ‘A group of order 8, 315, 553, 613, 086, 720, 000’, Bull. Lond. Math. Soc. 1 (1969), 79–88.Google Scholar
Conway, J. H. and Norton, S. P., ‘Monstrous moonshine’, Bull. Lond. Math. Soc. 11(3) (1979), 308–339.Google Scholar
Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, 3rd edn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290 (Springer, New York, 1999). With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Clarendon, Oxford, 1985). (Maximal Subgroups and Ordinary Characters for Simple Groups. With comput. assist. from J. G. Thackray).Google Scholar
Conway, J., McKay, J. and Sebbar, A., ‘On the discrete groups of moonshine’, Proc. Amer. Math. Soc. 132 (2004), 2233–2240.Google Scholar
Dixon, L., Ginsparg, P. and Harvey, J., ‘Beauty and the beast: superconformal symmetry in a monster module’, Comm. Math. Phys. 119(2) (1988), 221–241.Google Scholar
Dixon, L., Harvey, J. A., Vafa, C. and Witten, E., ‘Strings on orbifolds’, Nuclear Phys. B 261(4) (1985), 678–686.Google Scholar
Dixon, L., Harvey, J. A., Vafa, C. and Witten, E., ‘Strings on orbifolds II’, Nuclear Phys. B 274(2) (1986), 285–314.Google Scholar
Dong, C., ‘Vertex algebras associated with even lattices’, J. Algebra 161(1) (1993), 245–265.Google Scholar
Dong, C. and Lepowsky, J., Generalized Vertex Algebras and Relative Vertex Operators, Progress in Mathematics, 112 (Birkhäuser Boston, Inc., Boston, MA, 1993).Google Scholar
Dong, C., Li, H. and Mason, G., ‘Some twisted sectors for the moonshine module’, inMoonshine, The Monster, and Related Topics (South Hadley, MA, 1994), Contemporary Mathematics, 193 (American Mathematical Society, Providence, RI, 1996), 25–43.Google Scholar
Dong, C., Li, H. and Mason, G., ‘Twisted representations of vertex operator algebras’, Math. Ann. 310(3) (1998), 571–600.Google Scholar
Dong, C., Li, H. and Mason, G., ‘Modular invariance of trace functions in orbifold theory and generalized moonshine’, Comm. Math. Phys. 214 (2000), 1–56.Google Scholar
Dong, C., Li, H., Mason, G. and Montague, P. S., ‘The radical of a vertex operator algebra’, inThe Monster and Lie Algebras (Columbus, OH, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7 (de Gruyter, Berlin, 1998), 17–25.Google Scholar
Dong, C. and Mason, G., ‘Nonabelian orbifolds and the boson-fermion correspondence’, Comm. Math. Phys. 163(3) (1994), 523–559.Google Scholar
Dong, C. and Mason, G., ‘Rational vertex operator algebras and the effective central charge’, Int. Math. Res. Not. 56 (2004), 2989–3008.Google Scholar
Dong, C. and Mason, G., ‘Integrability of C 2 -cofinite vertex operator algebras’, Int. Math. Res. Not. (2006), 1–15. Art. ID 80468.Google Scholar
Dong, C. and Zhao, Z., ‘Twisted representations of vertex operator superalgebras’, Commun. Contemp. Math. 8(1) (2006), 101–121.Google Scholar
Duncan, J. F., ‘Super-moonshine for Conway’s largest sporadic group’, Duke Math. J. 139(2) (2007), 255–315.Google Scholar
Duncan, J. F. R. and Frenkel, I. B., ‘Rademacher sums, moonshine and gravity’, Commun. Number Theory Phys. 5(4) (2011), 1–128.Google Scholar
Duncan, J. F. R. and Mack-Crane, S., Derived Equivalences of K3 Surfaces and Twined Elliptic Genera, to appear (2015).Google Scholar
Eguchi, T. and Hikami, K., ‘Note on twisted elliptic genus of K3 surface’, Phys. Lett. B 694 (2011), 446–455.Google Scholar
Eguchi, T., Ooguri, H. and Tachikawa, Y., ‘Notes on the K3 surface and the Mathieu group M 24’, Exp. Math. 20 (2011), 91–96.Google Scholar
Eguchi, T. and Taormina, A., ‘On the unitary representations of N = 2 and N = 4 superconformal algebras’, Phys. Lett. B 210(1–2) (1988), 125–132.Google Scholar
Eichler, M. and Zagier, D., The Theory of Jacobi Forms (Birkhäuser, Boston, MA, 1985).Google Scholar
Ferenbaugh, C. R., ‘The genus-zero problem for n|h-type groups’, Duke Math. J. 72(1) (1993), 31–63.Google Scholar
Ford, D., McKay, J. and Norton, S., ‘More on replicable functions’, Comm. Algebra 22(13) (1994), 5175–5193.Google Scholar
Frenkel, I. B., ‘Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory’, J. Funct. Anal. 44(3) (1981), 259–327.Google Scholar
Frenkel, E. and Szczesny, M., ‘Twisted modules over vertex algebras on algebraic curves’, Adv. Math. 187(1) (2004), 195–227.Google Scholar
Frenkel, I. B. and Kac, V. G., ‘Basic representations of affine Lie algebras and dual resonance models’, Invent. Math. 62(1) (1980/81), 23–66.Google Scholar
Frenkel, I. B., Huang, Y.-Z. and Lepowsky, J., ‘On axiomatic approaches to vertex operator algebras and modules’, Mem. Amer. Math. Soc. 104(494) (1993).Google Scholar
Frenkel, I. B., Lepowsky, J. and Meurman, A., ‘A natural representation of the Fischer–Griess monster with the modular function J as character’, Proc. Natl Acad. Sci. USA 81(10) (1984), 3256–3260.Google Scholar
Frenkel, I. B., Lepowsky, J. and Meurman, A., ‘A moonshine module for the monster’, inVertex Operators in Mathematics and Physics (Berkeley, CA, 1983), Mathematical Sciences Research Institute Publications, 3 (Springer, New York, 1985), 231–273.Google Scholar
Frenkel, I. B., Lepowsky, J. and Meurman, A., Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, 134 (Academic, Boston, MA, 1988).Google Scholar
Frenkel, E. and Zvi, D. B., Vertex Algebras and Algebraic Curves, 2nd edn, Mathematical Surveys and Monographs, 88 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Gaberdiel, M. R., ‘An introduction to conformal field theory’, Rep. Progr. Phys. 63 (2000), 607–667.Google Scholar
Gaberdiel, M. R., ‘2D conformal field theory and vertex operator algebras’, Preprint, 2005, arXiv:hep-th/0509027.Google Scholar
Gaberdiel, M. R., Hohenegger, S. and Volpato, R., ‘Mathieu moonshine in the elliptic genus of K3’, JHEP 1010 (2010) 062.Google Scholar
Gaberdiel, M. R., Hohenegger, S. and Volpato, R., ‘Mathieu twining characters for K3’, JHEP 1009 (2010) 058 19 pages.Google Scholar
Gaberdiel, M. R., Hohenegger, S. and Volpato, R., ‘Symmetries of K3 sigma models’, Commun. Number Theory Phys. 6(1) (2012), 1–50.Google Scholar
Gaberdiel, M. R. and Volpato, R., ‘Mathieu moonshine and orbifold K3s’, Contrib. Math. Comput. Sci. 8 (2014), 109–141.Google Scholar
Gannon, T., ‘Much ado about Mathieu’, Preprint, 2012, arXiv:1211.5531 [math.RT].Google Scholar
Golay, M. J. E., ‘Notes on digital coding’, Proc. IRE 37 (1949), 657.Google Scholar
Griess, R. L. Jr., ‘The structure of the ‘monster’ simple group’, inProceedings of the Conference on Finite Groups (University of Utah, Park City, UT, 1975) (Academic, New York, 1976), 113–118.Google Scholar
Griess, R. L. Jr., ‘A construction of F 1 as automorphisms of a 196, 883-dimensional algebra’, Proc. Natl Acad. Sci. USA 78(2, Part 1) (1981), 686–691.Google Scholar
Griess, R. L. Jr., ‘The friendly giant’, Invent. Math. 69(1) (1982), 1–102.Google Scholar
Griffiths, P. A., Introduction to Algebraic Curves, Translations of Mathematical Monographs, 76 (American Mathematical Society, Providence, RI, 1989). Translated from the Chinese by Kuniko Weltin.Google Scholar
Hoehn, G., ‘Generalized moonshine for the baby monster’, 2003, available at http://www.math.ksu.edu/ gerald/papers/.Google Scholar
Hoehn, G., ‘Selbstduale vertexoperatorsuperalgebren und das babymonster (self-dual vertex operator super algebras and the baby monster)’, inBonner Mathematische Schriften (Bonn, 1996) Vol. 286 (Library of Mathematics, Bonn, 2007), 1–85.Google Scholar
Holmes, P. E. and Wilson, R. A., ‘A new computer construction of the Monster using 2-local subgroups’, J. Lond. Math. Soc. (2) 67(2) (2003), 349–364.Google Scholar
Huybrechts, D., ‘On derived categories of K3 surfaces, symplectic automorphisms and the Conway group’, Preprint, 2013, arXiv:1309.6528 [math.AG].Google Scholar
Ivanov, R. and Tuite, M., ‘Rational generalised moonshine from abelian orbifoldings of the moonshine module’, Nuclear Phys. B 635(3) (2002), 435–472.Google Scholar
Ivanov, R. and Tuite, M., ‘Some irrational generalised moonshine from orbifolds’, Nuclear Phys. B 635(3) (2002), 473–491.Google Scholar
Kac, V., Vertex Algebras for Beginners, 2nd edn, University Lecture Series, 10 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Kac, V. and Wang, W., ‘Vertex operator superalgebras and their representations’, inMathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemporary Mathematics, 175 (American Mathematical Society, Providence, RI, 1994), 161–191.Google Scholar
Knapp, A. W., Elliptic Curves, Mathematical Notes, 40 (Princeton University Press, Princeton, NJ, 1992).Google Scholar
Kondo, T., ‘The automorphism group of Leech lattice and elliptic modular functions’, J. Math. Soc. Japan 37(2) (1985), 337–362.Google Scholar
P. S. Landweber (Ed.), Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, 1326 (Springer, Berlin 1988).Google Scholar
Leech, J., ‘Some sphere packings in higher space’, Canad. J. Math. 16 (1964), 657–682.Google Scholar
Leech, J., ‘Notes on sphere packings’, Canad. J. Math. 19 (1967), 251–267.Google Scholar
Lepowsky, J. and Li, H., Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, 227 (Birkhäuser Boston Inc., Boston, MA, 2004).Google Scholar
Lepowsky, J. and Wilson, R. L., ‘Construction of the affine Lie algebra A 1(1)’, Comm. Math. Phys. 62(1) (1978), 43–53.CrossRefGoogle Scholar
Li, H. S., ‘Symmetric invariant bilinear forms on vertex operator algebras’, J. Pure Appl. Algebra 96(3) (1994), 279–297.Google Scholar
Li, H.-S., ‘Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules’, inMoonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Contemporary Mathematics, 193 (American Mathematical Society, Providence, RI, 1996), 203–236.Google Scholar
Linton, S., Parker, R., Walsh, P. and Wilson, R., ‘Computer construction of the monster’, J. Group Theory 1(4) (1998), 307–337.Google Scholar
Mandelstam, S., ‘Dual-resonance models’, Phys. Rep. 13 (1974), 259–353.Google Scholar
Mason, G., ‘Finite groups and modular functions’, inThe Arcata Conference on Representations of Finite Groups (Arcata, CA, 1986), Proceedings of Symposia in Pure Mathematics, 47 (American Mathematical Society, Providence, RI, 1987), 181–210. With an appendix by S. P. Norton.Google Scholar
Mathieu, É., ‘Mémoire sur l’étude des fonctions de plusiers quantités, sur la manière de les former et sur les substitutions qui les laissent invariables’, J. Math. Pures Appl. (9) 6 (1861), 241–323.Google Scholar
Mathieu, É., ‘Sur la fonction cinq fois transitive de 24 quantités’, J. Math. Pures Appl. (9) 18 (1873), 25–46.Google Scholar
Mukai, S., ‘Finite groups of automorphisms of K3 surfaces and the Mathieu group’, Invent. Math. 94(1) (1988), 183–221.Google Scholar
Norton, S., ‘Generalized moonshine’, Proc. Sympos. Pure Math. 47 (1987), 208–209.Google Scholar
Norton, S., ‘From moonshine to the monster’, inProceedings on Moonshine and Related Topics (Montréal, QC, 1999), CRM Proceedings & Lecture Notes, 30 (American Mathematical Society, Providence, RI, 2001), 163–171.Google Scholar
Ono, K., Rolen, L. and Trebat-Leder, S., ‘Classical and umbral moonshine: connections and $p$-adic properties’, Preprint, 2014, arXiv:1403.3712 [math.NT].Google Scholar
Queen, L., ‘Modular functions arising from some finite groups’, Math. Comp. 37(156) (1981), 547–580.Google Scholar
Rademacher, H., ‘The Fourier series and the functional equation of the absolute modular invariant J (𝜏)’, Amer. J. Math. 61(1) (1939), 237–248.Google Scholar
Rains, E. M. and Sloane, N. J. A., ‘Self-dual codes’, inHandbook of Coding Theory, Vol. I, II (North-Holland, Amsterdam, 1998), 177–294.Google Scholar
Scheithauer, N. R., ‘Vertex algebras, Lie algebras, and superstrings’, J. Algebra 200(2) (1998), 363–403.Google Scholar
Schwarz, J. H., ‘Dual-resonance theory’, Phys. Rep. 8 (1973), 269–335.Google Scholar
Shimura, G., Introduction to The Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, 11. Kanô Memorial Lectures, 1 (Iwanami Shoten, Tokyo, 1971).Google Scholar
Smith, S. D., ‘On the head characters of the monster simple group’, inFinite Groups – Coming of Age (Montreal, QC, 1982), Contemporary Mathematics, 45 (American Mathematical Society, Providence, RI, 1985), 303–313.Google Scholar
Thompson, J. G., ‘Finite groups and modular functions’, Bull. Lond. Math. Soc. 11(3) (1979), 347–351.Google Scholar
Thompson, J. G., ‘Some numerology between the Fischer–Griess monster and the elliptic modular function’, Bull. Lond. Math. Soc. 11(3) (1979), 352–353.Google Scholar
Volpato, R., ‘On symmetries of N = (4, 4) sigma models on T4’, JHEP 1408 (2014) 094.Google Scholar