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Book Information:
Author:
Terry Gannon
Title:
Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics
Additional book information:
Cambridge Monographs on Mathematical Physics,
Cambridge University Press,
Cambridge, Massachusetts,
2006,
492 pp.,
ISBN 978-0-521-83531-2,
US$130.00$
C. J. Cummins and T. Gannon, Modular equations and the genus zero property of moonshine functions, Invent. Math. 129 (1997), no. 3, 413–443. MR 1465329, DOI 10.1007/s002220050167
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI 10.1112/blms/11.3.308
John H. Conway, Simon P. Norton, and Leonard H. Soicher, The Bimonster, the group $Y_{555}$, and the projective plane of order $3$, Computers in algebra (Chicago, IL, 1985) Lecture Notes in Pure and Appl. Math., vol. 111, Dekker, New York, 1988, pp. 27–50. MR 1060755
Chongying Dong, Haisheng Li, and Geoffrey Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), no. 1, 1–56. MR 1794264, DOI 10.1007/s002200000242
[D] Duncan, John F. Moonshine for Rudvalis's sporadic group I, arXiv:math/0609449
I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260. MR 747596, DOI 10.1073/pnas.81.10.3256
Robert L. Griess Jr., The friendly giant, Invent. Math. 69 (1982), no. 1, 1–102. MR 671653, DOI 10.1007/BF01389186
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136, DOI 10.1007/978-3-663-14045-0
Mark Mahowald and Mike Hopkins, The structure of 24 dimensional manifolds having normal bundles which lift to $B\textrm {O}[8]$, Recent progress in homotopy theory (Baltimore, MD, 2000) Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 89–110. MR 1887530, DOI 10.1090/conm/293/04944
A. A. Ivanov, A geometric characterization of the Monster, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 46–62. MR 1200249, DOI 10.1017/CBO9780511629259.007
V. G. Kac, Infinite-dimensional algebras, Dedekind’s $\eta$-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85–136. MR 513845, DOI 10.1016/0001-8708(78)90033-6
Yasuyuki Kawahigashi and Roberto Longo, Local conformal nets arising from framed vertex operator algebras, Adv. Math. 206 (2006), no. 2, 729–751. MR 2263720, DOI 10.1016/j.aim.2005.11.003
Ching Hung Lam and Masahiko Miyamoto, Niemeier lattices, Coxeter elements, and McKay’s $E_8$-observation on the Monster simple group, Int. Math. Res. Not. , posted on (2006), Art. ID 35967, 27. MR 2219232, DOI 10.1155/IMRN/2006/35967
Masahiko Miyamoto, $21$ involutions acting on the Moonshine module, J. Algebra 175 (1995), no. 3, 941–965. MR 1341752, DOI 10.1006/jabr.1995.1220
Urmie Ray, Automorphic forms and Lie superalgebras, Algebra and Applications, vol. 5, Springer, Dordrecht, 2006. MR 2286867
A. J. E. Ryba, Modular Moonshine?, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 307–336. MR 1372729, DOI 10.1090/conm/193/02378
Stephen D. Smith, On the head characters of the Monster simple group, Finite groups—coming of age (Montreal, Que., 1982) Contemp. Math., vol. 45, Amer. Math. Soc., Providence, RI, 1985, pp. 303–313. MR 822245, DOI 10.1090/conm/045/822245
J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR 554402, DOI 10.1112/blms/11.3.352
[W] Witten, E. Three-Dimensional Gravity Revisited, arXiv:0706.3359
- [CG]
- Cummins, C. J.; Gannon, T. Modular equations and the genus zero property of moonshine functions. Invent. Math. 129 (1997), no. 3, 413-443. MR 1465329
- [CN]
- Conway, J. H.; Norton, S. P. Monstrous moonshine. Bull. London Math. Soc. 11 (1979), no. 3, 308-339. MR 0554399
- [CNS]
- Conway, John H.; Norton, Simon P.; Soicher, Leonard H. The Bimonster, the group
, and the projective plane of order 3. Computers in algebra (Chicago, IL, 1985), 27-50, Lecture Notes in Pure and Appl. Math., 111, Dekker, New York, 1988. MR 1060755
- [DLM]
- Dong, Chongying; Li, Haisheng; Mason, Geoffrey. Modular-invariance of trace functions in orbifold theory and generalized Moonshine. Comm. Math. Phys. 214 (2000), no. 1, 1-56. MR 1794264
- [D]
- Duncan, John F. Moonshine for Rudvalis's sporadic group I, arXiv:math/0609449
- [FLM]
- Frenkel, I. B.; Lepowsky, J.; Meurman, A. A natural representation of the Fischer-Griess Monster with the modular function
as character. Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, Phys. Sci., 3256-3260. MR 0747596
- [G]
- Griess, Robert L., Jr. The friendly giant. Invent. Math. 69 (1982), no. 1, 1-102. MR 0671653
- [H]
- Hirzebruch, Friedrich; Berger, Thomas; Jung, Rainer. Manifolds and modular forms. Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. xii+211 pp. ISBN: 3-528-06414-5 MR 1189136
- [HM]
- Mahowald, Mark; Hopkins, Mike. The structure of 24 dimensional manifolds having normal bundles which lift to BO[8]. Recent progress in homotopy theory (Baltimore, MD, 2000), 89-110, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002. MR 1887530
- [I]
- Ivanov, A. A. A geometric characterization of the Monster. Groups, combinatorics & geometry (Durham, 1990), 46-62, London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge, 1992. MR 1200249
- [K]
- Kac, V. G. An elucidation of: ``Infinite-dimensional algebras, Dedekind's
-function, classical Möbius function and the very strange formula''. 
and the cube root of the modular invariant 
. Adv. in Math. 35 (1980), no. 3, 264-273. MR 0513845
- [KL]
- Kawahigashi, Yasuyuki; Longo, Roberto. Local conformal nets arising from framed vertex operator algebras. (English summary) Adv. Math. 206 (2006), no. 2, 729-751. MR 2263720
- [LM]
- Lam, Ching Hung; Miyamoto, Masahiko. Niemeier lattices, Coxeter elements, and McKay's
-observation on the Monster simple group. Int. Math. Res. Not. 2006, Art. ID 35967 MR 2219232
- [M]
- Miyamoto, Masahiko. 21 involutions acting on the Moonshine module. (English summary) J. Algebra 175 (1995), no. 3, 941-965. MR 1341752
- [Ra]
- Ray, U. Automorphic Forms and Lie Superalgebras (Algebra and Applications) Springer; 1 edition (December 2006) ISBN 978-1402050091 MR 2286867
- [Ry]
- Ryba, A. J. E. Modular Moonshine? Moonshine, the Monster, and related topics (South Hadley, MA, 1994), 307-336, Contemp. Math., 193, Amer. Math. Soc., Providence, RI, 1996. MR 1372729
- [S]
- Smith, Stephen D. On the head characters of the Monster simple group. Finite groups--coming of age (Montreal, Que., 1982), 303-313, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985. MR 0822245
- [T]
- Thompson, J. G. Some numerology between the Fischer-Griess Monster and the elliptic modular function. Bull. London Math. Soc. 11 (1979), no. 3, 352-353. MR 0554402
- [W]
- Witten, E. Three-Dimensional Gravity Revisited, arXiv:0706.3359
Review Information:
Reviewer:
R. E. Borcherds
Affiliation:
University of California at Berkeley
Journal:
Bull. Amer. Math. Soc.
45 (2008), 675-679
DOI:
https://doi.org/10.1090/S0273-0979-08-01209-3
Published electronically:
June 25, 2008
Review copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
