##
**Super-moonshine for Conway’s largest sporadic group.**
*(English)*
Zbl 1171.17011

In this paper the author constructs a super vertex operator algebra (SVOA) \(A^{f\natural}\) over the field \(\mathbb{R}\) of real numbers whose full symmetry group is Conway’s largest sporadic simple group \(\mathrm{Co}_1\). An SVOA \(V\) is said to be nice if it is \(C_2\)-cofinite, the eigenvalues of the zero mode \(L(0)\) of the conformal element \(\omega\) are contained in \(\frac{1}{2}\mathbb{Z}_{\geq 0}\), and the degree zero subspace \(V_0\) is spanned by the vacuum vector. An \(N=1\) SVOA is an SVOA which contains a degree \(3/2\) element \(\tau\) such that the component operators \(G(n+1/2) = \tau_{n+1}\) generate a Neveu-Schwarz superalgebra. Such an element \(\tau\) is called a super conformal element. The SVOA \(A^{f\natural}\) over \(\mathbb{R}\) constructed in the paper is a nice and rational \(N=1\) SVOA with the properties (1) self-dual, (2) rank \(12\), and (3) trivial degree \(1/2\) subspace. An SVOA is said to be self-dual if it is simple and has no irreducible modules other than itself. As a real vector space, \(A^{f\natural}\) is constructed by using a Clifford module SVOA \(A(\mathfrak{l})\) associated to a \(24\)-dimensional Euclidean space \(\mathfrak{l}\) and its twisted module \(A(\mathfrak{l})_\theta = A(\mathfrak{l})_{\theta,G}\), where \(\theta\) is a canonical involution and \(G\) is a subgroup of \(\mathrm{Spin}(\mathfrak{l})\) corresponding to the Golay code. In fact, \(A^{f\natural}\) is defined to be \(A(\mathfrak{l})^0 \oplus A(\mathfrak{l})^0_\theta\), where \(A(\mathfrak{l})^0\) and \(A(\mathfrak{l})^0_\theta\) denote the subspace of fixed points of \(\theta\) in \(A(\mathfrak{l})\) and \(A(\mathfrak{l})_\theta\), respectively.

The first main result is that the real vector space \(A^{f\natural}\) admits a structure of self-dual rational \(N=1\) SVOA (Theorem 4.5). Let \(\tau_A\) be a super conformal element of \(A^{f\natural}\). Then the second main result is that the stabilizer of \(\tau_A\) in the full group of SVOA automorphisms of \(A^{f\natural}\) is isomorphic to Conway’s largest simple group \(\mathrm{Co}_1\) (Theorem 4.11). The fact that the perfect double cover \(\mathrm{Co}_0\) of \(\mathrm{Co}_1\) is a maximal finite subgroup of \(\mathrm{SO}_{24}(\mathbb{R})\) (Proposition 4.9) is used. Next, the author studies the uniqueness of \(A^{f\natural}\). Suppose \(V\) is a nice and rational \(N=1\) SVOA satisfying the three conditions (1), (2) and (3). Then the character of \(V\) is uniquely determined (Proposition 5.7). In particular, the degree \(1\) subspace \(V_1\) is a \(276\) dimensional Lie algebra.

Using a similar method as C. Dong and G. Mason [Pac. J. Math. 213, No. 2, 253–266 (2004; Zbl 1100.17010); Int. Math. Res. Not. 2004, No. 56, 2989–3008 (2004; Zbl 1106.17032)], the author shows that \(V_1\) is isomorphic to a simple Lie algebra of type \(D_{12}\) and \(V\) is isomorphic to a lattice SVOA associated to the lattice \(D_{12}^+\), which is also isomorphic to the complexification of \(A^{f\natural}\) (Theorem 5.15). Moreover, all super conformal elements in \(A^{f\natural}\) are conjugate under the action of \(\mathrm{Spin}(\mathfrak{l})\). Combining these facts, the third main result is obtained, namely, \(A^{f\natural}\) is a unique SVOA over \(\mathbb{R}\) which has the above mentioned properties (Theorem 5.28). Some of the properties of \(A^{f\natural}\), including the uniqueness, are closely related to the properties of the Golay code. The McKay-Thompson series, that is, the trace function for \(A^{f\natural}\) associated to any element of \(\mathrm{Co}_1\) is given explicitly.

The construction of \(A^{f\natural}\) is purely fermionic [cf. A. Feingold, I. Frenkel and J. Ries [Spinor construction of vertex operator algebras, triality, and \(E_ 8^{(1)}\). Contemporary Mathematics. 121. Providence, RI: American Mathematical Society (1991; Zbl 0743.17029)]. Another SVOA based on an \(E_8\)-lattice on which \(Co_1\) acts, was studied by I. Frenkel, J. Lepowsky and A. Meurman [Publ., Math. Sci. Res. Inst. 3, 231–273 (1985; Zbl 0556.17008)] and R. E. Borcherds and A. J. E. Ryba [Duke. Math. J. 83, No. 2, 435–459 (1996; Zbl 1032.17049)]. An explicit isomorphism between \(A^{f\natural}\) and the real form \(V^{f\natural}\) of the SVOA based on an \(E_8\)-lattice is also obtained.

The first main result is that the real vector space \(A^{f\natural}\) admits a structure of self-dual rational \(N=1\) SVOA (Theorem 4.5). Let \(\tau_A\) be a super conformal element of \(A^{f\natural}\). Then the second main result is that the stabilizer of \(\tau_A\) in the full group of SVOA automorphisms of \(A^{f\natural}\) is isomorphic to Conway’s largest simple group \(\mathrm{Co}_1\) (Theorem 4.11). The fact that the perfect double cover \(\mathrm{Co}_0\) of \(\mathrm{Co}_1\) is a maximal finite subgroup of \(\mathrm{SO}_{24}(\mathbb{R})\) (Proposition 4.9) is used. Next, the author studies the uniqueness of \(A^{f\natural}\). Suppose \(V\) is a nice and rational \(N=1\) SVOA satisfying the three conditions (1), (2) and (3). Then the character of \(V\) is uniquely determined (Proposition 5.7). In particular, the degree \(1\) subspace \(V_1\) is a \(276\) dimensional Lie algebra.

Using a similar method as C. Dong and G. Mason [Pac. J. Math. 213, No. 2, 253–266 (2004; Zbl 1100.17010); Int. Math. Res. Not. 2004, No. 56, 2989–3008 (2004; Zbl 1106.17032)], the author shows that \(V_1\) is isomorphic to a simple Lie algebra of type \(D_{12}\) and \(V\) is isomorphic to a lattice SVOA associated to the lattice \(D_{12}^+\), which is also isomorphic to the complexification of \(A^{f\natural}\) (Theorem 5.15). Moreover, all super conformal elements in \(A^{f\natural}\) are conjugate under the action of \(\mathrm{Spin}(\mathfrak{l})\). Combining these facts, the third main result is obtained, namely, \(A^{f\natural}\) is a unique SVOA over \(\mathbb{R}\) which has the above mentioned properties (Theorem 5.28). Some of the properties of \(A^{f\natural}\), including the uniqueness, are closely related to the properties of the Golay code. The McKay-Thompson series, that is, the trace function for \(A^{f\natural}\) associated to any element of \(\mathrm{Co}_1\) is given explicitly.

The construction of \(A^{f\natural}\) is purely fermionic [cf. A. Feingold, I. Frenkel and J. Ries [Spinor construction of vertex operator algebras, triality, and \(E_ 8^{(1)}\). Contemporary Mathematics. 121. Providence, RI: American Mathematical Society (1991; Zbl 0743.17029)]. Another SVOA based on an \(E_8\)-lattice on which \(Co_1\) acts, was studied by I. Frenkel, J. Lepowsky and A. Meurman [Publ., Math. Sci. Res. Inst. 3, 231–273 (1985; Zbl 0556.17008)] and R. E. Borcherds and A. J. E. Ryba [Duke. Math. J. 83, No. 2, 435–459 (1996; Zbl 1032.17049)]. An explicit isomorphism between \(A^{f\natural}\) and the real form \(V^{f\natural}\) of the SVOA based on an \(E_8\)-lattice is also obtained.

Reviewer: Hiromichi Yamada (Tokyo)

### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

20D08 | Simple groups: sporadic groups |

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\textit{J. F. Duncan}, Duke Math. J. 139, No. 2, 255--315 (2007; Zbl 1171.17011)

### References:

[1] | M. Aschbacher, Finite Group Theory , 2nd ed., Cambridge Stud. Adv. Math. 10 , Cambridge Univ. Press, Cambridge, 2000. · Zbl 0997.20001 |

[2] | R. E. Borcherds and A. J. E. Ryba, Modular m oonshine, II, Duke Math. J. 83 (1996), 435–459. · Zbl 1032.17049 |

[3] | J. H. Conway, A group of order \(8,315,553,613,086,720,000\) , Bull. London Math. Soc. 1 (1969), 79–88. · Zbl 0186.32304 |

[4] | -, “Three lectures on exceptional groups” in Finite Simple Groups (Oxford, 1969) , Academic Press, London, 1971, 215–247. |

[5] | 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. · Zbl 0568.20001 |

[6] | J. H. Conway and S. P. Norton, Monstrous moonshine , Bull. London Math. Soc. 11 (1979), 308–339. · Zbl 0424.20010 |

[7] | J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups , 2nd ed., Grundlehren Math. Wiss. 290 , Springer, New York, 1993. · Zbl 0785.11036 |

[8] | L. Dixon, P. Ginsparg, and J. Harvey, Beauty and the beast: Superconformal symmetry in a Monster module , Comm. Math. Phys. 119 (1988), 221–241. · Zbl 0657.17011 |

[9] | C. Dong, Vertex algebras associated with even lattices , J. Algebra 161 (1993), 245–265. · Zbl 0807.17022 |

[10] | C. Dong, R. L. Griesso Jr., and G. HöHn, Framed vertex operator algebras, codes and the m oonshine module, Comm. Math. Phys. 193 (1998), 407–448. · Zbl 0908.17018 |

[11] | C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators , Progr. Math. 112 , Birkhäuser, Boston, 1993. · Zbl 0803.17009 |

[12] | C. Dong, H. Li, and G. Mason, Simple currents and extensions of vertex operator algebras , Comm. Math. Phys. 180 (1996), 671–707. · Zbl 0873.17027 |

[13] | -, Twisted representations of vertex operator algebras , Math. Ann. 310 (1998), 571–600. · Zbl 0890.17029 |

[14] | C. Dong and G. Mason, Nonabelian orbifolds and the boson-fermion correspondence , Comm. Math. Phys. 163 (1994), 523–559. · Zbl 0808.17019 |

[15] | -, Holomorphic vertex operator algebras of small central charge , Pacific J. Math. 213 (2004), 253–266. · Zbl 1100.17010 |

[16] | -, Rational vertex operator algebras and the effective central charge , Int. Math. Res. Not. 2004 , no. 56, 2989–3008. · Zbl 1106.17032 |

[17] | -, Integrability of \(C_ 2\) -cofinite vertex operator algebras, Int. Math. Res. Not. 2006 , no. 80468. |

[18] | 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. · Zbl 0953.17014 |

[19] | C. Dong and Z. Zhao, Modularity in orbifold theory for vertex operator superalgebras , Comm. Math. Phys. 260 (2005), 227–256. · Zbl 1133.17016 |

[20] | A. W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups , Ann. of Math. (2) 102 (1975), 291–325. JSTOR: · Zbl 0315.20007 |

[21] | A. J. Feingold, I. B. Frenkel, and J. F. X. Ries, Spinor Construction of Vertex Operator Algebras, Triality, and \(E^ (1)_ 8\) , Contemp. Math. 121 , Amer. Math. Soc., Providence, 1991. · Zbl 0743.17029 |

[22] | 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. · Zbl 0479.17003 |

[23] | 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. · Zbl 0789.17022 |

[24] | 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. · Zbl 0556.17008 |

[25] | -, Vertex Operator Algebras and the Monster , Pure Appl. Math. 134 , Academic Press, Boston, 1988. · Zbl 0674.17001 |

[26] | A. FröHlich and M. J. Taylor, Algebraic Number Theory , Cambridge Stud. Adv. Math. 27 , Cambridge Univ. Press, Cambridge, 1993. |

[27] | G. HöHn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster , Bonner Math. Schriften 286 , Mathematisches Institut, Univ. Bonn, Bonn, 1996. |

[28] | -, personal communication, May 2005. |

[29] | 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. · Zbl 0844.17020 |

[30] | H.-K. Hwang, Limit theorems for the number of summands in integer partitions , J. Combin. Theory Ser. A 96 (2001), 89–126. · Zbl 1029.60013 |

[31] | 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. · Zbl 0838.17035 |

[32] | A. Kapustin and D. Orlov, Vertex algebras, mirror symmetry, and D-branes: The case of complex tori , Comm. Math. Phys. 233 (2003), 79–136. · Zbl 1051.17017 |

[33] | G. Meinardus, Asymptotische Aussagen über Partitionen , Math. Z. 59 (1954), 388–398. · Zbl 0055.03806 |

[34] | G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups , Des. Codes Cryptogr. 24 (2001), 99–121. · Zbl 1002.11057 |

[35] | R. A. Rankin, Modular Forms and Functions , Cambridge Univ. Press, Cambridge, 1977. · Zbl 0376.10020 |

[36] | N. R. Scheithauer, Vertex algebras, Lie algebras, and superstrings , J. Algebra 200 (1998), 363–403. · Zbl 0947.17010 |

[37] | J.-P. Serre, Linear Representations of Finite Groups , Grad. Texts in Math. 42 , Springer, New York, 1977. · Zbl 0355.20006 |

[38] | P. H. Tiep, Globally irreducible representations of finite groups and integral lattices , Geom. Dedicata 64 (1997), 85–123. · Zbl 0867.20005 |

[39] | Y. Zhu, Modular invariance of characters of vertex operator algebras , J. Amer. Math. Soc. 9 (1996), 237–302. JSTOR: · Zbl 0854.17034 |

[40] | -, Vertex operator algebras, elliptic functions, and modular forms , Ph.D. dissertation, Yale University, New Haven, 1990. |

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