## 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.

### MSC:

 17B69 Vertex operators; vertex operator algebras and related structures 20D08 Simple groups: sporadic groups
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### References:

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