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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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