Super-moonshine for Conway's largest sporadic group

Abstract

Abstract. We study a self-dual N = 1 super vertex operator algebra and prove that the full symmetry group is Conway's largest sporadic simple group. We verify a uniqueness result which is analogous to that conjectured to characterize the Moonshine vertex operator algebra. The action of the automorphism group is sufficiently transparent that one can derive explicit expressions for all the McKay-Thompson series. A corollary of the construction is that the perfect double cover of the Conway group may be characterized as a pointstabilizer in a spin module for the Spin group associated to a 24 dimensional Euclidean space. IntroductionThe preeminent example of the structure of vertex operator algebra (VOA) is the Moonshine VOA denoted V ♮ which was first constructed in [FLM88] and whose full automorphism group is the Monster sporadic group.Following [Höh96] we say that a VOA is nice when it is C 2 -cofinite and satisfies a certain natural grading condition (see §2.1), and we make a similar definition for super vertex operator algebras (SVOAs). We say that a VOA is rational when all of its modules are completely reducible (see §2.2). Then conjecturally V ♮ may be characterized among nice rational VOAs by the following properties:• self-dual • rank 24• no small elements where a self-dual VOA is one that has no non-trivial irreducible modules other than itself, and we write "no small elements" to mean no non-trivial vectors in the degree one subspace, since in a nice VOA this is the L(0)-homogeneous subspace with smallest degree that can be trivial. In this note we study what may be viewed as a super analogue of V ♮ . More specifically, we study an object A f ♮ characterized among nice rational N = 1 SVOAs by the following properties.• self-dual • rank 12• no small elements An N = 1 SVOA is an SVOA which admits a representation of the Neveu-Schwarz superalgebra, and now "no small elements" means no non-trivial vectors with degree 1/2. (We define an SVOA to be rational when its even sub-VOA is rational, and a self-dual SVOA is an SVOA with no irreducible modules other than itself.) The earliest evidence in the mathematical literature that there might be an object such as A f ♮ was given in [FLM85] where it was suggested to study a Z/2-L is a self-dual SVOA, and one finds that the graded character ofand is a Hauptmodul for a certain genus zero subgroup of the Modular groupΓ = PSL(2, Z). One can check that, but for the constant term, these coefficients exhibit moonshine phenomena for Co 1 . For example, we have that 276 is the dimension of an irreducible module for Co 1 , and 2048 = 1+276+1771 is a possible decomposition of the degree 3/2 subspace into irreducibles for Co 1 . The only problem being that there is no irreducible representation of Co 1 of dimension 8, and the space corresponding to the constant term in the character of V f L would have to be a sum of trivial modules were it a Co 1 -module at all. As observed in; that is, a space with the correct character for Co 1 . We note here that the existence of V f ♮ has...

Moonshine

“…(9.20)). It's construction as a Co 1 -module was sketched first in §15 of [103], described later in §5 of [22], and subsequently studied in detail in [83]. The Co 0 -module structure on V s is mentioned in [83], following [22], but it seems that the modular properties of the trace functions associated to the Co 0 -action were not considered until [87].…”

K3 string theory, lattices and moonshine

“…The Conway moonshine module is a chiral superconformal field theory with c = 12 and symmetry group Co 0 , which was first discussed in [49] and further studied in [40,41]. Using the Conway module, the authors of [42] associate two (possibly coinciding) Jacobi forms to each conjugacy class of Co 0 and conjecture that this set constitutes a complete list of possible K 3 twining genera.…”

Heterotic sigma models on T 8 and the Borcherds automorphic form Φ12

“…One can circumvent this vanishing by inserting powers of F in the trace to soak up zero modes, as in [14]; however, this will alter the modularity properties and spoil the connection we intend to make to Φ 12 . Instead, we consider, following [15], a 'twisted' computation where we trace over left-moving excitations but consider the ground states of the Z 2 orbifold of the right-moving sector -that is, we consider the right-movers to live in the Ramond sector of the Conway module constructed by Duncan [16]. 3 The orbifold action kills the extra zero modes, but leaves the left-moving sector untouched.…”