vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one. This is known as the Frenkel–Kac (or Frenkel–Kac–Segal) construction, and is based on the earlier construction by Sergio Fubini and Gabriele Veneziano of the tachyonic vertex operator in the dual resonance model. Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to Jacques Tits. In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices. And this is commonly considered the simplest way to construct the 248-dimensional group E8. Additional examples The monster vertex algebra (also called the "moonshine module"), the key to Borcherds's proof of the Monstrous moonshine conjectures, was constructed by Frenkel, Lepowsky, and Meurman in 1988. It is notable because its partition function is the modular invariant j–744, and its automorphism group is the largest sporadic simple group, known as the monster group. It is constructed by orbifolding the Leech lattice VOA by the order 2 automorphism induced by reflecting the Leech lattice in the origin. That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution. Frenkel, Lepowsky, and Meurman conjectured in 1988 that is the unique holomorphic vertex operator algebra with central charge 24, and partition function j–744. This conjecture is still open. Chiral de Rham complex: Malikov, Schechtman, and Vaintrob showed that by a method of localization, one may canonically attach a bcβγ (boson–fermion superfield) system to a smooth complex manifold. This complex of sheaves has a distinguished differential, and the global cohomology is a vertex superalgebra. Ben-Zvi, Heluani, and Szczesny showed that a Riemannian metric on the manifold induces an N=1 superconformal structure, which is promoted to an N=2 structure if the metric is Kähler and Ricci-flat, and a hyperKähler structure induces an N=4 structure. Borisov and Libgober showed that one may obtain the two-variable elliptic genus of a compact complex manifold from the cohomology of Chiral de Rham – if the manifold is Calabi–Yau, then this genus is a weak Jacobi form. Modules Much like ordinary rings, vertex algebras admit a notion of module, or representation.