标题： 自对偶顶点算子超代数与小怪物 (自对偶顶点算子超代数和 Babymonster) 显示英文标题

标题： Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)

摘要： 我们研究了自对偶顶点算子代数（VOA）和超代数（SVOA）。利用亏格一相关函数，表明仅对于半整数中心荷，自对偶SVOA才存在。描述了如何从具有更大中心荷的自对偶VOA构造自对偶SVOA。强调了与整格和二元码类比的关系。一个主要结果是构建了较短的月光模，即中心荷为23.5的自对偶SVOA，Baby怪物群——第二大散在单群——通过自同构作用于该模上。较短的月光模的特征为q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2)+...)，它是月光模的“较短表亲”。它的格和码类比是较短的Leech格和较短的Golay码。我们推测较短的月光模是具有这一特征的唯一SVOA。最后一章引入了极值VOA和SVOA的概念。这些是自对偶（S）VOA，其特征具有与相同中心荷的Virasoro代数真空表示相同的前几个系数。我们证明了极值VOA至少存在于中心荷8, 16, 24, 32, 40时，并且极值SVOA仅存在于中心荷c=0.5, 1, ..., 7.5, 8, 12, 14, 15, 15.5, 23.5和24时。中心荷为24（或23.5）的例子是（较短的）月光模。再次强调，我们的结果类似于已知的关于码和格的结果。 显示英文摘要

摘要： We investigate self-dual vertex operator algebras (VOAs) and super algebras (SVOAs). Using the genus one correlation functions, it is shown that self-dual SVOAs exist only for half-integral central charges. It is described how self-dual SVOAs can be constructed from self-dual VOAs of larger central charge. The analogy with integral lattices and binary codes is emphasized. One main result is the construction of the shorter Moonshine module, a self-dual SVOA of central charge 23.5 on which the Baby monster - the second largest sporadic simple group - acts by automorphisms. The shorter Moonshine module has the character q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2) +...) and is the "shorter cousin" of the Moonshine module. Its lattice and code analog are the shorter Leech lattice and shorter Golay code. We conjecture that the shorter Moonshine module is the unique SVOA with this character. The final chapter introduces the notion of extremal VOAs and SVOAs. These are self-dual (S)VOAs with character having the same first few coefficients as the vacuum representation of the Virasoro algebra of the same central charge. We show that extremal VOAs exist at least for the central charges 8, 16, 24, 32, 40 and that extremal SVOAs exist only for the central charges c=0.5, 1, ..., 7.5, 8, 12, 14, 15, 15.5, 23.5 and 24. Examples for c=24 (resp. 23.5) are the (shorter) Moonshine module. Again, our results are similar to results known for codes and lattices.