全纯辛费米

摘要：

设V是r对辛费米子的顶点算子超代数的偶部分。最多两个图，我们证明了V承认一个唯一的全纯扩展，如果r是8的倍数，否则没有全纯扩展。本文得到的两个结果表明了这一点：1）当r是8的倍数时，由格顶点算子代数给出了具有位移应力张量的偶自对偶格$D_r^+$的一种可能的全纯扩张. 2)我们在SF（h）中对拉格朗日代数进行了分类，SF（h）是与辛费米子相关的带状范畴。V的全纯扩张的分类由1）和2）得出，如果假设SF（h）与Rep（V）是带等价的，并且V的扩张的单模与SF（h）中相应的交换代数的单模有一一对应的关系。

英文摘要：

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: 1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice $D_r^+$ with shifted stress tensor. 2) We classify Lagrangian algebras in SF(h), a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from 1) and 2) if one assumes that SF(h) is ribbon equivalent to Rep(V), and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in SF(h).

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