Permutation orbifolds and associative algebras

Let V be a vertex operator algebra and g = (1 2…k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V ⊗k. It is proved that Dong-Li-Mason’s associated associative algebra Ag(V ⊗k) is isomorphic to Zhu’s algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V ⊗k-modules and irreducible V -modules.

设\(V\)是一个顶点算子代数，\(g=(1\ 2\cdots k)\)是一个\(k\)-循环，它被看作顶点算子代数\(V^{\otimes k}\)的一个自同构。证明了董 - 李 - 梅森相关的结合代数\(A_g(V^{\otimes k})\)明确同构于朱的代数\(A(V)\)。这个结果恢复了先前的一个结果，即不可约\(g\)-扭曲\(V^{\otimes k}\)-模和不可约\(V\)-模之间存在一一对应。