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Vertex operator algebras whose dimensions of weight one spaces are 8 and 16

Abstract

We classify simple vertex operator algebras (VOAs) V of CFT type, whose spaces generated by characters are equal to spaces solutions of monic modular linear differential equations of third order, which have two parameters; “expected central charge” c and “conformal weight” h. Among VOAs with these properties above, we focus on ones such that the weight one space V1 is either 8 or 16-dimensional, respectively. The typical examples of such VOAs are lattice VOAs associated with the √ 2E8 lattice for c = 8 and the Barnes-Walls lattice (denoted by Λ16) for c = 16, respectively. This is because central charges and dim V1 of lattice VOAs are equal to ranks of the corresponding lattices. Of course, we could study VOAs with central charge 8 and 16, which is certainly more natural. However, it is would be (was) not easy to characterize such VOAs since there exist solutions that are independent of one of two parameters c and h. If the character of V is free of a parameter, we cannot determine expected central charge. We may consider the characters of such V -modules, in fact we did try to figure out these cases. But it was not very successful since first coefficients of V -modules are not always 1. (In our method this property is crucial.) At first, we have shown that VOAs whose space of characters in our interest is equal to ones of the √ 2E8 lattice VOA for dim V1 = 8 up to integral multiples and of the Barnes-Walls lattice VOA VΛ16 for dim V1 = 16, respectively. Moreover, we showed that V is isomorphic to V√ 2E8 for c = 8 and V is isomorphic to VΛ16 for c = 16 under a mild condition. I will certainly use a black board instead of slides, and promise that I will finish my talk in expected time (except discussions). Finally, this is a joint work with Xingyu Jiao, Yuich Sakai and Hiroki Shimakura.