Integral forms for tensor products of Virasoro vertex operator algebras and their modules

In this paper, we study integral forms of the tensor products of Virasoro vertex operator algebras ⊗i∈IL(cpi,qi,0)⊗ni and their modules. On the one hand, with the tool of binary linear codes, we show the existence and give constructions of integral forms of the tensor product vertex operator algebras that contain the conformal vector. Several interesting and explicit examples of codes are presented. On the other hand, we investigate the integral form theory of modules over the vertex operator algebra ⊗i∈IL(cpi,qi,0)⊗ni. More precisely, we construct integral forms of modules and contragredient modules for these tensor product vertex operator algebras. The integrality of intertwining operators among integral forms of ⊗i∈IL(cpi,qi,0)⊗ni-modules are also obtained.