Modular framed vertex operator algebras and \(\mathbb{Z}[{1\over 2}]\)-forms

We investigate modular framed vertex operator algebras over an algebraically closed field \(\mathbb{F}\) $$\mathbb{F}$$ whose characteristic is different from 2 and 7. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a \(\mathbb{Z}[{1\over 2}]\) $$\mathbb{Z}[\frac{1}{2}]$$ -form for any framed vertex operator algebra over ℂ is constructed. As a result, one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over ℂ.