Modular framed vertex operator algebras

Framed vertex operator algebras over any algebraically closed field whose characteristic is different from 2 and 7 are studied. In particular, the rationality of framed vertex operator algebras is established. For a code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a 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.