Q-structure of Conformal Field Theory with Gauge Symmetries

Introduction It is well-known that abelian conformal eld theory has rich arithmetic structure (see, for example [DKK], [KSU1], [KSU2], [KUS3]). It is natural to ask whether this is also the case for no-abelian conformal eld theory. For the rst step to study arithmetic properties of non-abelian conformal eld theory, in the present notes we shall study the eld of de nition of conformal eld theory. Namely we shall show that conformal eld theory with gauge symmetries developed in [TUY] can be de ned over the rational number eld Q. This means that the sheaf of vacua (or conformal blocks) over the moduli space of pointed stable curves with formal neighbourhoods is de ned over Q and the projectively at connection on it is also de ned over Q. More precisely, if a family F = ( : C ! B; s 1 ; : : : ; s N ; 1 ; : : : ; N ) of N -pointed stable curves with formal neighbourhoods is de ned over a eld K of