Q-Structure of Conformal Field Theory with Gauge Symmetries

Abstract

It is well-known that abelian conformal field theory has a rich arithmetic structure (see, for example [DKK], [KSU1], [KSU2], [KSU3]). It is natural to ask whether this is also the case for non-abelian conformal field theory. As a first step to study arithmetic properties of non-abelian conformal field theory, in the present paper we shall study the field of definition of conformal field theory. Namely we shall show that conformal field theory with gauge symmetries as developed in [TUY] can be defined over the rational number field Q.This means that the sheaf of vacua (or conformal blocks) over the rnoduli space of pointed stable curves with formal neighbourhoods is defined over Q and the projectively flat connection on it, is also defined over Q. More precisely, if a family F = (π : C → B;s₁…,s_N;n₁,…,n_N)of N-pointed stable curves with formal neighbourhoods is defined over a field K of characteristic zero, then the sheaf of vacua $\nu _{\vec \lambda }^\dag$ (F) is defined over the field K and the projective flat connection on it is also defined over K. This agrees with the fact that for $\vec \lambda$ = $\vec 0$ the space of vacua is isomorphic to the space of generalized theta functions H°(M, L^ℓ) which is defined over Q. (See, for example [BL].)