Modular Invariance

Abstract

We have assumed until now, implicitly or not, that conformal field theories were defined on the whole complex plane. On the infinite plane the holomorphic and antiholomorphic (or left and right) sectors of a conformal theory completely decouple and may be studied separately. In fact, the two sectors may constitute distinct theories on their own since they do not interfere: Correlation functions factorize into holomorphic and antiholomorphic factors with a priori different properties. However, this situation is very unphysical. The decoupling exists only at the fixed point in parameter space (the conformally invariant point) and in the infinite-plane geometry. The physical spectrum of the theory should be continuously deformed as we leave the critical point, and the coupling between right and left sectors away from this point should lead to some constraints on the left and right content of the theory at the fixed point. In operator language, this implies that not every left-right combination of Verma modules is physically sound.