Continuously varying exponents and the value of the central charge

The author shows that the conformal field theories describing two-dimensional critical behaviour with continuously varying exponents must have central charge c=1 if (a) there are no conserved spin-2 currents other than the stress tensor and (b) the marginal operator responsible for the line of fixed points does not mix with other operators. In such cases one may show the existence of a fixed line knowing only the four-point function of the marginal operator. The author applies this to the Ashkin-Teller model and the XY model with fourfold symmetry breaking.